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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/texts/XML/archimedes/la/varro_demot_044_la_1584_old.xml Fri Dec 07 17:05:22 2012 +0100 @@ -0,0 +1,2021 @@ +<?xml version="1.0"?> +<archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > + <info> + <author>Varro, Michel</author> + <title>De motu tractatus, old version (54 pages)</title> + <date>1584</date> + <place>Geneva</place> + <translator></translator> + <lang>la</lang> + <cvs_file>varro_demot_044_la_1584.xml</cvs_file> + <cvs_version></cvs_version> + <locator>044.xml</locator> + <echodir>/permanent/archimedes/varro_demot_044_la_1584</echodir> + </info> + <text> + <front> + <section> + <pb xlink:href="044/01/001.jpg"></pb> + <p type="head"> + <s>TERRA <lb></lb> MACHINIS MOTA <lb></lb> DISSERTATIONES <lb></lb> GEOMETRICAE, MECHANICAE + <lb></lb> PHYSICAE, HYDROSTATICAE <lb></lb> In quibus <lb></lb> Machinarum Coniugatarum + uires inter ſe comparantur: <lb></lb> Multiplici Noua Methodo Terrae + magnitudo et <lb></lb> Grauitas inueſtigatur: ARCHIMEDES <lb></lb> terrae + motionem ſpondens ab arrogantia <lb></lb> ſuſpicione + uindicatur. <lb></lb></s> + <s>AVTHORE <lb></lb> PAVLO CASATO <lb></lb> E SOCIETATE IESV. <lb></lb></s> + <s>ROMAE. <lb></lb></s> + <s>Ex Typographia Ignatij de Lazaris.</s> + <s>M.DC.LVIII.<lb></lb></s> + <s>SVPERIORVM PERMISSV.<lb></lb></s> + </p> + <pb xlink:href="044/01/002.jpg"></pb> + <pb xlink:href="044/01/003.jpg" id="p.0001"></pb> + <p type="head"> + + + <s>ILLVSTRI ET IN <lb></lb> PRIMIS GENEROSO DOMINO, <lb></lb> DOMINO CAROLO BARONI <lb></lb> + Azerotin, Nameſtij Roſicij <lb></lb> Brandeiſij Domino, <lb></lb> + MICH. VARRO. S. P. D.</s> + </p> + <p type="main"> + + + <s><emph type="italics"></emph>CVM ab Ineunte ætate in Ma<lb></lb> thematicis + me exercuiſſem, ſem<lb></lb> per animo meo in + hæſit hoc deſide<lb></lb> rium, vt quæ de Archimede + Syra<lb></lb> cuſano referuntur, ea & ratione <lb></lb> + demonſtrare & experimento comprobare poſſem. <lb></lb> + </s> + <s>Cumque multa in eo genere meditatus eſſem, poſt<lb></lb> + quam ad iuris ciuilis ſtudium tranſii, ac tandem <lb></lb> ad + Rempub. nunquam mihi tantum otii contigit, <lb></lb> vt ea in + ordinem redigere potuerim. </s> + <s>At cum pau<lb></lb> cis retrò annis ſatis longam peregrinationem + per <lb></lb> Sarmatiam ſuſcepiſſem, + eſſetque vacuus curis a-<emph.end type="italics"></emph.end> + <pb xlink:href="044/01/004.jpg"></pb><emph type="italics"></emph>nimus, atque ad earum + rerum meditationem me <lb></lb> impelleret genius meus, cœpi aliquid de + his ſcribe<lb></lb> re, quantum ipſe iter faciendo meo marte + aſſequi <lb></lb> potui cum libris deſtituerer. </s> + <s>Scriptum illud cum <lb></lb> in maximè ardua & difficìli + contemplatione ver<lb></lb> ſetur, nec ab eo tempore ex quo + adſolitas occupa<lb></lb> tiones redii ei extremam manum apponere + mihi <lb></lb> licuerit, decreueram inter priuatas meas muſas <lb></lb> + aſſeruare Verebar enim vulgi <expan abbr="iudiciũ">iudicium</expan>, (cuius <expan abbr="ta-mẽ">ta<lb></lb> + men</expan> rationem viro politico <expan abbr="habendã">habendam</expan> eſſe cenſeo) ne<lb></lb> que + ſolum vulgi, verum etiam <expan abbr="eorũ">eorum</expan> qui + doctorum <lb></lb> nomine gaudent, cum inaudita quædam interdum <lb></lb> + quæque ab eorum opinione recedere <expan abbr="videãtur">videantur</expan> at<lb></lb> tingam. </s> + <s><expan abbr="Cupiebã">Cupiebam</expan> etiam in eo argumento + vlteriùs <lb></lb> progredi, antè quam quidquam ederem, atque ea <lb></lb> + quæ in meis aduerſariis ea de re ſparſa habeo, iis + <lb></lb> quæ <expan abbr="congeſſerã">congeſſeram</expan> ſubiicere, vt iuſtum <expan abbr="volumẽ">volumen</expan> fieret.<emph.end type="italics"></emph.end></s> + </p> + <p type="main"> + + <s><emph type="italics"></emph>Poſtquam verò te diſceſſum + ex hac nostra ci<lb></lb> uitate parare intellexi, in qua dum + verſatus es. <lb></lb> + </s> + <s>non ſolùm ei <expan abbr="ornamẽto">ornamento</expan> + fuiſti, ſed etiam optimos <lb></lb> quoſque virtutum tuarum + admiratione tibi con<lb></lb> ciliasti: mihi verò peculiariter + innumera beneuo-<emph.end type="italics"></emph.end> + <pb xlink:href="044/01/005.jpg"></pb><emph type="italics"></emph>lentiæ + ſigna exhibuiſti, non potui tantum virum, <lb></lb> meíque + <expan abbr="amãtem">amantem</expan> ſine aliquo munere + aut<emph.end type="italics"></emph.end> + <foreign lang="grc">μνημοσύνῳ</foreign><lb></lb> + <emph type="italics"></emph>dimittere. </s> + <s>Itaque libellum hunc tibi dicare, & <lb></lb> ſub tuo nomine in + publicum edere ſtatui. </s> + <s>Squalli<lb></lb> dus quidem prodit, & rudis, mole etiam + perexi<lb></lb> guus: ſplendidius aliquid & magis elaboratum + <lb></lb> postularet tua dignitas. </s> + <s>Atqui ea eſt argumenti <lb></lb> illius magnitudo, vt in eo aliquid + voluiſſe ſatis <lb></lb> ſit. </s> + <s>Militare ſanè munus eſt, eóque nomine tibi <lb></lb> + conueniens: ſi quis enim hac cognitione inſtructus <lb></lb> fuerit, + non minùs ea in re bellica vti poterit, <lb></lb> quam Archimedes in + Syracuſis defendendis <lb></lb> aduerſus Marcum Marcellum vſus + est. </s> + <s>Eá<lb></lb> que mirari deſinet quæ de eo Plutarchus + nar<lb></lb> rat. </s> + <s>Id lucidius <expan abbr="demõſtrare">demonſtrare</expan> + potuiſſem, ſi mihi hæc <lb></lb> vlterius + perſequendi otium fuiſſet: Jeiuna enim <lb></lb> per ſe + videntur, niſi quis ſuccum qui ſub corti<lb></lb> ce + latet, eliciat. </s> + <s>Malui tamen hoc veluti pro<lb></lb> gymnaſmate alios quibus plus est + otii & in<lb></lb> genii ad eiuſmodi inquiſitionem hortari + & ac<lb></lb> cendere, quàm præclaram illam + cognitio<lb></lb> nem diutius ſepultam relinquere cum ne-<emph.end type="italics"></emph.end> + <pb xlink:href="044/01/006.jpg"></pb> + <emph type="italics"></emph>minem hodie animaduertam qui ei inſudet, <lb></lb> + neque mihi ſpes vlla affulgeat diuturni otii, cuius <lb></lb> ope in hoc + ſtadio pergere poßim: Quòd ſi fortè <lb></lb> mihi + aliquando contingat, riuos plurimos ex his <lb></lb> fontibus me educturum; + <expan abbr="cõfido">confido</expan>; quibus multum <lb></lb> + <expan abbr="cõmoditàtis">commoditàtis</expan> rebus humanis + accedat. </s> + <s>Quæ enim <lb></lb> hic traduntur, ad motus omnes atque ad omnis ge<lb></lb> + neris virium comparationes accommodari poſ<lb></lb> ſunt ad + fines propoſitos aſſequendos. </s> + <s>Quod quàm <lb></lb> latè pateat, dici non poteſt. </s> + <s>Eſt enim eorum vſus <lb></lb> non ſolùm in mechanicis, in + quibus tamen eſt ma <lb></lb> ximus, ſed etiam in politicis & + œconomicis: ſunt <lb></lb> enim in illis motus, ſunt vires + & reſistentiæ. </s> + <s>In <lb></lb> arte medica & reliquis phyſices partibus pra<lb></lb> + cticis, quantus ſit, nemo fando exprimere poßit. <lb></lb> + </s> + <s>Quia vero ea quæ praxim docent gratiora ſunt <lb></lb> ipſa + contemplatione, & cauſæ propter effecta in<lb></lb> + quirantur,<emph.end type="italics"></emph.end> + <foreign lang="grc">Ἀλφήστας</foreign> + <emph type="italics"></emph>omnes hortor vt quod ego in<lb></lb> numeris aliis + cur is implicitus addere non poſ<lb></lb> ſum, ipſi + addant. </s> + <s>Abundat nunc Europa præ<lb></lb> ſtantibus ingeniis, ſi + Mœcenates adeſſent. <lb></lb> + </s> + <s>Ea autem quæ præter hæc meditatus ſum, ac fer-<emph.end type="italics"></emph.end><pb xlink:href="044/01/007.jpg"></pb> + <emph type="italics"></emph>mè parata habeo, hæc ſunt. </s> + <s>Tractatus de iactu. <lb></lb> + </s> + <s>De continuitate eiuſque ſolutione. </s> + <s>De <expan abbr="cõdenſa-tione">condenſa<lb></lb> + tione</expan> & rarefactione earumque cauſis & + effecti<lb></lb> bus: <expan abbr="Itẽ">Item</expan> in Mechanicis, + tractatus de variis ma<lb></lb> chinis ad motus <expan abbr="ciẽdos">ciendos</expan>, ac de perfectißimæ + cuiuſ<lb></lb> que ad id quod propoſitum fuerit moliendum + in<lb></lb> ueſtigatione. </s> + <s>Nonnulla <expan abbr="etiã">etiam</expan> de <expan abbr="Rerumpublicarũ">Rerumpublicarum</expan> + <lb></lb> motu tum interno <expan abbr="tũ">tum</expan> externo notaui, + quæ <expan abbr="eodẽ">eodem</expan> or<lb></lb> dine tradere + optarem: quorum omnium principia <lb></lb> hic ſi quis diligenter + animaduertat tradita ſunt. <lb></lb> + </s> + <s>Si quis mihi in his <expan abbr="palmã">palmam</expan> præripuerit, + meque an<lb></lb> teuerterit ei maximas gratias agam. </s> + <s>Sunt tamen <lb></lb> inter hæc quædam quæ vulgò pandere + nephas est. <lb></lb> + </s> + <s>Itaque theoriæ magis inſiſtendum puto, in qua ſi <lb></lb> + quis exercitatus fuerit, nullo negotio illam in opus <lb></lb> educere poterit, + idque ſine periculo fiet, cum vul<lb></lb> go non pateat. </s> + <s>Alioqui <expan abbr="periculũ">periculum</expan> eſt, ne ſi + particu<lb></lb> laria tradantur iis contenti homines, vt fieri + ſolet <lb></lb> vniuerſalem cognitionem & cauſarum + inquiſi<lb></lb> tionem negligant, pereatque ſcientia. </s> + <s>Eaque de <lb></lb> cauſa nihil quicquam de his quæ fecerat + Archi-<emph.end type="italics"></emph.end> + <pb xlink:href="044/01/008.jpg"></pb> + <emph type="italics"></emph>medes ſcriptum relinquere voluit. </s> + <s><expan abbr="Exiſtimãs">Exiſtimans</expan> eos <lb></lb> qui in + iis quæ<emph.end type="italics"></emph.end> + <foreign lang="grc">θεωρητικῶς</foreign> + <emph type="italics"></emph>tradiderat, diligentem ope<lb></lb> ram ponere + <expan abbr="vellẽt">vellent</expan>, multò maiora quoties + opus fo<lb></lb> ret præſtituros. </s> + <s>Equidem mihi perſuadeo non de<lb></lb> futuros qui varia iudicia de + hoc noſtro ſcripto fe<lb></lb> rant & quaſi nouum + athletam in arenàm pro<lb></lb> deuntem mirentur. </s> + <s>Quibus vno verbo reſpon<lb></lb> ſum volo, me nullius vnquam in + verba ma<lb></lb> gistri iuraſſe, ſed liberrimè + ſemper philoſo<lb></lb> phatum eſſe: ita vt etiam + in principia ab aliis <lb></lb> ſtatuta animaduertere mihi licere + putauerim. <lb></lb> + </s> + <s>Quod eò liberiùs feci, poſtquam magnam <expan abbr="partẽ">partem</expan> + <lb></lb> vulgo receptarum opinionum falſam eſſe re ipſa + <lb></lb> deprehendi. </s> + <s>Omnibus placuero, ſi tibi placuero. </s> + <s>Si <lb></lb> quid in his obſcurius fuerit, habes <expan abbr="clarißimũ">clarißimum</expan> vi<lb></lb> rum + Wenceſlaum lauinium tuum, abſtruſioris <lb></lb> + philoſophiæ indagatorem <expan abbr="ſummũ">ſummum</expan>, qui tibi omnia <lb></lb> explicabit. </s> + <s>Hoc igitur <expan abbr="munuſculũ">munuſculum</expan> vt + ſerena <expan abbr="frõ-te">fron<lb></lb> te</expan> + ſuſcipias rogo. </s> + <s>Vale. </s> + <s>V I. </s> + <s>K al. Jun. Anno <lb></lb> Christi Domini<emph.end type="italics"></emph.end> M. D. LXXXIV. </s> + + + + </p> + </section> + <pb xlink:href="044/01/009.jpg" pagenum="1"></pb> + <section> + <p type="head"> + + <s>M. VARRONIS DE <lb></lb> MOTV TRACTATVS.</s> + </p> + </section> + </front> + <body> + <chap> + <p type="head"> + + <s><emph type="italics"></emph>PROBLEMA.<emph.end type="italics"></emph.end></s> + </p> + <p type="head"> + + <s><emph type="italics"></emph>Data vi datum pondus mouere.<emph.end type="italics"></emph.end></s> + </p> + <p type="main"> + + <s>Hoc problema prima quidem <lb></lb> fronte mirabile videtur, vt ſcilicet + <lb></lb> pondus vel maximum viribus ta<lb></lb> men minimis, vt puta ſi + dicas vnius <lb></lb> hominis, vel etiam imbecillioris a<lb></lb> licuius + animalis viribus totum ter<lb></lb> ræ globum loco moueri + poſſe. </s> + <s>Quod Archimedem <lb></lb> Syracuſanum iactaſſe referunt hoc + dicto, <foreign lang="grc">δὸς ποῦ στῷ <lb></lb> τὰν γὰν κινῷ. + </foreign></s> + <s>Si quis tamen in Geometrico puluere <lb></lb> verſatus fuerit, + preſertim verò in iis quæ ab ipſo in <lb></lb> ſuo + <foreign lang="grc">ἰσοροπικῶν</foreign> libello <expan abbr="tradũtur">traduntur</expan>. </s> + <s>Illud cæteráque eiuſ<lb></lb> modi, ludicra Geometriæ, + vt & ipſe facere ſolitus di<lb></lb> citur, appellabit. </s> + <s>Ex huius autem problematis expli<lb></lb> catione, omnium machinarum quę ad + motus cien<lb></lb> dos excogitari poſſunt, ratio pendet. </s> + <s>Etſi verò <expan abbr="totũ">totum</expan> + <lb></lb> Geometrica demonſtratione expediri poſſit, quo + <pb xlink:href="044/01/010.jpg" pagenum="2"></pb>niam tamen hęc + conſideratio, quæ Græcis <foreign lang="grc">σταθμικὴ</foreign> di<lb></lb> citur Geometriæ tantùm eſt + <foreign lang="grc">ὑπάλληλος</foreign> mixta ſcili<lb></lb> + cet ex phyſica & Geometrica eo quòd illius + ſubie<lb></lb> ctum ſit motus. </s> + <s>Ideò paulò craſſiori Minerua & <lb></lb> præter + Geometricam ſimplicitatem <foreign lang="grc">φυσικῶς</foreign> + etiam <lb></lb> tractanda eſt: alioqui Archimedicis + demonſtratio<lb></lb> nibus ſtandum eſſet, + quæ in eo mancæ ſunt, quòd il<lb></lb> læ + propoſitiones quæ ex phyſicis peti debent, inde<lb></lb> + monſtratæ manent, nec explicantur, ſed pro + confeſ<lb></lb> ſis principiorum loco poſtulantur. </s> + <s>Eas igitur, qua<lb></lb> tenus ad huius problematis explicationem faciunt, + <lb></lb> hic diſcutiemus. </s> + <s>Neque enim Euclidis <expan abbr="librũ">librum</expan> de + gra<lb></lb> ui & leui, in quo hoc argumentum perſequi + voluiſ<lb></lb> ſe videtur, integrum habemus. </s> + <s>Ariſtoteles verò in <lb></lb> eo quod ab ipſo ſcriptum + extat de Mechanicis fra<lb></lb> gmento pręter ſuum morem, + cùm alio qui in omni<lb></lb> bus exactiſſimus ſit, + hanc quæſtionem potiùs nota<lb></lb> uit quàm + explicuit in ſexto & ſeptimo phyſicorum <lb></lb> libro + multa prætermiſit. </s> + <s>Vt igitur ad rem aggredia<lb></lb> mur, primùm voces, quibus vtendum + eſt, definie<lb></lb> mus, vt intelligatur quo ſenſu eas + accipiamus. </s> + </p> + <p type="head"> + + <s>DEFINITIO I.