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]><archimedes> <info> <author>Varro, Michael</author> <title>De Motu Tractatus</title> <date>1584</date> <place></place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk> <locator>0000000044</locator> </info> <text> <front> </front> <body> <chap> <pb id="p.0001"/>

<s>ILLVSTRI ET IN <lb/>PRIMIS GENEROSO DOMINO, <lb/>DOMINO CAROLO BARONI <lb/>Azerotin, Name&longs;tij Ro&longs;icij <lb/>Brandei&longs;ij Domino, <lb/>MICH. VARRO. S. P. D.</s>

<s><emph type="italics"/>CVM ab Ineunte ætate in Ma<lb/>thematicis me exercui&longs;&longs;em, &longs;em<lb/>per animo meo in hæ&longs;it hoc de&longs;ide<lb/>rium, vt quæ de Archimede Syra<lb/>cu&longs;ano referuntur, ea & ratione <lb/>demon&longs;trare & experimento comprobare po&longs;&longs;em. <lb/>Cumque multa in eo genere meditatus e&longs;&longs;em, po&longs;t<lb/>quam ad iuris ciuilis &longs;tudium tran&longs;ii, ac tandem <lb/>ad Rempub. </s>

<s>nunquam mihi tantum otii contigit, <lb/>vt ea in ordinem redigere potuerim. </s>

<s>At cum pau<lb/>cis retrò annis &longs;atis longam peregrinationem per <lb/>Sarmatiam &longs;u&longs;cepi&longs;&longs;em, e&longs;&longs;etque vacuus curis a-<emph.end type="italics"/><pb/><emph type="italics"/>nimus, atque ad earum rerum meditationem me <lb/>impelleret genius meus, c&oelig;pi aliquid de his &longs;cribe<lb/>re, quantum ip&longs;e iter faciendo meo marte a&longs;&longs;equi <lb/>potui cum libris de&longs;tituerer. </s>

<s>Scriptum illud cum <lb/>in maximè ardua & difficìli contemplatione ver|| <lb/>&longs;etur, nec ab eo tempore ex quo ad&longs;olitas occupa<lb/>tiones redii ei extremam manum apponere mihi <lb/>licuerit, decreueram inter priuatas meas mu&longs;as <lb/>a&longs;&longs;eruare Verebar enim vulgi <expan abbr="iudici&utilde;">iudicium</expan>, (cuius <expan abbr="ta-m&etilde;">ta<lb/>men</expan> rationem viro politico <expan abbr="habendã">habendam</expan> e&longs;&longs;e cen&longs;eo) ne<lb/>que &longs;olum vulgi, verum etiam <expan abbr="eor&utilde;">eorum</expan> qui doctorum <lb/>nomine gaudent, cum inaudita quædam interdum <lb/>quæque ab eorum opinione recedere <expan abbr="videãtur">videantur</expan> at<lb/>tingam. </s>

<s><expan abbr="Cupiebã">Cupiebam</expan> etiam in eo argumento vlteriùs <lb/>progredi, antè quam quidquam ederem, atque ea <lb/>quæ in meis aduer&longs;ariis ea de re &longs;par&longs;a habeo, iis <lb/>quæ <expan abbr="conge&longs;&longs;erã">conge&longs;&longs;eram</expan> &longs;ubiicere, vt iu&longs;tum <expan abbr="volum&etilde;">volumen</expan> fieret.<emph.end type="italics"/></s>

<s><emph type="italics"/>Po&longs;tquam verò te di&longs;ce&longs;&longs;um ex hac nostra ci<lb/>uitate parare intellexi, in qua dum ver&longs;atus es. <lb/>non &longs;olùm ei <expan abbr="ornam&etilde;to">ornamento</expan> fui&longs;ti, &longs;ed etiam optimos <lb/>quo&longs;que virtutum tuarum admiratione tibi con<lb/>ciliasti: mihi verò peculiariter innumera beneuo-<emph.end type="italics"/><pb/><emph type="italics"/>lentiæ &longs;igna exhibui&longs;ti, non potui tantum virum, <lb/>meíque <expan abbr="amãtem">amantem</expan> &longs;ine aliquo munere aut<emph.end type="italics"/> <foreign lang="greek">mnhmosu/nw|</foreign><lb/><emph type="italics"/>dimittere. </s>

<s>Itaque libellum hunc tibi dicare, & <lb/>&longs;ub tuo nomine in publicum edere &longs;tatui. </s>

<s>Squalli<lb/>dus quidem prodit, & rudis, mole etiam perexi<lb/>guus: &longs;plendidius aliquid & magis elaboratum <lb/>postularet tua dignitas. </s>

<s>Atqui ea e&longs;t argumenti <lb/>illius magnitudo, vt in eo aliquid volui&longs;&longs;e &longs;atis <lb/>&longs;it. </s>

<s>Militare &longs;anè munus e&longs;t, eóque nomine tibi <lb/>conueniens: &longs;i quis enim hac cognitione in&longs;tructus <lb/>fuerit, non minùs ea in re bellica vti poterit, <lb/>quam Archimedes in Syracu&longs;is defendendis <lb/>aduer&longs;us Marcum Marcellum v&longs;us est. </s>

<s>Eá<lb/>que mirari de&longs;inet quæ de eo Plutarchus nar<lb/>rat. </s>

<s>Id lucidius <expan abbr="demõ&longs;trare">demon&longs;trare</expan> potui&longs;&longs;em, &longs;i mihi hæc <lb/>vlterius per&longs;equendi otium fui&longs;&longs;et: Jeiuna enim <lb/>per &longs;e videntur, ni&longs;i quis &longs;uccum qui &longs;ub corti<lb/>ce latet, eliciat. </s>

<s>Malui tamen hoc veluti pro<lb/>gymna&longs;mate alios quibus plus est otii & in<lb/>genii ad eiu&longs;modi inqui&longs;itionem hortari & ac<lb/>cendere, quàm præclaram illam cognitio<lb/>nem diutius &longs;epultam relinquere cum ne-<emph.end type="italics"/><pb/><emph type="italics"/>minem hodie animaduertam qui ei in&longs;udet, <lb/>neque mihi &longs;pes vlla affulgeat diuturni otii, cuius <lb/>ope in hoc &longs;tadio pergere poßim: Quòd &longs;i fortè <lb/>mihi aliquando contingat, riuos plurimos ex his <lb/>fontibus me educturum; <expan abbr="cõfido">confido</expan>; quibus multum <lb/><expan abbr="cõmoditàtis">commoditàtis</expan> rebus humanis accedat. </s>

<s>Quæ enim <lb/>hic traduntur, ad motus omnes atque ad omnis ge|| <lb/>neris virium comparationes accommodari po&longs;<lb/>&longs;unt ad fines propo&longs;itos a&longs;&longs;equendos. </s>

<s>Quod quàm <lb/>latè pateat, dici non pote&longs;t. </s>

<s>E&longs;t enim eorum v&longs;us <lb/>non &longs;olùm in mechanicis, in quibus tamen e&longs;t ma|| <lb/>ximus, &longs;ed etiam in politicis & &oelig;conomicis: &longs;unt <lb/>enim in illis motus, &longs;unt vires & re&longs;istentiæ. </s>

<s>In <lb/>arte medica & reliquis phy&longs;ices partibus pra<lb/>cticis, quantus &longs;it, nemo fando exprimere poßit. <lb/>Quia vero ea quæ praxim docent gratiora &longs;unt <lb/>ip&longs;a contemplatione, & cau&longs;æ propter effecta in<lb/>quirantur,<emph.end type="italics"/> <foreign lang="greek">*a)lfh/stas</foreign> <emph type="italics"/>omnes hortor vt quod ego in<lb/>numeris aliis cur is implicitus addere non po&longs;<lb/>&longs;um, ip&longs;i addant. </s>

<s>Abundat nunc Europa præ<lb/>&longs;tantibus ingeniis, &longs;i M&oelig;cenates ade&longs;&longs;ent. <lb/>Ea autem quæ præter hæc meditatus &longs;um, ac fer-<emph.end type="italics"/><pb/><emph type="italics"/>mè parata habeo, hæc &longs;unt. </s>

<s>Tractatus de iactu. <lb/>De continuitate eiu&longs;que &longs;olutione. </s>

<s>De <expan abbr="cõden&longs;a-tione">conden&longs;a<lb/>tione</expan> & rarefactione earumque cau&longs;is & effecti<lb/>bus: <expan abbr="It&etilde;">Item</expan> in Mechanicis, tractatus de variis ma<lb/>chinis ad motus <expan abbr="ci&etilde;dos">ciendos</expan>, ac de perfectißimæ cuiu&longs;<lb/>que ad id quod propo&longs;itum fuerit moliendum in<lb/>ue&longs;tigatione. </s>

<s>Nonnulla <expan abbr="etiã">etiam</expan> de <expan abbr="Rerumpublicar&utilde;">Rerumpublicarum</expan> <lb/>motu tum interno <expan abbr="t&utilde;">tum</expan> externo notaui, quæ <expan abbr="eod&etilde;">eodem</expan> or<lb/>dine tradere optarem: quorum omnium principia <lb/>hic &longs;i quis diligenter animaduertat tradita &longs;unt. <lb/>Si quis mihi in his <expan abbr="palmã">palmam</expan> præripuerit, meque an<lb/>teuerterit ei maximas gratias agam. </s>

<s>Sunt tamen <lb/>inter hæc quædam quæ vulgò pandere nephas est. <lb/>Itaque theoriæ magis in&longs;i&longs;tendum puto, in qua &longs;i <lb/>quis exercitatus fuerit, nullo negotio illam in opus <lb/>educere poterit, idque &longs;ine periculo fiet, cum vul<lb/>go non pateat. </s>

<s>Alioqui <expan abbr="pericul&utilde;">periculum</expan> e&longs;t, ne &longs;i particu<lb/>laria tradantur iis contenti homines, vt fieri &longs;olet <lb/>vniuer&longs;alem cognitionem & cau&longs;arum inqui&longs;i<lb/>tionem negligant, pereatque &longs;cientia. </s>

<s>Eaque de <lb/>cau&longs;a nihil quicquam de his quæ fecerat Archi-<emph.end type="italics"/><pb/><emph type="italics"/>medes &longs;criptum relinquere voluit. </s>

<s><expan abbr="Exi&longs;timãs">Exi&longs;timans</expan> eos <lb/>qui in iis quæ<emph.end type="italics"/> <foreign lang="greek">qewrhtikw=s</foreign> <emph type="italics"/>tradiderat, diligentem ope<lb/>ram ponere <expan abbr="vell&etilde;t">vellent</expan>, multò maiora quoties opus fo<lb/>ret præ&longs;tituros. </s>

<s>Equidem mihi per&longs;uadeo non de<lb/>futuros qui varia iudicia de hoc no&longs;tro &longs;cripto fe<lb/>rant & qua&longs;i nouum athletam in arenàm pro<lb/>deuntem mirentur. </s>

<s>Quibus vno verbo re&longs;pon<lb/>&longs;um volo, me nullius vnquam in verba ma<lb/>gistri iura&longs;&longs;e, &longs;ed liberrimè &longs;emper philo&longs;o<lb/>phatum e&longs;&longs;e: ita vt etiam in principia ab aliis <lb/>&longs;tatuta animaduertere mihi licere putauerim. <lb/>Quod eò liberiùs feci, po&longs;tquam magnam <expan abbr="part&etilde;">partem</expan> <lb/>vulgo receptarum opinionum fal&longs;am e&longs;&longs;e re ip&longs;a <lb/>deprehendi. </s>

<s>Omnibus placuero, &longs;i tibi placuero. </s>

<s>Si <lb/>quid in his ob&longs;curius fuerit, habes <expan abbr="clarißim&utilde;">clarißimum</expan> vi<lb/>rum Wence&longs;laum lauinium tuum, ab&longs;tru&longs;ioris <lb/>philo&longs;ophiæ indagatorem <expan abbr="&longs;umm&utilde;">&longs;ummum</expan>, qui tibi omnia <lb/>explicabit. </s>

<s>Hoc igitur <expan abbr="munu&longs;cul&utilde;">munu&longs;culum</expan> vt &longs;erena <expan abbr="frõ-te">fron<lb/>te</expan> &longs;u&longs;cipias rogo. </s>

<s>Anno <lb/>Christi Domini<emph.end type="italics"/> M. D. LXXXIV. </s><pb pagenum="1"/><figure></figure>

<s>M. VARRONIS DE <lb/>MOTV TRACTATVS.</s>

<s><emph type="italics"/>PROBLEMA.<emph.end type="italics"/></s>

<s><emph type="italics"/>Data vi datum pondus mouere.<emph.end type="italics"/></s>

<s>Hoc problema prima quidem <lb/>fronte mirabile videtur, vt &longs;cilicet <lb/>pondus vel maximum viribus ta<lb/>men minimis, vt puta &longs;i dicas vnius <lb/>hominis, vel etiam imbecillioris a<lb/>licuius animalis viribus totum ter<lb/>ræ globum loco moueri po&longs;&longs;e. </s>

<s>Quod Archimedem <lb/>Syracu&longs;anum iacta&longs;&longs;e referunt hoc dicto, <foreign lang="greek">do\s pou= stw=| <lb/>ta\n ga\n kinw=|. </foreign></s>

<s>Si quis tamen in Geometrico puluere <lb/>ver&longs;atus fuerit, pre&longs;ertim verò in iis quæ ab ip&longs;o in <lb/>&longs;uo <foreign lang="greek">i)soropikw=n</foreign> libello <expan abbr="trad&utilde;tur">traduntur</expan>. </s>

<s>Illud cæteráque eiu&longs;<lb/>modi, ludicra Geometriæ, vt & ip&longs;e facere &longs;olitus di|| <lb/>citur, appellabit. </s>

<s>Ex huius autem problematis expli|| <lb/>catione, omnium machinarum qu&ecedil; ad motus cien<lb/>dos excogitari po&longs;&longs;unt, ratio pendet. </s>

<s>Et&longs;i verò <expan abbr="tot&utilde;">totum</expan> <lb/>Geometrica demon&longs;tratione expediri po&longs;&longs;it, quo<pb pagenum="2"/>niam tamen h&ecedil;c con&longs;ideratio, quæ Græcis <foreign lang="greek">staqmikh\</foreign> di|| <lb/>citur Geometriæ tantùm e&longs;t <foreign lang="greek">u(pa/llhlos</foreign> mixta &longs;cili<lb/>cet ex phy&longs;ica & Geometrica eo quòd illius &longs;ubie<lb/>ctum &longs;it motus. </s>

<s>Ideò paulò cra&longs;&longs;iori Minerua & <lb/>præter Geometricam &longs;implicitatem <foreign lang="greek">fusikw=s</foreign> etiam <lb/>tractanda e&longs;t: alioqui Archimedicis demon&longs;tratio<lb/>nibus &longs;tandum e&longs;&longs;et, quæ in eo mancæ &longs;unt, quòd il<lb/>læ propo&longs;itiones quæ ex phy&longs;icis peti debent, inde<lb/>mon&longs;tratæ manent, nec explicantur, &longs;ed pro confe&longs;<lb/>&longs;is principiorum loco po&longs;tulantur. </s>

<s>Eas igitur, qua<lb/>tenus ad huius problematis explicationem faciunt, <lb/>hic di&longs;cutiemus. </s>

<s>Neque enim Euclidis <expan abbr="libr&utilde;">librum</expan> de gra<lb/>ui & leui, in quo hoc argumentum per&longs;equi volui&longs;<lb/>&longs;e videtur, integrum habemus. </s>

<s>Ari&longs;toteles verò in <lb/>eo quod ab ip&longs;o &longs;criptum extat de Mechanicis fra<lb/>gmento pr&ecedil;ter &longs;uum morem, cùm alio qui in omni<lb/>bus exacti&longs;&longs;imus &longs;it, hanc quæ&longs;tionem potiùs nota<lb/>uit quàm explicuit in &longs;exto & &longs;eptimo phy&longs;icorum <lb/>libro multa prætermi&longs;it. </s>

<s>Vt igitur ad rem aggredia<lb/>mur, primùm voces, quibus vtendum e&longs;t, definie<lb/>mus, vt intelligatur quo &longs;en&longs;u eas accipiamus. </s>

<s>DEFINITIO I.</s>

<s>Vis dicitur agendi aut agenti re&longs;i&longs;tendi <expan abbr="pot&etilde;tia">potentia</expan>, <lb/>præ&longs;ertim verò mouendi & mouenti re&longs;i&longs;tendi. </s>

<s>Vis &longs;ubiectum dicitur id quod vis mouet, vel <pb pagenum="3"/>quod à vi mouetur. </s>

<s>Hîc non agimus de vi primaria quæ virium o<lb/>mnium principium e&longs;t, mouétque omnia, nec <expan abbr="tam&etilde;">tamen</expan> <lb/>mouetur, &longs;ed de ea, qu&ecedil; dum mouet, <expan abbr="etiã">etiam</expan> cum &longs;ub<lb/>iecto, cui ine&longs;t, mouetur. </s>

<s>Subiecta verò quæ nullam vim habent, nec mo<lb/>uentur, nec mouenti <expan abbr="re&longs;i&longs;t&utilde;t">re&longs;i&longs;tunt</expan>: &longs;i qua verò vis illis ac<lb/>cedat, tum ab ea &longs;e moueri patiuntur. </s>

<s>Et&longs;i autem plura &longs;int virium genera, tot &longs;cilicet, <lb/>quot &longs;unt in rerum natura contrariorum, actionem <lb/>& pa&longs;&longs;ionem recipientium, vt leue graue, rarum <expan abbr="d&etilde;-&longs;um">den<lb/>&longs;um</expan>, plenum vacuum, durum molle, & cætera eiu&longs;<lb/>modi, quoniam tamen ea omnia hîc per&longs;equi no&longs;tri <lb/>non e&longs;t in&longs;tituti, cùm de ea tantùm qua motus fit a<lb/>gere &longs;tatuerimus. </s>

<s>Cùm de motu hîc agemus motum ad locum, <lb/>quem Græci <foreign lang="greek">fora\n</foreign> vocant, intelligi volumus. </s>

<s>Linea autem recta quæ e&longs;t ab eo loco à quo mo<lb/>tus fieri incipit ad illum ad quem tendit. </s>

<s>Illius vis <lb/>quæ motum efficit, nutus dicetur. </s>

<s><expan abbr="Ead&etilde;">Eadem</expan> verò linea <expan abbr="cõ&longs;iderata">con&longs;iderata</expan> à loco ad <expan abbr="qu&etilde;">quem</expan> <expan abbr="t&etilde;dit">tendit</expan> vis <lb/>ad <expan abbr="ill&utilde;">illum</expan> à quo motus fieri incipit, contra <expan abbr="nut&utilde;">nutum</expan> dicitur. </s>

<s>Itidem & omnes illi parallelæ. </s>

<s>Quæ verò lineæ vel rectæ, vel curuæ, nutus <expan abbr="lineã">lineam</expan> <pb pagenum="4"/>ad angulos inæquales &longs;ecant, illæ obliquè nutum <lb/>ver&longs;us, vel contra nutum ferri <expan abbr="dic&utilde;tur">dicuntur</expan>, habita ratio<lb/>ne partium, quas &longs;pectant. </s>

<s>Plurimùm autem à &longs;itu corporis humani deno<lb/>minationem accipiunt illæ partes, vt &longs;ur&longs;um, deor<lb/>&longs;um, dextror&longs;um, &longs;ini&longs;tror&longs;um, antè vel ponè di<lb/>cantur. </s>

<s>Quæ verò lineæ, nutus lineam ad angulos <lb/>rectos &longs;ecant, neque ver&longs;us nutum, neque contra <lb/>nutum ferri dicuntur, &longs;ed æquè di&longs;tant à loco natu<lb/>rali. </s>

<s><emph type="italics"/>Virium diui&longs;io.<emph.end type="italics"/></s>

<s>Vis autem omnis aut naturalis e&longs;t, aut non natu<lb/>ralis, aut mixta. </s>

<s>Naturalis vis e&longs;t, qua res quælibet <lb/>natura &longs;ua mouetur, aut mouenti re&longs;i&longs;tit habita ra<lb/>tione tum loci &longs;ui naturalis, tum etiam &longs;itus &longs;uarum <lb/>partium. </s>

<s>Non naturalis e&longs;t quæ nec ratione loci &longs;ui <lb/>naturalis, nec &longs;itus partium mouet. </s>

<s>E&longs;t autem hæc <lb/>duplex, fortuita &longs;cilicet & voluntaria: illa vt vis ven<lb/>torum & &longs;pirituum intellectu carentium, hæc vt a<lb/>nimalium & &longs;pirituum intelligentium & &longs;imilium. <lb/>Mixta dicitur, quæ partim naturalis e&longs;t, partim non <lb/>naturalis. </s>

<s>Locus autem naturalis cuiu&longs;que rei e&longs;t, in quo <lb/>exi&longs;tens ip&longs;a quie&longs;cit, & &longs;i inde remota &longs;it, vis ei à na|| <lb/>tura in&longs;ita eam illuc impellit. </s>

<s>Huius verò motus aut <lb/>quietis cau&longs;&longs;a nulla alia a&longs;&longs;ignari pote&longs;t præter pri-<pb pagenum="5"/>mam illam rerum omnium conditricem, qu&ecedil;, ne o<lb/>mnia in omnibus e&longs;&longs;ent &longs;ingulis partibus vniuer&longs;i <lb/>&longs;ingula loca attribuit circa quæ conglobantur, & i<lb/>bi hærent. </s>

<s>Quicquid enim terreum e&longs;t, in terræ glo<lb/>bum <expan abbr="cõfluxit">confluxit</expan>. </s>

<s>Ita quicquid &longs;oli homogenes in Solis <lb/>corpus, lunaria omnia in lunam, & &longs;ic de cæteris cor|| <lb/>poribus vniuer&longs;i. </s>

<s>Singulæ autem eorum partes &longs;uis <lb/>totis adhærent, nec inde &longs;ponte <expan abbr="mou&etilde;tur">mouentur</expan>: &longs;i verò in<lb/>de motæ fuerint, eò redire nituntur. </s>

<s>Quod enim in <lb/>terra fieri videmus, idem & in reliquis corporibus v<lb/>niuer&longs;i fieri dicere po&longs;&longs;umus, &longs;cilicet quod vnita cir|| <lb/>ca locum &longs;uum naturalem maneant: &longs;i enim partes <lb/>eorum &longs;ponte &longs;epararentur, & vi ab ip&longs;is &longs;eparatæ <lb/>ad locum &longs;uum non redirent, &longs;tatim tota di&longs;&longs;olue<lb/>rentur. </s>

<s>Vis igitur illa in loco &longs;uo quie&longs;cendi grauitas di<lb/>citur, cuius contrarium e&longs;t leuitas. </s>

<s>Res autem nulla <lb/>per &longs;e leuis dicitur, &longs;ed <expan abbr="tãtùm">tantùm</expan> habita ratione alterius <lb/>loci, quàm &longs;ui naturalis: vt puta &longs;i pars aliqua Solis <lb/>vi quapiam in terram inuecta e&longs;&longs;et, &longs;uóque arbitrio <lb/>committeretur, illa &longs;tatim Solem peteret: ita &longs;i ter<lb/>ræ pars in Solem inuecta e&longs;&longs;et, &longs;uóque arbitrio com|| <lb/>mitteretur, &longs;tatim à &longs;ole euolaret, & &longs;e per cælum <lb/>terram ver&longs;us proriperet. </s>

<s>Vt igitur terræ partes in <lb/>terra graues &longs;unt: &longs;ic Solis partes in &longs;ole graues &longs;unt: <lb/>in terra verò leues, terræ verò partes in Sole leues <pb pagenum="6"/>&longs;unt. </s>

<s>Idem de cæteris corporibus vniuer&longs;i dici po<lb/>te&longs;t. </s>

<s>Neque enim faciliùs ex globo lunæ particulam <lb/>ab&longs;trahas, quàm ex terra glebam. </s>

<s>Aer autem nullum proprium habet locum natu|| <lb/>ralem, &longs;ed vbique e&longs;&longs;e pote&longs;t, cùm rerum omnium <lb/>locus vniuer&longs;alis e&longs;&longs;e videatur: de quo dubitari po<lb/>te&longs;t an &longs;it infinitus, cùm omnia contineat, à nulla re <lb/>verò totus comprehendatur, &longs;ed hæc di&longs;putatio non <lb/>e&longs;t huius loci. </s>

<s>Et&longs;i autem nullum proprium habet locum natu|| <lb/>ralem, neque leuis aut grauis &longs;it: leuis tamen e&longs;&longs;e vi<lb/>detur, cùm grauis non &longs;it, nec ægrè &longs;e moueri patia<lb/>tur. </s>

<s>Itaque &longs;i quo modo ita con&longs;titutus &longs;it, vt rem a<lb/>liquam, quò minùs ad locum &longs;uum naturalem per<lb/>uenire po&longs;&longs;it, impediat, tum per expre&longs;&longs;ionem &longs;eu <lb/><foreign lang="greek">e)/kqliyin</foreign>, ab ea eiicietur, in eiú&longs;que locum &longs;uccedet, <lb/>vt fit in aqua, cùm ei introducitur. </s>

<s>Quæ quidem <foreign lang="greek">e)/k<lb/>qliyis</foreign> iis etiam accidit quæ loco naturali gaudent, <lb/>cùm &longs;e&longs;e mutuò impediunt, ne ad illum ferantur. <lb/>Præ&longs;ertim verò in liquidis locum habet, quorum <lb/>partes facilius mouentur, quàm rerum <expan abbr="cohærenti&utilde;">cohærentium</expan> <lb/>& compactarum. </s>

<s>Inde accidit vt quicquid liquidis <lb/>immergitur, tantò fiat in illis leuius quàm grauius, <lb/>e&longs;t moles eiu&longs;dem liquoris ip&longs;orum moli æqualis. <lb/>Sed hæc de loco naturali fu&longs;iùs à nobis alibi medita<lb/>ta, obiter hîc attigi&longs;&longs;e &longs;ufficiat, vt intelligatur quid <pb pagenum="7"/>&longs;it vis naturalis. </s>

<s>Vis autem voluntaria nullum certum nutum <lb/>habet, &longs;ed illum tantùm in quo mouentis voluntas <lb/>conquie&longs;cit. </s>

<s>Cúmque vis naturalis vnum tantùm <lb/>nutum habeat, &longs;cilicet à loco à naturali requie re<lb/>moto, ad ip&longs;am naturalem requiem, illa infinitos nu|| <lb/>tus habet & indeterminatos per &longs;e ac voluntatis tan <lb/>tùm decreto determinabiles. </s>

<s>Fortuita verò ip&longs;o tantùm ca&longs;u determinabilem <lb/>nutum habet: quò enim ip&longs;a tendit, eo munere dici<lb/>tur &longs;iue &longs;ur&longs;um, &longs;iue deor&longs;um, &longs;iue ad latera. </s>

<s>Itaque <lb/>& hic & ille nutus <foreign lang="greek">a)dio/ristos</foreign> dicitur. </s>

<s>Quemadmodum <lb/>& is quem habent vires illæ, quibus res à &longs;itu <expan abbr="parti&utilde;">partium</expan> <lb/>naturali remotæ ad illum redeunt: prout enim ab <lb/>eo motæ &longs;unt, ita ad illum redeunt, prout etiam huc <lb/>aut illuc obuer&longs;æ &longs;unt, vt vis arcus aut bali&longs;tæ. </s>

<s>DEFIN. V.</s>

<s>Vires autem contrariæ dicuntur, quæ contrarios <lb/>motus ciere po&longs;&longs;unt, vt ea qu&ecedil; &longs;ur&longs;um mouet & qu&ecedil;<lb/>deor&longs;um, & &longs;ic de cæteris. </s>

<s>Con&longs;ideratur autem in vi <expan abbr="quãtitas">quantitas</expan>, tum eo quòd <lb/>vis partibus &longs;uis con&longs;tet, in quas in infinitum diuidi <lb/>pote&longs;t, & rur&longs;us additione aut multiplicatione au<lb/>geri, tum quòd æqualitatis exce&longs;&longs;us & defectus <expan abbr="cõ-parationem">con<lb/>parationem</expan> recipiat. </s><pb pagenum="8"/>

<s>Ac quoniam vis e&longs;t mouendi potentia, vis par<lb/>tes erunt quæ motus partes efficient, & quæ erit mo|| <lb/>tus partium, men&longs;ura eadem erit & vis partium. </s>

<s>Et <lb/>motus quidem propria men&longs;ura e&longs;t linea &longs;eu <expan abbr="&longs;pati&utilde;">&longs;patium</expan>. <lb/>Quantum enim res quæpiam mota, &longs;patij percur<lb/>rit, tantùm mota e&longs;&longs;e dicitur. </s>

<s>Quoniam verò in mo|| <lb/>mento vel in&longs;tanti quod in&longs;tar puncti e&longs;t, & magni<lb/>tudine caret, nullus motus fieri pote&longs;t, &longs;ed motus o<lb/>mnis in tempore fit. </s>

<s>Ideò ad motus men&longs;uram tem<lb/>pus etiam adhibere oportet. </s>

<s>Illud enim cum &longs;patio <lb/>vel linea, motus dici facit æquales aut inæquales. </s>

<s>DEFIN. VI.</s>

<s>Æquales igitur motus dicuntur, qui æqualibus <lb/>temporibus æqualia &longs;patia percurrunt. </s>

<s>Qui autem æqualibus temporibus æqualia &longs;pa<lb/>tia permeant, illi iidem proportionales &longs;unt: hoc <lb/>e&longs;t, quæ e&longs;t ratio temporis, quo alter eorum fit ad <lb/>tempus quo fit alter, eadem e&longs;t &longs;patij quod percurrit <lb/>alter ad &longs;patium quod reliquus percurrit. </s>

<s>Si enim <lb/>duorum quorum &longs;inguli vna hora miliaris vnius i<lb/>ter conficiunt, alter eodem motu per tria miliaria <lb/>ferri pergat, alter verò per duo <expan abbr="tãtùm">tantùm</expan>. </s>

<s>Ille tribus ho<lb/>ris ea ab&longs;oluet, hic verò duabus, & æquè celeriter <lb/>ferri dicentur, licet &longs;patia inæqualia <expan abbr="percurrãt">percurrant</expan>, quo<lb/>niam illa &longs;unt temporibus proportionalia. </s><pb pagenum="9"/>

<s>Inæquales autem motus dicuntur, quorum tem<lb/>pora non &longs;unt &longs;patiis proportionalia. </s>

<s>Eorum autem <lb/>maior ille dicitur, cuius maior erit ratio &longs;patij ad &longs;pa|| <lb/>tium, quàm temporis ad tempus, quibus fiunt illi <lb/>motus. </s>

<s>Ex quo intelligitur etiam quis minor dica<lb/>tur. </s>

<s>Maior igitur dicetur qui celeriùs feretur, minor, <lb/>qui tardiùs. </s>

<s>Æquales igitur vires dicentur, quæ æqualibus <lb/>motibus &longs;ubiecta &longs;ua mouebunt. </s>

<s>Maior verò quæ <lb/>&longs;ubiectum &longs;uum magis vel celeriùs mouebit. </s>

<s>Mi<lb/>nor, quæ minùs vel tardiùs. </s>

<s><emph type="italics"/>Motuum diui&longs;io.<emph.end type="italics"/></s>

<s>Comparantur autem motus &longs;ecundum omnes <lb/>comparationis gradus, hoc e&longs;t, æqualitatem, exce&longs;<lb/>&longs;um & defectum: tum &longs;ibi ip&longs;is, cùm totus &longs;uis par<lb/>tibus confertur, tum alij aliis. </s>

<s>Quatenus autem &longs;ibi ip&longs;i motus <expan abbr="cõparatur">comparatur</expan>, aut <lb/>e&longs;t <foreign lang="greek">o(moiomerh\s</foreign> &longs;eu æquabilis aut <foreign lang="greek">a)nomoiomerh\s</foreign>, id e&longs;t inæ<lb/>quabilis. </s>

<s>Aequabilis, cuius omnes partes tum mini<lb/>mæ tum maiores tempora habent &longs;patiis per quæ <lb/>cientur proportionalia. </s>

<s>Inæquabilis verò, in quo illa <lb/>non &longs;unt proportionalia. </s>

<s>Inæquabilium autem motuum duo &longs;unt genera, <pb pagenum="10"/>cre&longs;cens &longs;cilicet & decre&longs;cens, ídque vel continuè, <lb/>vel per interualla, ordinatim vel <foreign lang="greek">a)taktw=s. </foreign></s>

<s>Cre&longs;cens dicitur, quando partium po&longs;teriorum <lb/>ip&longs;ius maior e&longs;t celeritas, quàm priorum: <expan abbr="decre&longs;c&etilde;s">decre&longs;cens</expan>, <lb/>quando minor. </s>

<s>Continuè, quando nulla pars illius vel minima <lb/>alteri parti e&longs;t æqualis &longs;eu &longs;ibi ip&longs;i æquabilis. </s>

<s>Per interualla verò, quando partes quidem illius <lb/>&longs;ibi ip&longs;is æquabiles &longs;unt, aliæ verò aliis comparatæ <lb/>inæquabiles. </s>

<s>Ordinatim, quando incrementum aut <expan abbr="decrem&etilde;-tum">decremen<lb/>tum</expan> illud certa aliqua proportionis progre&longs;&longs;ione <lb/>fit, vt &longs;i in motu continuè cre&longs;cente, diui&longs;o toto mo|| <lb/>tus &longs;patio in aliquot partes æquales eadem &longs;it ratio <lb/>celeritatis, finis primi &longs;patij ad celeritatem finis &longs;e<lb/>cundi, quæ finis &longs;ecundi ad finem tertij, & &longs;ic dein<lb/>ceps: vel &longs;i finis &longs;ecundi &longs;patij duplo citiùs feratur, <lb/>quàm finis primi & finis tertij, triplo quàm primi: <lb/>finis verò quarti quadruplo, & &longs;ic deinceps. </s>

<s>Idem &longs;i <lb/>in quauis alia proportionis progre&longs;&longs;ione illarum <lb/>partium celeritas, aliarum ad alias &longs;e habeat. </s>

<s>In mo<lb/>tu verò per interualla cre&longs;cente <expan abbr="augm&etilde;tum">augmentum</expan> illud or<lb/>dinatim fieri dicetur, &longs;i interuallorum proportio a<lb/>liquem progre&longs;&longs;ionis ordinem &longs;eruet: puta &longs;i <expan abbr="prim&utilde;">primum</expan> <lb/>interuallum duplo tardiùs <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> moueatur, triplo <lb/>verò tardiùs tertio, &c. </s><pb pagenum="11"/>

<s>Inordinatè autem cre&longs;cere vel decre&longs;cere dicetur <lb/>motus, &longs;i incrementum illud aut decrementum nul|| <lb/>la certa & ordinata proportione fiat. </s>

<s>Comparantur autem motus alij aliis, æquabiles <lb/>&longs;cilicet & inæquabiles ordinatim progredientes: de <lb/>inordinatorum enim tum inter &longs;e, tum cum aliis <expan abbr="cõ||">con||</expan> <lb/>paratione nihil certò &longs;tatui pote&longs;t. </s>

<s>Comparantur igitur æquabiles cum æquabili<lb/>bus aut cum inæquabilibus, ac etiam inæquabiles <lb/>cum inæquabilibus. </s>

<s>Æquabiles autem cum æquabilibus compara<lb/>tionibus &longs;uis partibus corre&longs;pondentibus &longs;unt pro<lb/>portionales. </s>

<s>Si enim <expan abbr="proponãtur">proponantur</expan> duo motus æqua<lb/>biles, quorum alter altero maior &longs;it, quæ erit ratio <lb/>dimidiæ partis vnius ad dimidiam partem alterius, <lb/>eadem erit tertiæ ad tertiam, & &longs;ic deinceps. </s>

<s>Æquabilium verò motuum cum inæquabilibus, <lb/>cum iis &longs;cilicet qui per interualla cre&longs;cunt, compa<lb/>ratio fit tanquam cum pluribus diuer&longs;is motibus æ<lb/>quabilibus, &longs;unt enim interuallorum <expan abbr="illor&utilde;">illorum</expan> motus <lb/>æquabiles. </s>

<s>At cum continuè <expan abbr="cre&longs;c&etilde;tibus">cre&longs;centibus</expan> aut decre<lb/>&longs;centibus difficile e&longs;t comparationis modum defi<lb/>nire, cùm ne momento quidem &longs;tabilis maneat par|| <lb/>tium illarum quantitas. </s>

<s>E&longs;t tamen motus æquabilis <lb/>illorum men&longs;ura: tanti enim e&longs;&longs;e dicuntur, quanti <lb/>forent, &longs;i in ea celeritatis men&longs;ura ad quam perue<pb pagenum="12"/>nerunt, æquabiliter pergerent. </s>

<s>Cre&longs;centes verò cum cre&longs;centibus continuè, <expan abbr="cõ-parati">con<lb/>parati</expan> &longs;iquidem eadem progre&longs;&longs;ionis proportione <lb/>cre&longs;cant, æqualia &longs;patia ab eorum principiis nume<lb/>rata, æqualibus temporibus emen&longs;a habebunt, idem <lb/>in decre&longs;centibus. </s>

<s>Sed h&ecedil;c hactenus inqui&longs;iui&longs;&longs;e &longs;uf<lb/>ficiat: &longs;i quis enim particulatim omnia <expan abbr="exp&etilde;dere">expendere</expan> vel<lb/>let, in infinitum &longs;e&longs;e extenderet h&ecedil;c partium & pro<lb/>portionum motuum &longs;upputatio. </s>

<s>Iam videamus quæ vires quos horum motuum <lb/>cieant. </s>

<s>Et vis quidem ea qua res quælibet ad &longs;itum <lb/>naturalem &longs;uarum partium redit, motu <expan abbr="cõtinuè">continuè</expan> de<lb/>cre&longs;cente mouet: quo enim propiùs ad &longs;itum natu<lb/>ralem partium accedit, eo tardior & languidior e&longs;t. <lb/>Arcus enim cùm primùm laxari incipit, celeriùs <lb/>mouetur, quàm cùm motus &longs;ui finis propior e&longs;t. </s>

<s>Vis verò naturalis, qua res qu&ecedil;libet ad <expan abbr="loc&utilde;">locum</expan> &longs;uum <lb/>naturalem tendit, &longs;ubiectum &longs;uum, motu continuè <lb/>& ordinatim cre&longs;cente, mouet. </s>

<s>Illius autem motus <lb/>cau&longs;a e&longs;t quòd faciliùs id moueatur, quod in motu <lb/>e&longs;t, quàm quod quie&longs;cit. </s>

<s>Vis igitur eadem, <expan abbr="&longs;ubiect&utilde;">&longs;ubiectum</expan> <lb/>quod iam in motu e&longs;t premens, illud magis moue<lb/>bit, quàm &longs;i quie&longs;cat, & magis motum, magis etiam <lb/>mouebit: ita vt eadem vis motione maior fiat, quàm <lb/>per &longs;e &longs;it. </s>

<s>Et hæc e&longs;t cau&longs;a cur ictus, quo magis ab al<lb/>tero venit, eo vehementior &longs;it. </s>

<s>Motus autem huius <pb pagenum="13"/>&longs;patia hanc celeritatis proportionem &longs;eruant, vt qu&ecedil;<lb/>e&longs;t ratio totius &longs;patij, per quod fit ille motus ad par<lb/>tem ip&longs;ius (vtriu&longs;que initio inde &longs;umpto, vbi e&longs;t mo|| <lb/>tus initium) eadem &longs;it celeritatis ad celeritatem. <lb/><figure id="fig1"></figure><lb/>Exempli gratia, &longs;i vis aliqua per lineam ABC <lb/>mouerit, &longs;itque AB illius lineæ pars, quæ erit <lb/>ratio AC ad AB, eadem erit celeritatis motus <lb/>in puncto C ad <expan abbr="celeritat&etilde;">celeritatem</expan> motus in puncto B. <lb/>Cuiu&longs;modi proportio ob&longs;eruatur in paralle<lb/>lis triangulum &longs;ecantibus. </s>

<s>Vt enim &longs;e habet <lb/><figure id="fig2"></figure><lb/>AC ad AB, &longs;ic CG ad BF, & vt AD ad <lb/>AC, &longs;ic DH ad CG. </s>

<s>Itaque &longs;i in &longs;patia ali<lb/>quot æqualia diuidatur totius motus &longs;pa|| <lb/>tium, finis &longs;ecundi duplo citiùs feretur, <lb/>quàm finis primi: finis verò tertijtriplo <lb/>citiùs quàm finis primi, & &longs;ic deinceps. <lb/>Hac autem ratione fit, vt &longs;patiorum illo<lb/>rum initio maxima &longs;it celeritatis <expan abbr="differ&etilde;-tia">differen<lb/>tia</expan>: progre&longs;&longs;u verò &longs;emper minuatur, ac tandem fer<lb/>mè eadem &longs;it, vt fit in trianguli lateribus, quæ lon<lb/>gi&longs;&longs;imè producta æquè di&longs;tare videntur. </s>

<s>Eáque e&longs;t <lb/>ratio cur Solis & Lunæ radij, etiam&longs;i concurrant (in <lb/>ip&longs;orum &longs;cilicet corporibus, aut vltra res quas <expan abbr="illu-&longs;trãt">illu<lb/>&longs;trant</expan>) paralleli <expan abbr="tam&etilde;">tamen</expan> <expan abbr="appareãt">appareant</expan>. </s>

<s><expan abbr="Ead&etilde;">Eadem</expan> <expan abbr="etiã">etiam</expan> cau&longs;a e&longs;t cur <lb/>line&ecedil; omnes ad <expan abbr="perp&etilde;dicul&utilde;">perpendiculum</expan> in <expan abbr="terrã">terram</expan> cadentes, paral<lb/>lelæ videantur, cùm <expan abbr="tam&etilde;">tamen</expan> in centro terræ <expan abbr="cõcurrant">concurrant</expan>. </s><pb pagenum="14"/>

<s>Hunc igitur motum vis naturalis efficit, modò <lb/>nulla quies intercedat. </s>

<s>Vis autem voluntaria motum omnem ciere apta <lb/>e&longs;t. </s>

<s>Fortuita verò inordinatum tantùm motum effi<lb/>cit. </s>

<s>Et hæc de variis virium motibus, <expan abbr="eorúmq;">eorúmque</expan> pro<lb/>portione & partibus. </s>

<s>E&longs;t verò etiam virium partitio quædam, quæ ex <lb/>eorum quibus <expan abbr="inhær&etilde;t">inhærent</expan> &longs;ubiectorum diui&longs;ione, aut <lb/>alterius ad alterum proportione re&longs;ultat. </s>

<s>In &longs;ubie<lb/>ctis enim, quæ <foreign lang="greek">o(moiomerh=</foreign> &longs;unt, vis naturalis æqualiter <lb/>per omnes partes diffu&longs;a e&longs;t: ita vt quæ e&longs;t ratio mo<lb/>lis ad molem &longs;ub eadem figura, <expan abbr="ead&etilde;">eadem</expan> &longs;it vis ad vim: <lb/>vt &longs;i globus alicuius metalli, ad alium globum eiu&longs;<lb/>dem metalli mole quadruplus &longs;it, illius quoque <expan abbr="põ-dus">pon<lb/>dus</expan> ad huius pondus quadruplum erit. </s>

<s>Hactenus vim <expan abbr="cõ&longs;iderauimus">con&longs;iderauimus</expan> quatenus mouet, <lb/>&longs;upere&longs;t vt eam con&longs;ideremus, quatenus mouenti <lb/>re&longs;i&longs;tit. </s>

<s>Habet autem locum re&longs;i&longs;tentia, vbi duæ vi<lb/>res contrariæ committuntur: &longs;i enim vtraque <expan abbr="eod&etilde;">eodem</expan> <lb/>motu moueat, nulla erit re&longs;i&longs;tentia, &longs;ed altera alteri <lb/>addita maiorem vim con&longs;tituet. </s>

<s>Committi autem dicuntur vires, quando ita ap<lb/>plicantur & <expan abbr="connect&utilde;tur">connectuntur</expan>, vt altera nutu &longs;uo moueri <lb/>non po&longs;&longs;it, quin altera contra nutum &longs;uum mouea<lb/>tur. </s>

<s>Viribus autem ita connexis, accidit vt altera in <lb/>alterius &longs;ubiectum agat, & altera in &longs;ubiecto &longs;uo exi-<pb pagenum="15"/>&longs;tens, alteri in altero &longs;ubiecto exi&longs;tenti re&longs;i&longs;tat, ita vt <lb/>eorum &longs;ubiecta quodammodo reciproca fiant. </s>

<s>CONCLVSIO I.</s>

<s>Illam autem re&longs;i&longs;tentiam vi momenti in eodem <lb/>&longs;ubiecto æqualem, aut eandem cum ea e&longs;&longs;e, ex ip&longs;ius <lb/>definitione con&longs;tat. </s>

<s>E&longs;t enim re&longs;i&longs;tere <expan abbr="nõ">non</expan> pati &longs;e mo<lb/>ueri. </s>

<s>Quantum autem vis quælibet <expan abbr="&longs;ubiect&utilde;">&longs;ubiectum</expan> &longs;uum <lb/>nutu &longs;uo mouet, tantum illud contra nutum &longs;uum <lb/>moueri non patitur. </s>

<s>Mouendi igitur & <expan abbr="mou&etilde;ti">mouenti</expan> re<lb/>&longs;i&longs;tendi potentia in eodem &longs;ubiecto æquales &longs;unt. </s>

<s>Quemadmodum autem in eodem &longs;ubiecto ma<lb/>ior vis ine&longs;&longs;e dicitur, in quo e&longs;t maioris motus po<lb/>tentia, &longs;ic maior re&longs;i&longs;tentia erit maioris motus con<lb/>trarij impatientia. </s>

<s>Eadem igitur vis magis mouere <lb/>nitenti <expan abbr="cõtra">contra</expan> ip&longs;ius nutum magis re&longs;i&longs;tet, minus ve<lb/>rò nitenti minùs re&longs;i&longs;tet. </s>

<s>Et quo maior erit motus contrarius, eo magis re<lb/>&longs;i&longs;tet: id e&longs;t, quo celerius vis quælibet à nutu &longs;uo re<lb/>uelletur, eo magis re&longs;i&longs;tet. </s>

<s>Duplo igitur citiùs re<lb/>uellenti duplo re&longs;i&longs;tet, triplo citius triplo re&longs;i&longs;tet, & <lb/>&longs;ic <foreign lang="greek">a)nalo/gws</foreign> in omni proportione. </s>

<s>Quæ igitur erit <lb/>ratio motus ad motum, contra nutum eadem erit <lb/>in vna & eadem vi ratio re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam. </s>

<s>Si verò plures vires comparentur. </s><pb pagenum="16"/>

<s>Æqualium quidem virium æqualibus motibus <lb/>æquales erunt re&longs;i&longs;tentiæ. </s>

<s>Si enim æquales vires æ<lb/>qualiter à &longs;uis nutibus reuellantur, æqualiter re&longs;i<lb/>&longs;tent. </s>

<s>Inæqualibus verò motibus, earum re&longs;i&longs;tenti&ecedil; in<lb/>æquales erunt, & motuum proportionem &longs;equen<lb/>tur. </s>

<s>Exempli gratia, &longs;int duæ vires A & B, quarum v<lb/>traque vna hora miliare vnum percurrat, illæ æqua<lb/>les erunt. </s>

<s>Si igitur vtraque quarta parte vnius milia<lb/>ris, vel dimidio miliari à &longs;uo nutu eodem tempore <lb/>reuellatur, eadem erit vtriu&longs;que illi motui re&longs;i&longs;ten<lb/>tia. </s>

<s>Sin verò altera quidem quarta parte vnius milia<lb/>ris à nutu &longs;uo reuellatur, altera verò eodem tempo<lb/>re integro miliari reuellatur: hæc re&longs;i&longs;tentia ad il<lb/>lam quadrupla erit, & &longs;ic de cæteris motuum pro<lb/>portionibus. </s>

<s>Inæqualium verò virium re&longs;i&longs;tentiæ æqualibus <lb/>motibus ip&longs;arum virium proportionem &longs;equentur. <lb/>Quo maior enim e&longs;t vis, eo magis eodem tempore <lb/>eodémque &longs;patio contra nutum mouenti re&longs;i&longs;tit. <lb/><expan abbr="Ex&etilde;pli">Exempli</expan> gratia, &longs;it vis A quadrupla ad vim B, vis A mo<lb/>uenti &longs;e vno miliari quadruplo re&longs;i&longs;tet, quàm B mo<lb/>uenti &longs;e vno miliari eodem tempore. </s><pb pagenum="17"/>

<s>Inæqualium verò virium inæqualibus motibus <lb/>re&longs;i&longs;tentiæ, quando motus eam inter &longs;e proportio<lb/>nem &longs;eruabunt, quæ e&longs;t inter eas <foreign lang="greek">a)ntipeponqw=s</foreign>, &longs;eu <lb/>reciprocè, æquales erunt. </s>

<s>Sit enim vis A quæ ad vim <lb/>B eandem habeat <expan abbr="ration&etilde;">rationem</expan> quam C ad D, puta quam <lb/>3 ad 1, aut 4 ad 1, aut quæcunque alia &longs;it, <expan abbr="moueatúrq;">moueatúrque</expan> <lb/>vtraque &longs;patio E contra nutum &longs;uum eodem tem<lb/>pore, re&longs;i&longs;tentia A ad re&longs;i&longs;tentiam B &longs;e habebit, vt C <lb/>ad D per proximam conclu&longs;ionem. </s>

<s>Si verò B moue<lb/>atur eodem tempore &longs;patio aliquo quod &longs;e habeat <lb/>ad &longs;patium E vt C ad D, re&longs;i&longs;tentia B huic motui &longs;e <lb/>habebit ad re&longs;i&longs;tentiam priori motui, vt C ad D per <lb/>quartam <expan abbr="conclu&longs;ion&etilde;">conclu&longs;ionem</expan>. </s>

<s>Duæ ergo re&longs;i&longs;tentiæ ad ean<lb/>dem, eandem habebunt rationem, ergo æquales e<lb/>runt per <emph type="italics"/>9<emph.end type="italics"/> prop. <emph type="italics"/>5.<emph.end type="italics"/> Elem. </s>

<s>Si verò motuum virium inæqualium proportio <lb/>non &longs;it eadem quæ e&longs;t ip&longs;arum virium reciprocè, vis <lb/>illius quæ ad alteram maiorem habebit rationem, <lb/>quàm motus alterius ad motum ip&longs;ius, re&longs;i&longs;tentia <lb/>maior erit, altera verò minor. </s>

<s>Exempli gratia, &longs;i vis <lb/>A &longs;it tripla ad vim B, & motus quo B à nutu &longs;uo reuel|| <lb/>letur, &longs;it minor quàm triplus ad motum A, re&longs;i&longs;ten<lb/>tia A erit maior quàm re&longs;i&longs;tentia B. </s>

<s>Idem erit, &longs;i ma<lb/>ior &longs;it ratio A ad B, quàm 3 ado, motus verò B ad mo<pb pagenum="18"/>tum A &longs;it triplus. </s>

<s>Contrà verò &longs;i A ad B &longs;it tripla: mo<lb/>tus verò B ad motum A &longs;it maior quàm triplus, ma<lb/>ior erit re&longs;i&longs;tentia B quàm A. </s>

<s>Idem accidet, &longs;i A ad B <lb/>ratio minor &longs;it quàm 3 ad 1, motus verò B ad motum <lb/>A &longs;it triplus. </s>

<s>Hunc autem exce&longs;&longs;um & defectum <expan abbr="nõ">non</expan> <lb/>vlteriùs &longs;crutabimur, vt ip&longs;ius quantitatis <expan abbr="men&longs;urã">men&longs;uram</expan> <lb/>a&longs;&longs;equamur: alioqui in infinitum fieret progre&longs;&longs;us. </s>

<s>Ex his colligimus <expan abbr="re&longs;i&longs;t&etilde;tias">re&longs;i&longs;tentias</expan> tribus modis æqua<lb/>les aut inæquales dici, per &longs;e &longs;cilicet aut motione, aut <lb/>vtroque modo. </s>

<s>Per &longs;e quidem cùm &longs;ubiectorum <lb/>vires &longs;unt æquales (id e&longs;t æqualiter &longs;ua &longs;ubiecta mo<lb/>uent) aut inæquales. </s>

<s>Motione verò quando per &longs;e <lb/>quidem in æquales &longs;unt, motione verò æquales <expan abbr="fi&utilde;t">fiunt</expan>: <lb/>aut quando per &longs;e æquales &longs;unt, motione verò inæ<lb/>quales <expan abbr="fi&utilde;t">fiunt</expan> eo, quo dictum e&longs;t, modo. </s>

<s>Vtroque mo|| <lb/>do æquales aut inæquales <expan abbr="dic&utilde;tur">dicuntur</expan>, quando tum per <lb/>&longs;e, tum motione tales fiunt. </s>

<s>His explicatis videamus <lb/>quis eorum &longs;it effectus, vbi vires committentur. </s>

<s>Primùm ex prima conclu&longs;ione &longs;equitur qu&ecedil; erit <lb/>ratio re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam, eandem fore re&longs;i<lb/>&longs;tentiæ ad vim cuius e&longs;t altera re&longs;i&longs;tentia: &longs;unt enim <lb/>vis & <expan abbr="re&longs;i&longs;t&etilde;tia">re&longs;i&longs;tentia</expan> in <expan abbr="eod&etilde;">eodem</expan> &longs;ubiecto æquales. </s>

<s>Ergo mul<lb/>to magis &longs;equitur, &longs;i <expan abbr="re&longs;i&longs;t&etilde;tia">re&longs;i&longs;tentia</expan> altera maior, altera mi|| <lb/>nor fuerit, illam huius vi <expan abbr="maior&etilde;">maiorem</expan> fore, vel illius vim <lb/>hac re&longs;i&longs;tentia maiorem, &longs;i æqualis æqualem. </s><pb pagenum="19"/>

<s>Re&longs;i&longs;tentia autem vi contrariæ commi&longs;&longs;a <expan abbr="tãtum">tantum</expan> <lb/>de ea tollit quanta e&longs;t ip&longs;a re&longs;i&longs;tentia. </s>

<s>Sublata autem <lb/>vi tollitur motus: &longs;ublata verò re&longs;i&longs;tentia, &longs;i vis ad&longs;it, <lb/>&longs;equitur motus. </s>

<s>Vbi igitur vis & re&longs;i&longs;tentia in&ecedil;qua<lb/>les committentur, &longs;i vis maior &longs;it quàm re&longs;i&longs;tentia, <lb/>fiet motus &longs;ecundum vis illius nutum, & contra nu<lb/>tum vis illius quæ re&longs;i&longs;tit. </s>

<s>Si verò re&longs;i&longs;tentia maior <lb/>fuerit, tum ip&longs;a fiet vis mouens, & vim contrariam <lb/>contra nutum ip&longs;ius mouebit, dum ip&longs;a nutu &longs;uo <lb/>mouebitur. </s>

<s>Hinc &longs;equuntur duo theoremata, circa <lb/>quæ totius huius con&longs;iderationis cardo vertitur. </s>

<s>THEOREMA I.</s>

<s>Duarum virium connexarum, quarum (&longs;i mo<lb/>ueantur) motus erunt ip&longs;is <foreign lang="greek">a)ntipeponqw=s</foreign> proportiona<lb/>les neutra alteram mouebit, &longs;ed æquilibrium <expan abbr="faci&etilde;t">facient</expan>. </s>

<s>Sit vis A commi&longs;&longs;a cum vi B, &longs;itque vis A ad vim <lb/>B ratio per &longs;e, vt C ad D <expan abbr="quæc&utilde;que">quæcunque</expan> illa &longs;it, &longs;iue dupla, <lb/>&longs;iue tripla, &longs;iue alia. </s>

<s>Sit etiam <expan abbr="ead&etilde;">eadem</expan> ratio motus quo <lb/>B mouebitur, &longs;i ita, vt connexa &longs;unt, moueantur) ad <lb/>motum quo A mouebitur, quæ e&longs;t C ad D, dico mo<lb/>tum non &longs;equuturum, &longs;ed factum iri æquilibrium. <lb/>Aut enim A & B vires per &longs;e erunt æquales aut inæ<lb/>quales: &longs;i æquales, ergo & ip&longs;arum motus æquales e<lb/>runt: &longs;unt enim ex hypothe&longs;i ip&longs;is proportionales: <pb pagenum="20"/>ergo per quartam conclu&longs;ionem earum re&longs;i&longs;tentiæ <lb/>æquales erunt, ergo per decimam conclu&longs;ionem <expan abbr="nõ">non</expan> <lb/>fiet motus. </s>

<s>Si verò &longs;unt inæquales, cùm earum mo<lb/>tus ex hypothe&longs;i &longs;int <foreign lang="greek">a)ntipeponqw=s</foreign> proportionales per <lb/>&longs;eptimam conclu&longs;ionem æquales erunt etiam re&longs;i<lb/>&longs;tentiæ. </s>

<s>Ergo nec motus fiet: nullo igitur modo fiet <lb/>earum motus. </s>

<s>Quod demon&longs;trandum erat. </s>

<s>THEOREMA II.</s>

<s>Quarum verò ita connexarum (&longs;i <expan abbr="moueãtur">moueantur</expan>) mo|| <lb/>tus, ip&longs;is proportionales non erunt: illa alteram mo<lb/>uebit, cuius ad alteram ratio maior erit, quàm huius <lb/>motus ad illius motum. </s>

<s>Sit vis A cum vi B commi&longs;&longs;a, &longs;itque A ad B ratio <lb/>per &longs;e, vt C ad D: ita verò connexæ &longs;int, vt &longs;i mouean<lb/>tur, minor &longs;it ratio motus quo B mouebitur ad mo<lb/>tum, quo A mouebitur, quum C ad D dico A motu<lb/>ram B: erit enim per octauam conclu&longs;ionem re&longs;i<lb/>&longs;tentia B minor re&longs;i&longs;tentia A: ergo per nonam con<lb/>clu&longs;ionem re&longs;i&longs;tentia B minor vi A: vis igitur A vim <lb/>B mouebit per vndecimam conclu&longs;ionem. </s>

<s>Quod <lb/>demon&longs;trandum erat. </s>

<s>Hinc po&longs;&longs;emus ad problematis no&longs;tri demon<lb/>&longs;trationem rectà pergere: ante tamen cra&longs;&longs;iùs ali<lb/>quanto hæc explicanda &longs;unt, vt à quouis faciliùs in<lb/>telligantur. </s>

<s>Duæ &longs;unt motus men&longs;uræ, locus &longs;cilicet <lb/>& tempus: vtroque igitur, tempore videlicet & loco <pb pagenum="21"/>maior, minor aut æqualis dicitur: & quo minori <expan abbr="t&etilde;-pore">ten<lb/>pore</expan> idem &longs;patium ab&longs;oluitur, eo maior e&longs;t, vel quo <lb/>maius &longs;patium eodem tempore. </s>

<s>Vt igitur motus magnus dicatur, perinde e&longs;t &longs;i <lb/>paruo tempore fiat, aut magno &longs;patio. </s>

<s>Quod autem maiorem <expan abbr="mot&utilde;">motum</expan> ciere pote&longs;t illud, <lb/>vis maior dicitur, quòd minorem minor. </s>

<s>Vis autem &longs;eu mouendi potentia in eodem &longs;ub<lb/>iecto certa & finita e&longs;t. </s>

<s>Qu&ecedil;libet enim res vi natura<lb/>li pr&ecedil;dita, &longs;i à loco naturali ab&longs;it, &longs;uóque arbitrio <expan abbr="cõ-mittatur">con<lb/>mittatur</expan>, certo tempore eò redit. </s>

<s>E&longs;t enim certum <lb/>in rerum natura quanto tempore libræ vnius <expan abbr="põdus">pondus</expan> <lb/>deor&longs;um &longs;ponte &longs;ua delatum, miliare vnum aut duo <lb/>conficiat pro ratione materi&ecedil;, vel quantum &longs;patij v<lb/>na vel duabus horis percurrat. </s>

<s>Id verò quantum &longs;it, <lb/>hominum indu&longs;tria nondum quod &longs;ciam explora<lb/>tum e&longs;t: aliâs autem id demon&longs;trare conabimur. </s>

<s>Mouendi verò potentia in alieno &longs;ubiecto infi<lb/>nita e&longs;t, hoc e&longs;t, in <expan abbr="infinit&utilde;">infinitum</expan> augeri vel minui pote&longs;t, <lb/>quoniam in finita auctione & diminutione e&longs;t re&longs;i<lb/>&longs;tentia: tanta enim e&longs;t, quanta e&longs;t in eodem &longs;ubiecto <lb/>vis: <expan abbr="quantú&longs;q;">quantú&longs;que</expan> motus illius e&longs;t vis alterius re&longs;pectu. <lb/>Quo igitur vis quæ alienum &longs;ubiectum mouere ni<lb/>titur, illud magis mouere nitetur, eo minùs illud <lb/>mouere poterit, maior enim erit illius re&longs;i&longs;tentia. <lb/>Quemadmodum enim quod magis nutu &longs;uo mo<pb pagenum="22"/>uetur, maiorem vim mouendi habet. </s>

<s>Ita illud idem <lb/>quod magis contra nutum &longs;uum mouetur, <expan abbr="maior&etilde;">maiorem</expan> <lb/>vim re&longs;i&longs;tendi habet. </s>

<s>E contrario verò quo vis quæ<lb/>libet minorem motum in alieno &longs;ubiecto ciere ni<lb/>tetur, eo faciliùs illud mouebit. </s>