</s> + </p> + <p type="main"> + + <s>Vis dicitur agendi aut agenti reſiſtendi <expan abbr="potẽtia">potentia</expan>, <lb></lb> præſertim + verò mouendi & mouenti reſiſtendi. </s> + </p> + <p type="head"> + + <s>II.</s> + </p> + <p type="main"> + + <s>Vis ſubiectum dicitur id quod vis mouet, vel <pb xlink:href="044/01/011.jpg" pagenum="3"></pb>quod à vi mouetur. </s> + </p> + <p type="main"> + + <s>Hîc non agimus de vi primaria quæ virium o<lb></lb> mnium + principium eſt, mouétque omnia, nec <expan abbr="tamẽ">tamen</expan> + <lb></lb> mouetur, ſed de ea, quę dum mouet, <expan abbr="etiã">etiam</expan> cum ſub<lb></lb> iecto, cui ineſt, mouetur. + </s> + </p> + <p type="main"> + + <s>Subiecta verò quæ nullam vim habent, nec mo<lb></lb> uentur, nec + mouenti <expan abbr="reſiſtũt">reſiſtunt</expan>: ſi qua verò vis illis + ac<lb></lb> cedat, tum ab ea ſe moueri patiuntur. </s> + </p> + <p type="main"> + + <s>Etſi autem plura ſint virium genera, tot ſcilicet, <lb></lb> + quot ſunt in rerum natura contrariorum, actionem <lb></lb> & + paſſionem recipientium, vt leue graue, rarum <expan abbr="dẽ-ſum">den<lb></lb> ſum</expan>, plenum + vacuum, durum molle, & cætera eiuſ<lb></lb> modi, quoniam + tamen ea omnia hîc perſequi noſtri <lb></lb> non eſt + inſtituti, cùm de ea tantùm qua motus fit a<lb></lb> gere + ſtatuerimus. </s> + </p> + <p type="head"> + + <s>III.</s> + </p> + <p type="main"> + + <s>Cùm de motu hîc agemus motum ad locum, <lb></lb> quem Græci + <foreign lang="grc">φορὰν</foreign> vocant, intelligi volumus. </s> + </p> + <p type="head"> + + <s>IIII.</s> + </p> + <p type="main"> + + <s>Linea autem recta quæ eſt ab eo loco à quo mo<lb></lb> tus + fieri incipit ad illum ad quem tendit. </s> + <s>Illius vis <lb></lb> quæ motum efficit, nutus dicetur. </s> + </p> + <p type="main"> + + <s><expan abbr="Eadẽ">Eadem</expan> verò linea <expan abbr="cõſiderata">conſiderata</expan> à loco ad + <expan abbr="quẽ">quem</expan> + <expan abbr="tẽdit">tendit</expan> vis <lb></lb> ad <expan abbr="illũ">illum</expan> à quo motus fieri incipit, contra + <expan abbr="nutũ">nutum</expan> dicitur. </s> + </p> + <p type="main"> + + <s>Itidem & omnes illi parallelæ. </s> + </p> + <p type="main"> + + <s>Quæ verò lineæ vel rectæ, vel curuæ, nutus <expan abbr="lineã">lineam</expan> + <pb xlink:href="044/01/012.jpg" pagenum="4"></pb>ad angulos inæquales + ſecant, illæ obliquè nutum <lb></lb> verſus, vel contra + nutum ferri <expan abbr="dicũtur">dicuntur</expan>, habita + ratio<lb></lb> ne partium, quas ſpectant. </s> + </p> + <p type="main"> + + <s>Plurimùm autem à ſitu corporis humani deno<lb></lb> + minationem accipiunt illæ partes, vt ſurſum, + deor<lb></lb> ſum, dextrorſum, ſiniſtrorſum, + antè vel ponè di<lb></lb> cantur. </s> + <s>Quæ verò lineæ, nutus lineam ad angulos <lb></lb> rectos + ſecant, neque verſus nutum, neque contra <lb></lb> nutum ferri + dicuntur, ſed æquè diſtant à loco natu<lb></lb> + rali. </s> + </p> + </chap> + <chap> + <p type="head"> + + <s><emph type="italics"></emph>Virium diuiſio.<emph.end type="italics"></emph.end></s> + </p> + <p type="main"> + + <s>Vis autem omnis aut naturalis eſt, aut non natu<lb></lb> ralis, aut + mixta. </s> + <s>Naturalis vis eſt, qua res quælibet <lb></lb> natura ſua + mouetur, aut mouenti reſiſtit habita ra<lb></lb> tione tum + loci ſui naturalis, tum etiam ſitus ſuarum <lb></lb> partium. </s> + <s>Non naturalis eſt quæ nec ratione loci ſui <lb></lb> naturalis, + nec ſitus partium mouet. </s> + <s>Eſt autem hæc <lb></lb> duplex, fortuita ſcilicet & + voluntaria: illa vt vis ven<lb></lb> torum & ſpirituum + intellectu carentium, hæc vt a<lb></lb> nimalium & + ſpirituum intelligentium & ſimilium. <lb></lb> + </s> + <s>Mixta dicitur, quæ partim naturalis eſt, partim non <lb></lb> + naturalis. </s> + </p> + <p type="main"> + + <s>Locus autem naturalis cuiuſque rei eſt, in quo <lb></lb> + exiſtens ipſa quieſcit, & ſi inde remota + ſit, vis ei à na<lb></lb> tura inſita eam illuc impellit. </s> + <s>Huius verò motus aut <lb></lb> quietis cauſſa nulla alia + aſſignari poteſt præter pri- <pb xlink:href="044/01/013.jpg" pagenum="5"></pb>mam illam rerum omnium + conditricem, quę, ne o<lb></lb> mnia in omnibus eſſent + ſingulis partibus vniuerſi <lb></lb> ſingula loca attribuit + circa quæ conglobantur, & i<lb></lb> bi hærent. </s> + <s>Quicquid enim terreum eſt, in terræ glo<lb></lb> bum <expan abbr="cõfluxit">confluxit</expan>. </s> + <s>Ita quicquid ſoli homogenes in Solis <lb></lb> corpus, lunaria omnia in + lunam, & ſic de cæteris cor<lb></lb> poribus vniuerſi. </s> + <s>Singulæ autem eorum partes ſuis <lb></lb> totis adhærent, nec + inde ſponte <expan abbr="mouẽtur">mouentur</expan>: ſi + verò in<lb></lb> de motæ fuerint, eò redire nituntur. </s> + <s>Quod enim in <lb></lb> terra fieri videmus, idem & in reliquis corporibus + v<lb></lb> niuerſi fieri dicere poſſumus, ſcilicet + quod vnita cir<lb></lb> ca locum ſuum naturalem maneant: ſi enim + partes <lb></lb> eorum ſponte ſepararentur, & vi ab ipſis + ſeparatæ <lb></lb> ad locum ſuum non redirent, ſtatim + tota diſſolue<lb></lb> rentur. </s> + </p> + <p type="main"> + + <s>Vis igitur illa in loco ſuo quieſcendi grauitas di<lb></lb> + citur, cuius contrarium eſt leuitas. </s> + <s>Res autem nulla <lb></lb> per ſe leuis dicitur, ſed <expan abbr="tãtùm">tantùm</expan> habita ratione alterius <lb></lb> + loci, quàm ſui naturalis: vt puta ſi pars aliqua Solis + <lb></lb> vi quapiam in terram inuecta eſſet, ſuóque + arbitrio <lb></lb> committeretur, illa ſtatim Solem peteret: ita ſi + ter<lb></lb> ræ pars in Solem inuecta eſſet, + ſuóque arbitrio com<lb></lb> mitteretur, ſtatim à + ſole euolaret, & ſe per cælum <lb></lb> terram verſus + proriperet. </s> + <s>Vt igitur terræ partes in <lb></lb> terra graues ſunt: ſic Solis + partes in ſole graues ſunt: <lb></lb> in terra verò leues, + terræ verò partes in Sole leues <pb xlink:href="044/01/014.jpg" pagenum="6"></pb>ſunt. </s> + <s>Idem de cæteris corporibus vniuerſi dici po<lb></lb> teſt. </s> + <s>Neque enim faciliùs ex globo lunæ particulam <lb></lb> abſtrahas, + quàm ex terra glebam. </s> + </p> + <p type="main"> + + <s>Aer autem nullum proprium habet locum natu<lb></lb> ralem, ſed vbique + eſſe poteſt, cùm rerum omnium <lb></lb> locus + vniuerſalis eſſe videatur: de quo dubitari po<lb></lb> + teſt an ſit infinitus, cùm omnia contineat, à nulla re + <lb></lb> verò totus comprehendatur, ſed hæc diſputatio + non <lb></lb> eſt huius loci. </s> + </p> + <p type="main"> + + <s>Etſi autem nullum proprium habet locum natu<lb></lb> ralem, neque leuis aut + grauis ſit: leuis tamen eſſe vi<lb></lb> detur, cùm + grauis non ſit, nec ægrè ſe moueri patia<lb></lb> + tur. </s> + <s>Itaque ſi quo modo ita conſtitutus ſit, vt rem a<lb></lb> + liquam, quò minùs ad locum ſuum naturalem per<lb></lb> + uenire poſſit, impediat, tum per expreſſionem + ſeu <lb></lb> + <foreign lang="grc">ἔκθλιψιν</foreign>, ab ea eiicietur, in + eiúſque locum ſuccedet, <lb></lb> vt fit in aqua, cùm ei + introducitur. </s> + <s>Quæ quidem <foreign lang="grc">ἔκ<lb></lb> θλιψις</foreign> iis + etiam accidit quæ loco naturali gaudent, <lb></lb> cùm + ſeſe mutuò impediunt, ne ad illum ferantur. <lb></lb> + </s> + <s>Præſertim verò in liquidis locum habet, quorum <lb></lb> partes + facilius mouentur, quàm rerum <expan abbr="cohærentiũ">cohærentium</expan> + <lb></lb> & compactarum. </s> + <s>Inde accidit vt quicquid liquidis <lb></lb> immergitur, tantò fiat in illis + leuius quàm grauius, <lb></lb> eſt moles eiuſdem liquoris + ipſorum moli æqualis. <lb></lb> + </s> + <s>Sed hæc de loco naturali fuſiùs à nobis alibi + medita<lb></lb> ta, obiter hîc attigiſſe ſufficiat, + vt intelligatur quid <pb xlink:href="044/01/015.jpg" pagenum="7"></pb>ſit + vis naturalis. </s> + </p> + <p type="main"> + + <s>Vis autem voluntaria nullum certum nutum <lb></lb> habet, ſed illum + tantùm in quo mouentis voluntas <lb></lb> conquieſcit. </s> + <s>Cúmque vis naturalis vnum tantùm <lb></lb> nutum habeat, ſcilicet + à loco à naturali requie re<lb></lb> moto, ad ipſam + naturalem requiem, illa infinitos nu <lb></lb> tus habet & indeterminatos + per ſe ac voluntatis tan <lb></lb> tùm decreto determinabiles. </s> + </p> + <p type="main"> + + <s>Fortuita verò ipſo tantùm caſu determinabilem <lb></lb> + nutum habet: quò enim ipſa tendit, eo munere dici<lb></lb> tur + ſiue ſurſum, ſiue deorſum, ſiue ad + latera. </s> + <s>Itaque <lb></lb> & hic & ille nutus <foreign lang="grc">ἀδιόριστος</foreign> dicitur. </s> + <s>Quemadmodum <lb></lb> & is quem habent vires illæ, quibus res à + ſitu <expan abbr="partiũ">partium</expan> + <lb></lb> naturali remotæ ad illum redeunt: prout enim ab <lb></lb> eo + motæ ſunt, ita ad illum redeunt, prout etiam huc <lb></lb> aut illuc + obuerſæ ſunt, vt vis arcus aut baliſtæ. </s> + </p> + <p type="head"> + + <s>DEFIN. V.</s> + </p> + <p type="main"> + + <s>Vires autem contrariæ dicuntur, quæ contrarios <lb></lb> motus ciere + poſſunt, vt ea <expan abbr="quę">quæ</expan> + ſurſum mouet & <expan abbr="quę">quæ</expan><lb></lb> + deorſum, & ſic de cæteris. </s> + </p> + <p type="main"> + + <s>Conſideratur autem in vi <expan abbr="quãtitas">quantitas</expan>, + tum eo quòd <lb></lb> vis partibus ſuis conſtet, in quas in + infinitum diuidi <lb></lb> poteſt, & rurſus additione aut + multiplicatione au<lb></lb> geri, tum quòd æqualitatis + exceſſus & defectus <expan abbr="cõ-parationem">con<lb></lb> parationem</expan> recipiat. </s> + </p> + + + <pb xlink:href="044/01/016.jpg" pagenum="8"></pb> + <p type="main"> + + <s>Ac quoniam vis eſt mouendi potentia, vis par<lb></lb> tes erunt + quæ motus partes efficient, & quæ erit mo<lb></lb> tus partium, + menſura eadem erit & vis partium. </s> + <s>Et <lb></lb> motus quidem propria menſura eſt linea ſeu <expan abbr="ſpatiũ">ſpatium</expan>. <lb></lb> + </s> + <s>Quantum enim res quæpiam mota, ſpatij percur<lb></lb> rit, + tantùm mota eſſe dicitur. </s> + <s>Quoniam verò in mo<lb></lb> mento vel inſtanti quod inſtar + puncti eſt, & magni<lb></lb> tudine caret, nullus motus fieri + poteſt, ſed motus o<lb></lb> mnis in tempore fit. </s> + <s>Ideò ad motus menſuram tem<lb></lb> pus etiam adhibere oportet. </s> + <s>Illud enim cum ſpatio <lb></lb> vel linea, motus dici facit æquales + aut inæquales. </s> + </p> + <p type="head"> + + <s>DEFIN. VI.</s> + </p> + <p type="main"> + + <s>Æquales igitur motus dicuntur, qui æqualibus <lb></lb> temporibus + æqualia ſpatia percurrunt. </s> + </p> + <p type="main"> + + <s>Qui autem æqualibus temporibus æqualia ſpa<lb></lb> tia + permeant, illi iidem proportionales ſunt: hoc <lb></lb> eſt, + quæ eſt ratio temporis, quo alter eorum fit ad <lb></lb> tempus quo + fit alter, eadem eſt ſpatij quod percurrit <lb></lb> alter ad + ſpatium quod reliquus percurrit. </s> + <s>Si enim <lb></lb> duorum quorum ſinguli vna hora miliaris vnius + i<lb></lb> ter conficiunt, alter eodem motu per tria miliaria <lb></lb> ferri + pergat, alter verò per duo <expan abbr="tãtùm">tantùm</expan>. </s> + <s>Ille tribus ho<lb></lb> ris ea abſoluet, hic verò duabus, & + æquè celeriter <lb></lb> ferri dicentur, licet ſpatia + inæqualia <expan abbr="percurrãt">percurrant</expan>, + quo<lb></lb> niam illa ſunt temporibus proportionalia. </s> + </p> + + + <pb xlink:href="044/01/017.jpg" pagenum="9"></pb> + <p type="head"> + + <s>VII.</s> + </p> + <p type="main"> + + <s>Inæquales autem motus dicuntur, quorum tem<lb></lb> pora non + ſunt ſpatiis proportionalia. </s> + <s>Eorum autem <lb></lb> maior ille dicitur, cuius maior erit ratio ſpatij ad + ſpa<lb></lb> tium, quàm temporis ad tempus, quibus fiunt illi <lb></lb> + motus. </s> + <s>Ex quo intelligitur etiam quis minor dica<lb></lb> tur. </s> + <s>Maior igitur dicetur qui celeriùs feretur, minor, <lb></lb> qui + tardiùs. </s> + </p> + <p type="head"> + + <s>VIII.</s> + </p> + <p type="main"> + + <s>Æquales igitur vires dicentur, quæ æqualibus <lb></lb> motibus + ſubiecta ſua mouebunt. </s> + <s>Maior verò quæ <lb></lb> ſubiectum ſuum magis vel + celeriùs mouebit. </s> + <s>Mi<lb></lb> nor, quæ minùs vel tardiùs. </s> + </p> + </chap> + <chap> + <p type="head"> + + <s><emph type="italics"></emph>Motuum diuiſio.<emph.end type="italics"></emph.end></s> + </p> + <p type="main"> + + <s>Comparantur autem motus ſecundum omnes <lb></lb> comparationis gradus, hoc + eſt, æqualitatem, exceſ<lb></lb> ſum & + defectum: tum ſibi ipſis, cùm totus ſuis + par<lb></lb> tibus confertur, tum alij aliis. </s> + </p> + <p type="main"> + + <s>Quatenus autem ſibi ipſi motus <expan abbr="cõparatur">comparatur</expan>, aut <lb></lb> eſt <foreign lang="grc">ὁμοιομερὴς</foreign> ſeu æquabilis aut <foreign lang="grc">ἀνομοιομερὴς</foreign>, id eſt inæ<lb></lb> + quabilis. </s> + <s>Aequabilis, cuius omnes partes tum mini<lb></lb> mæ tum maiores + tempora habent ſpatiis per quæ <lb></lb> cientur proportionalia. </s> + <s>Inæquabilis verò, in quo illa <lb></lb> non ſunt proportionalia. + </s> + </p> + <p type="main"> + + <s>Inæquabilium autem motuum duo ſunt genera, <pb xlink:href="044/01/018.jpg" pagenum="10"></pb>creſcens ſcilicet + & decreſcens, ídque vel continuè, <lb></lb> vel per + interualla, ordinatim vel <foreign lang="grc">ἀτακτῶς. </foreign></s> + </p> + <p type="main"> + + <s>Creſcens dicitur, quando partium poſteriorum <lb></lb> ipſius + maior eſt celeritas, quàm priorum: <expan abbr="decreſcẽs">decreſcens</expan>, <lb></lb> quando + minor. </s> + </p> + <p type="main"> + + <s>Continuè, quando nulla pars illius vel minima <lb></lb> alteri parti + eſt æqualis ſeu ſibi ipſi æquabilis. </s> + </p> + <p type="main"> + + <s>Per interualla verò, quando partes quidem illius <lb></lb> ſibi + ipſis æquabiles ſunt, aliæ verò aliis + comparatæ <lb></lb> inæquabiles. </s> + </p> + <p type="main"> + + <s>Ordinatim, quando incrementum aut <expan abbr="decremẽ-tum">decremen<lb></lb> tum</expan> illud certa aliqua proportionis + progreſſione <lb></lb> fit, vt ſi in motu continuè + creſcente, diuiſo toto mo<lb></lb> tus ſpatio in aliquot + partes æquales eadem ſit ratio <lb></lb> celeritatis, finis primi + ſpatij ad celeritatem finis ſe<lb></lb> cundi, quæ finis + ſecundi ad finem tertij, & ſic dein<lb></lb> ceps: vel + ſi finis ſecundi ſpatij duplo citiùs feratur, <lb></lb> + quàm finis primi & finis tertij, triplo quàm primi: <lb></lb> + finis verò quarti quadruplo, & ſic deinceps. </s> + <s>Idem ſi <lb></lb> in quauis alia proportionis progreſſione + illarum <lb></lb> partium celeritas, aliarum ad alias ſe habeat. </s> + <s>In mo<lb></lb> tu verò per interualla creſcente <expan abbr="augmẽtum">augmentum</expan> illud or<lb></lb> dinatim + fieri dicetur, ſi interuallorum proportio a<lb></lb> liquem + progreſſionis ordinem ſeruet: puta ſi <expan abbr="primũ">primum</expan> + <lb></lb> interuallum duplo tardiùs <expan abbr="ſecũdo">ſecundo</expan> moueatur, triplo <lb></lb> verò tardiùs + tertio, &c. </s> + </p> + + + <pb xlink:href="044/01/019.jpg" pagenum="11"></pb> + <p type="main"> + + <s>Inordinatè autem creſcere vel decreſcere dicetur <lb></lb> + motus, ſi incrementum illud aut decrementum nul<lb></lb> la certa & + ordinata proportione fiat. </s> + </p> + <p type="main"> + + <s>Comparantur autem motus alij aliis, æquabiles <lb></lb> ſcilicet & + inæquabiles ordinatim progredientes: de <lb></lb> inordinatorum enim tum + inter ſe, tum cum aliis <expan abbr="cõ||">com</expan><lb></lb> + paratione nihil certò ſtatui poteſt. </s> + </p> + <p type="main"> + + <s>Comparantur igitur æquabiles cum æquabili<lb></lb> bus aut cum + inæquabilibus, ac etiam inæquabiles <lb></lb> cum inæquabilibus. + </s> + </p> + <p type="main"> + + <s>Æquabiles autem cum æquabilibus compara<lb></lb> tionibus + ſuis partibus correſpondentibus ſunt pro<lb></lb> + portionales. </s> + <s>Si enim <expan abbr="proponãtur">proponantur</expan> duo motus + æqua<lb></lb> biles, quorum alter altero maior ſit, quæ + erit ratio <lb></lb> dimidiæ partis vnius ad dimidiam partem alterius, + <lb></lb> eadem erit tertiæ ad tertiam, & ſic deinceps. </s> + </p> + <p type="main"> + + <s>Æquabilium verò motuum cum inæquabilibus, <lb></lb> cum iis + ſcilicet qui per interualla creſcunt, compa<lb></lb> ratio fit + tanquam cum pluribus diuerſis motibus æ<lb></lb> quabilibus, + ſunt enim interuallorum <expan abbr="illorũ">illorum</expan> + motus <lb></lb> æquabiles. </s> + <s>At cum continuè <expan abbr="creſcẽtibus">creſcentibus</expan> aut decre<lb></lb> ſcentibus + difficile eſt comparationis modum defi<lb></lb> nire, cùm ne + momento quidem ſtabilis maneat par<lb></lb> tium illarum quantitas. </s> + <s>Eſt tamen motus æquabilis <lb></lb> illorum menſura: tanti enim + eſſe dicuntur, quanti <lb></lb> forent, ſi in ea celeritatis + menſura ad quam perue <pb xlink:href="044/01/020.jpg" pagenum="12"></pb>nerunt, æquabiliter pergerent. </s> + </p> + <p type="main"> + + <s>Creſcentes verò cum creſcentibus continuè, <expan abbr="cõ-parati">con<lb></lb> parati</expan> ſiquidem eadem + progreſſionis proportione <lb></lb> creſcant, æqualia + ſpatia ab eorum principiis nume<lb></lb> rata, æqualibus + temporibus emenſa habebunt, idem <lb></lb> in decreſcentibus. </s> + <s>Sed hęc hactenus inquiſiuiſſe ſuf<lb></lb> + ficiat: ſi quis enim particulatim omnia <expan abbr="expẽdere">expendere</expan> vel<lb></lb> let, in infinitum ſeſe + extenderet hęc partium & pro<lb></lb> portionum motuum + ſupputatio. </s> + </p> + <p type="main"> + + <s>Iam videamus quæ vires quos horum motuum <lb></lb> cieant. </s> + <s>Et vis quidem ea qua res quælibet ad ſitum <lb></lb> naturalem + ſuarum partium redit, motu <expan abbr="cõtinuè">continuè</expan> de<lb></lb> creſcente mouet: quo enim + propiùs ad ſitum natu<lb></lb> ralem partium accedit, eo + tardior & languidior eſt. <lb></lb> + </s> + <s>Arcus enim cùm primùm laxari incipit, celeriùs <lb></lb> mouetur, + quàm cùm motus ſui finis propior eſt. </s> + </p> + <p type="main"> + + <s>Vis verò naturalis, qua res quęlibet ad <expan abbr="locũ">locum</expan> ſuum <lb></lb> naturalem tendit, ſubiectum + ſuum, motu continuè <lb></lb> & ordinatim creſcente, mouet. </s> + <s>Illius autem motus <lb></lb> cauſa eſt quòd faciliùs id + moueatur, quod in motu <lb></lb> eſt, quàm quod quieſcit. </s> + <s>Vis igitur eadem, <expan abbr="ſubiectũ">ſubiectum</expan> + <lb></lb> quod iam in motu eſt premens, illud magis moue<lb></lb> bit, + quàm ſi quieſcat, & magis motum, magis etiam <lb></lb> + mouebit: ita vt eadem vis motione maior fiat, quàm <lb></lb> per ſe + ſit. </s> + <s>Et hæc eſt cauſa cur ictus, quo magis ab al<lb></lb> tero + venit, eo vehementior ſit. </s> + <s>Motus autem huius <pb xlink:href="044/01/021.jpg" pagenum="13"></pb>ſpatia + hanc celeritatis proportionem ſeruant, vt <expan abbr="quę">quæ</expan> + <lb></lb> eſt ratio totius ſpatij, per quod fit ille motus ad + par<lb></lb> tem ipſius (vtriuſque initio inde ſumpto, + vbi eſt mo<lb></lb> tus initium) eadem ſit celeritatis ad + celeritatem. <lb></lb> + <figure id="id.044.01.021.1.jpg" xlink:href="044/01/021/1.jpg"></figure><lb></lb> + Exempli gratia, ſi vis aliqua per lineam ABC <lb></lb> mouerit, + ſitque AB illius lineæ pars, quæ erit <lb></lb> ratio AC ad AB, + eadem erit celeritatis motus <lb></lb> in puncto C ad <expan abbr="celeritatẽ">celeritatem</expan> motus in puncto B. <lb></lb> + </s> + <s>Cuiuſmodi proportio obſeruatur in paralle<lb></lb> lis + triangulum ſecantibus. </s> + <s>Vt enim ſe habet <lb></lb> + <figure id="id.044.01.021.2.jpg" xlink:href="044/01/021/2.jpg"></figure><lb></lb> AC ad + AB, ſic CG ad BF, & vt AD ad <lb></lb> AC, ſic DH ad CG. + </s> + <s>Itaque ſi in ſpatia ali<lb></lb> quot æqualia diuidatur + totius motus ſpa<lb></lb> tium, finis ſecundi duplo citiùs + feretur, <lb></lb> quàm finis primi: finis verò tertijtriplo <lb></lb> + citiùs quàm finis primi, & ſic deinceps. <lb></lb> + </s> + <s>Hac autem ratione fit, vt ſpatiorum illo<lb></lb> rum initio maxima + ſit celeritatis <expan abbr="differẽ-tia">differen<lb></lb> + tia</expan>: progreſſu verò ſemper minuatur, ac + tandem fer<lb></lb> mè eadem ſit, vt fit in trianguli + lateribus, quæ lon<lb></lb> giſſimè producta + æquè diſtare videntur. </s> + <s>Eáque eſt <lb></lb> ratio cur Solis & Lunæ radij, + etiamſi concurrant (in <lb></lb> ipſorum ſcilicet corporibus, + aut vltra res quas <expan abbr="illu-ſtrãt">illu<lb></lb> + ſtrant</expan>) paralleli <expan abbr="tamẽ">tamen</expan> + <expan abbr="appareãt">appareant</expan>. </s> + <s><expan abbr="Eadẽ">Eadem</expan> + <expan abbr="etiã">etiam</expan> cauſa eſt cur <lb></lb> + lineę omnes ad <expan abbr="perpẽdiculũ">perpendiculum</expan> in <expan abbr="terrã">terram</expan> + cadentes, paral<lb></lb> lelæ videantur, cùm <expan abbr="tamẽ">tamen</expan> in centro terræ <expan abbr="cõcurrant">concurrant</expan>. </s> + </p> + + + <pb xlink:href="044/01/022.jpg" pagenum="14"></pb> + <p type="main"> + + <s>Hunc igitur motum vis