<s>Tarditate igitur motus, re&longs;i&longs;tentia in infinitum <lb/>minui pote&longs;t: minuta verò re&longs;i&longs;tentia vis contrariæ <lb/>effectus augetur, ita vt vis quæ per &longs;e minima e&longs;t, in <lb/>contrariam cui plurimum diminuta &longs;it, re&longs;i&longs;tentia <lb/>maximè agat. </s>

<s>Perinde igitur e&longs;t, &longs;i vis mouens ma<lb/>gna &longs;it, mouenda verò parua, ac &longs;i illa celeriter fera<lb/>tur, hæc verò tardè: <expan abbr="quãtum">quantum</expan> enim vis mouens &longs;i ma<lb/>gna fuerit in mouendam minorem poterit, tantum <lb/>vis parua celeriter mota in magnam tardè motam <lb/>poterit. </s>

<s>Si igitur velimus vt vis parua magnam mo<lb/>ueat, eas ita collocare oportet, vt quantum magna <lb/>paruam &longs;uperat, tantum illi de motus celeritate de<lb/>trahatur, & aliquid ampliùs. </s>

<s>Si enim tanta &longs;it tardi<lb/>tas motus vis vnius, re&longs;pectu motus vis alterius, <expan abbr="quã-ta">quan<lb/>ta</expan> e&longs;t proportio vis illius, ad hanc non fiet motus: vt <lb/>&longs;i pondus A quatuor librarum cum pondere B libræ <lb/>vnius committatur: &longs;intque ita connexa, vt dum <lb/>A vno &longs;patio mouebitur, B quatuor &longs;patiis mouea<lb/>tur, ita vt motus A quadruplo tardior &longs;it motu B <expan abbr="nõ">non</expan> <lb/>fiet motus, quia quantum A excedit B pondere tan<lb/>tùm deficit motus tarditate. </s><pb pagenum="23"/>

<s>Tantum enim e&longs;t libram vnam quatuor &longs;patiis <lb/>moueri, quantum libras quatuor vno &longs;patio eodem <lb/>tempore: &longs;i igitur alterutri eorum ita <expan abbr="con&longs;titutor&utilde;">con&longs;titutorum</expan>, <lb/>vel momentum vis addatur, id cui additum fuerit, <lb/>alterum mouebit. </s>

<s>Idem fiet &longs;i A ita connectatur vt <lb/>vel momento citiùs quàm moueri po&longs;itum e&longs;t, mo|| <lb/>ueatur vel B tardiùs. </s>

<s>Id etiam alia ratione o&longs;tendi pote&longs;t, &longs;i vis A qua<lb/>drupla &longs;it ad vim B, erunt in A quatuor partes, ip&longs;i B <lb/>æquales. </s>

<s>Si igitur B cum &longs;ingulis illis committatur, <lb/>ita vt æqualiter moueantur, non fiet motus: &longs;i verò <lb/>alterutri aut ip&longs;i B, aut &longs;ingulis illis partibus vis vel <lb/>motus momentum adiiciatur vel detrahatur, illa cui <lb/>adiectum fuerit, aut cui non detractum fuerit, nutu <lb/>&longs;uo mouebitur, & aliam contra nutum eius moue<lb/>bit. </s>

<s>Addito igitur ip&longs;i B momento, dum vno &longs;patio <lb/>mouebitur, &longs;ingulas illas partes vno &longs;patio moue<lb/>bit. </s>

<s>Vbi igitur &longs;ingulas &longs;emel mouerit, ip&longs;a quater <lb/>mota erit. </s>

<s>Dum verò &longs;ingulæ &longs;emel motæ erunt, to<lb/>tum ex illis con&longs;tans &longs;emel motum intelligetur: po<lb/>terit igitur B addito ip&longs;i momento, dum quater mo|| <lb/>uebitur, &longs;emel totum A mouere. </s>

<s>Tertio modo id ip&longs;um concludere po&longs;&longs;umus. </s>

<s>Si <lb/>duæ vires æquales connectantur, ita vt motæ, æqua<lb/>liter moueantur, altera in <expan abbr="alterã">alteram</expan> non aget. </s>

<s>Si verò ita <lb/>connectantur vt motæ inæqualiter moueantur, <pb pagenum="24"/>quantum altera alteram celeritate &longs;uperabit, <expan abbr="tãtum">tantum</expan> <lb/>& vi &longs;uperabit. </s>

<s>Vt &longs;i vis A vi B æqualis &longs;it, ac ita <expan abbr="con-nectãtur">con<lb/>nectantur</expan>, vt B quatuor &longs;patiis moueatur, dum A vno <lb/>&longs;patio mouebitur, B quadruplam vim habebit ad A, <lb/>quia eam in motibus &longs;uis proportionem &longs;eruant, <lb/>quam &longs;i <expan abbr="&longs;eruar&etilde;t">&longs;eruarent</expan> arbitrio &longs;uo commi&longs;&longs;æ, B ad A qua<lb/>drupla e&longs;&longs;et. </s>

<s>Si igitur illis ita connexis, ip&longs;i A adda<lb/>tur vis triplo maior quàm ip&longs;a &longs;it, B illis quatuor re<lb/>&longs;i&longs;tet, nec fiet motus. </s>

<s>Duarum igitur virium comparatarum, quanto <lb/>altera &longs;ubiectum &longs;uum celeriùs mouebit, quàm al<lb/>tera: tanto illa hanc celeriùs mouere poterit, quàm <lb/>ip&longs;a moueatur, &longs;i illi vis <expan abbr="mom&etilde;tum">momentum</expan> additum fuerit. <lb/>Ita vt quæ erit proportio vis ad vim, eadem &longs;it mo<lb/>tus, quem illa in hac ciere pote&longs;t ad motum quo ip&longs;a <lb/>mouebitur. </s>

<s>Et è conuer&longs;o quæ erit ratio motus ad <lb/>motum, eadem erit vis cui additum fuerit momen<lb/>tum ad eam quam ip&longs;a mouere poterit ratio reci<lb/>procè. </s>

<s>Iam ad problematis no&longs;tri demon&longs;tratio<lb/>nem veniamus. </s>

<s><emph type="italics"/>Propo&longs;iti problematis demon&longs;tratio.<emph.end type="italics"/></s>

<s>Sit data vis A quantacunque illa &longs;it magna vel <lb/>parua: datum verò pondus B quantumquantum il<lb/>lud &longs;it, dico me vi A pondus B tollere po&longs;&longs;e. </s>

<s>Id &longs;ic de<lb/>mon&longs;tro. </s><pb pagenum="25"/>

<s>Primùm enim ex doctrina &longs;ecundi <expan abbr="l&etilde;matis">lemmatis</expan>, quod <lb/>inferiùs demon&longs;trabitur, &longs;ciam proportionem pro<lb/>ximè maiorem, quàm &longs;it A ad B proportio. </s>

<s>Deinde <lb/>ex doctrina primi lemmatis ita connectam A & B, <lb/>vt quando ambo mouebuntur, nunc &longs;it ratio motus <lb/>quo B mouebitur ad <expan abbr="mot&utilde;">motum</expan> quo A mouebitur, quàm <lb/>&longs;it A ad B. </s>

<s>His peractis &longs;equitur vim A pondus B mo<lb/>turam ex &longs;ecundo &longs;uperiùs demon&longs;trato theorema<lb/>te. </s>

<s>Quod erat propo&longs;itum. </s>

<s>LEMMA I.</s>

<s>Duas vires ita connectere, vt &longs;i moueantur, <expan abbr="ear&utilde;">earum</expan> <lb/>motus, in data ratione alter ad alterum &longs;e habeant <lb/>vires contrariæ, aut medio aliquo, aut per &longs;e ab&longs;que <lb/>vllo medio committuntur. </s>

<s>Si ab&longs;que medio com<lb/>mittant, eodem motu mouebuntur, maior enim mi|| <lb/>norem eodem motu, quo ip&longs;a moueri poterit, mo<lb/>uebit: aut æquilibrium <expan abbr="faci&etilde;t">facient</expan>, &longs;i æquales &longs;int: vt &longs;i le<lb/>ue graui committatur, &longs;iquidem leuitas grauitate <lb/>maior &longs;it, attolletur graue: &longs;in verò grauitas maior <lb/>&longs;it, leue deprimetur: &longs;i æqualia &longs;int, non mouebun<lb/>tur. </s>

<s>Si verò medio aliquo connectantur mediorum <lb/>varia &longs;unt genera. </s>

<s>Aut enim medium e&longs;t flexibile & <lb/><foreign lang="greek">o(moiomere\s</foreign>, vt funis, catena, &c. </s>

<s>aut e&longs;t inflexibile, il<lb/>lúdque aut rectum, aut curuum, vt recta linea vel <lb/>curua vel angulus. </s>

<s>Atque hæc omnia aut continua <pb pagenum="26"/>&longs;unt aut diui&longs;a, &longs;implicia aut compo&longs;ita. </s>

<s>Horum autem mediorum opera fit vt vires illis <lb/>applicatæ, quarum iidem &longs;unt nutus, contrariis mo<lb/>tibus moueantur. </s>

<s>Id <expan abbr="aut&etilde;">autem</expan> fit cùm in mediis illis inter <lb/><expan abbr="eor&utilde;">eorum</expan> extrema interiacet quies vna vel plures. </s>

<s><expan abbr="Ex&etilde;pli">Exempli</expan> <lb/>gratia, &longs;i duo pondera funis extremitatibus alligata <lb/>&longs;int, & funis clauo fixo & immobili incumbat pro<lb/>pter illam quietem inter <expan abbr="vtrumq;">vtrumque</expan> pondus <expan abbr="po&longs;itã">po&longs;itam</expan> <expan abbr="nõ">non</expan> <lb/>poterit <expan abbr="alter&utilde;">alterum</expan> deor&longs;um moueri, quin <expan abbr="alter&utilde;">alterum</expan> &longs;ur&longs;um <lb/>moueatur. </s>

<s><expan abbr="Id&etilde;">Idem</expan> fiet in linea recta, &longs;i enim illius extre<lb/>mitatibus pondera duo annexa &longs;int, & inter ea &longs;it in <lb/>illa <expan abbr="punct&utilde;">punctum</expan> aliquod quie&longs;cens, dum alterum ex illis <lb/>ponderibus deor&longs;um feretur, alterum a&longs;cendet. </s>

<s><expan abbr="P&utilde;-ctum">Pun<lb/>ctum</expan> autem illud quie&longs;cens in linea illa recta, Gr&ecedil;cis <lb/>hypomochlium dicitur, eò quòd vecti, quem <foreign lang="greek">mo/xlon</foreign><lb/>vocant, &longs;ubiiciatur. </s>

<s>Huius autem hypomochlij, in <lb/>recta linea &longs;e vecte collocatio faciet, vt lineæ extre<lb/>ma &longs;ecundum datam rationem moueantur. </s>

<s>Si enim <lb/>recta linea in datam rationem diui&longs;a fuerit, hoc e&longs;t, <lb/>vt pars altera ad <expan abbr="alterã">alteram</expan> eam habeat <expan abbr="ration&etilde;">rationem</expan>, <expan abbr="quã">quam</expan> quis <lb/>voluerit. (Quod <expan abbr="quid&etilde;">quidem</expan> quo modo fiat docet Euc.) & <lb/>in puncto diui&longs;ionis collocetur <expan abbr="hypomochli&utilde;">hypomochlium</expan>, illius <lb/>lineæ extrema &longs;ecundum <expan abbr="illã">illam</expan> <expan abbr="ration&etilde;">rationem</expan> mouebuntur: <lb/>&longs;iue enim con&longs;ideretur <expan abbr="circulor&utilde;">circulorum</expan>, quos illa extrema <lb/>de&longs;cribent, proportio &longs;iue &longs;patium quod illa in linea <lb/>perpendiculari notabunt vtroque modo illi motus, <pb pagenum="27"/><expan abbr="parti&utilde;">partium</expan> <expan abbr="illar&utilde;">illarum</expan> <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="&longs;eruab&utilde;t">&longs;eruabunt</expan>. </s>

<s>Sit <expan abbr="ex&etilde;pli">exempli</expan> gratia <lb/>linea AB, quæ in puncto C in datam rationem &longs;ecta <lb/>&longs;it, puta vt pars AC quadrupla &longs;it ad partem CB, mo<lb/>ueatúrque circa centrum C, punctum A de&longs;cribet cir|| <lb/>culum <expan abbr="quadrupl&utilde;">quadruplum</expan> ad illum quem B <lb/>de&longs;cribet. </s>

<s>E&longs;t enim <expan abbr="ead&etilde;">eadem</expan> ratio in cir<lb/>culo diametri ad <expan abbr="diametr&utilde;">diametrum</expan>, quæ e&longs;t <lb/>circunferentiæ ad <expan abbr="circunferentiã">circunferentiam</expan> (vt <lb/>alibi demon&longs;trauimus.) Hac igitur <lb/>ratione A puncti motus quadruplus <lb/><figure id="fig3"></figure><lb/>erit ad puncti B <expan abbr="mot&utilde;">motum</expan>. </s>

<s>Si verò ponamus AD <expan abbr="perpen-dicular&etilde;">perpen<lb/>dicularem</expan> e&longs;&longs;e, & linea AB illi primùm <expan abbr="coincid&etilde;s">coincidens</expan> circa <lb/>punctum C, moueatur donec A ad D perueniat: tum <lb/><expan abbr="eod&etilde;">eodem</expan> momento B perueniet ad E: motum igitur erit <lb/>A in linea <expan abbr="perp&etilde;diculari">perpendiculari</expan> tota circuli maioris diame<lb/>tro, quæ e&longs;t AD:B verò in <expan abbr="ead&etilde;">eadem</expan> linea, minoris <expan abbr="tãtùm">tantùm</expan> <lb/>circuli diametro <expan abbr="mot&utilde;">motum</expan> erit, qu&ecedil; e&longs;t BE. </s>

<s>Atqui diame|| <lb/>ter AD quadrupla e&longs;t ad BE, quia ex hypothe&longs;i <expan abbr="&longs;emi-diametror&utilde;">&longs;emi<lb/>diametrorum</expan> <expan abbr="illor&utilde;">illorum</expan> <expan abbr="circulor&utilde;">circulorum</expan> proportio e&longs;t, vt 4 ad 1. <lb/>Motus igitur <expan abbr="p&utilde;cti">puncti</expan> in linea A <expan abbr="perp&etilde;diculari">perpendiculari</expan> ad <expan abbr="mot&utilde;">motum</expan> <lb/><expan abbr="p&utilde;cti">puncti</expan> B quadruplus erit: <expan abbr="Id&etilde;">Idem</expan> dicetur &longs;i in data aliqua <lb/>alia ratione &longs;ecta &longs;it linea AB. <expan abbr="Demõ&longs;trat&utilde;">Demon&longs;tratum</expan> igitur e&longs;t <lb/>quomodo fieri po&longs;&longs;it vt rectæ line&ecedil; extrema <expan abbr="&longs;ec&utilde;d&utilde;">&longs;ecundum</expan> <lb/><expan abbr="datã">datam</expan> <expan abbr="ration&etilde;">rationem</expan> moueantur. </s>

<s>Si igitur illis extremis duæ <lb/>vires applicentur, <expan abbr="moueb&utilde;tur">mouebuntur</expan> <expan abbr="eod&etilde;">eodem</expan> ip&longs;o motu: ergo <lb/>&longs;ecundum datam vel propo&longs;itam rationem. </s>

<s>Quod <lb/>a&longs;&longs;umptum erat. </s><pb pagenum="28"/>

<s>Quod autem in vecte demon&longs;tratum e&longs;t, illud e<lb/>tiam in reliquis mediis demon&longs;trandum erit, et&longs;i <lb/>lemmati &longs;atisfactum e&longs;t, dum in vno exemplo id <lb/>probatum e&longs;t. </s>

<s>Ante igitur &longs;ecundum lemma de<lb/>mon&longs;trabimus. </s>

<s>LEMMA II.</s>

<s>Proportionem proximè maiorem vel minorem, <lb/>quàm &longs;it datæ vis ad datum pondus proportio, de<lb/>terminare vis cuiuslibet quantitas ex motu ciere po|| <lb/>te&longs;t, metienda e&longs;t. </s>

<s>Motum autem ciere pote&longs;t vel in <lb/>&longs;ubiecto &longs;uo, vel in alieno. </s>

<s>Vis autem men&longs;ura non <lb/>&longs;umitur ex eo motu quem in &longs;ubiecto &longs;uo ciere po<lb/>te&longs;t, eo quòd licet vis quælibet certum motum ha<lb/>beat & <expan abbr="determinat&utilde;">determinatum</expan> quo &longs;ubiectum &longs;uum mouet, <lb/>illius tamen quantitas, vt &longs;uprà diximus, nondum <lb/>demon&longs;trata e&longs;t. </s>

<s>Supere&longs;t igitur vt vires motu illo <lb/>metiamur, quem in alieno &longs;ubiecto ciere po&longs;&longs;unt, vt <lb/>id fiat, quærenda nobis &longs;unt &longs;ubiecta quæ in homi<lb/>num pote&longs;tate &longs;int, & cum vi qualibet committi <lb/>po&longs;&longs;int. </s>

<s>Omnium autem mobilium &longs;ubiectorum, <lb/>maximè in hominum pote&longs;tate &longs;unt grauia: leuia e<lb/>nim coercere vix po&longs;&longs;umus. </s>

<s>Ideo vires illis metiri <lb/>&longs;olemus, &longs;ed grauibus quæ vel vel figuris &longs;uis & com|| <lb/>page vel va&longs;e aliquo coercentur. </s>

<s>Itaque v&longs;us homi<lb/>num certas qua&longs;dam ponderum men&longs;uras &longs;ibi &longs;ta <lb/>tui, ponderibus &longs;cilicet quibu&longs;dam certa <expan abbr="quãtitate">quantitate</expan> <pb pagenum="29"/>con&longs;tantibus inditis nominibus, vt e&longs;&longs;ent <expan abbr="ponder&utilde;">ponderum</expan> <lb/>omnium communes men&longs;uræ, vt &longs;unt libræ, vncia, <lb/>drachm&ecedil;, &c. </s>

<s>quas famo&longs;as men&longs;uras <expan abbr="vocãt">vocant</expan>. </s>

<s>Quem<lb/>admodum igitur numeros numeris, &longs;ic pondera <expan abbr="põ-deribus">pon<lb/>deribus</expan> metimur. </s>

<s><expan abbr="Tãtum">Tantum</expan> enim pondus e&longs;&longs;e dicitur, <lb/>quot libras vncias drachmas æqualiter mouere po<lb/>te&longs;t dempto momento. </s>

<s>Nec tantum pondera hoc <lb/>modo metimur, &longs;ed etiam alias omnes vires motum <lb/>ad locum cientes. </s>

<s>Quot enim libras vir aut aliud a<lb/>nimal vel <expan abbr="v&etilde;tus">ventus</expan> aut ignis, aut aliqua alia vis dempto <lb/>momento mouere poterit, tot libris illam æqualem <lb/>e&longs;&longs;e dicemus. </s>

<s>Si igitur data vel propo&longs;ita vis metien<lb/>da &longs;it, &longs;iquidem naturalis &longs;it, quoniam docuimus <lb/>vim naturalem per totum &longs;ubiectum diffu&longs;am e&longs;&longs;e <lb/>in rebus homogeneis: id e&longs;t, vt quæ e&longs;t proportio <lb/>molis ad molem, eadem &longs;it ponderis ad pondus: &longs;u<lb/>memus partem aliquam illi homogeneam, aut ex i<lb/>p&longs;o &longs;ubiecto, aut ex alio ip&longs;i homogeneo, eámque <lb/>famo&longs;a aliqua men&longs;ura metiemur, vtramque &longs;cili<lb/>cet committendo & ob&longs;eruando, quem motum al<lb/>tera in altera ciere po&longs;&longs;it, vbi enim æquilibrium fa<lb/>cient motibus extremorum, quibus affixæ fuerint, <lb/>proportionales erunt per 1 theorema: motus autem <lb/>illi linearum dimen&longs;ione quam Geometria docet, <lb/>noti erunt, & eorum proportio, nota igitur erit & <lb/>virium proportio. </s>

<s>Atqui men&longs;uræ famo&longs;æ nota, per <pb pagenum="30"/>&longs;e e&longs;t quantitas: duorum autem proportione cogni<lb/>ta & alterius quantitate, &longs;tatim & reliqui quantitas <lb/>innote&longs;cit per &longs;ecundam da. </s>

<s>nota igitur vis <expan abbr="quã-titate">quan<lb/>titate</expan> quæ parti illi ine&longs;t, no&longs;cetur & vis <expan abbr="quãtitas">quantitas</expan> qu&ecedil;<lb/>toti inerit: quæ enim erit ratio molis &longs;ubiecti vis da<lb/>tæ ad molem particul&ecedil; &longs;umpt&ecedil;, eadem erit vis totius <lb/>ad vim partis. </s>

<s>Hîc igitur erunt quatuor proportio<lb/>nalia, &longs;cilicet vt moles ad molem: &longs;ic vis ad vim ex <lb/>quibus tria nota erunt: moles enim metiri Geome<lb/>tria nos docuit, præterea vis partis, vt demon&longs;traui<lb/>mus, nota e&longs;t, ergo & vis totius quantitas per deci<lb/>mam &longs;eptimam &longs;eptimi Elem. </s>

<s>Si verò &longs;ubiectum non &longs;it homogeneum, vi ta<lb/>men naturali &longs;it præditum, &longs;i quidem data vis quam <lb/>metiri volumus, ea &longs;itque motum ciere volumus, <lb/>tum con&longs;iderabimus quæ pars in illa, vis minimum <lb/>habeat: & ex ea totam ip&longs;ius molem æ&longs;timabimus: <lb/>&longs;i verò &longs;it vis mouenda, &longs;tatuemus qua&longs;i tota &longs;it ho<lb/>mogenea ip&longs;ius parti, quæ vis plurimum in &longs;e habe<lb/>bit, & ex ea totam ip&longs;ius molem æ&longs;timabimus. </s>

<s>Ita<lb/>que tum illius quanta minima, tum huius quanta <lb/>maxima e&longs;&longs;e pote&longs;t, vis quantitas nobis nota erit per <lb/>proximè demon&longs;tratam rationem. </s>

<s>Notis autem vi<lb/>rium quantitatibus, nota erit & earum proportio: <lb/>ergo & proportio ip&longs;a proximè maior vel mi<lb/>nor: addita enim vel detracta ip&longs;ius denomi-<pb pagenum="31"/>nationi, vnitate erit proximè maior vel mi<lb/>nor. </s>

<s>Si verò vis data non &longs;it naturalis, <expan abbr="voluntariã">voluntariam</expan> qui<lb/>dem ita æ&longs;timare po&longs;&longs;umus, qualis vt plurimum <lb/>e&longs;t, & &longs;i quidem ea &longs;it, qua mouere volumus, eam <lb/>&longs;tatuemus, quanta minima in eiu&longs;dem generis &longs;ub<lb/>iecto e&longs;&longs;e pote&longs;t, vt &longs;i vim hominis quinquaginta li<lb/>bris æqualem ponamus, vim equi centum: &longs;i verò <lb/>ea &longs;it quam mouere volumus, &longs;tatuemus eam <expan abbr="quã-ta">quan<lb/>ta</expan> maxima e&longs;&longs;e pote&longs;t, vt vim hominis 300 libra<lb/>rum, vim equi 500 librarum, & &longs;ic dé cæteris: ita vt <lb/>nullum &longs;it dubium quin illa minor &longs;it, quam &longs;tatue<lb/>rimus, hæc verò maior. </s>

<s>Vis verò fortuitæ quantitatem nulla certa con<lb/>iectura a&longs;&longs;equi po&longs;&longs;umus: ita vt qua&longs;cunque ma<lb/>chinas ei aptemus, modò moueat, modò non mo<lb/>ueat, neque ad no&longs;trum <expan abbr="in&longs;titut&utilde;">in&longs;titutum</expan> magnoperè per<lb/>tinet illa inqui&longs;itio: cum fortuitorum nulla &longs;it di&longs;ci<lb/>plina. </s>

<s>His igitur modis virium duarum datarum pro<lb/>portio proximè maior nota fiet, quod in &longs;ecundo <lb/>lemmate demon&longs;trandum &longs;ump&longs;eramus. </s>

<s>Iam redeamus ad mediorum, quibus vi<lb/>res annectuntur, con&longs;ide<lb/>rationem. </s><pb pagenum="32"/><figure></figure>

<s>Docuimus quis &longs;it &longs;implicis ve<lb/>ctis effectus, &longs;implex autem ve<lb/>ctis &longs;emicirculi conuer&longs;ione &longs;uam <lb/>operationem ab&longs;ouit, ita vt &longs;i <lb/>vlteriùs F moueatur in in alio &longs;e<lb/>micirculo motus prioribus contra<lb/>rios cieat: vt exempli gratia, &longs;it <lb/>vectis AB, cuius hypomochium <lb/>&longs;it C, dum A punctum de&longs;cribet &longs;emicirculum AFD, <lb/>motus ille deor&longs;um erit: interea verò B punctum <lb/>de&longs;cribet &longs;emicirculum BGE a&longs;cendendo: &longs;i verò A <lb/>tran&longs;cendat, D incipiet a&longs;cendere: B verò tran&longs;cen<lb/>dens, E <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan>: ideo excogitata e&longs;t vectis ratio per<lb/>petua ex plurium vectium &longs;ucce&longs;&longs;ione circa idem <lb/>hypomochlium: e&longs;t autem illa tum in ergatis aut &longs;u<lb/>culis, tum in duorum tympanorum homocentrico<lb/>rum, &longs;eu eadem axe transfixorum in planis parallelis <lb/>aptatione, quorum &longs;emidiametri &longs;int in eadem pro<lb/>portione quæ in vecte ad propo&longs;itum motum cien<lb/>dum nece&longs;&longs;aria e&longs;t: centrum verò eorum &longs;eu axis fi<lb/>xa &longs;it, ac ita vires aptentur, vt maior minorem, mi<lb/>nor verò maiorem tympanum impellat. </s>

<s>Quemad<lb/>modum autem horum tympanorum homocentri<lb/>corum opera vectis perpetui ratio <expan abbr="inu&etilde;ta">inuenta</expan> e&longs;t, ita eo<lb/>rum multiplicatione motus, & mouentis & <expan abbr="mou&etilde;-di">mouen<lb/>di</expan>, proportio in infinitum augeri pote&longs;t. </s>

<s>Cuius rei <pb pagenum="33"/>maximus e&longs;t v&longs;us: nec enim materia ad vectem, cu<lb/>ius longitudo &longs;tadij vnius requireretur, idonea in<lb/>ueniri po&longs;&longs;et: plurium autem tympanorum propor<lb/>tionalium aptatione fiet machina tractabilis, cuius <lb/>vis maior erit quàm vectis, cuius longitudo &longs;tadij v<lb/>nius e&longs;&longs;et. </s>

<s>Si enim duo tympani homocentrici a<lb/>ptentur, quorum proportio &longs;it alterius ad alterum, <lb/>decupla, vis quæ libram vnam æquabit, vim decem <lb/>libris æqualem dempto momento mouebit, &longs;i con<lb/>gruè illis tympanis aptentur: &longs;i verò adhuc duo alij <lb/>tympani fiant, quorum alterius ad alterum decupla <lb/>&longs;it proportio, ac minor <expan abbr="illor&utilde;">illorum</expan> ita aptetur, vt <expan abbr="maior&etilde;">maiorem</expan> <lb/>ex prioribus moueat, appendatur deinde minori ex <lb/>prioribus tympanis vis centum libris æqualis, maio<lb/>ri verò ex po&longs;terioribus vis vni libræ æqualis, tum <lb/>hæc illam mouebit, & &longs;ic in infinitum motus extre<lb/>morum proportio multiplicari pote&longs;t. </s>

<s>Flexibili ve<lb/>rò medio quies ita aptari pote&longs;t, vt duo eius extre<lb/>ma diuer&longs;is motibus moueantur, & quidem &longs;ecun<lb/>dum datam rationem. </s>

<s>Exemplum habemus in orga|| <lb/>nis poly&longs;pactis, &longs;eu trochleis, in quibus altero funis <lb/>extremo immobili manente, reliquum funis circa <lb/>plures trochleas conuoluitur, quarum aliæ centris <lb/>immobilibus fixæ &longs;unt, aliæ verò ip&longs;is contrariæ cen <lb/>tris mobilibus. </s>

<s>Atque ita circa illas conuoluitur fu<lb/>nis, vt inter eas &longs;it &longs;patium tantum, quanta e&longs;t linea, <pb pagenum="34"/>per quam motum ciere volumus. </s>

<s>Quoties igitur fu<lb/>nis extremum quod moueri pote&longs;t, trahitur, &longs;ingul&ecedil;<lb/>reuolutiones æqualiter minuuntur, eáque ratione, <lb/><expan abbr="di&longs;tãtia">di&longs;tantia</expan> quæ e&longs;t inter trochleas contrarias minuitur, <lb/>ita &longs;cilicet vt quot &longs;unt reuolutiones, in tot partes <lb/>di&longs;tributum &longs;it motus &longs;patium. </s>

<s>Quot igitur reuolu|| <lb/>tiones erunt, totuplex erit motus extremitatis funis <lb/>ad motum trochlearum mobilium ver&longs;us fixas. </s>

<s>Si i<lb/>gitur motus i&longs;tius extrema con&longs;tituantur, alterum <lb/><expan abbr="quid&etilde;">quidem</expan> funis illa extremitas quæ mouetur, <expan abbr="alter&utilde;">alterum</expan> ve<lb/>rò terminus &longs;patij illius, quod e&longs;t à trochleis fixis ad <lb/>mobiles contrarias: quæ erit proportio numeri con<lb/>uolutionum funis ad vnitatem, eadem erit motus, <lb/>quo funis extremum mouebitur ad motum quo al<lb/>terum extremum mouebitur: pote&longs;t autem in infi<lb/>nitum augeri conuolutionum numerus, ergo & mo|| <lb/>tus illius proportio. </s>

<s>Angulus autem ad motum ciendum ita ad<lb/>hibetur. </s>

<s>Diximus motus men&longs;uram in nutus li<lb/>nea &longs;umi, quantum igitur vis aliqua per eam ver&longs;us <lb/>locum naturalem mouetur, tanta e&longs;t, quantum verò <lb/>per eam à loco naturali reuellitur, tanta e&longs;t eius re&longs;i<lb/>&longs;tentia. </s>

<s>Quod verò per lineam à loco naturali æquè <lb/>di&longs;tantem (id e&longs;t, per eam quæ nutus lineas &longs;ecat ad <lb/>angulos rectos) mouetur, illud mouenti non re&longs;i<lb/>&longs;tit, omnium autem linearum inter illas intercepta-<pb pagenum="35"/>rum, ac cum illis in earum inter&longs;ectionis puncto <expan abbr="cõ-currentium">con<lb/>currentium</expan>, quæ obliquè nutum ver&longs;us, aut contra <lb/>nutum ferri dicuntur, quo propiùs quælibet ad nu<lb/>tus lineam accedit, per illam rei motæ vis aut <expan abbr="re&longs;i&longs;t&etilde;-tia">re&longs;i&longs;ten<lb/>tia</expan> maior e&longs;t: quò verò propiùs ad lineam à loco na<lb/>turali æqui di&longs;tantem accedit, eò minor e&longs;t. </s>

<s><expan abbr="Omni&utilde;">Omnium</expan> <lb/>autem maxima e&longs;t in nutus linea, æquidi&longs;tans verò <lb/>à loco naturali motui per lineam nutus omnino op|| <lb/>po&longs;ita e&longs;t, obliquæ verò non ita quia &longs;ecundum illas <lb/><expan abbr="eod&etilde;">eodem</expan> &longs;patio delata vis propiùs ad <expan abbr="loc&utilde;">locum</expan> <expan abbr="natural&etilde;">naturalem</expan> acce<lb/>dit, aut ab eo recedit, quàm e&longs;&longs;et, cùm moueri c&oelig;pit. </s><figure></figure>

<s>Sit exempli gratia AB linea <lb/>nutus, vis alicuius, puta ponde<lb/>ris, &longs;itque A &longs;ur&longs;um & contra nu|| <lb/>tum: B verò deor&longs;um & nutum <lb/>ver&longs;us, de&longs;cribatúrque circulus, <lb/>cuius AB &longs;it diameter, quàm CD, <lb/>alia diameter &longs;ecet ad angulos <lb/>rectos in centro E: omnes igitur <lb/>lineæ à centro E ad circunferentiam circuli ductæ, <lb/>quæ <expan abbr="cad&etilde;t">cadent</expan> intra <expan abbr="&longs;emicircul&utilde;">&longs;emicirculum</expan> CAD, contra <expan abbr="nut&utilde;">nutum</expan> <expan abbr="a&longs;c&etilde;-dere">a&longs;cen<lb/>dere</expan> dicentur, quatenus circunferentiam &longs;pectant: <lb/>quatenus verò centrum &longs;pectant, de&longs;cendere dicen<lb/>tur: <expan abbr="cõtra">contra</expan> verò omnes in &longs;emicirculo CBD à centro <lb/>ad circunferentiam ductæ de&longs;cendere circunferen<lb/>tiam ver&longs;us, & centrum ver&longs;us a&longs;cendere dicentur: <pb pagenum="36"/>illæ igitur erunt, quæ obliquè nutum ver&longs;us aut con|| <lb/>tra nutum ferri <expan abbr="dic&utilde;tur">dicuntur</expan>: linea verò CED, neque <expan abbr="a&longs;c&etilde;-det">a&longs;cen<lb/>det</expan>, neque de&longs;cendet: lineæ verò in ip&longs;am ad angu<lb/>los rectos incidentes nutus lineæ erunt, quoniam li<lb/>neæ AB parallelæ erunt: &longs;i igitur à centro E <expan abbr="ducãtur">ducantur</expan> <lb/>lineæ ad circunferentiam inter A & D, puta EF, EG, <lb/>EH, quarum EF &longs;it proxima lineæ AB:EH verò pro<lb/>xima lineæ CD, ac per illas moueantur contra <expan abbr="nut&utilde;">nutum</expan> <lb/>tres vires æquales eodem tempore, ita vt prima per <lb/>lineam EF perueniat ad punctum F: &longs;ecunda verò <lb/>per EG perueniat ad G, tertia per EH perueniat ad H, <lb/>dico vis motæ per EF maiorem fore re&longs;i&longs;tentiam, <lb/>quàm illius quæ per EG aut EH, mouebitur & illius <lb/>quæ per EG mouebitur, maiorem quàm eius quæ <lb/>per EH mouebitur. </s>

<s>Ducantur enim à punctis FGH <lb/>in lineam ED perpendiculares FK, GL, HM, illæ <expan abbr="er&utilde;t">erunt</expan> <lb/>nutus lineæ: quanta igitur erit FK, tantum vis prima <lb/>mota cen&longs;ebitur, quanta verò GH, tantum vis &longs;ecun|| <lb/>da: quanta verò HM, tantum vis tertia mota cen&longs;ebi<lb/>tur: at qui quæ e&longs;t ratio motus ad motum in viribus <lb/>æqualibus, per quartam <expan abbr="cõclu&longs;ionem">conclu&longs;ionem</expan> huius tracta<lb/>tus, eadem e&longs;t re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam: quæ igitur <lb/>erit ratio linearum illarum perpendicularium, inter <lb/>&longs;e eadem erit & <expan abbr="re&longs;i&longs;t&etilde;tiarum">re&longs;i&longs;tentiarum</expan> virium per lineas ob<lb/>liquas motarum, à quibus illæ perpendiculares du<lb/>ctæ &longs;unt, atqui quo lineæ illæ perpendiculares pro-<pb pagenum="37"/>piùs ad AB circuli diametrum accedunt, eò &longs;unt ma|| <lb/>iores per decimam quartam tertij Elem. </s>

<s>ergo & <lb/>vires per eas lineas delat&ecedil; à <expan abbr="quar&utilde;">quarum</expan> extremitatibus du|| <lb/><expan abbr="c&etilde;tur">centur</expan>, maiores <expan abbr="re&longs;i&longs;t&etilde;tias">re&longs;i&longs;tentias</expan> <expan abbr="habeb&utilde;t">habebunt</expan>: at qui quò magis <lb/>ad AB, accedunt eo magis ab ED recedunt: quò igi<lb/>tur magis ad ED accedent, eò minores erunt <expan abbr="re&longs;i&longs;t&etilde;-tiæ">re&longs;i&longs;ten<lb/>tiæ</expan>, hinc &longs;equitur tanquam corollarium. </s>

<s>Si duæ vi<lb/>res per&longs;e æquales triangulo rectangulo aptentur, cu|| <lb/>ius latus alterum, angulum rectum <expan abbr="con&longs;titu&etilde;tium">con&longs;tituentium</expan>: <lb/>&longs;it earum nutus linea, ac per illud vis altera mouea<lb/>tur, altera verò per latus angulo recto oppo&longs;itum, <lb/>quæ erit ratio huius lateris ad illud, eadem erit re&longs;i<lb/>&longs;tentia vis illius ad vis huius <expan abbr="re&longs;i&longs;t&etilde;tiam">re&longs;i&longs;tentiam</expan>. </s>

<s>Atqui duo<lb/>rum illorum laterum proportio in infinitum auge<lb/>ri vel minui pote&longs;t, ergo & re&longs;i&longs;tentia virium illis ap|| <lb/>plicatarum. </s>

<s>Hoc igitur modo po&longs;&longs;umus vti angulo <lb/>&longs;eu triangulo ad motus ciendos, nempe eo immoto <lb/>vires per eius latera mouendo. </s>

<s>Sed & alia ratione eo vti po&longs;&longs;umus, ip&longs;um &longs;cilicet <lb/>triangulum mouendo, qui tunc cuneus dicitur. </s>

<s>Vt <lb/>autem eo hac ratione vtamur, vires ita di&longs;ponere o<lb/>portet, vt altera illarum vni ex lateribus angulum <lb/>rectum con&longs;tituentibus incumbat, altera verò lateri <lb/>ip&longs;um &longs;ubtendenti. </s>

<s>Illa enim tantùm mouebitur, <lb/>quantum latus cui altera incumbit. </s>

<s>Sit exempli gra<pb pagenum="38"/><figure id="fig4"></figure><lb/>tia triangulus ABC, cuius angulus B <lb/>rectus &longs;it, latus verò illum &longs;ubtendens <lb/>&longs;it AC, incumbátque vis D lateri AB, <lb/>vis verò E lateri AC, &longs;itque vis D nutus <lb/>linea BC, vis verò E nutus &longs;it linea AB, erigatúrque à <lb/>puncto C linea CF <expan abbr="perp&etilde;dicularis">perpendicularis</expan> ad BC æqualis AB, <lb/>à qua vis E <expan abbr="nõ">non</expan> di&longs;cedat. </s>

<s>Si triangulum illud in plano <lb/>fixo moueatur, donec AB perueniat ad CF, mota e<lb/>rit vis D nutu &longs;uo tantùm, quanta e&longs;t linea BC, vis ve<lb/>rò E tantum, quanta e&longs;t linea AB. </s>

<s>Cùm autem po&longs;&longs;it <lb/>in infinitum augeri & minui, laterum illorum pro<lb/>portio, po&longs;&longs;unt etiam duorum illorum <expan abbr="extremor&utilde;">extremorum</expan> <lb/>motus in data ratione con&longs;titui. </s>

<s>Quanta enim erit <lb/>BC ad AB, tantus erit motus vis D ad motum vis E: <lb/>ergo & in hoc medio primum lemma no&longs;trum de<lb/>mon&longs;tratum e&longs;t. </s>

<s>In hoc autem medij genere hoc diuer&longs;um ab <lb/>aliis mediis accidit, quòd non tam facilè vtrin<lb/>que motus eo cietur, ac in illis, in quibus &longs;i virium <lb/>proportio momento vel &longs;uperet, vel minor &longs;it pro<lb/>portione motus <expan abbr="extremor&utilde;">extremorum</expan> medij, tum motus hinc <lb/>vel inde cietur. </s>

<s>Atqui in hoc propter &longs;uperficie<lb/>rum contactum, quarum pori vel a&longs;peritates &longs;e<lb/>&longs;e mutuò &longs;ubingrediuntur, & ita inuicem adhæ<lb/>rent, fit vt ægriùs cieatur motus, faciliùs ve-<pb pagenum="39"/>rò mouebuntur, &longs;i illæ &longs;uperficies leues fue<lb/>rint, vt &longs;i pinguibus inungantur, vel ex materia <lb/>leui con&longs;tent. </s>

<s>Ideò enim reliqua faciliùs mo<lb/>uentur, quòd circa puncta veluti quædam mo<lb/>ueantur. </s>

<s>Simplicis autem trianguli rectilinei aut cu<lb/>nei in diuturnis motibus ciendis rarus e&longs;t v&longs;us, <lb/>tum ob illud quod notauimus incommodum ex <lb/>vitio materiæ, quo fit vt in eo qua&longs;&longs;atione opus <lb/>&longs;it, tum etiam quòd breui eius operatio termine<lb/>tur. </s>

<s>Ideo illo vtimur aut cùm &longs;olutionem con<lb/>tinui molimur, quæ breui tempore fit, vel <lb/>cùm aliquid di&longs;tendere aut aliquid figere volu<lb/>mus. </s>

<s>Anguli verò curui linei magnus e&longs;t v&longs;us, præ<lb/>&longs;ertim helicis cylindricæ. </s>

<s>Nihil enim aliud e&longs;t he<lb/>lix quàm triangulus curuus: &longs;i enim alteram ex <lb/>lineis rectis angulum con&longs;tituentibus, cylindri <lb/>ba&longs;i ip&longs;i lineæ æquali obuolueris, reliquam ve<lb/>rò &longs;eruato eodem quem con&longs;tituunt angulo, &longs;u<lb/>per cylindri &longs;uperficiem curuaueris, habebis he<lb/>licem cylindricam, quam &longs;i iterum &longs;eruato eo<lb/>dem angulo in rectum extendas, habebis trian<lb/>gulum rectilineum. </s>

<s>Hæc autem helix commo<lb/>di&longs;&longs;ima e&longs;t, tum quòd in parua mole triangulum <pb pagenum="40"/>longi&longs;&longs;imum obuolutum contineat tum quod par<lb/>tes eius omnes &longs;ibi inuicem <expan abbr="cõgruant">congruant</expan>: omnes enim <lb/>partes helicis cylindricæ, aut circa eundem, aut circa <lb/>æquales cylindros de&longs;criptæ, &longs;eruato eodem angulo <lb/>&longs;ibi inuicem &longs;uppo&longs;itæ congruunt. </s>

<s>Quo fit, vt &longs;i ca<lb/>ui cylindri interiori &longs;uperficiei impre&longs;&longs;a &longs;it helix, a<lb/>lia verò cylindri &longs;uperficiei connex&ecedil; ip&longs;i cau&ecedil; æqua<lb/>li, &longs;eruato eodem qui in illo e&longs;t, angulo, &longs;ibi inui<lb/>cem & omnes vnius omnibus alterius partibus con|| <lb/>gruant. </s>

<s>Huius autem medij cùm plures &longs;int partes, con|| <lb/>&longs;tat enim duabus &longs;uperficiebus, pluribus etiam mo|| <lb/>dis variari pote&longs;t. </s>

<s>In &longs;umma autem eius affectus hic <lb/>e&longs;t, vt cylindri ba&longs;i in &longs;uperficie immobili circum a<lb/>xem conuer&longs;a, vis mouenda helicem premat: dum <lb/>enim cylindrus circum axem conuertetur, vis mo<lb/>uenda qualibet reuolutione tantùm &longs;ecundum cy<lb/>lindri longitudinem mouebitur, quanta e&longs;t di&longs;tan<lb/>tia duarum helicis &longs;pirarum. </s>

<s>Quæ igitur erit propor|| <lb/>tio circunferentiæ circuli ba&longs;im cylindri <expan abbr="con&longs;titu&etilde;-tis">con&longs;tituen<lb/>tis</expan> ad illam <expan abbr="di&longs;tãtiam">di&longs;tantiam</expan>, eadem erit motus orbicularis <lb/>cuiu&longs;libet puncti in cylindri &longs;uperficie &longs;ignati ad mo|| <lb/>tum rectum vis helicem prementis. </s>

<s>Illud igitur medium ex duobus compo&longs;itum e&longs;t, <lb/>recto &longs;cilicet & curuo: ita igitur vires illi aptandæ <lb/>&longs;unt, vt eius quam mouere volumus, nutus linea &longs;it <pb pagenum="41"/>cylindri longitudo: illa verò qua mouere volumus, <lb/>in orbem moueatur, aut &longs;i eius nutus linea recta &longs;it, <lb/>circa cylindrum fluens illum moueat contingendo. </s>

<s>Et hæc quidem de mediis in &longs;uo genere <expan abbr="con&longs;i&longs;t&etilde;-tibus">con&longs;i&longs;ten<lb/>tibus</expan>. </s>

<s>Po&longs;&longs;unt autem fieri eorum Syzygiæ vectis &longs;ci<lb/>licet cum trochleis, helicis cum vecte, aut cum tro<lb/>chleis, aut <expan abbr="omni&utilde;">omnium</expan> &longs;imul. </s>

<s>Vectis cum trochleis &longs;i fu<lb/>nis illud extremum, quod mouetur in trochleis, er<lb/>gatis aut &longs;uculis inuoluatur: helix cum vecte <expan abbr="coni&utilde;-getur">coniun<lb/>getur</expan>, &longs;i tympanus circum axem moueatur, &longs;itque i<lb/>ta denticulatus, vt dentes ip&longs;ius helicem in cylindro <lb/>excauatam ingrediantur, quam machinam helicem <lb/>perpetuam vulgò vocant, eò quòd eius conuer&longs;io <lb/>perpetua e&longs;&longs;e po&longs;&longs;it, cum helicis &longs;implicis operatio <lb/>non excedat ip&longs;ius longitudinem. </s>

<s>Quorum <expan abbr="omni&utilde;">omnium</expan> <lb/>tum inter &longs;e, tum ad &longs;ubiecta mouenda accommo<lb/>datio adeò varia e&longs;t, vt &longs;cripto <expan abbr="cõpreh&etilde;di">comprehendi</expan> vix po&longs;&longs;it. <lb/>Ex his autem quæ dicta &longs;unt, mediocris ingenij me<lb/>chanicus poterit ea prout ip&longs;i nece&longs;&longs;e erit aptare. </s>

<s>In <lb/>machinis autem omnibus hæc cautio <expan abbr="adhib&etilde;da">adhibenda</expan> e&longs;t, <lb/>vt earum &longs;tructura firma &longs;it, præ&longs;ertim verò vbi cir<lb/>cum axes aliquas fit motus, deinde vt vincula, qui<lb/>bus illis vires <expan abbr="affing&utilde;tur">affinguntur</expan>, valida &longs;int, illarum enim o<lb/>mnium vis in &longs;uo &longs;tatu manendi vtrique vi aptandæ <lb/>æqualis e&longs;&longs;e debet: aget enim in illas vis <expan abbr="vtraq;">vtraque</expan>, quòd <lb/>&longs;i medium firmum non &longs;it, motus in ip&longs;a machina <pb pagenum="42"/>ciebitur. </s>

<s>Itaque di&longs;&longs;oluetur, ideò manca videtur pe<lb/>titio illa Archimedis in hoc problemate <foreign lang="greek">do\s pou= stw= ta\n <lb/>ga\n kinw=</foreign>, quòd locum <expan abbr="tãtùm">tantùm</expan> in quo con&longs;i&longs;tat, &longs;ibi da<lb/>ri po&longs;tulet, cùm pr&ecedil;terea vincula quibus terra à loco <lb/>&longs;uo naturali remota &longs;u&longs;tineri po&longs;&longs;et, petere debue<lb/>rit: id <expan abbr="aut&etilde;">autem</expan> vt & ea quæ de motu in infinitum augen<lb/>do vel minuendo diximus, ita intelligenda &longs;unt, vt <lb/>&longs;ciamus infinita hominum pote&longs;tati, quacunque ar<lb/>te iuuetur, non &longs;ube&longs;&longs;e: quamuis enim Geometri<lb/>ca con&longs;ideratio in infinitum &longs;e&longs;e <expan abbr="ext&etilde;dat">extendat</expan>, &longs;unt ta<lb/>men certi fines, vltra quos natura rerum nos pro<lb/>gredi non patitur: &longs;unt præterea vitia materiæ, <lb/>quæ Geometra non con&longs;iderat, illa tamen non <lb/>ob&longs;tant quò minùs id quod proponitur, <expan abbr="ver&utilde;">verum</expan> &longs;it in <lb/>intellectu. </s>

<s>An verò id quod proponitur tale &longs;it, vt in <lb/>opus educi po&longs;&longs;it, con&longs;iderabit Geometer habita ra<lb/>tione circun&longs;tantiarum, præ&longs;ertim verò temporis, <lb/>quod ip&longs;i præ&longs;cribetur, & &longs;umptuum quos facere <lb/>poterit is qui aliquid faciendum proponet, quæ &longs;i <lb/>abundè &longs;uppetant, nihil non fieri poterit. </s>

<s>Vt igitur hunc tractatulum concludamus, ac ve<lb/>lut in &longs;ummam contrahamus: In motibus ciendis <lb/>tria &longs;unt con&longs;ideranda. </s>

<s>Vis qua motum ciere volu<lb/>mus, vis quam mouere volumus, & motum quo <lb/>mouere volumus: duo enim quælibet ex illis ter<lb/>tium determinant. </s>

<s>Si enim vi parua vim magnam <pb pagenum="43"/>mouere volumus, id nonni&longs;i paruo motu facere <lb/>po&longs;&longs;umus: &longs;i verò vim aliquam magno motu mo<lb/>uere velimus, vi magna mouente ad id opus e&longs;t. </s>

<s>Si <lb/>vi parua magnum motum ciere volumus, mini<lb/>mam vim mouendam e&longs;&longs;e oportet: vt puta, &longs;i libra <lb/>vna centum libras mouere velimus, oportet motum <lb/>illius, motu huius <expan abbr="c&etilde;tuplo">centuplo</expan> maiorem e&longs;&longs;e. </s>

<s>Si verò ve<lb/>limus libra vna aliam vim ita mouere, vt ea cen<lb/>tuplo citiùs moueatur, quàm libræ illius pondus, il<lb/>lam cente&longs;imam tantùm libræ vnius partem e&longs;&longs;e ne<lb/>ce&longs;&longs;e e&longs;t: &longs;i verò libram vnam ita mouere velimus, vt <lb/>centuplo citius moueatur, quàm vis quæ illam mo<lb/>uebit, vi centum libris maiore ad id opus erit. </s>

<s><expan abbr="Neq;">Neque</expan> <lb/>patitur natura &longs;ibi in his vim fieri: &longs;i enim eiu&longs;modi <lb/>proportio aliquo modo infringi po&longs;&longs;et, &longs;tat<gap/> da<lb/>retur <foreign lang="greek">au)tw/ma e)nde/lexes</foreign>, vel vt <expan abbr="vocãt">vocant</expan>, motus perpetuus <lb/>in perpetua materia. </s>

<s>Ex his igitur fundamentis inuentæ &longs;unt machinæ <lb/>omnes, quotquot antehac &longs;unt excogitat&ecedil;, & quot<lb/>quot po&longs;thac excogitari poterunt, ad ea referentur. </s>

<s>Itaque &longs;i propo&longs;itæ cuiu&longs;cunque machinæ effe<lb/>ctum &longs;cire velimus, con&longs;ideranda &longs;unt duo eius <lb/>extrema, quibus vires applicantur: quæ enim erit <lb/>ratio motus vnius ex illis extremis, ad motum <lb/>alterius eadem erit & virium, quæ illis extre<lb/>mis ad motum ciendum applicari poterunt, <pb pagenum="44"/>addito aut dempto momento, vt &longs;i dum <expan abbr="alter&utilde;">alterum</expan> ma<lb/>chinæ extremum palmo vno mouetur, alterum cen|| <lb/>tum palmis moueatur vis quælibet huic annexa, al<lb/>teram alteri annexam centuplam momento minùs, <lb/>mouebit: &longs;ed motu centuplo: mouebit autem & re<lb/>liquas omnes vires, quæ infra centuplam proportio<lb/>nem ad eam habebunt: &longs;i verò non vim centuplam <lb/>mouere, &longs;ed in data vi motum centuplum ciere veli<lb/>mus, eam quidem in extremo quod centuplo citiùs <lb/>mouetur, locare oportebit, alteram verò alteri extre|| <lb/>mo centuplo maiorem adhibere nece&longs;&longs;e erit. </s>

<s>Hinc oriuntur tria problemata, videlicet data vi <lb/>datum pondus mouere, quod iam <expan abbr="demõ&longs;trauimus">demon&longs;trauimus</expan>: <lb/>item data vi datum motum ciere, quod ex <expan abbr="præced&etilde;-tis">præceden<lb/>tis</expan> demon&longs;tratione ab&longs;oluitur: tertium, datam vim, <lb/>dato motu mouere, quod quidem faciliùs demon<lb/>&longs;tratur, quàm ab&longs;oluitur. </s>

<s>Scimus enim, vt id fiat, <lb/>vim aliam proportionalem (vt docuimus) requiri, <lb/>quo modo autem illam habere po&longs;&longs;imus, explicare <lb/>hoc opus, hic labor e&longs;t. </s>

<s>Non enim quemadmodum <lb/>organa ad motus in data proportione ciendos ha<lb/>bere po&longs;&longs;umus: ita & vires infinitæ magnitudinis po|| <lb/>te&longs;tati no&longs;træ &longs;ub&longs;unt. </s>

<s>Quod enim ad vires in grauitatis & leuitatis gene|| <lb/>re con&longs;i&longs;tentes attinet, eas vt moueant antè moueri <lb/>oportet à loco vel &longs;itu naturali, vi <expan abbr="aut&etilde;">autem</expan> aliqua id fie-<pb pagenum="45"/>ri oportet, quæ iam in promptu &longs;it. </s>

<s>Quæ &longs;i tanta e&longs;t, <lb/>vt po&longs;&longs;it motum propo&longs;itum ciere, fru&longs;trà fiat, &longs;i alij <lb/>à loco naturali remouendæ, qua po&longs;tea ad motum <lb/>vtendum &longs;it, adhibeatur. </s>

<s>Nullum igitur ex eiu&longs;modi viribus commodum <lb/>percipi pote&longs;t, ni&longs;i quis in futurum &longs;ibi pro&longs;piciens, <lb/>multa à loco vel &longs;itu naturali ex otio remoueat, vt <lb/>iis, cùm opus erit ad motus ciendos vtatur. </s>

<s>Hac ra<lb/>tione quantas vires in promptuario habebit, tantos <lb/>motus ciere poterit. </s>

<s>Præcipuum igitur quod ad hu<lb/>ius problematis con&longs;tructionem pertinet, e&longs;t, vt vi<lb/>res quàm maximas pote&longs;tati no&longs;træ &longs;ubiiciamus, vo|| <lb/>luntariis aut fortuitis viribus naturales præparando, <lb/>aut à natura præparatas, quæ pote&longs;tati no&longs;træ &longs;ub<lb/>&longs;unt, accipiendo. </s>

<s>Qualia multa &longs;i mortales aduerte<lb/>rent, fieri po&longs;&longs;ent, vt aliàs, Deo duce, docebimus. </s>

<s>Earum autem virium quæ in raritatis & den&longs;ita<lb/>tis proportione con&longs;i&longs;tunt, &longs;ubiecta plurimùm in <lb/>no&longs;tra pote&longs;tate &longs;unt: multa enim &longs;unt naturalia &longs;ub <lb/>iecta actu, den&longs;a potentia verò rara. </s>

<s>Si quod igitur <lb/>eorum potentia proxima &longs;it rari&longs;&longs;imum, ita vt nullo <lb/>negotio actus ille raritatis induci po&longs;&longs;it, <expan abbr="concluda-túrq;">concluda<lb/>túrque</expan> loco aliquo angu&longs;to, po&longs;tea inducatur ille a<lb/>ctus, cùm rara <expan abbr="maior&etilde;">maiorem</expan> locum occupent, quum den<lb/>&longs;a, fiet vt locus in omnem partem di&longs;tendatur, illius <lb/>autem partes minùs cohærentes, tantum impellen<pb pagenum="46"/>tur, quanta e&longs;t proportio molis rei rarefactæ ad mo<lb/>lem illius cùm den&longs;a e&longs;&longs;et: illa autem raritatis po<lb/>tentia proxima e&longs;t in compo&longs;itione &longs;ulphuris <lb/>& nitri: ea igitur & &longs;imilibus &longs;ubiectis, <lb/>in data vi datus motus <lb/>cieri pote&longs;t. </s>

<s>FINIS.</s><figure></figure>

<s>Errata quæ inter imprimendum irrep&longs;erunt, <lb/>&longs;ic corrigito.</s>

<s><emph type="italics"/>Pagina<emph.end type="italics"/> 7. <emph type="italics"/>linea<emph.end type="italics"/> 10, <emph type="italics"/>munere, lege nuere. </s>

<s>pag.<emph.end type="italics"/> 11. <emph type="italics"/>lin.<emph.end type="italics"/> 12, <emph type="italics"/>comparationibus, lege comparati, ommi<lb/>bus. </s>

<s>pag.<emph.end type="italics"/> 12. <emph type="italics"/>lin.<emph.end type="italics"/> 16. <emph type="italics"/>finis, lege fini. </s>

<s>eadem. </s>

<s>penult. </s>

<s>Altero, lege Alto. </s>

<s>pag.<emph.end type="italics"/> 25. <emph type="italics"/>lin.<emph.end type="italics"/> 5. <emph type="italics"/>nunc &longs;it ratio, lege minor &longs;it. </s>

<s>linea<emph.end type="italics"/> 13. <emph type="italics"/>Vi<lb/>res maiori litera<emph.end type="italics"/> (<emph type="italics"/>e&longs;t enim &longs;ectionis initium. </s>

<s>lin.<emph.end type="italics"/> 15. <emph type="italics"/>mittant, lege mittantur. </s>

<s>pag.<emph.end type="italics"/> 27. <emph type="italics"/>li.<emph.end type="italics"/> 20. <lb/><emph type="italics"/>puncti in linea<emph.end type="italics"/> A, <emph type="italics"/>lege puncti<emph.end type="italics"/> A <emph type="italics"/>in linea. </s>

<s>pag.<emph.end type="italics"/> 28. <emph type="italics"/>lin.<emph.end type="italics"/> 9. <emph type="italics"/>vis, lege V is, maiori litera<emph.end type="italics"/> (<emph type="italics"/>e&longs;t enim <lb/>&longs;ectionis initium. </s>

<s>e. motu ciere, leg. </s>

<s>motu quem ciere. </s>

<s>lin.<emph.end type="italics"/> 23. <emph type="italics"/>metiri &longs;olemus, lege non <lb/>&longs;olemus. </s>

<s>&longs;tatui, lege &longs;tatuit. </s>

<s>pag.<emph.end type="italics"/> 32. <emph type="italics"/>lin.<emph.end type="italics"/> 4. <emph type="italics"/>ab&longs;ouit, lege ab&longs;oluit. </s>

<s>lin.<emph.end type="italics"/> 5. <emph type="italics"/>dele<emph.end type="italics"/> F. <emph type="italics"/>lin.<emph.end type="italics"/><lb/>12. <emph type="italics"/>&<emph.end type="italics"/> 13, <emph type="italics"/>pone virgulam po&longs;t<emph.end type="italics"/> D <emph type="italics"/>&<emph.end type="italics"/> E. <emph type="italics"/>lin.<emph.end type="italics"/> 8. <emph type="italics"/>hypomochium, lege hypomochlium. </s>

<s>pag.<emph.end type="italics"/> 37. <emph type="italics"/>li.<emph.end type="italics"/><lb/>13. <emph type="italics"/>re&longs;i&longs;tentia, lege re&longs;i&longs;tentiæ. </s>

<s>pag.<emph.end type="italics"/> 40. <emph type="italics"/>linea<emph.end type="italics"/> 5. <emph type="italics"/>&longs;uppo&longs;itæ, lege <lb/>&longs;uperpo&longs;itæ li.<emph.end type="italics"/> 7. <emph type="italics"/>connexæ, lege conuexæ. </s>

<s>lin.<emph.end type="italics"/> 13. <emph type="italics"/>affectus, lege effectus.<emph.end type="italics"/></s> </chap> </body> <back></back> </text></archimedes>