naturalis efficit, modò <lb></lb> nulla quies + intercedat. </s> + </p> + <p type="main"> + + <s>Vis autem voluntaria motum omnem ciere apta <lb></lb> eſt. </s> + <s>Fortuita verò inordinatum tantùm motum effi<lb></lb> cit. </s> + <s>Et hæc de variis virium motibus, <expan abbr="eorúmq;">eorúmque</expan> pro<lb></lb> portione & partibus. </s> + </p> + <p type="main"> + + <s>Eſt verò etiam virium partitio quædam, quæ ex <lb></lb> + eorum quibus <expan abbr="inhærẽt">inhærent</expan> + ſubiectorum diuiſione, aut <lb></lb> alterius ad alterum proportione + reſultat. </s> + <s>In ſubie<lb></lb> ctis enim, quæ <foreign lang="grc">ὁμοιομερῆ</foreign> ſunt, vis naturalis æqualiter <lb></lb> + per omnes partes diffuſa eſt: ita vt quæ eſt ratio + mo<lb></lb> lis ad molem ſub eadem figura, <expan abbr="eadẽ">eadem</expan> ſit vis ad vim: <lb></lb> vt ſi globus alicuius + metalli, ad alium globum eiuſ<lb></lb> dem metalli mole quadruplus + ſit, illius quoque <expan abbr="põ-dus">pon<lb></lb> + dus</expan> ad huius pondus quadruplum erit. </s> + </p> + <p type="main"> + + <s>Hactenus vim <expan abbr="cõſiderauimus">conſiderauimus</expan> quatenus mouet, <lb></lb> ſupereſt + vt eam conſideremus, quatenus mouenti <lb></lb> reſiſtit. </s> + <s>Habet autem locum reſiſtentia, vbi duæ vi<lb></lb> res + contrariæ committuntur: ſi enim vtraque <expan abbr="eodẽ">eodem</expan> + <lb></lb> motu moueat, nulla erit reſiſtentia, ſed altera + alteri <lb></lb> addita maiorem vim conſtituet. </s> + </p> + <p type="main"> + + <s>Committi autem dicuntur vires, quando ita ap<lb></lb> plicantur & + <expan abbr="connectũtur">connectuntur</expan>, vt altera nutu + ſuo moueri <lb></lb> non poſſit, quin altera contra nutum + ſuum mouea<lb></lb> tur. </s> + <s>Viribus autem ita connexis, accidit vt altera in <lb></lb> alterius + ſubiectum agat, & altera in ſubiecto ſuo exi- <pb xlink:href="044/01/023.jpg" pagenum="15"></pb>ſtens, alteri in altero + ſubiecto exiſtenti reſiſtat, ita vt <lb></lb> eorum + ſubiecta quodammodo reciproca fiant. </s> + </p> + <p type="head"> + + <s>CONCLVSIO I.</s> + </p> + <p type="main"> + + <s>Illam autem reſiſtentiam vi momenti in eodem <lb></lb> ſubiecto + æqualem, aut eandem cum ea eſſe, ex ipſius <lb></lb> + definitione conſtat. </s> + <s>Eſt enim reſiſtere <expan abbr="nõ">non</expan> pati + ſe mo<lb></lb> ueri. </s> + <s>Quantum autem vis quælibet <expan abbr="ſubiectũ">ſubiectum</expan> ſuum <lb></lb> nutu ſuo mouet, tantum + illud contra nutum ſuum <lb></lb> moueri non patitur. </s> + <s>Mouendi igitur & <expan abbr="mouẽti">mouenti</expan> + re<lb></lb> ſiſtendi potentia in eodem ſubiecto + æquales ſunt. </s> + </p> + <p type="head"> + + <s>II.</s> + </p> + <p type="main"> + + <s>Quemadmodum autem in eodem ſubiecto ma<lb></lb> ior vis + ineſſe dicitur, in quo eſt maioris motus po<lb></lb> + tentia, ſic maior reſiſtentia erit maioris motus + con<lb></lb> trarij impatientia. </s> + <s>Eadem igitur vis magis mouere <lb></lb> nitenti <expan abbr="cõtra">contra</expan> ipſius nutum magis reſiſtet, minus + ve<lb></lb> rò nitenti minùs reſiſtet. </s> + </p> + <p type="head"> + + <s>III.</s> + </p> + <p type="main"> + + <s>Et quo maior erit motus contrarius, eo magis re<lb></lb> + ſiſtet: id eſt, quo celerius vis quælibet à nutu + ſuo re<lb></lb> uelletur, eo magis reſiſtet. </s> + <s>Duplo igitur citiùs re<lb></lb> uellenti duplo reſiſtet, + triplo citius triplo reſiſtet, & <lb></lb> ſic <foreign lang="grc">ἀναλόγως</foreign> in omni proportione. </s> + <s>Quæ igitur erit <lb></lb> ratio motus ad motum, contra nutum eadem erit + <lb></lb> in vna & eadem vi ratio reſiſtentiæ ad + reſiſtentiam. </s> + </p> + <p type="main"> + + <s>Si verò plures vires comparentur. </s> + </p> + + + <pb xlink:href="044/01/024.jpg" pagenum="16"></pb> + <p type="head"> + + <s>IIII.</s> + </p> + <p type="main"> + + <s>Æqualium quidem virium æqualibus motibus <lb></lb> æquales erunt + reſiſtentiæ. </s> + <s>Si enim æquales vires æ<lb></lb> qualiter à ſuis + nutibus reuellantur, æqualiter reſi<lb></lb> ſtent. </s> + </p> + <p type="head"> + + <s>V.</s> + </p> + <p type="main"> + + <s>Inæqualibus verò motibus, earum reſiſtentię + in<lb></lb> æquales erunt, & motuum proportionem + ſequen<lb></lb> tur. </s> + <s>Exempli gratia, ſint duæ vires A & B, quarum v<lb></lb> + traque vna hora miliare vnum percurrat, illæ æqua<lb></lb> les + erunt. </s> + <s>Si igitur vtraque quarta parte vnius milia<lb></lb> ris, vel dimidio + miliari à ſuo nutu eodem tempore <lb></lb> reuellatur, eadem erit + vtriuſque illi motui reſiſten<lb></lb> tia. </s> + <s>Sin verò altera quidem quarta parte vnius milia<lb></lb> ris à + nutu ſuo reuellatur, altera verò eodem tempo<lb></lb> re + integro miliari reuellatur: hæc reſiſtentia ad + il<lb></lb> lam quadrupla erit, & ſic de cæteris motuum + pro<lb></lb> portionibus. </s> + </p> + <p type="head"> + + <s>VI.</s> + </p> + <p type="main"> + + <s>Inæqualium verò virium reſiſtentiæ æqualibus + <lb></lb> motibus ipſarum virium proportionem ſequentur. <lb></lb> + </s> + <s>Quo maior enim eſt vis, eo magis eodem tempore <lb></lb> eodémque + ſpatio contra nutum mouenti reſiſtit. <lb></lb> + </s> + <s><expan abbr="Exẽpli">Exempli</expan> gratia, ſit vis A quadrupla + ad vim B, vis A mo<lb></lb> uenti ſe vno miliari quadruplo + reſiſtet, quàm B mo<lb></lb> uenti ſe vno miliari + eodem tempore. </s> + </p> + + + <pb xlink:href="044/01/025.jpg" pagenum="17"></pb> + <p type="main"> + + <s>Inæqualium verò virium inæqualibus motibus <lb></lb> + reſiſtentiæ, quando motus eam inter ſe + proportio<lb></lb> nem ſeruabunt, quæ eſt inter eas + <foreign lang="grc">ἀντιπεπονθῶς</foreign>, ſeu <lb></lb> + reciprocè, æquales erunt. </s> + <s>Sit enim vis A quæ ad vim <lb></lb> B eandem habeat <expan abbr="rationẽ">rationem</expan> quam C ad D, puta quam <lb></lb> 3 ad + 1, aut 4 ad 1, aut quæcunque alia ſit, <expan abbr="moueatúrq;">moueatúrque</expan> + <lb></lb> vtraque ſpatio E contra nutum ſuum eodem tem<lb></lb> + pore, reſiſtentia A ad reſiſtentiam B ſe + habebit, vt C <lb></lb> ad D per proximam concluſionem. </s> + <s>Si verò B moue<lb></lb> atur eodem tempore ſpatio aliquo quod + ſe habeat <lb></lb> ad ſpatium E vt C ad D, reſiſtentia + B huic motui ſe <lb></lb> habebit ad reſiſtentiam priori + motui, vt C ad D per <lb></lb> quartam <expan abbr="concluſionẽ">concluſionem</expan>. </s> + <s>Duæ ergo reſiſtentiæ ad ean<lb></lb> dem, eandem + habebunt rationem, ergo æquales e<lb></lb> runt per <emph type="italics"></emph>9<emph.end type="italics"></emph.end> prop. <emph type="italics"></emph>5.<emph.end type="italics"></emph.end> Elem. Eucl. + </s> + </p> + <p type="head"> + + <s>VIII.