<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Varro, Michel</author> <title>De motu tractatus</title> <date>1584</date> <place>Geneva</place> <translator/> <lang>la</lang> <cvs_file>varro_demot_01_la_1584</cvs_file> <cvs_version/> <locator>044.xml</locator> </info> <text><front><section> <pb xlink:href="044/01/001.jpg" id="p.0001"/> <s>ILLVSTRI ET IN <lb/> PRIMIS GENEROSO DOMINO, <lb/> DOMINO CAROLO BARONI <lb/> Azerotin, Name&longs;tij Ro&longs;icij <lb/> Brandei&longs;ij Domino, <lb/> MICH. VARRO. S. P. D.</s> <s><emph type="italics"/>CVM ab Ineunte ætate in Ma<lb/> thematicis me exercui&longs;&longs;em, &longs;em<lb/> per animo meo in hæ&longs;it hoc de&longs;ide<lb/> rium, vt quæ de Archimede Syra<lb/> cu&longs;ano referuntur, ea & ratione <lb/> demon&longs;trare & experimento comprobare po&longs;&longs;em. <lb/> </s><s>Cumque multa in eo genere meditatus e&longs;&longs;em, po&longs;t<lb/> quam ad iuris ciuilis &longs;tudium tran&longs;ii, ac tandem <lb/> ad Rempub. nunquam mihi tantum otii contigit, <lb/> vt ea in ordinem redigere potuerim. </s><s>At cum pau<lb/> cis retrò annis &longs;atis longam peregrinationem per <lb/> Sarmatiam &longs;u&longs;cepi&longs;&longs;em, e&longs;&longs;etque vacuus curis a-<emph.end type="italics"/> <pb xlink:href="044/01/002.jpg"/><emph type="italics"/>nimus, atque ad earum rerum meditationem me <lb/> impelleret genius meus, c&oelig;pi aliquid de his &longs;cribe<lb/> re, quantum ip&longs;e iter faciendo meo marte a&longs;&longs;equi <lb/> potui cum libris de&longs;tituerer. </s><s>Scriptum illud cum <lb/> in maximè ardua & difficìli contemplatione ver<lb/> &longs;etur, nec ab eo tempore ex quo ad&longs;olitas occupa<lb/> tiones redii ei extremam manum apponere mihi <lb/> licuerit, decreueram inter priuatas meas mu&longs;as <lb/> a&longs;&longs;eruare Verebar enim vulgi <expan abbr="iudici&utilde;">iudicium</expan>, (cuius <expan abbr="ta-m&etilde;">ta<lb/> men</expan> rationem viro politico <expan abbr="habendã">habendam</expan> e&longs;&longs;e cen&longs;eo) ne<lb/> que &longs;olum vulgi, verum etiam <expan abbr="eor&utilde;">eorum</expan> qui doctorum <lb/> nomine gaudent, cum inaudita quædam interdum <lb/> quæque ab eorum opinione recedere <expan abbr="videãtur">videantur</expan> at<lb/> tingam. </s><s><expan abbr="Cupiebã">Cupiebam</expan> etiam in eo argumento vlteriùs <lb/> progredi, antè quam quidquam ederem, atque ea <lb/> quæ in meis aduer&longs;ariis ea de re &longs;par&longs;a habeo, iis <lb/> quæ <expan abbr="conge&longs;&longs;erã">conge&longs;&longs;eram</expan> &longs;ubiicere, vt iu&longs;tum <expan abbr="volum&etilde;">volumen</expan> fieret.<emph.end type="italics"/></s> <s><emph type="italics"/>Po&longs;tquam verò te di&longs;ce&longs;&longs;um ex hac nostra ci<lb/> uitate parare intellexi, in qua dum ver&longs;atus es. <lb/> </s><s>non &longs;olùm ei <expan abbr="ornam&etilde;to">ornamento</expan> fui&longs;ti, &longs;ed etiam optimos <lb/> quo&longs;que virtutum tuarum admiratione tibi con<lb/> ciliasti: mihi verò peculiariter innumera beneuo-<emph.end type="italics"/> <pb xlink:href="044/01/003.jpg"/><emph type="italics"/>lentiæ &longs;igna exhibui&longs;ti, non potui tantum virum, <lb/> meíque <expan abbr="amãtem">amantem</expan> &longs;ine aliquo munere aut<emph.end type="italics"/> <foreign lang="greek">mnhmosu/nw|</foreign><lb/> <emph type="italics"/>dimittere. </s><s>Itaque libellum hunc tibi dicare, & <lb/> &longs;ub tuo nomine in publicum edere &longs;tatui. </s><s>Squalli<lb/> dus quidem prodit, & rudis, mole etiam perexi<lb/> guus: &longs;plendidius aliquid & magis elaboratum <lb/> postularet tua dignitas. </s><s>Atqui ea e&longs;t argumenti <lb/> illius magnitudo, vt in eo aliquid volui&longs;&longs;e &longs;atis <lb/> &longs;it. </s><s>Militare &longs;anè munus e&longs;t, eóque nomine tibi <lb/> conueniens: &longs;i quis enim hac cognitione in&longs;tructus <lb/> fuerit, non minùs ea in re bellica vti poterit, <lb/> quam Archimedes in Syracu&longs;is defendendis <lb/> aduer&longs;us Marcum Marcellum v&longs;us est. </s><s>Eá<lb/> que mirari de&longs;inet quæ de eo Plutarchus nar<lb/> rat. </s><s>Id lucidius <expan abbr="demõ&longs;trare">demon&longs;trare</expan> potui&longs;&longs;em, &longs;i mihi hæc <lb/> vlterius per&longs;equendi otium fui&longs;&longs;et: Jeiuna enim <lb/> per &longs;e videntur, ni&longs;i quis &longs;uccum qui &longs;ub corti<lb/> ce latet, eliciat. </s><s>Malui tamen hoc veluti pro<lb/> gymna&longs;mate alios quibus plus est otii & in<lb/> genii ad eiu&longs;modi inqui&longs;itionem hortari & ac<lb/> cendere, quàm præclaram illam cognitio<lb/> nem diutius &longs;epultam relinquere cum ne-<emph.end type="italics"/> <pb xlink:href="044/01/004.jpg"/> <emph type="italics"/>minem hodie animaduertam qui ei in&longs;udet, <lb/> neque mihi &longs;pes vlla affulgeat diuturni otii, cuius <lb/> ope in hoc &longs;tadio pergere poßim: Quòd &longs;i fortè <lb/> mihi aliquando contingat, riuos plurimos ex his <lb/> fontibus me educturum; <expan abbr="cõfido">confido</expan>; quibus multum <lb/> <expan abbr="cõmoditàtis">commoditàtis</expan> rebus humanis accedat. </s><s>Quæ enim <lb/> hic traduntur, ad motus omnes atque ad omnis ge <lb/> neris virium comparationes accommodari po&longs;<lb/> &longs;unt ad fines propo&longs;itos a&longs;&longs;equendos. </s><s>Quod quàm <lb/> latè pateat, dici non pote&longs;t. </s><s>E&longs;t enim eorum v&longs;us <lb/> non &longs;olùm in mechanicis, in quibus tamen e&longs;t ma <lb/> ximus, &longs;ed etiam in politicis & &oelig;conomicis: &longs;unt <lb/> enim in illis motus, &longs;unt vires & re&longs;istentiæ. </s><s>In <lb/> arte medica & reliquis phy&longs;ices partibus pra<lb/> cticis, quantus &longs;it, nemo fando exprimere poßit. <lb/> </s><s>Quia vero ea quæ praxim docent gratiora &longs;unt <lb/> ip&longs;a contemplatione, & cau&longs;æ propter effecta in<lb/> quirantur,<emph.end type="italics"/> <foreign lang="greek">*a)lfh/stas</foreign> <emph type="italics"/>omnes hortor vt quod ego in<lb/> numeris aliis cur is implicitus addere non po&longs;<lb/> &longs;um, ip&longs;i addant. </s><s>Abundat nunc Europa præ<lb/> &longs;tantibus ingeniis, &longs;i M&oelig;cenates ade&longs;&longs;ent. <lb/> </s><s>Ea autem quæ præter hæc meditatus &longs;um, ac fer-<emph.end type="italics"/><pb xlink:href="044/01/005.jpg"/> <emph type="italics"/>mè parata habeo, hæc &longs;unt. </s><s>Tractatus de iactu. <lb/> </s><s>De continuitate eiu&longs;que &longs;olutione. </s><s>De <expan abbr="cõden&longs;a-tione">conden&longs;a<lb/> tione</expan> & rarefactione earumque cau&longs;is & effecti<lb/> bus: <expan abbr="It&etilde;">Item</expan> in Mechanicis, tractatus de variis ma<lb/> chinis ad motus <expan abbr="ci&etilde;dos">ciendos</expan>, ac de perfectißimæ cuiu&longs;<lb/> que ad id quod propo&longs;itum fuerit moliendum in<lb/> ue&longs;tigatione. </s><s>Nonnulla <expan abbr="etiã">etiam</expan> de <expan abbr="Rerumpublicar&utilde;">Rerumpublicarum</expan> <lb/> motu tum interno <expan abbr="t&utilde;">tum</expan> externo notaui, quæ <expan abbr="eod&etilde;">eodem</expan> or<lb/> dine tradere optarem: quorum omnium principia <lb/> hic &longs;i quis diligenter animaduertat tradita &longs;unt. <lb/> </s><s>Si quis mihi in his <expan abbr="palmã">palmam</expan> præripuerit, meque an<lb/> teuerterit ei maximas gratias agam. </s><s>Sunt tamen <lb/> inter hæc quædam quæ vulgò pandere nephas est. <lb/> </s><s>Itaque theoriæ magis in&longs;i&longs;tendum puto, in qua &longs;i <lb/> quis exercitatus fuerit, nullo negotio illam in opus <lb/> educere poterit, idque &longs;ine periculo fiet, cum vul<lb/> go non pateat. </s><s>Alioqui <expan abbr="pericul&utilde;">periculum</expan> e&longs;t, ne &longs;i particu<lb/> laria tradantur iis contenti homines, vt fieri &longs;olet <lb/> vniuer&longs;alem cognitionem & cau&longs;arum inqui&longs;i<lb/> tionem negligant, pereatque &longs;cientia. </s><s>Eaque de <lb/> cau&longs;a nihil quicquam de his quæ fecerat Archi-<emph.end type="italics"/> <pb xlink:href="044/01/006.jpg"/> <emph type="italics"/>medes &longs;criptum relinquere voluit. </s><s><expan abbr="Exi&longs;timãs">Exi&longs;timans</expan> eos <lb/> qui in iis quæ<emph.end type="italics"/> <foreign lang="greek">qewrhtikw=s</foreign> <emph type="italics"/>tradiderat, diligentem ope<lb/> ram ponere <expan abbr="vell&etilde;t">vellent</expan>, multò maiora quoties opus fo<lb/> ret præ&longs;tituros. </s><s>Equidem mihi per&longs;uadeo non de<lb/> futuros qui varia iudicia de hoc no&longs;tro &longs;cripto fe<lb/> rant & qua&longs;i nouum athletam in arenàm pro<lb/> deuntem mirentur. </s><s>Quibus vno verbo re&longs;pon<lb/> &longs;um volo, me nullius vnquam in verba ma<lb/> gistri iura&longs;&longs;e, &longs;ed liberrimè &longs;emper philo&longs;o<lb/> phatum e&longs;&longs;e: ita vt etiam in principia ab aliis <lb/> &longs;tatuta animaduertere mihi licere putauerim. <lb/> </s><s>Quod eò liberiùs feci, po&longs;tquam magnam <expan abbr="part&etilde;">partem</expan> <lb/> vulgo receptarum opinionum fal&longs;am e&longs;&longs;e re ip&longs;a <lb/> deprehendi. </s><s>Omnibus placuero, &longs;i tibi placuero. </s><s>Si <lb/> quid in his ob&longs;curius fuerit, habes <expan abbr="clarißim&utilde;">clarißimum</expan> vi<lb/> rum Wence&longs;laum lauinium tuum, ab&longs;tru&longs;ioris <lb/> philo&longs;ophiæ indagatorem <expan abbr="&longs;umm&utilde;">&longs;ummum</expan>, qui tibi omnia <lb/> explicabit. </s><s>Hoc igitur <expan abbr="munu&longs;cul&utilde;">munu&longs;culum</expan> vt &longs;erena <expan abbr="frõ-te">fron<lb/> te</expan> &longs;u&longs;cipias rogo. </s><s>Vale. </s><s>V I. </s><s>K al. Jun. Anno <lb/> Christi Domini<emph.end type="italics"/> M. D. LXXXIV. </s> </section><pb xlink:href="044/01/007.jpg" pagenum="1"/><section> <s>M. VARRONIS DE <lb/> MOTV TRACTATVS.</s></section></front><body><chap> <s><emph type="italics"/>PROBLEMA.<emph.end type="italics"/></s> <s><emph type="italics"/>Data vi datum pondus mouere.<emph.end type="italics"/></s> <s>Hoc problema prima quidem <lb/> fronte mirabile videtur, vt &longs;cilicet <lb/> pondus vel maximum viribus ta<lb/> men minimis, vt puta &longs;i dicas vnius <lb/> hominis, vel etiam imbecillioris a<lb/> licuius animalis viribus totum ter<lb/> ræ globum loco moueri po&longs;&longs;e. </s><s>Quod Archimedem <lb/> Syracu&longs;anum iacta&longs;&longs;e referunt hoc dicto, <foreign lang="greek">do\s pou= stw=| <lb/> ta\n ga\n kinw=|. </foreign></s><s>Si quis tamen in Geometrico puluere <lb/> ver&longs;atus fuerit, pre&longs;ertim verò in iis quæ ab ip&longs;o in <lb/> &longs;uo <foreign lang="greek">i)soropikw=n</foreign> libello <expan abbr="trad&utilde;tur">traduntur</expan>. </s><s>Illud cæteráque eiu&longs;<lb/> modi, ludicra Geometriæ, vt & ip&longs;e facere &longs;olitus di<lb/> citur, appellabit. </s><s>Ex huius autem problematis expli<lb/> catione, omnium machinarum qu&ecedil; ad motus cien<lb/> dos excogitari po&longs;&longs;unt, ratio pendet. </s><s>Et&longs;i verò <expan abbr="tot&utilde;">totum</expan> <lb/> Geometrica demon&longs;tratione expediri po&longs;&longs;it, quo <pb xlink:href="044/01/008.jpg" pagenum="2"/>niam tamen h&ecedil;c con&longs;ideratio, quæ Græcis <foreign lang="greek">staqmikh\</foreign> di <lb/> citur Geometriæ tantùm e&longs;t <foreign lang="greek">u(pa/llhlos</foreign> mixta &longs;cili<lb/> cet ex phy&longs;ica & Geometrica eo quòd illius &longs;ubie<lb/> ctum &longs;it motus. </s><s>Ideò paulò cra&longs;&longs;iori Minerua & <lb/> præter Geometricam &longs;implicitatem <foreign lang="greek">fusikw=s</foreign> etiam <lb/> tractanda e&longs;t: alioqui Archimedicis demon&longs;tratio<lb/> nibus &longs;tandum e&longs;&longs;et, quæ in eo mancæ &longs;unt, quòd il<lb/> læ propo&longs;itiones quæ ex phy&longs;icis peti debent, inde<lb/> mon&longs;tratæ manent, nec explicantur, &longs;ed pro confe&longs;<lb/> &longs;is principiorum loco po&longs;tulantur. </s><s>Eas igitur, qua<lb/> tenus ad huius problematis explicationem faciunt, <lb/> hic di&longs;cutiemus. </s><s>Neque enim Euclidis <expan abbr="libr&utilde;">librum</expan> de gra<lb/> ui & leui, in quo hoc argumentum per&longs;equi volui&longs;<lb/> &longs;e videtur, integrum habemus. </s><s>Ari&longs;toteles verò in <lb/> eo quod ab ip&longs;o &longs;criptum extat de Mechanicis fra<lb/> gmento pr&ecedil;ter &longs;uum morem, cùm alio qui in omni<lb/> bus exacti&longs;&longs;imus &longs;it, hanc quæ&longs;tionem potiùs nota<lb/> uit quàm explicuit in &longs;exto & &longs;eptimo phy&longs;icorum <lb/> libro multa prætermi&longs;it. </s><s>Vt igitur ad rem aggredia<lb/> mur, primùm voces, quibus vtendum e&longs;t, definie<lb/> mus, vt intelligatur quo &longs;en&longs;u eas accipiamus. </s> <s>DEFINITIO I.</s> <s>Vis dicitur agendi aut agenti re&longs;i&longs;tendi <expan abbr="pot&etilde;tia">potentia</expan>, <lb/> præ&longs;ertim verò mouendi & mouenti re&longs;i&longs;tendi. </s> <s>II.</s> <s>Vis &longs;ubiectum dicitur id quod vis mouet, vel <pb xlink:href="044/01/009.jpg" pagenum="3"/>quod à vi mouetur. </s> <s>Hîc non agimus de vi primaria quæ virium o<lb/> mnium principium e&longs;t, mouétque omnia, nec <expan abbr="tam&etilde;">tamen</expan> <lb/> mouetur, &longs;ed de ea, qu&ecedil; dum mouet, <expan abbr="etiã">etiam</expan> cum &longs;ub<lb/> iecto, cui ine&longs;t, mouetur. </s> <s>Subiecta verò quæ nullam vim habent, nec mo<lb/> uentur, nec mouenti <expan abbr="re&longs;i&longs;t&utilde;t">re&longs;i&longs;tunt</expan>: &longs;i qua verò vis illis ac<lb/> cedat, tum ab ea &longs;e moueri patiuntur. </s> <s>Et&longs;i autem plura &longs;int virium genera, tot &longs;cilicet, <lb/> quot &longs;unt in rerum natura contrariorum, actionem <lb/> & pa&longs;&longs;ionem recipientium, vt leue graue, rarum <expan abbr="d&etilde;-&longs;um">den<lb/> &longs;um</expan>, plenum vacuum, durum molle, & cætera eiu&longs;<lb/> modi, quoniam tamen ea omnia hîc per&longs;equi no&longs;tri <lb/> non e&longs;t in&longs;tituti, cùm de ea tantùm qua motus fit a<lb/> gere &longs;tatuerimus. </s> <s>III.</s> <s>Cùm de motu hîc agemus motum ad locum, <lb/> quem Græci <foreign lang="greek">fora\n</foreign> vocant, intelligi volumus. </s> <s>IIII.</s> <s>Linea autem recta quæ e&longs;t ab eo loco à quo mo<lb/> tus fieri incipit ad illum ad quem tendit. </s><s>Illius vis <lb/> quæ motum efficit, nutus dicetur. </s> <s><expan abbr="Ead&etilde;">Eadem</expan> verò linea <expan abbr="cõ&longs;iderata">con&longs;iderata</expan> à loco ad <expan abbr="qu&etilde;">quem</expan> <expan abbr="t&etilde;dit">tendit</expan> vis <lb/> ad <expan abbr="ill&utilde;">illum</expan> à quo motus fieri incipit, contra <expan abbr="nut&utilde;">nutum</expan> dicitur. </s> <s>Itidem & omnes illi parallelæ. </s> <s>Quæ verò lineæ vel rectæ, vel curuæ, nutus <expan abbr="lineã">lineam</expan> <pb xlink:href="044/01/010.jpg" pagenum="4"/>ad angulos inæquales &longs;ecant, illæ obliquè nutum <lb/> ver&longs;us, vel contra nutum ferri <expan abbr="dic&utilde;tur">dicuntur</expan>, habita ratio<lb/> ne partium, quas &longs;pectant. </s> <s>Plurimùm autem à &longs;itu corporis humani deno<lb/> minationem accipiunt illæ partes, vt &longs;ur&longs;um, deor<lb/> &longs;um, dextror&longs;um, &longs;ini&longs;tror&longs;um, antè vel ponè di<lb/> cantur. </s><s>Quæ verò lineæ, nutus lineam ad angulos <lb/> rectos &longs;ecant, neque ver&longs;us nutum, neque contra <lb/> nutum ferri dicuntur, &longs;ed æquè di&longs;tant à loco natu<lb/> rali. </s></chap><chap> <s><emph type="italics"/>Virium diui&longs;io.<emph.end type="italics"/></s> <s>Vis autem omnis aut naturalis e&longs;t, aut non natu<lb/> ralis, aut mixta. </s><s>Naturalis vis e&longs;t, qua res quælibet <lb/> natura &longs;ua mouetur, aut mouenti re&longs;i&longs;tit habita ra<lb/> tione tum loci &longs;ui naturalis, tum etiam &longs;itus &longs;uarum <lb/> partium. </s><s>Non naturalis e&longs;t quæ nec ratione loci &longs;ui <lb/> naturalis, nec &longs;itus partium mouet. </s><s>E&longs;t autem hæc <lb/> duplex, fortuita &longs;cilicet & voluntaria: illa vt vis ven<lb/> torum & &longs;pirituum intellectu carentium, hæc vt a<lb/> nimalium & &longs;pirituum intelligentium & &longs;imilium. <lb/> </s><s>Mixta dicitur, quæ partim naturalis e&longs;t, partim non <lb/> naturalis. </s> <s>Locus autem naturalis cuiu&longs;que rei e&longs;t, in quo <lb/> exi&longs;tens ip&longs;a quie&longs;cit, & &longs;i inde remota &longs;it, vis ei à na<lb/> tura in&longs;ita eam illuc impellit. </s><s>Huius verò motus aut <lb/> quietis cau&longs;&longs;a nulla alia a&longs;&longs;ignari pote&longs;t præter pri- <pb xlink:href="044/01/011.jpg" pagenum="5"/>mam illam rerum omnium conditricem, qu&ecedil;, ne o<lb/> mnia in omnibus e&longs;&longs;ent &longs;ingulis partibus vniuer&longs;i <lb/> &longs;ingula loca attribuit circa quæ conglobantur, & i<lb/> bi hærent. </s><s>Quicquid enim terreum e&longs;t, in terræ glo<lb/> bum <expan abbr="cõfluxit">confluxit</expan>. </s><s>Ita quicquid &longs;oli homogenes in Solis <lb/> corpus, lunaria omnia in lunam, & &longs;ic de cæteris cor<lb/> poribus vniuer&longs;i. </s><s>Singulæ autem eorum partes &longs;uis <lb/> totis adhærent, nec inde &longs;ponte <expan abbr="mou&etilde;tur">mouentur</expan>: &longs;i verò in<lb/> de motæ fuerint, eò redire nituntur. </s><s>Quod enim in <lb/> terra fieri videmus, idem & in reliquis corporibus v<lb/> niuer&longs;i fieri dicere po&longs;&longs;umus, &longs;cilicet quod vnita cir<lb/> ca locum &longs;uum naturalem maneant: &longs;i enim partes <lb/> eorum &longs;ponte &longs;epararentur, & vi ab ip&longs;is &longs;eparatæ <lb/> ad locum &longs;uum non redirent, &longs;tatim tota di&longs;&longs;olue<lb/> rentur. </s> <s>Vis igitur illa in loco &longs;uo quie&longs;cendi grauitas di<lb/> citur, cuius contrarium e&longs;t leuitas. </s><s>Res autem nulla <lb/> per &longs;e leuis dicitur, &longs;ed <expan abbr="tãtùm">tantùm</expan> habita ratione alterius <lb/> loci, quàm &longs;ui naturalis: vt puta &longs;i pars aliqua Solis <lb/> vi quapiam in terram inuecta e&longs;&longs;et, &longs;uóque arbitrio <lb/> committeretur, illa &longs;tatim Solem peteret: ita &longs;i ter<lb/> ræ pars in Solem inuecta e&longs;&longs;et, &longs;uóque arbitrio com<lb/> mitteretur, &longs;tatim à &longs;ole euolaret, & &longs;e per cælum <lb/> terram ver&longs;us proriperet. </s><s>Vt igitur terræ partes in <lb/> terra graues &longs;unt: &longs;ic Solis partes in &longs;ole graues &longs;unt: <lb/> in terra verò leues, terræ verò partes in Sole leues <pb xlink:href="044/01/012.jpg" pagenum="6"/>&longs;unt. </s><s>Idem de cæteris corporibus vniuer&longs;i dici po<lb/> te&longs;t. </s><s>Neque enim faciliùs ex globo lunæ particulam <lb/> ab&longs;trahas, quàm ex terra glebam. </s> <s>Aer autem nullum proprium habet locum natu<lb/> ralem, &longs;ed vbique e&longs;&longs;e pote&longs;t, cùm rerum omnium <lb/> locus vniuer&longs;alis e&longs;&longs;e videatur: de quo dubitari po<lb/> te&longs;t an &longs;it infinitus, cùm omnia contineat, à nulla re <lb/> verò totus comprehendatur, &longs;ed hæc di&longs;putatio non <lb/> e&longs;t huius loci. </s> <s>Et&longs;i autem nullum proprium habet locum natu<lb/> ralem, neque leuis aut grauis &longs;it: leuis tamen e&longs;&longs;e vi<lb/> detur, cùm grauis non &longs;it, nec ægrè &longs;e moueri patia<lb/> tur. </s><s>Itaque &longs;i quo modo ita con&longs;titutus &longs;it, vt rem a<lb/> liquam, quò minùs ad locum &longs;uum naturalem per<lb/> uenire po&longs;&longs;it, impediat, tum per expre&longs;&longs;ionem &longs;eu <lb/> <foreign lang="greek">e)/kqliyin</foreign>, ab ea eiicietur, in eiú&longs;que locum &longs;uccedet, <lb/> vt fit in aqua, cùm ei introducitur. </s><s>Quæ quidem <foreign lang="greek">e)/k<lb/> qliyis</foreign> iis etiam accidit quæ loco naturali gaudent, <lb/> cùm &longs;e&longs;e mutuò impediunt, ne ad illum ferantur. <lb/> </s><s>Præ&longs;ertim verò in liquidis locum habet, quorum <lb/> partes facilius mouentur, quàm rerum <expan abbr="cohærenti&utilde;">cohærentium</expan> <lb/> & compactarum. </s><s>Inde accidit vt quicquid liquidis <lb/> immergitur, tantò fiat in illis leuius quàm grauius, <lb/> e&longs;t moles eiu&longs;dem liquoris ip&longs;orum moli æqualis. <lb/> </s><s>Sed hæc de loco naturali fu&longs;iùs à nobis alibi medita<lb/> ta, obiter hîc attigi&longs;&longs;e &longs;ufficiat, vt intelligatur quid <pb xlink:href="044/01/013.jpg" pagenum="7"/>&longs;it vis naturalis. </s> <s>Vis autem voluntaria nullum certum nutum <lb/> habet, &longs;ed illum tantùm in quo mouentis voluntas <lb/> conquie&longs;cit. </s><s>Cúmque vis naturalis vnum tantùm <lb/> nutum habeat, &longs;cilicet à loco à naturali requie re<lb/> moto, ad ip&longs;am naturalem requiem, illa infinitos nu <lb/> tus habet & indeterminatos per &longs;e ac voluntatis tan <lb/> tùm decreto determinabiles. </s> <s>Fortuita verò ip&longs;o tantùm ca&longs;u determinabilem <lb/> nutum habet: quò enim ip&longs;a tendit, eo munere dici<lb/> tur &longs;iue &longs;ur&longs;um, &longs;iue deor&longs;um, &longs;iue ad latera. </s><s>Itaque <lb/> & hic & ille nutus <foreign lang="greek">a)dio/ristos</foreign> dicitur. </s><s>Quemadmodum <lb/> & is quem habent vires illæ, quibus res à &longs;itu <expan abbr="parti&utilde;">partium</expan> <lb/> naturali remotæ ad illum redeunt: prout enim ab <lb/> eo motæ &longs;unt, ita ad illum redeunt, prout etiam huc <lb/> aut illuc obuer&longs;æ &longs;unt, vt vis arcus aut bali&longs;tæ. </s> <s>DEFIN. V.</s> <s>Vires autem contrariæ dicuntur, quæ contrarios <lb/> motus ciere po&longs;&longs;unt, vt ea <expan abbr="qu&ecedil;">quæ</expan> &longs;ur&longs;um mouet & <expan abbr="qu&ecedil;">quæ</expan><lb/> deor&longs;um, & &longs;ic de cæteris. </s> <s>Con&longs;ideratur autem in vi <expan abbr="quãtitas">quantitas</expan>, tum eo quòd <lb/> vis partibus &longs;uis con&longs;tet, in quas in infinitum diuidi <lb/> pote&longs;t, & rur&longs;us additione aut multiplicatione au<lb/> geri, tum quòd æqualitatis exce&longs;&longs;us & defectus <expan abbr="cõ-parationem">con<lb/> parationem</expan> recipiat. </s> <pb xlink:href="044/01/014.jpg" pagenum="8"/> <s>Ac quoniam vis e&longs;t mouendi potentia, vis par<lb/> tes erunt quæ motus partes efficient, & quæ erit mo<lb/> tus partium, men&longs;ura eadem erit & vis partium. </s><s>Et <lb/> motus quidem propria men&longs;ura e&longs;t linea &longs;eu <expan abbr="&longs;pati&utilde;">&longs;patium</expan>. <lb/> </s><s>Quantum enim res quæpiam mota, &longs;patij percur<lb/> rit, tantùm mota e&longs;&longs;e dicitur. </s><s>Quoniam verò in mo <lb/> mento vel in&longs;tanti quod in&longs;tar puncti e&longs;t, & magni<lb/> tudine caret, nullus motus fieri pote&longs;t, &longs;ed motus o<lb/> mnis in tempore fit. </s><s>Ideò ad motus men&longs;uram tem<lb/> pus etiam adhibere oportet. </s><s>Illud enim cum &longs;patio <lb/> vel linea, motus dici facit æquales aut inæquales. </s> <s>DEFIN. VI.</s> <s>Æquales igitur motus dicuntur, qui æqualibus <lb/> temporibus æqualia &longs;patia percurrunt. </s> <s>Qui autem æqualibus temporibus æqualia &longs;pa<lb/> tia permeant, illi iidem proportionales &longs;unt: hoc <lb/> e&longs;t, quæ e&longs;t ratio temporis, quo alter eorum fit ad <lb/> tempus quo fit alter, eadem e&longs;t &longs;patij quod percurrit <lb/> alter ad &longs;patium quod reliquus percurrit. </s><s>Si enim <lb/> duorum quorum &longs;inguli vna hora miliaris vnius i<lb/> ter conficiunt, alter eodem motu per tria miliaria <lb/> ferri pergat, alter verò per duo <expan abbr="tãtùm">tantùm</expan>. </s><s>Ille tribus ho<lb/> ris ea ab&longs;oluet, hic verò duabus, & æquè celeriter <lb/> ferri dicentur, licet &longs;patia inæqualia <expan abbr="percurrãt">percurrant</expan>, quo<lb/> niam illa &longs;unt temporibus proportionalia. </s> <pb xlink:href="044/01/015.jpg" pagenum="9"/> <s>VII.</s> <s>Inæquales autem motus dicuntur, quorum tem<lb/> pora non &longs;unt &longs;patiis proportionalia. </s><s>Eorum autem <lb/> maior ille dicitur, cuius maior erit ratio &longs;patij ad &longs;pa<lb/> tium, quàm temporis ad tempus, quibus fiunt illi <lb/> motus. </s><s>Ex quo intelligitur etiam quis minor dica<lb/> tur. </s><s>Maior igitur dicetur qui celeriùs feretur, minor, <lb/> qui tardiùs. </s> <s>VIII.</s> <s>Æquales igitur vires dicentur, quæ æqualibus <lb/> motibus &longs;ubiecta &longs;ua mouebunt. </s><s>Maior verò quæ <lb/> &longs;ubiectum &longs;uum magis vel celeriùs mouebit. </s><s>Mi<lb/> nor, quæ minùs vel tardiùs. </s></chap><chap> <s><emph type="italics"/>Motuum diui&longs;io.<emph.end type="italics"/></s> <s>Comparantur autem motus &longs;ecundum omnes <lb/> comparationis gradus, hoc e&longs;t, æqualitatem, exce&longs;<lb/> &longs;um & defectum: tum &longs;ibi ip&longs;is, cùm totus &longs;uis par<lb/> tibus confertur, tum alij aliis. </s> <s>Quatenus autem &longs;ibi ip&longs;i motus <expan abbr="cõparatur">comparatur</expan>, aut <lb/> e&longs;t <foreign lang="greek">o(moiomerh\s</foreign> &longs;eu æquabilis aut <foreign lang="greek">a)nomoiomerh\s</foreign>, id e&longs;t inæ<lb/> quabilis. </s><s>Aequabilis, cuius omnes partes tum mini<lb/> mæ tum maiores tempora habent &longs;patiis per quæ <lb/> cientur proportionalia. </s><s>Inæquabilis verò, in quo illa <lb/> non &longs;unt proportionalia. </s> <s>Inæquabilium autem motuum duo &longs;unt genera, <pb xlink:href="044/01/016.jpg" pagenum="10"/>cre&longs;cens &longs;cilicet & decre&longs;cens, ídque vel continuè, <lb/> vel per interualla, ordinatim vel <foreign lang="greek">a)taktw=s. </foreign></s> <s>Cre&longs;cens dicitur, quando partium po&longs;teriorum <lb/> ip&longs;ius maior e&longs;t celeritas, quàm priorum: <expan abbr="decre&longs;c&etilde;s">decre&longs;cens</expan>, <lb/> quando minor. </s> <s>Continuè, quando nulla pars illius vel minima <lb/> alteri parti e&longs;t æqualis &longs;eu &longs;ibi ip&longs;i æquabilis. </s> <s>Per interualla verò, quando partes quidem illius <lb/> &longs;ibi ip&longs;is æquabiles &longs;unt, aliæ verò aliis comparatæ <lb/> inæquabiles. </s> <s>Ordinatim, quando incrementum aut <expan abbr="decrem&etilde;-tum">decremen<lb/> tum</expan> illud certa aliqua proportionis progre&longs;&longs;ione <lb/> fit, vt &longs;i in motu continuè cre&longs;cente, diui&longs;o toto mo<lb/> tus &longs;patio in aliquot partes æquales eadem &longs;it ratio <lb/> celeritatis, finis primi &longs;patij ad celeritatem finis &longs;e<lb/> cundi, quæ finis &longs;ecundi ad finem tertij, & &longs;ic dein<lb/> ceps: vel &longs;i finis &longs;ecundi &longs;patij duplo citiùs feratur, <lb/> quàm finis primi & finis tertij, triplo quàm primi: <lb/> finis verò quarti quadruplo, & &longs;ic deinceps. </s><s>Idem &longs;i <lb/> in quauis alia proportionis progre&longs;&longs;ione illarum <lb/> partium celeritas, aliarum ad alias &longs;e habeat. </s><s>In mo<lb/> tu verò per interualla cre&longs;cente <expan abbr="augm&etilde;tum">augmentum</expan> illud or<lb/> dinatim fieri dicetur, &longs;i interuallorum proportio a<lb/> liquem progre&longs;&longs;ionis ordinem &longs;eruet: puta &longs;i <expan abbr="prim&utilde;">primum</expan> <lb/> interuallum duplo tardiùs <expan abbr="&longs;ec&utilde;do">&longs;ecundo</expan> moueatur, triplo <lb/> verò tardiùs tertio, &c. </s> <pb xlink:href="044/01/017.jpg" pagenum="11"/> <s>Inordinatè autem cre&longs;cere vel decre&longs;cere dicetur <lb/> motus, &longs;i incrementum illud aut decrementum nul<lb/> la certa & ordinata proportione fiat. </s> <s>Comparantur autem motus alij aliis, æquabiles <lb/> &longs;cilicet & inæquabiles ordinatim progredientes: de <lb/> inordinatorum enim tum inter &longs;e, tum cum aliis <expan abbr="cõ||">com</expan><lb/> paratione nihil certò &longs;tatui pote&longs;t. </s> <s>Comparantur igitur æquabiles cum æquabili<lb/> bus aut cum inæquabilibus, ac etiam inæquabiles <lb/> cum inæquabilibus. </s> <s>Æquabiles autem cum æquabilibus compara<lb/> tionibus &longs;uis partibus corre&longs;pondentibus &longs;unt pro<lb/> portionales. </s><s>Si enim <expan abbr="proponãtur">proponantur</expan> duo motus æqua<lb/> biles, quorum alter altero maior &longs;it, quæ erit ratio <lb/> dimidiæ partis vnius ad dimidiam partem alterius, <lb/> eadem erit tertiæ ad tertiam, & &longs;ic deinceps. </s> <s>Æquabilium verò motuum cum inæquabilibus, <lb/> cum iis &longs;cilicet qui per interualla cre&longs;cunt, compa<lb/> ratio fit tanquam cum pluribus diuer&longs;is motibus æ<lb/> quabilibus, &longs;unt enim interuallorum <expan abbr="illor&utilde;">illorum</expan> motus <lb/> æquabiles. </s><s>At cum continuè <expan abbr="cre&longs;c&etilde;tibus">cre&longs;centibus</expan> aut decre<lb/> &longs;centibus difficile e&longs;t comparationis modum defi<lb/> nire, cùm ne momento quidem &longs;tabilis maneat par<lb/> tium illarum quantitas. </s><s>E&longs;t tamen motus æquabilis <lb/> illorum men&longs;ura: tanti enim e&longs;&longs;e dicuntur, quanti <lb/> forent, &longs;i in ea celeritatis men&longs;ura ad quam perue <pb xlink:href="044/01/018.jpg" pagenum="12"/>nerunt, æquabiliter pergerent. </s> <s>Cre&longs;centes verò cum cre&longs;centibus continuè, <expan abbr="cõ-parati">con<lb/> parati</expan> &longs;iquidem eadem progre&longs;&longs;ionis proportione <lb/> cre&longs;cant, æqualia &longs;patia ab eorum principiis nume<lb/> rata, æqualibus temporibus emen&longs;a habebunt, idem <lb/> in decre&longs;centibus. </s><s>Sed h&ecedil;c hactenus inqui&longs;iui&longs;&longs;e &longs;uf<lb/> ficiat: &longs;i quis enim particulatim omnia <expan abbr="exp&etilde;dere">expendere</expan> vel<lb/> let, in infinitum &longs;e&longs;e extenderet h&ecedil;c partium & pro<lb/> portionum motuum &longs;upputatio. </s> <s>Iam videamus quæ vires quos horum motuum <lb/> cieant. </s><s>Et vis quidem ea qua res quælibet ad &longs;itum <lb/> naturalem &longs;uarum partium redit, motu <expan abbr="cõtinuè">continuè</expan> de<lb/> cre&longs;cente mouet: quo enim propiùs ad &longs;itum natu<lb/> ralem partium accedit, eo tardior & languidior e&longs;t. <lb/> </s><s>Arcus enim cùm primùm laxari incipit, celeriùs <lb/> mouetur, quàm cùm motus &longs;ui finis propior e&longs;t. </s> <s>Vis verò naturalis, qua res qu&ecedil;libet ad <expan abbr="loc&utilde;">locum</expan> &longs;uum <lb/> naturalem tendit, &longs;ubiectum &longs;uum, motu continuè <lb/> & ordinatim cre&longs;cente, mouet. </s><s>Illius autem motus <lb/> cau&longs;a e&longs;t quòd faciliùs id moueatur, quod in motu <lb/> e&longs;t, quàm quod quie&longs;cit. </s><s>Vis igitur eadem, <expan abbr="&longs;ubiect&utilde;">&longs;ubiectum</expan> <lb/> quod iam in motu e&longs;t premens, illud magis moue<lb/> bit, quàm &longs;i quie&longs;cat, & magis motum, magis etiam <lb/> mouebit: ita vt eadem vis motione maior fiat, quàm <lb/> per &longs;e &longs;it. </s><s>Et hæc e&longs;t cau&longs;a cur ictus, quo magis ab al<lb/> tero venit, eo vehementior &longs;it. </s><s>Motus autem huius <pb xlink:href="044/01/019.jpg" pagenum="13"/>&longs;patia hanc celeritatis proportionem &longs;eruant, vt <expan abbr="qu&ecedil;">quæ</expan> <lb/> e&longs;t ratio totius &longs;patij, per quod fit ille motus ad par<lb/> tem ip&longs;ius (vtriu&longs;que initio inde &longs;umpto, vbi e&longs;t mo<lb/> tus initium) eadem &longs;it celeritatis ad celeritatem. <lb/> <figure id="id.044.01.019.1.jpg" xlink:href="044/01/019/1.jpg"/><lb/> Exempli gratia, &longs;i vis aliqua per lineam ABC <lb/> mouerit, &longs;itque AB illius lineæ pars, quæ erit <lb/> ratio AC ad AB, eadem erit celeritatis motus <lb/> in puncto C ad <expan abbr="celeritat&etilde;">celeritatem</expan> motus in puncto B. <lb/> </s><s>Cuiu&longs;modi proportio ob&longs;eruatur in paralle<lb/> lis triangulum &longs;ecantibus. </s><s>Vt enim &longs;e habet <lb/> <figure id="id.044.01.019.2.jpg" xlink:href="044/01/019/2.jpg"/><lb/> AC ad AB, &longs;ic CG ad BF, & vt AD ad <lb/> AC, &longs;ic DH ad CG. </s><s>Itaque &longs;i in &longs;patia ali<lb/> quot æqualia diuidatur totius motus &longs;pa<lb/> tium, finis &longs;ecundi duplo citiùs feretur, <lb/> quàm finis primi: finis verò tertijtriplo <lb/> citiùs quàm finis primi, & &longs;ic deinceps. <lb/> </s><s>Hac autem ratione fit, vt &longs;patiorum illo<lb/> rum initio maxima &longs;it celeritatis <expan abbr="differ&etilde;-tia">differen<lb/> tia</expan>: progre&longs;&longs;u verò &longs;emper minuatur, ac tandem fer<lb/> mè eadem &longs;it, vt fit in trianguli lateribus, quæ lon<lb/> gi&longs;&longs;imè producta æquè di&longs;tare videntur. </s><s>Eáque e&longs;t <lb/> ratio cur Solis & Lunæ radij, etiam&longs;i concurrant (in <lb/> ip&longs;orum &longs;cilicet corporibus, aut vltra res quas <expan abbr="illu-&longs;trãt">illu<lb/> &longs;trant</expan>) paralleli <expan abbr="tam&etilde;">tamen</expan> <expan abbr="appareãt">appareant</expan>. </s><s><expan abbr="Ead&etilde;">Eadem</expan> <expan abbr="etiã">etiam</expan> cau&longs;a e&longs;t cur <lb/> line&ecedil; omnes ad <expan abbr="perp&etilde;dicul&utilde;">perpendiculum</expan> in <expan abbr="terrã">terram</expan> cadentes, paral<lb/> lelæ videantur, cùm <expan abbr="tam&etilde;">tamen</expan> in centro terræ <expan abbr="cõcurrant">concurrant</expan>. </s> <pb xlink:href="044/01/020.jpg" pagenum="14"/> <s>Hunc igitur motum vis naturalis efficit, modò <lb/> nulla quies intercedat. </s> <s>Vis autem voluntaria motum omnem ciere apta <lb/> e&longs;t. </s><s>Fortuita verò inordinatum tantùm motum effi<lb/> cit. </s><s>Et hæc de variis virium motibus, <expan abbr="eorúmq;">eorúmque</expan> pro<lb/> portione & partibus. </s> <s>E&longs;t verò etiam virium partitio quædam, quæ ex <lb/> eorum quibus <expan abbr="inhær&etilde;t">inhærent</expan> &longs;ubiectorum diui&longs;ione, aut <lb/> alterius ad alterum proportione re&longs;ultat. </s><s>In &longs;ubie<lb/> ctis enim, quæ <foreign lang="greek">o(moiomerh=</foreign> &longs;unt, vis naturalis æqualiter <lb/> per omnes partes diffu&longs;a e&longs;t: ita vt quæ e&longs;t ratio mo<lb/> lis ad molem &longs;ub eadem figura, <expan abbr="ead&etilde;">eadem</expan> &longs;it vis ad vim: <lb/> vt &longs;i globus alicuius metalli, ad alium globum eiu&longs;<lb/> dem metalli mole quadruplus &longs;it, illius quoque <expan abbr="põ-dus">pon<lb/> dus</expan> ad huius pondus quadruplum erit. </s> <s>Hactenus vim <expan abbr="cõ&longs;iderauimus">con&longs;iderauimus</expan> quatenus mouet, <lb/> &longs;upere&longs;t vt eam con&longs;ideremus, quatenus mouenti <lb/> re&longs;i&longs;tit. </s><s>Habet autem locum re&longs;i&longs;tentia, vbi duæ vi<lb/> res contrariæ committuntur: &longs;i enim vtraque <expan abbr="eod&etilde;">eodem</expan> <lb/> motu moueat, nulla erit re&longs;i&longs;tentia, &longs;ed altera alteri <lb/> addita maiorem vim con&longs;tituet. </s> <s>Committi autem dicuntur vires, quando ita ap<lb/> plicantur & <expan abbr="connect&utilde;tur">connectuntur</expan>, vt altera nutu &longs;uo moueri <lb/> non po&longs;&longs;it, quin altera contra nutum &longs;uum mouea<lb/> tur. </s><s>Viribus autem ita connexis, accidit vt altera in <lb/> alterius &longs;ubiectum agat, & altera in &longs;ubiecto &longs;uo exi- <pb xlink:href="044/01/021.jpg" pagenum="15"/>&longs;tens, alteri in altero &longs;ubiecto exi&longs;tenti re&longs;i&longs;tat, ita vt <lb/> eorum &longs;ubiecta quodammodo reciproca fiant. </s> <s>CONCLVSIO I.</s> <s>Illam autem re&longs;i&longs;tentiam vi momenti in eodem <lb/> &longs;ubiecto æqualem, aut eandem cum ea e&longs;&longs;e, ex ip&longs;ius <lb/> definitione con&longs;tat. </s><s>E&longs;t enim re&longs;i&longs;tere <expan abbr="nõ">non</expan> pati &longs;e mo<lb/> ueri. </s><s>Quantum autem vis quælibet <expan abbr="&longs;ubiect&utilde;">&longs;ubiectum</expan> &longs;uum <lb/> nutu &longs;uo mouet, tantum illud contra nutum &longs;uum <lb/> moueri non patitur. </s><s>Mouendi igitur & <expan abbr="mou&etilde;ti">mouenti</expan> re<lb/> &longs;i&longs;tendi potentia in eodem &longs;ubiecto æquales &longs;unt. </s> <s>II.</s> <s>Quemadmodum autem in eodem &longs;ubiecto ma<lb/> ior vis ine&longs;&longs;e dicitur, in quo e&longs;t maioris motus po<lb/> tentia, &longs;ic maior re&longs;i&longs;tentia erit maioris motus con<lb/> trarij impatientia. </s><s>Eadem igitur vis magis mouere <lb/> nitenti <expan abbr="cõtra">contra</expan> ip&longs;ius nutum magis re&longs;i&longs;tet, minus ve<lb/> rò nitenti minùs re&longs;i&longs;tet. </s> <s>III.</s> <s>Et quo maior erit motus contrarius, eo magis re<lb/> &longs;i&longs;tet: id e&longs;t, quo celerius vis quælibet à nutu &longs;uo re<lb/> uelletur, eo magis re&longs;i&longs;tet. </s><s>Duplo igitur citiùs re<lb/> uellenti duplo re&longs;i&longs;tet, triplo citius triplo re&longs;i&longs;tet, & <lb/> &longs;ic <foreign lang="greek">a)nalo/gws</foreign> in omni proportione. </s><s>Quæ igitur erit <lb/> ratio motus ad motum, contra nutum eadem erit <lb/> in vna & eadem vi ratio re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam. </s> <s>Si verò plures vires comparentur. </s> <pb xlink:href="044/01/022.jpg" pagenum="16"/> <s>IIII.</s> <s>Æqualium quidem virium æqualibus motibus <lb/> æquales erunt re&longs;i&longs;tentiæ. </s><s>Si enim æquales vires æ<lb/> qualiter à &longs;uis nutibus reuellantur, æqualiter re&longs;i<lb/> &longs;tent. </s> <s>V.</s> <s>Inæqualibus verò motibus, earum re&longs;i&longs;tenti&ecedil; in<lb/> æquales erunt, & motuum proportionem &longs;equen<lb/> tur. </s><s>Exempli gratia, &longs;int duæ vires A & B, quarum v<lb/> traque vna hora miliare vnum percurrat, illæ æqua<lb/> les erunt. </s><s>Si igitur vtraque quarta parte vnius milia<lb/> ris, vel dimidio miliari à &longs;uo nutu eodem tempore <lb/> reuellatur, eadem erit vtriu&longs;que illi motui re&longs;i&longs;ten<lb/> tia. </s><s>Sin verò altera quidem quarta parte vnius milia<lb/> ris à nutu &longs;uo reuellatur, altera verò eodem tempo<lb/> re integro miliari reuellatur: hæc re&longs;i&longs;tentia ad il<lb/> lam quadrupla erit, & &longs;ic de cæteris motuum pro<lb/> portionibus. </s> <s>VI.</s> <s>Inæqualium verò virium re&longs;i&longs;tentiæ æqualibus <lb/> motibus ip&longs;arum virium proportionem &longs;equentur. <lb/> </s><s>Quo maior enim e&longs;t vis, eo magis eodem tempore <lb/> eodémque &longs;patio contra nutum mouenti re&longs;i&longs;tit. <lb/> </s><s><expan abbr="Ex&etilde;pli">Exempli</expan> gratia, &longs;it vis A quadrupla ad vim B, vis A mo<lb/> uenti &longs;e vno miliari quadruplo re&longs;i&longs;tet, quàm B mo<lb/> uenti &longs;e vno miliari eodem tempore. </s> <pb xlink:href="044/01/023.jpg" pagenum="17"/> <s>Inæqualium verò virium inæqualibus motibus <lb/> re&longs;i&longs;tentiæ, quando motus eam inter &longs;e proportio<lb/> nem &longs;eruabunt, quæ e&longs;t inter eas <foreign lang="greek">a)ntipeponqw=s</foreign>, &longs;eu <lb/> reciprocè, æquales erunt. </s><s>Sit enim vis A quæ ad vim <lb/> B eandem habeat <expan abbr="ration&etilde;">rationem</expan> quam C ad D, puta quam <lb/> 3 ad 1, aut 4 ad 1, aut quæcunque alia &longs;it, <expan abbr="moueatúrq;">moueatúrque</expan> <lb/> vtraque &longs;patio E contra nutum &longs;uum eodem tem<lb/> pore, re&longs;i&longs;tentia A ad re&longs;i&longs;tentiam B &longs;e habebit, vt C <lb/> ad D per proximam conclu&longs;ionem. </s><s>Si verò B moue<lb/> atur eodem tempore &longs;patio aliquo quod &longs;e habeat <lb/> ad &longs;patium E vt C ad D, re&longs;i&longs;tentia B huic motui &longs;e <lb/> habebit ad re&longs;i&longs;tentiam priori motui, vt C ad D per <lb/> quartam <expan abbr="conclu&longs;ion&etilde;">conclu&longs;ionem</expan>. </s><s>Duæ ergo re&longs;i&longs;tentiæ ad ean<lb/> dem, eandem habebunt rationem, ergo æquales e<lb/> runt per <emph type="italics"/>9<emph.end type="italics"/> prop. <emph type="italics"/>5.<emph.end type="italics"/> Elem. Eucl. </s> <s>VIII.</s> <s>Si verò motuum virium inæqualium proportio <lb/> non &longs;it eadem quæ e&longs;t ip&longs;arum virium reciprocè, vis <lb/> illius quæ ad alteram maiorem habebit rationem, <lb/> quàm motus alterius ad motum ip&longs;ius, re&longs;i&longs;tentia <lb/> maior erit, altera verò minor. </s><s>Exempli gratia, &longs;i vis <lb/> A &longs;it tripla ad vim B, & motus quo B à nutu &longs;uo reuel<lb/> letur, &longs;it minor quàm triplus ad motum A, re&longs;i&longs;ten<lb/> tia A erit maior quàm re&longs;i&longs;tentia B. </s><s>Idem erit, &longs;i ma<lb/> ior &longs;it ratio A ad B, quàm 3 ado, motus verò B ad mo <pb xlink:href="044/01/024.jpg" pagenum="18"/>tum A &longs;it triplus. </s><s>Contrà verò &longs;i A ad B &longs;it tripla: mo<lb/> tus verò B ad motum A &longs;it maior quàm triplus, ma<lb/> ior erit re&longs;i&longs;tentia B quàm A. </s><s>Idem accidet, &longs;i A ad B <lb/> ratio minor &longs;it quàm 3 ad 1, motus verò B ad motum <lb/> A &longs;it triplus. </s><s>Hunc autem exce&longs;&longs;um & defectum <expan abbr="nõ">non</expan> <lb/> vlteriùs &longs;crutabimur, vt ip&longs;ius quantitatis <expan abbr="men&longs;urã">men&longs;uram</expan> <lb/> a&longs;&longs;equamur: alioqui in infinitum fieret progre&longs;&longs;us. </s> <s>Ex his colligimus <expan abbr="re&longs;i&longs;t&etilde;tias">re&longs;i&longs;tentias</expan> tribus modis æqua<lb/> les aut inæquales dici, per &longs;e &longs;cilicet aut motione, aut <lb/> vtroque modo. </s><s>Per &longs;e quidem cùm &longs;ubiectorum <lb/> vires &longs;unt æquales (id e&longs;t æqualiter &longs;ua &longs;ubiecta mo<lb/> uent) aut inæquales. </s><s>Motione verò quando per &longs;e <lb/> quidem in æquales &longs;unt, motione verò æquales <expan abbr="fi&utilde;t">fiunt</expan>: <lb/> aut quando per &longs;e æquales &longs;unt, motione verò inæ<lb/> quales <expan abbr="fi&utilde;t">fiunt</expan> eo, quo dictum e&longs;t, modo. </s><s>Vtroque mo <lb/> do æquales aut inæquales <expan abbr="dic&utilde;tur">dicuntur</expan>, quando tum per <lb/> &longs;e, tum motione tales fiunt. </s><s>His explicatis videamus <lb/> quis eorum &longs;it effectus, vbi vires committentur. </s> <s>IX.</s> <s>Primùm ex prima conclu&longs;ione &longs;equitur qu&ecedil; erit <lb/> ratio re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam, eandem fore re&longs;i<lb/> &longs;tentiæ ad vim cuius e&longs;t altera re&longs;i&longs;tentia: &longs;unt enim <lb/> vis & <expan abbr="re&longs;i&longs;t&etilde;tia">re&longs;i&longs;tentia</expan> in <expan abbr="eod&etilde;">eodem</expan> &longs;ubiecto æquales. </s><s>Ergo mul<lb/> to magis &longs;equitur, &longs;i <expan abbr="re&longs;i&longs;t&etilde;tia">re&longs;i&longs;tentia</expan> altera maior, altera mi <lb/> nor fuerit, illam huius vi <expan abbr="maior&etilde;">maiorem</expan> fore, vel illius vim <lb/> hac re&longs;i&longs;tentia maiorem, &longs;i æqualis æqualem. </s> <pb xlink:href="044/01/025.jpg" pagenum="19"/> <s>X</s> <s>Re&longs;i&longs;tentia autem vi contrariæ commi&longs;&longs;a <expan abbr="tãtum">tantum</expan> <lb/> de ea tollit quanta e&longs;t ip&longs;a re&longs;i&longs;tentia. </s><s>Sublata autem <lb/> vi tollitur motus: &longs;ublata verò re&longs;i&longs;tentia, &longs;i vis ad&longs;it, <lb/> &longs;equitur motus. </s><s>Vbi igitur vis & re&longs;i&longs;tentia in&ecedil;qua<lb/> les committentur, &longs;i vis maior &longs;it quàm re&longs;i&longs;tentia, <lb/> fiet motus &longs;ecundum vis illius nutum, & contra nu<lb/> tum vis illius quæ re&longs;i&longs;tit. </s><s>Si verò re&longs;i&longs;tentia maior <lb/> fuerit, tum ip&longs;a fiet vis mouens, & vim contrariam <lb/> contra nutum ip&longs;ius mouebit, dum ip&longs;a nutu &longs;uo <lb/> mouebitur. </s><s>Hinc &longs;equuntur duo theoremata, circa <lb/> quæ totius huius con&longs;iderationis cardo vertitur. </s> <s>THEOREMA I.</s> <s>Duarum virium connexarum, quarum (&longs;i mo<lb/> ueantur) motus erunt ip&longs;is <foreign lang="greek">a)ntipeponqw=s</foreign> proportiona<lb/> les neutra alteram mouebit, &longs;ed æquilibrium <expan abbr="faci&etilde;t">facient</expan>. </s> <s>Sit vis A commi&longs;&longs;a cum vi B, &longs;itque vis A ad vim <lb/> B ratio per &longs;e, vt C ad D <expan abbr="quæc&utilde;que">quæcunque</expan> illa &longs;it, &longs;iue dupla, <lb/> &longs;iue tripla, &longs;iue alia. </s><s>Sit etiam <expan abbr="ead&etilde;">eadem</expan> ratio motus quo <lb/> B mouebitur, &longs;i ita, vt connexa &longs;unt, moueantur) ad <lb/> motum quo A mouebitur, quæ e&longs;t C ad D, dico mo<lb/> tum non &longs;equuturum, &longs;ed factum iri æquilibrium. <lb/> </s><s>Aut enim A & B vires per &longs;e erunt æquales aut inæ<lb/> quales: &longs;i æquales, ergo & ip&longs;arum motus æquales e<lb/> runt: &longs;unt enim ex hypothe&longs;i ip&longs;is proportionales: <pb xlink:href="044/01/026.jpg" pagenum="20"/>ergo per quartam conclu&longs;ionem earum re&longs;i&longs;tentiæ <lb/> æquales erunt, ergo per decimam conclu&longs;ionem <expan abbr="nõ">non</expan> <lb/> fiet motus. </s><s>Si verò &longs;unt inæquales, cùm earum mo<lb/> tus ex hypothe&longs;i &longs;int <foreign lang="greek">a)ntipeponqw=s</foreign> proportionales per <lb/> &longs;eptimam conclu&longs;ionem æquales erunt etiam re&longs;i<lb/> &longs;tentiæ. </s><s>Ergo nec motus fiet: nullo igitur modo fiet <lb/> earum motus. </s><s>Quod demon&longs;trandum erat. </s> <s>THEOREMA II.</s> <s>Quarum verò ita connexarum (&longs;i <expan abbr="moueãtur">moueantur</expan>) mo<lb/> tus, ip&longs;is proportionales non erunt: illa alteram mo<lb/> uebit, cuius ad alteram ratio maior erit, quàm huius <lb/> motus ad illius motum. </s> <s>Sit vis A cum vi B commi&longs;&longs;a, &longs;itque A ad B ratio <lb/> per &longs;e, vt C ad D: ita verò connexæ &longs;int, vt &longs;i mouean<lb/> tur, minor &longs;it ratio motus quo B mouebitur ad mo<lb/> tum, quo A mouebitur, quum C ad D dico A motu<lb/> ram B: erit enim per octauam conclu&longs;ionem re&longs;i<lb/> &longs;tentia B minor re&longs;i&longs;tentia A: ergo per nonam con<lb/> clu&longs;ionem re&longs;i&longs;tentia B minor vi A: vis igitur A vim <lb/> B mouebit per vndecimam conclu&longs;ionem. </s><s>Quod <lb/> demon&longs;trandum erat. </s> <s>Hinc po&longs;&longs;emus ad problematis no&longs;tri demon<lb/> &longs;trationem rectà pergere: ante tamen cra&longs;&longs;iùs ali<lb/> quanto hæc explicanda &longs;unt, vt à quouis faciliùs in<lb/> telligantur. </s><s>Duæ &longs;unt motus men&longs;uræ, locus &longs;cilicet <lb/> & tempus: vtroque igitur, tempore videlicet & loco <pb xlink:href="044/01/027.jpg" pagenum="21"/>maior, minor aut æqualis dicitur: & quo minori <expan abbr="t&etilde;-pore">tem<lb/> pore</expan> idem &longs;patium ab&longs;oluitur, eo maior e&longs;t, vel quo <lb/> maius &longs;patium eodem tempore. </s> <s>Vt igitur motus magnus dicatur, perinde e&longs;t &longs;i <lb/> paruo tempore fiat, aut magno &longs;patio. </s> <s>Quod autem maiorem <expan abbr="mot&utilde;">motum</expan> ciere pote&longs;t illud, <lb/> vis maior dicitur, quòd minorem minor. </s> <s>Vis autem &longs;eu mouendi potentia in eodem &longs;ub<lb/> iecto certa & finita e&longs;t. </s><s>Qu&ecedil;libet enim res vi natura<lb/> li pr&ecedil;dita, &longs;i à loco naturali ab&longs;it, &longs;uóque arbitrio <expan abbr="cõ-mittatur">con<lb/> mittatur</expan>, certo tempore eò redit. </s><s>E&longs;t enim certum <lb/> in rerum natura quanto tempore libræ vnius <expan abbr="põdus">pondus</expan> <lb/> deor&longs;um &longs;ponte &longs;ua delatum, miliare vnum aut duo <lb/> conficiat pro ratione materi&ecedil;, vel quantum &longs;patij v<lb/> na vel duabus horis percurrat. </s><s>Id verò quantum &longs;it, <lb/> hominum indu&longs;tria nondum quod &longs;ciam explora<lb/> tum e&longs;t: aliâs autem id demon&longs;trare conabimur. </s> <s>Mouendi verò potentia in alieno &longs;ubiecto infi<lb/> nita e&longs;t, hoc e&longs;t, in <expan abbr="infinit&utilde;">infinitum</expan> augeri vel minui pote&longs;t, <lb/> quoniam in finita auctione & diminutione e&longs;t re&longs;i<lb/> &longs;tentia: tanta enim e&longs;t, quanta e&longs;t in eodem &longs;ubiecto <lb/> vis: <expan abbr="quantú&longs;q;">quantú&longs;que</expan> motus illius e&longs;t vis alterius re&longs;pectu. <lb/> </s><s>Quo igitur vis quæ alienum &longs;ubiectum mouere ni<lb/> titur, illud magis mouere nitetur, eo minùs illud <lb/> mouere poterit, maior enim erit illius re&longs;i&longs;tentia. <lb/> </s><s>Quemadmodum enim quod magis nutu &longs;uo mo <pb xlink:href="044/01/028.jpg" pagenum="22"/>uetur, maiorem vim mouendi habet. </s><s>Ita illud idem <lb/> quod magis contra nutum &longs;uum mouetur, <expan abbr="maior&etilde;">maiorem</expan> <lb/> vim re&longs;i&longs;tendi habet. </s><s>E contrario verò quo vis quæ<lb/> libet minorem motum in alieno &longs;ubiecto ciere ni<lb/> tetur, eo faciliùs illud mouebit. </s> <s>Tarditate igitur motus, re&longs;i&longs;tentia in infinitum <lb/> minui pote&longs;t: minuta verò re&longs;i&longs;tentia vis contrariæ <lb/> effectus augetur, ita vt vis quæ per &longs;e minima e&longs;t, in <lb/> contrariam cui plurimum diminuta &longs;it, re&longs;i&longs;tentia <lb/> maximè agat. </s><s>Perinde igitur e&longs;t, &longs;i vis mouens ma<lb/> gna &longs;it, mouenda verò parua, ac &longs;i illa celeriter fera<lb/> tur, hæc verò tardè: <expan abbr="quãtum">quantum</expan> enim vis mouens &longs;i ma<lb/> gna fuerit in mouendam minorem poterit, tantum <lb/> vis parua celeriter mota in magnam tardè motam <lb/> poterit. </s><s>Si igitur velimus vt vis parua magnam mo<lb/> ueat, eas ita collocare oportet, vt quantum magna <lb/> paruam &longs;uperat, tantum illi de motus celeritate de<lb/> trahatur, & aliquid ampliùs. </s><s>Si enim tanta &longs;it tardi<lb/> tas motus vis vnius, re&longs;pectu motus vis alterius, <expan abbr="quã-ta">quan<lb/> ta</expan> e&longs;t proportio vis illius, ad hanc non fiet motus: vt <lb/> &longs;i pondus A quatuor librarum cum pondere B libræ <lb/> vnius committatur: &longs;intque ita connexa, vt dum <lb/> A vno &longs;patio mouebitur, B quatuor &longs;patiis mouea<lb/> tur, ita vt motus A quadruplo tardior &longs;it motu B <expan abbr="nõ">non</expan> <lb/> fiet motus, quia quantum A excedit B pondere tan<lb/> tùm deficit motus tarditate. </s> <pb xlink:href="044/01/029.jpg" pagenum="23"/> <s>Tantum enim e&longs;t libram vnam quatuor &longs;patiis <lb/> moueri, quantum libras quatuor vno &longs;patio eodem <lb/> tempore: &longs;i igitur alterutri eorum ita <expan abbr="con&longs;titutor&utilde;">con&longs;titutorum</expan>, <lb/> vel momentum vis addatur, id cui additum fuerit, <lb/> alterum mouebit. </s><s>Idem fiet &longs;i A ita connectatur vt <lb/> vel momento citiùs quàm moueri po&longs;itum e&longs;t, mo<lb/> ueatur vel B tardiùs. </s> <s>Id etiam alia ratione o&longs;tendi pote&longs;t, &longs;i vis A qua<lb/> drupla &longs;it ad vim B, erunt in A quatuor partes, ip&longs;i B <lb/> æquales. </s><s>Si igitur B cum &longs;ingulis illis committatur, <lb/> ita vt æqualiter moueantur, non fiet motus: &longs;i verò <lb/> alterutri aut ip&longs;i B, aut &longs;ingulis illis partibus vis vel <lb/> motus momentum adiiciatur vel detrahatur, illa cui <lb/> adiectum fuerit, aut cui non detractum fuerit, nutu <lb/> &longs;uo mouebitur, & aliam contra nutum eius moue<lb/> bit. </s><s>Addito igitur ip&longs;i B momento, dum vno &longs;patio <lb/> mouebitur, &longs;ingulas illas partes vno &longs;patio moue<lb/> bit. </s><s>Vbi igitur &longs;ingulas &longs;emel mouerit, ip&longs;a quater <lb/> mota erit. </s><s>Dum verò &longs;ingulæ &longs;emel motæ erunt, to<lb/> tum ex illis con&longs;tans &longs;emel motum intelligetur: po<lb/> terit igitur B addito ip&longs;i momento, dum quater mo<lb/> uebitur, &longs;emel totum A mouere. </s> <s>Tertio modo id ip&longs;um concludere po&longs;&longs;umus. </s><s>Si <lb/> duæ vires æquales connectantur, ita vt motæ, æqua<lb/> liter moueantur, altera in <expan abbr="alterã">alteram</expan> non aget. </s><s>Si verò ita <lb/> connectantur vt motæ inæqualiter moueantur, <pb xlink:href="044/01/030.jpg" pagenum="24"/>quantum altera alteram celeritate &longs;uperabit, <expan abbr="tãtum">tantum</expan> <lb/> & vi &longs;uperabit. </s><s>Vt &longs;i vis A vi B æqualis &longs;it, ac ita <expan abbr="con-nectãtur">con<lb/> nectantur</expan>, vt B quatuor &longs;patiis moueatur, dum A vno <lb/> &longs;patio mouebitur, B quadruplam vim habebit ad A, <lb/> quia eam in motibus &longs;uis proportionem &longs;eruant, <lb/> quam &longs;i <expan abbr="&longs;eruar&etilde;t">&longs;eruarent</expan> arbitrio &longs;uo commi&longs;&longs;æ, B ad A qua<lb/> drupla e&longs;&longs;et. </s><s>Si igitur illis ita connexis, ip&longs;i A adda<lb/> tur vis triplo maior quàm ip&longs;a &longs;it, B illis quatuor re<lb/> &longs;i&longs;tet, nec fiet motus. </s> <s>Duarum igitur virium comparatarum, quanto <lb/> altera &longs;ubiectum &longs;uum celeriùs mouebit, quàm al<lb/> tera: tanto illa hanc celeriùs mouere poterit, quàm <lb/> ip&longs;a moueatur, &longs;i illi vis <expan abbr="mom&etilde;tum">momentum</expan> additum fuerit. <lb/> </s><s>Ita vt quæ erit proportio vis ad vim, eadem &longs;it mo<lb/> tus, quem illa in hac ciere pote&longs;t ad motum quo ip&longs;a <lb/> mouebitur. </s><s>Et è conuer&longs;o quæ erit ratio motus ad <lb/> motum, eadem erit vis cui additum fuerit momen<lb/> tum ad eam quam ip&longs;a mouere poterit ratio reci<lb/> procè. </s><s>Iam ad problematis no&longs;tri demon&longs;tratio<lb/> nem veniamus. </s></chap><chap> <s><emph type="italics"/>Propo&longs;iti problematis demon&longs;tratio.<emph.end type="italics"/></s> <s>Sit data vis A quantacunque illa &longs;it magna vel <lb/> parua: datum verò pondus B quantumquantum il<lb/> lud &longs;it, dico me vi A pondus B tollere po&longs;&longs;e. </s><s>Id &longs;ic de<lb/> mon&longs;tro. </s> <pb xlink:href="044/01/031.jpg" pagenum="25"/> <s>Primùm enim ex doctrina &longs;ecundi <expan abbr="l&etilde;matis">lemmatis</expan>, quod <lb/> inferiùs demon&longs;trabitur, &longs;ciam proportionem pro<lb/> ximè maiorem, quàm &longs;it A ad B proportio. </s><s>Deinde <lb/> ex doctrina primi lemmatis ita connectam A & B, <lb/> vt quando ambo mouebuntur, nunc &longs;it ratio motus <lb/> quo B mouebitur ad <expan abbr="mot&utilde;">motum</expan> quo A mouebitur, quàm <lb/> &longs;it A ad B. </s><s>His peractis &longs;equitur vim A pondus B mo<lb/> turam ex &longs;ecundo &longs;uperiùs demon&longs;trato theorema<lb/> te. </s><s>Quod erat propo&longs;itum. </s> <s>LEMMA I.</s> <s>Duas vires ita connectere, vt &longs;i moueantur, <expan abbr="ear&utilde;">earum</expan> <lb/> motus, in data ratione alter ad alterum &longs;e habeant <lb/> vires contrariæ, aut medio aliquo, aut per &longs;e ab&longs;que <lb/> vllo medio committuntur. </s><s>Si ab&longs;que medio com<lb/> mittant, eodem motu mouebuntur, maior enim mi<lb/> norem eodem motu, quo ip&longs;a moueri poterit, mo<lb/> uebit: aut æquilibrium <expan abbr="faci&etilde;t">facient</expan>, &longs;i æquales &longs;int: vt &longs;i le<lb/> ue graui committatur, &longs;iquidem leuitas grauitate <lb/> maior &longs;it, attolletur graue: &longs;in verò grauitas maior <lb/> &longs;it, leue deprimetur: &longs;i æqualia &longs;int, non mouebun<lb/> tur. </s> <s>Si verò medio aliquo connectantur mediorum <lb/> varia &longs;unt genera. </s><s>Aut enim medium e&longs;t flexibile & <lb/> <foreign lang="greek">o(moiomere\s</foreign>, vt funis, catena, &c. </s><s>aut e&longs;t inflexibile, il<lb/> lúdque aut rectum, aut curuum, vt recta linea vel <lb/> curua vel angulus. </s><s>Atque hæc omnia aut continua <pb xlink:href="044/01/032.jpg" pagenum="26"/>&longs;unt aut diui&longs;a, &longs;implicia aut compo&longs;ita. </s> <s>Horum autem mediorum opera fit vt vires illis <lb/> applicatæ, quarum iidem &longs;unt nutus, contrariis mo<lb/> tibus moueantur. </s><s>Id <expan abbr="aut&etilde;">autem</expan> fit cùm in mediis illis inter <lb/> <expan abbr="eor&utilde;">eorum</expan> extrema interiacet quies vna vel plures. </s><s><expan abbr="Ex&etilde;pli">Exempli</expan> <lb/> gratia, &longs;i duo pondera funis extremitatibus alligata <lb/> &longs;int, & funis clauo fixo & immobili incumbat pro<lb/> pter illam quietem inter <expan abbr="vtrumq;">vtrumque</expan> pondus <expan abbr="po&longs;itã">po&longs;itam</expan> <expan abbr="nõ">non</expan> <lb/> poterit <expan abbr="alter&utilde;">alterum</expan> deor&longs;um moueri, quin <expan abbr="alter&utilde;">alterum</expan> &longs;ur&longs;um <lb/> moueatur. </s><s><expan abbr="Id&etilde;">Idem</expan> fiet in linea recta, &longs;i enim illius extre<lb/> mitatibus pondera duo annexa &longs;int, & inter ea &longs;it in <lb/> illa <expan abbr="punct&utilde;">punctum</expan> aliquod quie&longs;cens, dum alterum ex illis <lb/> ponderibus deor&longs;um feretur, alterum a&longs;cendet. </s><s><expan abbr="P&utilde;-ctum">Pun<lb/> ctum</expan> autem illud quie&longs;cens in linea illa recta, Gr&ecedil;cis <lb/> hypomochlium dicitur, eò quòd vecti, quem <foreign lang="greek">mo/xlon</foreign><lb/> vocant, &longs;ubiiciatur. </s><s>Huius autem hypomochlij, in <lb/> recta linea &longs;e vecte collocatio faciet, vt lineæ extre<lb/> ma &longs;ecundum datam rationem moueantur. </s><s>Si enim <lb/> recta linea in datam rationem diui&longs;a fuerit, hoc e&longs;t, <lb/> vt pars altera ad <expan abbr="alterã">alteram</expan> eam habeat <expan abbr="ration&etilde;">rationem</expan>, <expan abbr="quã">quam</expan> quis <lb/> voluerit. (Quod <expan abbr="quid&etilde;">quidem</expan> quo modo fiat docet Euc.) & <lb/> in puncto diui&longs;ionis collocetur <expan abbr="hypomochli&utilde;">hypomochlium</expan>, illius <lb/> lineæ extrema &longs;ecundum <expan abbr="illã">illam</expan> <expan abbr="ration&etilde;">rationem</expan> mouebuntur: <lb/> &longs;iue enim con&longs;ideretur <expan abbr="circulor&utilde;">circulorum</expan>, quos illa extrema <lb/> de&longs;cribent, proportio &longs;iue &longs;patium quod illa in linea <lb/> perpendiculari notabunt vtroque modo illi motus, <pb xlink:href="044/01/033.jpg" pagenum="27"/><expan abbr="parti&utilde;">partium</expan> <expan abbr="illar&utilde;">illarum</expan> <expan abbr="proportion&etilde;">proportionem</expan> <expan abbr="&longs;eruab&utilde;t">&longs;eruabunt</expan>. </s><s>Sit <expan abbr="ex&etilde;pli">exempli</expan> gratia <lb/> linea AB, quæ in puncto C in datam rationem &longs;ecta <lb/> &longs;it, puta vt pars AC quadrupla &longs;it ad partem CB, mo<lb/> ueatúrque circa centrum C, punctum A de&longs;cribet cir<lb/> culum <expan abbr="quadrupl&utilde;">quadruplum</expan> ad illum quem B <lb/> de&longs;cribet. </s><s>E&longs;t enim <expan abbr="ead&etilde;">eadem</expan> ratio in cir<lb/> culo diametri ad <expan abbr="diametr&utilde;">diametrum</expan>, quæ e&longs;t <lb/> circunferentiæ ad <expan abbr="circunferentiã">circunferentiam</expan> (vt <lb/> alibi demon&longs;trauimus.) Hac igitur <lb/> ratione A puncti motus quadruplus <lb/> <figure id="id.044.01.033.1.jpg" xlink:href="044/01/033/1.jpg"/><lb/> erit ad puncti B <expan abbr="mot&utilde;">motum</expan>. </s><s>Si verò ponamus AD <expan abbr="perpen-dicular&etilde;">perpen<lb/> dicularem</expan> e&longs;&longs;e, & linea AB illi primùm <expan abbr="coincid&etilde;s">coincidens</expan> circa <lb/> punctum C, moueatur donec A ad D perueniat: tum <lb/> <expan abbr="eod&etilde;">eodem</expan> momento B perueniet ad E: motum igitur erit <lb/> A in linea <expan abbr="perp&etilde;diculari">perpendiculari</expan> tota circuli maioris diame<lb/> tro, quæ e&longs;t AD:B verò in <expan abbr="ead&etilde;">eadem</expan> linea, minoris <expan abbr="tãtùm">tantùm</expan> <lb/> circuli diametro <expan abbr="mot&utilde;">motum</expan> erit, qu&ecedil; e&longs;t BE. </s><s>Atqui diame<lb/> ter AD quadrupla e&longs;t ad BE, quia ex hypothe&longs;i <expan abbr="&longs;emi-diametror&utilde;">&longs;emi<lb/> diametrorum</expan> <expan abbr="illor&utilde;">illorum</expan> <expan abbr="circulor&utilde;">circulorum</expan> proportio e&longs;t, vt 4 ad 1. <lb/> Motus igitur <expan abbr="p&utilde;cti">puncti</expan> in linea A <expan abbr="perp&etilde;diculari">perpendiculari</expan> ad <expan abbr="mot&utilde;">motum</expan> <lb/> <expan abbr="p&utilde;cti">puncti</expan> B quadruplus erit: <expan abbr="Id&etilde;">Idem</expan> dicetur &longs;i in data aliqua <lb/> alia ratione &longs;ecta &longs;it linea AB. <expan abbr="Demõ&longs;trat&utilde;">Demon&longs;tratum</expan> igitur e&longs;t <lb/> quomodo fieri po&longs;&longs;it vt rectæ line&ecedil; extrema <expan abbr="&longs;ec&utilde;d&utilde;">&longs;ecundum</expan> <lb/> <expan abbr="datã">datam</expan> <expan abbr="ration&etilde;">rationem</expan> moueantur. </s><s>Si igitur illis extremis duæ <lb/> vires applicentur, <expan abbr="moueb&utilde;tur">mouebuntur</expan> <expan abbr="eod&etilde;">eodem</expan> ip&longs;o motu: ergo <lb/> &longs;ecundum datam vel propo&longs;itam rationem. </s><s>Quod <lb/> a&longs;&longs;umptum erat. </s> <pb xlink:href="044/01/034.jpg" pagenum="28"/> <s>Quod autem in vecte demon&longs;tratum e&longs;t, illud e<lb/> tiam in reliquis mediis demon&longs;trandum erit, et&longs;i <lb/> lemmati &longs;atisfactum e&longs;t, dum in vno exemplo id <lb/> probatum e&longs;t. </s><s>Ante igitur &longs;ecundum lemma de<lb/> mon&longs;trabimus. </s> <s>LEMMA II.</s> <s>Proportionem proximè maiorem vel minorem, <lb/> quàm &longs;it datæ vis ad datum pondus proportio, de<lb/> terminare vis cuiuslibet quantitas ex motu ciere po|| <lb/> te&longs;t, metienda e&longs;t. </s><s>Motum autem ciere pote&longs;t vel in <lb/> &longs;ubiecto &longs;uo, vel in alieno. </s><s>Vis autem men&longs;ura non <lb/> &longs;umitur ex eo motu quem in &longs;ubiecto &longs;uo ciere po<lb/> te&longs;t, eo quòd licet vis quælibet certum motum ha<lb/> beat & <expan abbr="determinat&utilde;">determinatum</expan> quo &longs;ubiectum &longs;uum mouet, <lb/> illius tamen quantitas, vt &longs;uprà diximus, nondum <lb/> demon&longs;trata e&longs;t. </s><s>Supere&longs;t igitur vt vires motu illo <lb/> metiamur, quem in alieno &longs;ubiecto ciere po&longs;&longs;unt, vt <lb/> id fiat, quærenda nobis &longs;unt &longs;ubiecta quæ in homi<lb/> num pote&longs;tate &longs;int, & cum vi qualibet committi <lb/> po&longs;&longs;int. </s><s>Omnium autem mobilium &longs;ubiectorum, <lb/> maximè in hominum pote&longs;tate &longs;unt grauia: leuia e<lb/> nim coercere vix po&longs;&longs;umus. </s><s>Ideo vires illis metiri <lb/> &longs;olemus, &longs;ed grauibus quæ vel vel figuris &longs;uis & com<lb/> page vel va&longs;e aliquo coercentur. </s><s>Itaque v&longs;us homi<lb/> num certas qua&longs;dam ponderum men&longs;uras &longs;ibi &longs;ta <lb/> tui, ponderibus &longs;cilicet quibu&longs;dam certa <expan abbr="quãtitate">quantitate</expan> <pb xlink:href="044/01/035.jpg" pagenum="29"/>con&longs;tantibus inditis nominibus, vt e&longs;&longs;ent <expan abbr="ponder&utilde;">ponderum</expan> <lb/> omnium communes men&longs;uræ, vt &longs;unt libræ, vncia, <lb/> <expan abbr="drachm&ecedil;">drachmæ</expan>, &c. </s><s>quas famo&longs;as men&longs;uras <expan abbr="vocãt">vocant</expan>. </s><s>Quem<lb/> admodum igitur numeros numeris, &longs;ic pondera <expan abbr="põ-deribus">pon<lb/> deribus</expan> metimur. </s><s><expan abbr="Tãtum">Tantum</expan> enim pondus e&longs;&longs;e dicitur, <lb/> quot libras vncias drachmas æqualiter mouere po<lb/> te&longs;t dempto momento. </s><s>Nec tantum pondera hoc <lb/> modo metimur, &longs;ed etiam alias omnes vires motum <lb/> ad locum cientes. </s><s>Quot enim libras vir aut aliud a<lb/> nimal vel <expan abbr="v&etilde;tus">ventus</expan> aut ignis, aut aliqua alia vis dempto <lb/> momento mouere poterit, tot libris illam æqualem <lb/> e&longs;&longs;e dicemus. </s><s>Si igitur data vel propo&longs;ita vis metien<lb/> da &longs;it, &longs;iquidem naturalis &longs;it, quoniam docuimus <lb/> vim naturalem per totum &longs;ubiectum diffu&longs;am e&longs;&longs;e <lb/> in rebus homogeneis: id e&longs;t, vt quæ e&longs;t proportio <lb/> molis ad molem, eadem &longs;it ponderis ad pondus: &longs;u<lb/> memus partem aliquam illi homogeneam, aut ex i<lb/> p&longs;o &longs;ubiecto, aut ex alio ip&longs;i homogeneo, eámque <lb/> famo&longs;a aliqua men&longs;ura metiemur, vtramque &longs;cili<lb/> cet committendo & ob&longs;eruando, quem motum al<lb/> tera in altera ciere po&longs;&longs;it, vbi enim æquilibrium fa<lb/> cient motibus extremorum, quibus affixæ fuerint, <lb/> proportionales erunt per 1 theorema: motus autem <lb/> illi linearum dimen&longs;ione quam Geometria docet, <lb/> noti erunt, & eorum proportio, nota igitur erit & <lb/> virium proportio. </s><s>Atqui men&longs;uræ famo&longs;æ nota, per <pb xlink:href="044/01/036.jpg" pagenum="30"/>&longs;e e&longs;t quantitas: duorum autem proportione cogni<lb/> ta & alterius quantitate, &longs;tatim & reliqui quantitas <lb/> innote&longs;cit per &longs;ecundam da. Eu. nota igitur vis <expan abbr="quã-titate">quan<lb/> titate</expan> quæ parti illi ine&longs;t, no&longs;cetur & vis <expan abbr="quãtitas">quantitas</expan> <expan abbr="qu&ecedil;">quæ</expan><lb/> toti inerit: quæ enim erit ratio molis &longs;ubiecti vis da<lb/> tæ ad molem particul&ecedil; &longs;umpt&ecedil;, eadem erit vis totius <lb/> ad vim partis. </s> <s>Hîc igitur erunt quatuor proportio<lb/> nalia, &longs;cilicet vt moles ad molem: &longs;ic vis ad vim ex <lb/> quibus tria nota erunt: moles enim metiri Geome<lb/> tria nos docuit, præterea vis partis, vt demon&longs;traui<lb/> mus, nota e&longs;t, ergo & vis totius quantitas per deci<lb/> mam &longs;eptimam &longs;eptimi Elem. Eucl. </s> <s>Si verò &longs;ubiectum non &longs;it homogeneum, vi ta<lb/> men naturali &longs;it præditum, &longs;i quidem data vis quam <lb/> metiri volumus, ea &longs;itque motum ciere volumus, <lb/> tum con&longs;iderabimus quæ pars in illa, vis minimum <lb/> habeat: & ex ea totam ip&longs;ius molem æ&longs;timabimus: <lb/> &longs;i verò &longs;it vis mouenda, &longs;tatuemus qua&longs;i tota &longs;it ho<lb/> mogenea ip&longs;ius parti, quæ vis plurimum in &longs;e habe<lb/> bit, & ex ea totam ip&longs;ius molem æ&longs;timabimus. </s><s>Ita<lb/> que tum illius quanta minima, tum huius quanta <lb/> maxima e&longs;&longs;e pote&longs;t, vis quantitas nobis nota erit per <lb/> proximè demon&longs;tratam rationem. </s><s>Notis autem vi<lb/> rium quantitatibus, nota erit & earum proportio: <lb/> ergo & proportio ip&longs;a proximè maior vel mi<lb/> nor: addita enim vel detracta ip&longs;ius denomi- <pb xlink:href="044/01/037.jpg" pagenum="31"/>nationi, vnitate erit proximè maior vel mi<lb/> nor. </s> <s>Si verò vis data non &longs;it naturalis, <expan abbr="voluntariã">voluntariam</expan> qui<lb/> dem ita æ&longs;timare po&longs;&longs;umus, qualis vt plurimum <lb/> e&longs;t, & &longs;i quidem ea &longs;it, qua mouere volumus, eam <lb/> &longs;tatuemus, quanta minima in eiu&longs;dem generis &longs;ub<lb/> iecto e&longs;&longs;e pote&longs;t, vt &longs;i vim hominis quinquaginta li<lb/> bris æqualem ponamus, vim equi centum: &longs;i verò <lb/> ea &longs;it quam mouere volumus, &longs;tatuemus eam <expan abbr="quã-ta">quan<lb/> ta</expan> maxima e&longs;&longs;e pote&longs;t, vt vim hominis 300 libra<lb/> rum, vim equi 500 librarum, & &longs;ic dé cæteris: ita vt <lb/> nullum &longs;it dubium quin illa minor &longs;it, quam &longs;tatue<lb/> rimus, hæc verò maior. </s> <s>Vis verò fortuitæ quantitatem nulla certa con<lb/> iectura a&longs;&longs;equi po&longs;&longs;umus: ita vt qua&longs;cunque ma<lb/> chinas ei aptemus, modò moueat, modò non mo<lb/> ueat, neque ad no&longs;trum <expan abbr="in&longs;titut&utilde;">in&longs;titutum</expan> magnoperè per<lb/> tinet illa inqui&longs;itio: cum fortuitorum nulla &longs;it di&longs;ci<lb/> plina. </s> <s>His igitur modis virium duarum datarum pro<lb/> portio proximè maior nota fiet, quod in &longs;ecundo <lb/> lemmate demon&longs;trandum &longs;ump&longs;eramus. </s> <s>Iam redeamus ad mediorum, quibus vi<lb/> res annectuntur, con&longs;ide<lb/> rationem. </s> <pb xlink:href="044/01/038.jpg" pagenum="32"/><figure id="id.044.01.038.1.jpg" xlink:href="044/01/038/1.jpg"/> <s>Docuimus quis &longs;it &longs;implicis ve<lb/> ctis effectus, &longs;implex autem ve<lb/> ctis &longs;emicirculi conuer&longs;ione &longs;uam <lb/> operationem ab&longs;ouit, ita vt &longs;i <lb/> vlteriùs F moueatur in in alio &longs;e<lb/> micirculo motus prioribus contra<lb/> rios cieat: vt exempli gratia, &longs;it <lb/> vectis AB, cuius hypomochium <lb/> &longs;it C, dum A punctum de&longs;cribet &longs;emicirculum AFD, <lb/> motus ille deor&longs;um erit: interea verò B punctum <lb/> de&longs;cribet &longs;emicirculum BGE a&longs;cendendo: &longs;i verò A <lb/> tran&longs;cendat, D incipiet a&longs;cendere: B verò tran&longs;cen<lb/> dens, E <expan abbr="de&longs;c&etilde;det">de&longs;cendet</expan>: ideo excogitata e&longs;t vectis ratio per<lb/> petua ex plurium vectium &longs;ucce&longs;&longs;ione circa idem <lb/> hypomochlium: e&longs;t autem illa tum in ergatis aut &longs;u<lb/> culis, tum in duorum tympanorum homocentrico<lb/> rum, &longs;eu eadem axe transfixorum in planis parallelis <lb/> aptatione, quorum &longs;emidiametri &longs;int in eadem pro<lb/> portione quæ in vecte ad propo&longs;itum motum cien<lb/> dum nece&longs;&longs;aria e&longs;t: centrum verò eorum &longs;eu axis fi<lb/> xa &longs;it, ac ita vires aptentur, vt maior minorem, mi<lb/> nor verò maiorem tympanum impellat. </s><s>Quemad<lb/> modum autem horum tympanorum homocentri<lb/> corum opera vectis perpetui ratio <expan abbr="inu&etilde;ta">inuenta</expan> e&longs;t, ita eo<lb/> rum multiplicatione motus, & mouentis & <expan abbr="mou&etilde;-di">mouen<lb/> di</expan>, proportio in infinitum augeri pote&longs;t. </s><s>Cuius rei <pb xlink:href="044/01/039.jpg" pagenum="33"/>maximus e&longs;t v&longs;us: nec enim materia ad vectem, cu<lb/> ius longitudo &longs;tadij vnius requireretur, idonea in<lb/> ueniri po&longs;&longs;et: plurium autem tympanorum propor<lb/> tionalium aptatione fiet machina tractabilis, cuius <lb/> vis maior erit quàm vectis, cuius longitudo &longs;tadij v<lb/> nius e&longs;&longs;et. </s><s>Si enim duo tympani homocentrici a<lb/> ptentur, quorum proportio &longs;it alterius ad alterum, <lb/> decupla, vis quæ libram vnam æquabit, vim decem <lb/> libris æqualem dempto momento mouebit, &longs;i con<lb/> gruè illis tympanis aptentur: &longs;i verò adhuc duo alij <lb/> tympani fiant, quorum alterius ad alterum decupla <lb/> &longs;it proportio, ac minor <expan abbr="illor&utilde;">illorum</expan> ita aptetur, vt <expan abbr="maior&etilde;">maiorem</expan> <lb/> ex prioribus moueat, appendatur deinde minori ex <lb/> prioribus tympanis vis centum libris æqualis, maio<lb/> ri verò ex po&longs;terioribus vis vni libræ æqualis, tum <lb/> hæc illam mouebit, & &longs;ic in infinitum motus extre<lb/> morum proportio multiplicari pote&longs;t. </s><s>Flexibili ve<lb/> rò medio quies ita aptari pote&longs;t, vt duo eius extre<lb/> ma diuer&longs;is motibus moueantur, & quidem &longs;ecun<lb/> dum datam rationem. </s><s>Exemplum habemus in orga <lb/> nis poly&longs;pactis, &longs;eu trochleis, in quibus altero funis <lb/> extremo immobili manente, reliquum funis circa <lb/> plures trochleas conuoluitur, quarum aliæ centris <lb/> immobilibus fixæ &longs;unt, aliæ verò ip&longs;is contrariæ cen <lb/> tris mobilibus. </s><s>Atque ita circa illas conuoluitur fu<lb/> nis, vt inter eas &longs;it &longs;patium tantum, quanta e&longs;t linea, <pb xlink:href="044/01/040.jpg" pagenum="34"/>per quam motum ciere volumus. </s><s>Quoties igitur fu<lb/> nis extremum quod moueri pote&longs;t, trahitur, <expan abbr="&longs;ingul&ecedil;">&longs;ingulæ</expan><lb/> reuolutiones æqualiter minuuntur, eáque ratione, <lb/> <expan abbr="di&longs;tãtia">di&longs;tantia</expan> quæ e&longs;t inter trochleas contrarias minuitur, <lb/> ita &longs;cilicet vt quot &longs;unt reuolutiones, in tot partes <lb/> di&longs;tributum &longs;it motus &longs;patium. </s><s>Quot igitur reuolu<lb/> tiones erunt, totuplex erit motus extremitatis funis <lb/> ad motum trochlearum mobilium ver&longs;us fixas. </s><s>Si i<lb/> gitur motus i&longs;tius extrema con&longs;tituantur, alterum <lb/> <expan abbr="quid&etilde;">quidem</expan> funis illa extremitas quæ mouetur, <expan abbr="alter&utilde;">alterum</expan> ve<lb/> rò terminus &longs;patij illius, quod e&longs;t à trochleis fixis ad <lb/> mobiles contrarias: quæ erit proportio numeri con<lb/> uolutionum funis ad vnitatem, eadem erit motus, <lb/> quo funis extremum mouebitur ad motum quo al<lb/> terum extremum mouebitur: pote&longs;t autem in infi<lb/> nitum augeri conuolutionum numerus, ergo & mo<lb/> tus illius proportio. </s> <s>Angulus autem ad motum ciendum ita ad<lb/> hibetur. </s><s>Diximus motus men&longs;uram in nutus li<lb/> nea &longs;umi, quantum igitur vis aliqua per eam ver&longs;us <lb/> locum naturalem mouetur, tanta e&longs;t, quantum verò <lb/> per eam à loco naturali reuellitur, tanta e&longs;t eius re&longs;i<lb/> &longs;tentia. </s><s>Quod verò per lineam à loco naturali æquè <lb/> di&longs;tantem (id e&longs;t, per eam quæ nutus lineas &longs;ecat ad <lb/> angulos rectos) mouetur, illud mouenti non re&longs;i<lb/> &longs;tit, omnium autem linearum inter illas intercepta- <pb xlink:href="044/01/041.jpg" pagenum="35"/>rum, ac cum illis in earum inter&longs;ectionis puncto <expan abbr="cõ-currentium">con<lb/> currentium</expan>, quæ obliquè nutum ver&longs;us, aut contra <lb/> nutum ferri dicuntur, quo propiùs quælibet ad nu<lb/> tus lineam accedit, per illam rei motæ vis aut <expan abbr="re&longs;i&longs;t&etilde;-tia">re&longs;i&longs;ten<lb/> tia</expan> maior e&longs;t: quò verò propiùs ad lineam à loco na<lb/> turali æqui di&longs;tantem accedit, eò minor e&longs;t. </s><s><expan abbr="Omni&utilde;">Omnium</expan> <lb/> autem maxima e&longs;t in nutus linea, æquidi&longs;tans verò <lb/> à loco naturali motui per lineam nutus omnino op<lb/> po&longs;ita e&longs;t, obliquæ verò non ita quia &longs;ecundum illas <lb/> <expan abbr="eod&etilde;">eodem</expan> &longs;patio delata vis propiùs ad <expan abbr="loc&utilde;">locum</expan> <expan abbr="natural&etilde;">naturalem</expan> acce<lb/> dit, aut ab eo recedit, quàm e&longs;&longs;et, cùm moueri c&oelig;pit. </s><figure id="id.044.01.041.1.jpg" xlink:href="044/01/041/1.jpg"/><s>Sit exempli gratia AB linea <lb/> nutus, vis alicuius, puta ponde<lb/> ris, &longs;itque A &longs;ur&longs;um & contra nu <lb/> tum: B verò deor&longs;um & nutum <lb/> ver&longs;us, de&longs;cribatúrque circulus, <lb/> cuius AB &longs;it diameter, quàm CD, <lb/> alia diameter &longs;ecet ad angulos <lb/> rectos in centro E: omnes igitur <lb/> lineæ à centro E ad circunferentiam circuli ductæ, <lb/> quæ <expan abbr="cad&etilde;t">cadent</expan> intra <expan abbr="&longs;emicircul&utilde;">&longs;emicirculum</expan> CAD, contra <expan abbr="nut&utilde;">nutum</expan> <expan abbr="a&longs;c&etilde;-dere">a&longs;cen<lb/> dere</expan> dicentur, quatenus circunferentiam &longs;pectant: <lb/> quatenus verò centrum &longs;pectant, de&longs;cendere dicen<lb/> tur: <expan abbr="cõtra">contra</expan> verò omnes in &longs;emicirculo CBD à centro <lb/> ad circunferentiam ductæ de&longs;cendere circunferen<lb/> tiam ver&longs;us, & centrum ver&longs;us a&longs;cendere dicentur: <pb xlink:href="044/01/042.jpg" pagenum="36"/>illæ igitur erunt, quæ obliquè nutum ver&longs;us aut con <lb/> tra nutum ferri <expan abbr="dic&utilde;tur">dicuntur</expan>: linea verò CED, neque <expan abbr="a&longs;c&etilde;-det">a&longs;cen<lb/> det</expan>, neque de&longs;cendet: lineæ verò in ip&longs;am ad angu<lb/> los rectos incidentes nutus lineæ erunt, quoniam li<lb/> neæ AB parallelæ erunt: &longs;i igitur à centro E <expan abbr="ducãtur">ducantur</expan> <lb/> lineæ ad circunferentiam inter A & D, puta EF, EG, <lb/> EH, quarum EF &longs;it proxima lineæ AB:EH verò pro<lb/> xima lineæ CD, ac per illas moueantur contra <expan abbr="nut&utilde;">nutum</expan> <lb/> tres vires æquales eodem tempore, ita vt prima per <lb/> lineam EF perueniat ad punctum F: &longs;ecunda verò <lb/> per EG perueniat ad G, tertia per EH perueniat ad H, <lb/> dico vis motæ per EF maiorem fore re&longs;i&longs;tentiam, <lb/> quàm illius quæ per EG aut EH, mouebitur & illius <lb/> quæ per EG mouebitur, maiorem quàm eius quæ <lb/> per EH mouebitur. </s><s>Ducantur enim à punctis FGH <lb/> in lineam ED perpendiculares FK, GL, HM, illæ <expan abbr="er&utilde;t">erunt</expan> <lb/> nutus lineæ: quanta igitur erit FK, tantum vis prima <lb/> mota cen&longs;ebitur, quanta verò GH, tantum vis &longs;ecun<lb/> da: quanta verò HM, tantum vis tertia mota cen&longs;ebi<lb/> tur: at qui quæ e&longs;t ratio motus ad motum in viribus <lb/> æqualibus, per quartam <expan abbr="cõclu&longs;ionem">conclu&longs;ionem</expan> huius tracta<lb/> tus, eadem e&longs;t re&longs;i&longs;tentiæ ad re&longs;i&longs;tentiam: quæ igitur <lb/> erit ratio linearum illarum perpendicularium, inter <lb/> &longs;e eadem erit & <expan abbr="re&longs;i&longs;t&etilde;tiarum">re&longs;i&longs;tentiarum</expan> virium per lineas ob<lb/> liquas motarum, à quibus illæ perpendiculares du<lb/> ctæ &longs;unt, atqui quo lineæ illæ perpendiculares pro- <pb xlink:href="044/01/043.jpg" pagenum="37"/>piùs ad AB circuli diametrum accedunt, eò &longs;unt ma<lb/> iores per decimam quartam tertij Elem. Eu. ergo & <lb/> vires per eas lineas delat&ecedil; à <expan abbr="quar&utilde;">quarum</expan> extremitatibus du<lb/> <expan abbr="c&etilde;tur">centur</expan>, maiores <expan abbr="re&longs;i&longs;t&etilde;tias">re&longs;i&longs;tentias</expan> <expan abbr="habeb&utilde;t">habebunt</expan>: at qui quò magis <lb/> ad AB, accedunt eo magis ab ED recedunt: quò igi<lb/> tur magis ad ED accedent, eò minores erunt <expan abbr="re&longs;i&longs;t&etilde;-tiæ">re&longs;i&longs;ten<lb/> tiæ</expan>, hinc &longs;equitur tanquam corollarium. </s> <s>Si duæ vi<lb/> res per&longs;e æquales triangulo rectangulo aptentur, cu<lb/> ius latus alterum, angulum rectum <expan abbr="con&longs;titu&etilde;tium">con&longs;tituentium</expan>: <lb/> &longs;it earum nutus linea, ac per illud vis altera mouea<lb/> tur, altera verò per latus angulo recto oppo&longs;itum, <lb/> quæ erit ratio huius lateris ad illud, eadem erit re&longs;i<lb/> &longs;tentia vis illius ad vis huius <expan abbr="re&longs;i&longs;t&etilde;tiam">re&longs;i&longs;tentiam</expan>. </s><s>Atqui duo<lb/> rum illorum laterum proportio in infinitum auge<lb/> ri vel minui pote&longs;t, ergo & re&longs;i&longs;tentia virium illis ap<lb/> plicatarum. </s><s>Hoc igitur modo po&longs;&longs;umus vti angulo <lb/> &longs;eu triangulo ad motus ciendos, nempe eo immoto <lb/> vires per eius latera mouendo. </s> <s>Sed & alia ratione eo vti po&longs;&longs;umus, ip&longs;um &longs;cilicet <lb/> triangulum mouendo, qui tunc cuneus dicitur. </s><s>Vt <lb/> autem eo hac ratione vtamur, vires ita di&longs;ponere o<lb/> portet, vt altera illarum vni ex lateribus angulum <lb/> rectum con&longs;tituentibus incumbat, altera verò lateri <lb/> ip&longs;um &longs;ubtendenti. </s><s>Illa enim tantùm mouebitur, <lb/> quantum latus cui altera incumbit. </s><s>Sit exempli gra <pb xlink:href="044/01/044.jpg" pagenum="38"/><figure id="id.044.01.044.1.jpg" xlink:href="044/01/044/1.jpg"/><lb/> tia triangulus ABC, cuius angulus B <lb/> rectus &longs;it, latus verò illum &longs;ubtendens <lb/> &longs;it AC, incumbátque vis D lateri AB, <lb/> vis verò E lateri AC, &longs;itque vis D nutus <lb/> linea BC, vis verò E nutus &longs;it linea AB, erigatúrque à <lb/> puncto C linea CF <expan abbr="perp&etilde;dicularis">perpendicularis</expan> ad BC æqualis AB, <lb/> à qua vis E <expan abbr="nõ">non</expan> di&longs;cedat. </s><s>Si triangulum illud in plano <lb/> fixo moueatur, donec AB perueniat ad CF, mota e<lb/> rit vis D nutu &longs;uo tantùm, quanta e&longs;t linea BC, vis ve<lb/> rò E tantum, quanta e&longs;t linea AB. </s><s>Cùm autem po&longs;&longs;it <lb/> in infinitum augeri & minui, laterum illorum pro<lb/> portio, po&longs;&longs;unt etiam duorum illorum <expan abbr="extremor&utilde;">extremorum</expan> <lb/> motus in data ratione con&longs;titui. </s><s>Quanta enim erit <lb/> BC ad AB, tantus erit motus vis D ad motum vis E: <lb/> ergo & in hoc medio primum lemma no&longs;trum de<lb/> mon&longs;tratum e&longs;t. </s> <s>In hoc autem medij genere hoc diuer&longs;um ab <lb/> aliis mediis accidit, quòd non tam facilè vtrin<lb/> que motus eo cietur, ac in illis, in quibus &longs;i virium <lb/> proportio momento vel &longs;uperet, vel minor &longs;it pro<lb/> portione motus <expan abbr="extremor&utilde;">extremorum</expan> medij, tum motus hinc <lb/> vel inde cietur. </s><s>Atqui in hoc propter &longs;uperficie<lb/> rum contactum, quarum pori vel a&longs;peritates &longs;e<lb/> &longs;e mutuò &longs;ubingrediuntur, & ita inuicem adhæ<lb/> rent, fit vt ægriùs cieatur motus, faciliùs ve- <pb xlink:href="044/01/045.jpg" pagenum="39"/>rò mouebuntur, &longs;i illæ &longs;uperficies leues fue<lb/> rint, vt &longs;i pinguibus inungantur, vel ex materia <lb/> leui con&longs;tent. </s><s>Ideò enim reliqua faciliùs mo<lb/> uentur, quòd circa puncta veluti quædam mo<lb/> ueantur. </s> <s>Simplicis autem trianguli rectilinei aut cu<lb/> nei in diuturnis motibus ciendis rarus e&longs;t v&longs;us, <lb/> tum ob illud quod notauimus incommodum ex <lb/> vitio materiæ, quo fit vt in eo qua&longs;&longs;atione opus <lb/> &longs;it, tum etiam quòd breui eius operatio termine<lb/> tur. </s><s>Ideo illo vtimur aut cùm &longs;olutionem con<lb/> tinui molimur, quæ breui tempore fit, vel <lb/> cùm aliquid di&longs;tendere aut aliquid figere volu<lb/> mus. </s> <s>Anguli verò curui linei magnus e&longs;t v&longs;us, præ<lb/> &longs;ertim helicis cylindricæ. </s><s>Nihil enim aliud e&longs;t he<lb/> lix quàm triangulus curuus: &longs;i enim alteram ex <lb/> lineis rectis angulum con&longs;tituentibus, cylindri <lb/> ba&longs;i ip&longs;i lineæ æquali obuolueris, reliquam ve<lb/> rò &longs;eruato eodem quem con&longs;tituunt angulo, &longs;u<lb/> per cylindri &longs;uperficiem curuaueris, habebis he<lb/> licem cylindricam, quam &longs;i iterum &longs;eruato eo<lb/> dem angulo in rectum extendas, habebis trian<lb/> gulum rectilineum. </s><s>Hæc autem helix commo<lb/> di&longs;&longs;ima e&longs;t, tum quòd in parua mole triangulum <pb xlink:href="044/01/046.jpg" pagenum="40"/>longi&longs;&longs;imum obuolutum contineat tum quod par<lb/> tes eius omnes &longs;ibi inuicem <expan abbr="cõgruant">congruant</expan>: omnes enim <lb/> partes helicis cylindricæ, aut circa eundem, aut circa <lb/> æquales cylindros de&longs;criptæ, &longs;eruato eodem angulo <lb/> &longs;ibi inuicem &longs;uppo&longs;itæ congruunt. </s><s>Quo fit, vt &longs;i ca<lb/> ui cylindri interiori &longs;uperficiei impre&longs;&longs;a &longs;it helix, a<lb/> lia verò cylindri &longs;uperficiei connex&ecedil; ip&longs;i cau&ecedil; æqua<lb/> li, &longs;eruato eodem qui in illo e&longs;t, angulo, &longs;ibi inui<lb/> cem & omnes vnius omnibus alterius partibus con<lb/> gruant. </s> <s>Huius autem medij cùm plures &longs;int partes, con<lb/> &longs;tat enim duabus &longs;uperficiebus, pluribus etiam mo<lb/> dis variari pote&longs;t. </s><s>In &longs;umma autem eius affectus hic <lb/> e&longs;t, vt cylindri ba&longs;i in &longs;uperficie immobili circum a<lb/> xem conuer&longs;a, vis mouenda helicem premat: dum <lb/> enim cylindrus circum axem conuertetur, vis mo<lb/> uenda qualibet reuolutione tantùm &longs;ecundum cy<lb/> lindri longitudinem mouebitur, quanta e&longs;t di&longs;tan<lb/> tia duarum helicis &longs;pirarum. </s><s>Quæ igitur erit propor<lb/> tio circunferentiæ circuli ba&longs;im cylindri <expan abbr="con&longs;titu&etilde;-tis">con&longs;tituen<lb/> tis</expan> ad illam <expan abbr="di&longs;tãtiam">di&longs;tantiam</expan>, eadem erit motus orbicularis <lb/> cuiu&longs;libet puncti in cylindri &longs;uperficie &longs;ignati ad mo<lb/> tum rectum vis helicem prementis. </s> <s>Illud igitur medium ex duobus compo&longs;itum e&longs;t, <lb/> recto &longs;cilicet & curuo: ita igitur vires illi aptandæ <lb/> &longs;unt, vt eius quam mouere volumus, nutus linea &longs;it <pb xlink:href="044/01/047.jpg" pagenum="41"/>cylindri longitudo: illa verò qua mouere volumus, <lb/> in orbem moueatur, aut &longs;i eius nutus linea recta &longs;it, <lb/> circa cylindrum fluens illum moueat contingendo. </s> <s>Et hæc quidem de mediis in &longs;uo genere <expan abbr="con&longs;i&longs;t&etilde;-tibus">con&longs;i&longs;ten<lb/> tibus</expan>. </s><s>Po&longs;&longs;unt autem fieri eorum Syzygiæ vectis &longs;ci<lb/> licet cum trochleis, helicis cum vecte, aut cum tro<lb/> chleis, aut <expan abbr="omni&utilde;">omnium</expan> &longs;imul. </s><s>Vectis cum trochleis &longs;i fu<lb/> nis illud extremum, quod mouetur in trochleis, er<lb/> gatis aut &longs;uculis inuoluatur: helix cum vecte <expan abbr="coni&utilde;-getur">coniun<lb/> getur</expan>, &longs;i tympanus circum axem moueatur, &longs;itque i<lb/> ta denticulatus, vt dentes ip&longs;ius helicem in cylindro <lb/> excauatam ingrediantur, quam machinam helicem <lb/> perpetuam vulgò vocant, eò quòd eius conuer&longs;io <lb/> perpetua e&longs;&longs;e po&longs;&longs;it, cum helicis &longs;implicis operatio <lb/> non excedat ip&longs;ius longitudinem. </s><s>Quorum <expan abbr="omni&utilde;">omnium</expan> <lb/> tum inter &longs;e, tum ad &longs;ubiecta mouenda accommo<lb/> datio adeò varia e&longs;t, vt &longs;cripto <expan abbr="cõpreh&etilde;di">comprehendi</expan> vix po&longs;&longs;it. <lb/> </s><s>Ex his autem quæ dicta &longs;unt, mediocris ingenij me<lb/> chanicus poterit ea prout ip&longs;i nece&longs;&longs;e erit aptare. </s><s>In <lb/> machinis autem omnibus hæc cautio <expan abbr="adhib&etilde;da">adhibenda</expan> e&longs;t, <lb/> vt earum &longs;tructura firma &longs;it, præ&longs;ertim verò vbi cir<lb/> cum axes aliquas fit motus, deinde vt vincula, qui<lb/> bus illis vires <expan abbr="affing&utilde;tur">affinguntur</expan>, valida &longs;int, illarum enim o<lb/> mnium vis in &longs;uo &longs;tatu manendi vtrique vi aptandæ <lb/> æqualis e&longs;&longs;e debet: aget enim in illas vis <expan abbr="vtraq;">vtraque</expan>, quòd <lb/> &longs;i medium firmum non &longs;it, motus in ip&longs;a machina <pb xlink:href="044/01/048.jpg" pagenum="42"/>ciebitur. </s><s>Itaque di&longs;&longs;oluetur, ideò manca videtur pe<lb/> titio illa Archimedis in hoc problemate <foreign lang="greek">do\s pou= stw= ta\n <lb/> ga\n kinw=</foreign>, quòd locum <expan abbr="tãtùm">tantùm</expan> in quo con&longs;i&longs;tat, &longs;ibi da<lb/> ri po&longs;tulet, cùm pr&ecedil;terea vincula quibus terra à loco <lb/> &longs;uo naturali remota &longs;u&longs;tineri po&longs;&longs;et, petere debue<lb/> rit: id <expan abbr="aut&etilde;">autem</expan> vt & ea quæ de motu in infinitum augen<lb/> do vel minuendo diximus, ita intelligenda &longs;unt, vt <lb/> &longs;ciamus infinita hominum pote&longs;tati, quacunque ar<lb/> te iuuetur, non &longs;ube&longs;&longs;e: quamuis enim Geometri<lb/> ca con&longs;ideratio in infinitum &longs;e&longs;e <expan abbr="ext&etilde;dat">extendat</expan>, &longs;unt ta<lb/> men certi fines, vltra quos natura rerum nos pro<lb/> gredi non patitur: &longs;unt præterea vitia materiæ, <lb/> quæ Geometra non con&longs;iderat, illa tamen non <lb/> ob&longs;tant quò minùs id quod proponitur, <expan abbr="ver&utilde;">verum</expan> &longs;it in <lb/> intellectu. </s><s>An verò id quod proponitur tale &longs;it, vt in <lb/> opus educi po&longs;&longs;it, con&longs;iderabit Geometer habita ra<lb/> tione circun&longs;tantiarum, præ&longs;ertim verò temporis, <lb/> quod ip&longs;i præ&longs;cribetur, & &longs;umptuum quos facere <lb/> poterit is qui aliquid faciendum proponet, quæ &longs;i <lb/> abundè &longs;uppetant, nihil non fieri poterit. </s> <s>Vt igitur hunc tractatulum concludamus, ac ve<lb/> lut in &longs;ummam contrahamus: In motibus ciendis <lb/> tria &longs;unt con&longs;ideranda. </s><s>Vis qua motum ciere volu<lb/> mus, vis quam mouere volumus, & motum quo <lb/> mouere volumus: duo enim quælibet ex illis ter<lb/> tium determinant. </s><s>Si enim vi parua vim magnam <pb xlink:href="044/01/049.jpg" pagenum="43"/>mouere volumus, id nonni&longs;i paruo motu facere <lb/> po&longs;&longs;umus: &longs;i verò vim aliquam magno motu mo<lb/> uere velimus, vi magna mouente ad id opus e&longs;t. </s><s>Si <lb/> vi parua magnum motum ciere volumus, mini<lb/> mam vim mouendam e&longs;&longs;e oportet: vt puta, &longs;i libra <lb/> vna centum libras mouere velimus, oportet motum <lb/> illius, motu huius <expan abbr="c&etilde;tuplo">centuplo</expan> maiorem e&longs;&longs;e. </s><s>Si verò ve<lb/> limus libra vna aliam vim ita mouere, vt ea cen<lb/> tuplo citiùs moueatur, quàm libræ illius pondus, il<lb/> lam cente&longs;imam tantùm libræ vnius partem e&longs;&longs;e ne<lb/> ce&longs;&longs;e e&longs;t: &longs;i verò libram vnam ita mouere velimus, vt <lb/> centuplo citius moueatur, quàm vis quæ illam mo<lb/> uebit, vi centum libris maiore ad id opus erit. </s><s><expan abbr="Neq;">Neque</expan> <lb/> patitur natura &longs;ibi in his vim fieri: &longs;i enim eiu&longs;modi <lb/> proportio aliquo modo infringi po&longs;&longs;et, &longs;tatim da<lb/> retur <foreign lang="greek">au)tw/ma e)nde/lexes</foreign>, vel vt <expan abbr="vocãt">vocant</expan>, motus perpetuus <lb/> in perpetua materia. </s> <s>Ex his igitur fundamentis inuentæ &longs;unt machinæ <lb/> omnes, quotquot antehac &longs;unt excogitat&ecedil;, & quot<lb/> quot po&longs;thac excogitari poterunt, ad ea referentur. </s> <s>Itaque &longs;i propo&longs;itæ cuiu&longs;cunque machinæ effe<lb/> ctum &longs;cire velimus, con&longs;ideranda &longs;unt duo eius <lb/> extrema, quibus vires applicantur: quæ enim erit <lb/> ratio motus vnius ex illis extremis, ad motum <lb/> alterius eadem erit & virium, quæ illis extre<lb/> mis ad motum ciendum applicari poterunt, <pb xlink:href="044/01/050.jpg" pagenum="44"/>addito aut dempto momento, vt &longs;i dum <expan abbr="alter&utilde;">alterum</expan> ma<lb/> chinæ extremum palmo vno mouetur, alterum cen <lb/> tum palmis moueatur vis quælibet huic annexa, al<lb/> teram alteri annexam centuplam momento minùs, <lb/> mouebit: &longs;ed motu centuplo: mouebit autem & re<lb/> liquas omnes vires, quæ infra centuplam proportio<lb/> nem ad eam habebunt: &longs;i verò non vim centuplam <lb/> mouere, &longs;ed in data vi motum centuplum ciere veli<lb/> mus, eam quidem in extremo quod centuplo citiùs <lb/> mouetur, locare oportebit, alteram verò alteri extre <lb/> mo centuplo maiorem adhibere nece&longs;&longs;e erit. </s> <s>Hinc oriuntur tria problemata, videlicet data vi <lb/> datum pondus mouere, quod iam <expan abbr="demõ&longs;trauimus">demon&longs;trauimus</expan>: <lb/> item data vi datum motum ciere, quod ex <expan abbr="præced&etilde;-tis">præceden<lb/> tis</expan> demon&longs;tratione ab&longs;oluitur: tertium, datam vim, <lb/> dato motu mouere, quod quidem faciliùs demon<lb/> &longs;tratur, quàm ab&longs;oluitur. </s><s>Scimus enim, vt id fiat, <lb/> vim aliam proportionalem (vt docuimus) requiri, <lb/> quo modo autem illam habere po&longs;&longs;imus, explicare <lb/> hoc opus, hic labor e&longs;t. </s><s>Non enim quemadmodum <lb/> organa ad motus in data proportione ciendos ha<lb/> bere po&longs;&longs;umus: ita & vires infinitæ magnitudinis po<lb/> te&longs;tati no&longs;træ &longs;ub&longs;unt. </s> <s>Quod enim ad vires in grauitatis & leuitatis gene <lb/> re con&longs;i&longs;tentes attinet, eas vt moueant antè moueri <lb/> oportet à loco vel &longs;itu naturali, vi <expan abbr="aut&etilde;">autem</expan> aliqua id fie- <pb xlink:href="044/01/051.jpg" pagenum="45"/>ri oportet, quæ iam in promptu &longs;it. </s><s>Quæ &longs;i tanta e&longs;t, <lb/> vt po&longs;&longs;it motum propo&longs;itum ciere, fru&longs;trà fiat, &longs;i alij <lb/> à loco naturali remouendæ, qua po&longs;tea ad motum <lb/> vtendum &longs;it, adhibeatur. </s> <s>Nullum igitur ex eiu&longs;modi viribus commodum <lb/> percipi pote&longs;t, ni&longs;i quis in futurum &longs;ibi pro&longs;piciens, <lb/> multa à loco vel &longs;itu naturali ex otio remoueat, vt <lb/> iis, cùm opus erit ad motus ciendos vtatur. </s><s>Hac ra<lb/> tione quantas vires in promptuario habebit, tantos <lb/> motus ciere poterit. </s><s>Præcipuum igitur quod ad hu<lb/> ius problematis con&longs;tructionem pertinet, e&longs;t, vt vi<lb/> res quàm maximas pote&longs;tati no&longs;træ &longs;ubiiciamus, vo<lb/> luntariis aut fortuitis viribus naturales præparando, <lb/> aut à natura præparatas, quæ pote&longs;tati no&longs;træ &longs;ub<lb/> &longs;unt, accipiendo. </s><s>Qualia multa &longs;i mortales aduerte<lb/> rent, fieri po&longs;&longs;ent, vt aliàs, Deo duce, docebimus. </s> <s>Earum autem virium quæ in raritatis & den&longs;ita<lb/> tis proportione con&longs;i&longs;tunt, &longs;ubiecta plurimùm in <lb/> no&longs;tra pote&longs;tate &longs;unt: multa enim &longs;unt naturalia &longs;ub <lb/> iecta actu, den&longs;a potentia verò rara. </s><s>Si quod igitur <lb/> eorum potentia proxima &longs;it rari&longs;&longs;imum, ita vt nullo <lb/> negotio actus ille raritatis induci po&longs;&longs;it, <expan abbr="concluda-túrq;">concluda<lb/> túrque</expan> loco aliquo angu&longs;to, po&longs;tea inducatur ille a<lb/> ctus, cùm rara <expan abbr="maior&etilde;">maiorem</expan> locum occupent, quum den<lb/> &longs;a, fiet vt locus in omnem partem di&longs;tendatur, illius <lb/> autem partes minùs cohærentes, tantum impellen <pb xlink:href="044/01/052.jpg" pagenum="46"/>tur, quanta e&longs;t proportio molis rei rarefactæ ad mo<lb/> lem illius cùm den&longs;a e&longs;&longs;et: illa autem raritatis po<lb/> tentia proxima e&longs;t in compo&longs;itione &longs;ulphuris <lb/> & nitri: ea igitur & &longs;imilibus &longs;ubiectis, <lb/> in data vi datus motus <lb/> cieri pote&longs;t. </s> <s>FINIS.</s> <s>Errata quæ inter imprimendum irrep&longs;erunt, <lb/> &longs;ic corrigito.</s> <s><emph type="italics"/>Pagina<emph.end type="italics"/> 7. <emph type="italics"/>linea<emph.end type="italics"/> 10, <emph type="italics"/>munere, lege nuere. </s> <s>pag.<emph.end type="italics"/> 11. <emph type="italics"/>lin.<emph.end type="italics"/> 12, <emph type="italics"/>comparationibus, lege comparati, omni<lb/> bus. </s> <s>pag.<emph.end type="italics"/> 12. <emph type="italics"/>lin.<emph.end type="italics"/> 16. <emph type="italics"/>finis, lege fini. </s> <s>pag. eadem. lin. penult. Altero, lege Alto.</s> <s>pag.<emph.end type="italics"/> 20. <lb/> <emph type="italics"/>lin.<emph.end type="italics"/> 16. <emph type="italics"/>quum<emph.end type="italics"/> C <emph type="italics"/>ad<emph.end type="italics"/> L. <emph type="italics"/>lege quam. </s> <s>pag.<emph.end type="italics"/> 25. <emph type="italics"/>lin.<emph.end type="italics"/> 5. <emph type="italics"/>nunc &longs;it ratio, lege minor &longs;it. </s> <s>linea<emph.end type="italics"/> 13. <emph type="italics"/>Vi<lb/> res maiori litera<emph.end type="italics"/> (<emph type="italics"/>e&longs;t enim &longs;ectionis initium. </s> <s>lin.<emph.end type="italics"/> 15. <emph type="italics"/>mittant, lege mittantur. </s> <s>pag.<emph.end type="italics"/> 27. <emph type="italics"/>li.<emph.end type="italics"/> 20. <lb/> <emph type="italics"/>puncti in linea<emph.end type="italics"/> A, <emph type="italics"/>lege puncti<emph.end type="italics"/> A <emph type="italics"/>in linea. </s> <s>pag.<emph.end type="italics"/> 28. <emph type="italics"/>lin.<emph.end type="italics"/> 9. <emph type="italics"/>vis, lege Vis, maiori litera<emph.end type="italics"/> (<emph type="italics"/>e&longs;t enim <lb/> &longs;ectionis initium. </s> <s>lin. e. motu ciere, leg. motu quem ciere. </s> <s>lin.<emph.end type="italics"/> 23. <emph type="italics"/>metiri &longs;olemus, lege non <lb/> &longs;olemus. </s> <s>lin. vlt. &longs;tatui, lege &longs;tatuit.</s> <s>pag.<emph.end type="italics"/> 32. <emph type="italics"/>lin.<emph.end type="italics"/> 4. <emph type="italics"/>ab&longs;ouit, lege ab&longs;oluit. </s> <s>lin.<emph.end type="italics"/> 5. <emph type="italics"/>dele<emph.end type="italics"/> F. </s> <s><emph type="italics"/>lin.<emph.end type="italics"/><lb/> 12. <emph type="italics"/>&<emph.end type="italics"/> 13, <emph type="italics"/>pone virgulam po&longs;t<emph.end type="italics"/> D <emph type="italics"/>&<emph.end type="italics"/> E.</s> <s> <emph type="italics"/>lin.<emph.end type="italics"/> 8. <emph type="italics"/>hypomochium, lege hypomochlium. </s> <s>pag.<emph.end type="italics"/> 37. <emph type="italics"/>li.<emph.end type="italics"/><lb/> 13. <emph type="italics"/>re&longs;i&longs;tentia, lege re&longs;i&longs;tentiæ. </s> <s>pag.<emph.end type="italics"/> 39. <emph type="italics"/>lin.<emph.end type="italics"/> 3. <emph type="italics"/>leui, lege læui. </s> <s>pag.<emph.end type="italics"/> 40. <emph type="italics"/>linea<emph.end type="italics"/> 5. <emph type="italics"/>&longs;uppo&longs;itæ, lege <lb/> &longs;uperpo&longs;itæ</s> <s>li.<emph.end type="italics"/> 7. <emph type="italics"/>connexæ, lege conuexæ. </s> <s>lin.<emph.end type="italics"/> 13. <emph type="italics"/>affectus, lege effectus.<emph.end type="italics"/></s> </chap> </body> <back/> </text></archimedes>

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