</s> + </p> + <p type="main"> + + <s>Si verò motuum virium inæqualium proportio <lb></lb> non ſit + eadem quæ eſt ipſarum virium reciprocè, vis <lb></lb> + illius quæ ad alteram maiorem habebit rationem, <lb></lb> quàm motus + alterius ad motum ipſius, reſiſtentia <lb></lb> maior erit, + altera verò minor. </s> + <s>Exempli gratia, ſi vis <lb></lb> A ſit tripla ad vim B, & motus + quo B à nutu ſuo reuel<lb></lb> letur, ſit minor quàm + triplus ad motum A, reſiſten<lb></lb> tia A erit maior + quàm reſiſtentia B. </s> + <s>Idem erit, ſi ma<lb></lb> ior ſit ratio A ad B, quàm 3 + ado, motus verò B ad mo <pb xlink:href="044/01/026.jpg" pagenum="18"></pb>tum A ſit triplus. </s> + <s>Contrà verò ſi A ad B ſit tripla: mo<lb></lb> tus + verò B ad motum A ſit maior quàm triplus, ma<lb></lb> ior + erit reſiſtentia B quàm A. </s> + <s>Idem accidet, ſi A ad B <lb></lb> ratio minor ſit quàm 3 ad 1, + motus verò B ad motum <lb></lb> A ſit triplus. </s> + <s>Hunc autem exceſſum & defectum <expan abbr="nõ">non</expan> + <lb></lb> vlteriùs ſcrutabimur, vt ipſius quantitatis <expan abbr="menſurã">menſuram</expan> + <lb></lb> aſſequamur: alioqui in infinitum fieret + progreſſus. </s> + </p> + <p type="main"> + + <s>Ex his colligimus <expan abbr="reſiſtẽtias">reſiſtentias</expan> tribus modis æqua<lb></lb> les + aut inæquales dici, per ſe ſcilicet aut motione, aut <lb></lb> + vtroque modo. </s> + <s>Per ſe quidem cùm ſubiectorum <lb></lb> vires ſunt + æquales (id eſt æqualiter ſua ſubiecta + mo<lb></lb> uent) aut inæquales. </s> + <s>Motione verò quando per ſe <lb></lb> quidem in æquales + ſunt, motione verò æquales <expan abbr="fiũt">fiunt</expan>: <lb></lb> aut quando per ſe æquales ſunt, + motione verò inæ<lb></lb> quales <expan abbr="fiũt">fiunt</expan> eo, quo dictum eſt, modo. </s> + <s>Vtroque mo <lb></lb> do æquales aut inæquales <expan abbr="dicũtur">dicuntur</expan>, quando tum per <lb></lb> ſe, + tum motione tales fiunt. </s> + <s>His explicatis videamus <lb></lb> quis eorum ſit effectus, vbi vires + committentur. </s> + </p> + <p type="head"> + + <s>IX.</s> + </p> + <p type="main"> + + <s>Primùm ex prima concluſione ſequitur quę erit <lb></lb> + ratio reſiſtentiæ ad reſiſtentiam, eandem fore + reſi<lb></lb> ſtentiæ ad vim cuius eſt altera + reſiſtentia: ſunt enim <lb></lb> vis & <expan abbr="reſiſtẽtia">reſiſtentia</expan> in + <expan abbr="eodẽ">eodem</expan> ſubiecto æquales. </s> + <s>Ergo mul<lb></lb> to magis ſequitur, ſi <expan abbr="reſiſtẽtia">reſiſtentia</expan> + altera maior, altera mi <lb></lb> nor fuerit, illam huius vi <expan abbr="maiorẽ">maiorem</expan> fore, vel illius vim <lb></lb> hac + reſiſtentia maiorem, ſi æqualis æqualem. </s> + </p> + + + <pb xlink:href="044/01/027.jpg" pagenum="19"></pb> + <p type="head"> + + <s>X</s> + </p> + <p type="main"> + + <s>Reſiſtentia autem vi contrariæ commiſſa <expan abbr="tãtum">tantum</expan> + <lb></lb> de ea tollit quanta eſt ipſa reſiſtentia. </s> + <s>Sublata autem <lb></lb> vi tollitur motus: ſublata verò + reſiſtentia, ſi vis adſit, <lb></lb> ſequitur + motus. </s> + <s>Vbi igitur vis & reſiſtentia inęqua<lb></lb> les + committentur, ſi vis maior ſit quàm + reſiſtentia, <lb></lb> fiet motus ſecundum vis illius nutum, + & contra nu<lb></lb> tum vis illius quæ reſiſtit. </s> + <s>Si verò reſiſtentia maior <lb></lb> fuerit, tum ipſa fiet + vis mouens, & vim contrariam <lb></lb> contra nutum ipſius mouebit, + dum ipſa nutu ſuo <lb></lb> mouebitur. </s> + <s>Hinc ſequuntur duo theoremata, circa <lb></lb> quæ totius huius + conſiderationis cardo vertitur. </s> + </p> + <p type="head"> + + <s>THEOREMA I.</s> + </p> + <p type="main"> + + <s>Duarum virium connexarum, quarum (ſi mo<lb></lb> ueantur) motus erunt + ipſis <foreign lang="grc">ἀντιπεπονθῶς</foreign> + proportiona<lb></lb> les neutra alteram mouebit, ſed + æquilibrium <expan abbr="faciẽt">facient</expan>. </s> + </p> + <p type="main"> + + <s>Sit vis A commiſſa cum vi B, ſitque vis A ad vim <lb></lb> B + ratio per ſe, vt C ad D <expan abbr="quæcũque">quæcunque</expan> illa ſit, ſiue dupla, <lb></lb> + ſiue tripla, ſiue alia. </s> + <s>Sit etiam <expan abbr="eadẽ">eadem</expan> ratio motus quo <lb></lb> B + mouebitur, ſi ita, vt connexa ſunt, moueantur) ad <lb></lb> motum + quo A mouebitur, quæ eſt C ad D, dico mo<lb></lb> tum non + ſequuturum, ſed factum iri æquilibrium. <lb></lb> + </s> + <s>Aut enim A & B vires per ſe erunt æquales aut + inæ<lb></lb> quales: ſi æquales, ergo & ipſarum + motus æquales e<lb></lb> runt: ſunt enim ex hypotheſi + ipſis proportionales: <pb xlink:href="044/01/028.jpg" pagenum="20"></pb>ergo per quartam concluſionem earum reſiſtentiæ + <lb></lb> æquales erunt, ergo per decimam concluſionem <expan abbr="nõ">non</expan> + <lb></lb> fiet motus. </s> + <s>Si verò ſunt inæquales, cùm earum mo<lb></lb> tus ex + hypotheſi ſint <foreign lang="grc">ἀντιπεπονθῶς</foreign> + proportionales per <lb></lb> ſeptimam concluſionem æquales + erunt etiam reſi<lb></lb> ſtentiæ. </s> + <s>Ergo nec motus fiet: nullo igitur modo fiet <lb></lb> earum motus. </s> + <s>Quod demonſtrandum erat. </s> + </p> + <p type="head"> + + <s>THEOREMA II.</s> + </p> + <p type="main"> + + <s>Quarum verò ita connexarum (ſi <expan abbr="moueãtur">moueantur</expan>) mo<lb></lb> tus, ipſis proportionales non erunt: + illa alteram mo<lb></lb> uebit, cuius ad alteram ratio maior erit, + quàm huius <lb></lb> motus ad illius motum. </s> + </p> + <p type="main"> + + <s>Sit vis A cum vi B commiſſa, ſitque A ad B ratio <lb></lb> per + ſe, vt C ad D: ita verò connexæ ſint, vt ſi + mouean<lb></lb> tur, minor ſit ratio motus quo B mouebitur ad + mo<lb></lb> tum, quo A mouebitur, quum C ad D dico A motu<lb></lb> ram + B: erit enim per octauam concluſionem reſi<lb></lb> + ſtentia B minor reſiſtentia A: ergo per nonam + con<lb></lb> cluſionem reſiſtentia B minor vi A: vis + igitur A vim <lb></lb> B mouebit per vndecimam concluſionem. </s> + <s>Quod <lb></lb> demonſtrandum erat. </s> + </p> + <p type="main"> + + <s>Hinc poſſemus ad problematis noſtri demon<lb></lb> + ſtrationem rectà pergere: ante tamen craſſiùs + ali<lb></lb> quanto hæc explicanda ſunt, vt à quouis + faciliùs in<lb></lb> telligantur. </s> + <s>Duæ ſunt motus menſuræ, locus ſcilicet <lb></lb> + & tempus: vtroque igitur, tempore videlicet & loco <pb xlink:href="044/01/029.jpg" pagenum="21"></pb>maior, minor aut æqualis + dicitur: & quo minori <expan abbr="tẽ-pore">tem<lb></lb> + pore</expan> idem ſpatium abſoluitur, eo maior eſt, + vel quo <lb></lb> maius ſpatium eodem tempore. </s> + </p> + <p type="main"> + + <s>Vt igitur motus magnus dicatur, perinde eſt ſi <lb></lb> paruo + tempore fiat, aut magno ſpatio. </s> + </p> + <p type="main"> + + <s>Quod autem maiorem <expan abbr="motũ">motum</expan> ciere poteſt + illud, <lb></lb> vis maior dicitur, quòd minorem minor. </s> + </p> + <p type="main"> + + <s>Vis autem ſeu mouendi potentia in eodem ſub<lb></lb> iecto + certa & finita eſt. </s> + <s>Quęlibet enim res vi natura<lb></lb> li prędita, ſi + à loco naturali abſit, ſuóque arbitrio <expan abbr="cõ-mittatur">con<lb></lb> mittatur</expan>, certo tempore + eò redit. </s> + <s>Eſt enim certum <lb></lb> in rerum natura quanto tempore libræ vnius + <expan abbr="põdus">pondus</expan> + <lb></lb> deorſum ſponte ſua delatum, miliare vnum aut duo + <lb></lb> conficiat pro ratione materię, vel quantum ſpatij + v<lb></lb> na vel duabus horis percurrat. </s> + <s>Id verò quantum ſit, <lb></lb> hominum induſtria nondum quod + ſciam explora<lb></lb> tum eſt: aliâs autem id + demonſtrare conabimur. </s> + </p> + <p type="main"> + + <s>Mouendi verò potentia in alieno ſubiecto infi<lb></lb> nita + eſt, hoc eſt, in <expan abbr="infinitũ">infinitum</expan> + augeri vel minui poteſt, <lb></lb> quoniam in finita auctione & + diminutione eſt reſi<lb></lb> ſtentia: tanta enim + eſt, quanta eſt in eodem ſubiecto <lb></lb> vis: <expan abbr="quantúſq;">quantúſque</expan> motus illius + eſt vis alterius reſpectu. <lb></lb> + </s> + <s>Quo igitur vis quæ alienum ſubiectum mouere ni<lb></lb> titur, + illud magis mouere nitetur, eo minùs illud <lb></lb> mouere poterit, maior + enim erit illius reſiſtentia. <lb></lb> + </s> + <s>Quemadmodum enim quod magis nutu ſuo mo <pb xlink:href="044/01/030.jpg" pagenum="22"></pb>uetur, maiorem vim mouendi + habet. </s> + <s>Ita illud idem <lb></lb> quod magis contra nutum ſuum mouetur, <expan abbr="maiorẽ">maiorem</expan> + <lb></lb> vim reſiſtendi habet. </s> + <s>E contrario verò quo vis quæ<lb></lb> libet minorem motum in + alieno ſubiecto ciere ni<lb></lb> tetur, eo faciliùs illud + mouebit. </s> + </p> + <p type="main"> + + <s>Tarditate igitur motus, reſiſtentia in infinitum <lb></lb> minui + poteſt: minuta verò reſiſtentia vis contrariæ + <lb></lb> effectus augetur, ita vt vis quæ per ſe minima eſt, + in <lb></lb> contrariam cui plurimum diminuta ſit, reſiſtentia + <lb></lb> maximè agat. </s> + <s>Perinde igitur eſt, ſi vis mouens ma<lb></lb> gna ſit, + mouenda verò parua, ac ſi illa celeriter fera<lb></lb> tur, + hæc verò tardè: <expan abbr="quãtum">quantum</expan> + enim vis mouens ſi ma<lb></lb> gna fuerit in mouendam minorem + poterit, tantum <lb></lb> vis parua celeriter mota in magnam tardè motam + <lb></lb> poterit. </s> + <s>Si igitur velimus vt vis parua magnam mo<lb></lb> ueat, eas ita collocare + oportet, vt quantum magna <lb></lb> paruam ſuperat, tantum illi de motus + celeritate de<lb></lb> trahatur, & aliquid ampliùs. </s> + <s>Si enim tanta ſit tardi<lb></lb> tas motus vis vnius, reſpectu + motus vis alterius, <expan abbr="quã-ta">quan<lb></lb> ta</expan> + eſt proportio vis illius, ad hanc non fiet motus: vt <lb></lb> ſi + pondus A quatuor librarum cum pondere B libræ <lb></lb> vnius committatur: + ſintque ita connexa, vt dum <lb></lb> A vno ſpatio mouebitur, B + quatuor ſpatiis mouea<lb></lb> tur, ita vt motus A quadruplo tardior + ſit motu B <expan abbr="nõ">non</expan> + <lb></lb> fiet motus, quia quantum A excedit B pondere tan<lb></lb> tùm + deficit motus tarditate. </s> + </p> + + + <pb xlink:href="044/01/031.jpg" pagenum="23"></pb> + <p type="main"> + + <s>Tantum enim eſt libram vnam quatuor ſpatiis <lb></lb> moueri, quantum + libras quatuor vno ſpatio eodem <lb></lb> tempore: ſi igitur + alterutri eorum ita <expan abbr="conſtitutorũ">conſtitutorum</expan>, <lb></lb> vel momentum vis addatur, id cui + additum fuerit, <lb></lb> alterum mouebit. </s> + <s>Idem fiet ſi A ita connectatur vt <lb></lb> vel momento citiùs + quàm moueri poſitum eſt, mo<lb></lb> ueatur vel B tardiùs. + </s> + </p> + <p type="main"> + + <s>Id etiam alia ratione oſtendi poteſt, ſi vis A + qua<lb></lb> drupla ſit ad vim B, erunt in A quatuor partes, + ipſi B <lb></lb> æquales. </s> + <s>Si igitur B cum ſingulis illis committatur, <lb></lb> ita vt æqualiter + moueantur, non fiet motus: ſi verò <lb></lb> alterutri aut ipſi + B, aut ſingulis illis partibus vis vel <lb></lb> motus momentum adiiciatur + vel detrahatur, illa cui <lb></lb> adiectum fuerit, aut cui non detractum + fuerit, nutu <lb></lb> ſuo mouebitur, & aliam contra nutum eius + moue<lb></lb> bit. </s> + <s>Addito igitur ipſi B momento, dum vno ſpatio <lb></lb> mouebitur, + ſingulas illas partes vno ſpatio moue<lb></lb> bit. </s> + <s>Vbi igitur ſingulas ſemel mouerit, ipſa quater <lb></lb> mota + erit. </s> + <s>Dum verò ſingulæ ſemel motæ erunt, to<lb></lb> + tum ex illis conſtans ſemel motum intelligetur: po<lb></lb> + terit igitur B addito ipſi momento, dum quater mo<lb></lb> uebitur, + ſemel totum A mouere. </s> + </p> + <p type="main"> + + <s>Tertio modo id ipſum concludere poſſumus. </s> + <s>Si <lb></lb> duæ vires æquales connectantur, ita vt motæ, + æqua<lb></lb> liter moueantur, altera in <expan abbr="alterã">alteram</expan> non aget. </s> + <s>Si verò ita <lb></lb> connectantur vt motæ inæqualiter moueantur, + <pb xlink:href="044/01/032.jpg" pagenum="24"></pb>quantum altera alteram + celeritate ſuperabit, <expan abbr="tãtum">tantum</expan> + <lb></lb> & vi ſuperabit. </s> + <s>Vt ſi vis A vi B æqualis ſit, ac ita <expan abbr="con-nectãtur">con<lb></lb> nectantur</expan>, vt B quatuor + ſpatiis moueatur, dum A vno <lb></lb> ſpatio mouebitur, B quadruplam + vim habebit ad A, <lb></lb> quia eam in motibus ſuis proportionem + ſeruant, <lb></lb> quam ſi <expan abbr="ſeruarẽt">ſeruarent</expan> arbitrio ſuo commiſſæ, B + ad A qua<lb></lb> drupla eſſet. </s> + <s>Si igitur illis ita connexis, ipſi A adda<lb></lb> tur vis triplo + maior quàm ipſa ſit, B illis quatuor re<lb></lb> + ſiſtet, nec fiet motus. </s> + </p> + <p type="main"> + + <s>Duarum igitur virium comparatarum, quanto <lb></lb> altera ſubiectum + ſuum celeriùs mouebit, quàm al<lb></lb> tera: tanto illa + hanc celeriùs mouere poterit, quàm <lb></lb> ipſa moueatur, + ſi illi vis <expan abbr="momẽtum">momentum</expan> additum + fuerit. <lb></lb> + </s> + <s>Ita vt quæ erit proportio vis ad vim, eadem ſit mo<lb></lb> tus, + quem illa in hac ciere poteſt ad motum quo ipſa <lb></lb> mouebitur. </s> + <s>Et è conuerſo quæ erit ratio motus ad <lb></lb> motum, eadem erit + vis cui additum fuerit momen<lb></lb> tum ad eam quam ipſa mouere + poterit ratio reci<lb></lb> procè. </s> + <s>Iam ad problematis noſtri demonſtratio<lb></lb> nem veniamus. + </s> + </p> + </chap> + <chap> + <p type="head"> + + <s><emph type="italics"></emph>Propoſiti problematis + demonſtratio.<emph.end type="italics"></emph.end></s> + </p> + <p type="main"> + + <s>Sit data vis A quantacunque illa ſit magna vel <lb></lb> parua: datum + verò pondus B quantumquantum il<lb></lb> lud ſit, dico me vi A + pondus B tollere poſſe. </s> + <s>Id ſic de<lb></lb> monſtro. </s> + </p> + + + <pb xlink:href="044/01/033.jpg" pagenum="25"></pb> + <p type="main"> + + <s>Primùm enim ex doctrina ſecundi <expan abbr="lẽmatis">lemmatis</expan>, quod <lb></lb> inferiùs demonſtrabitur, + ſciam proportionem pro<lb></lb> ximè maiorem, quàm + ſit A ad B proportio. </s> + <s>Deinde <lb></lb> ex doctrina primi lemmatis ita connectam A & B, <lb></lb> vt + quando ambo mouebuntur, nunc ſit ratio motus <lb></lb> quo B mouebitur ad + <expan abbr="motũ">motum</expan> quo A mouebitur, quàm <lb></lb> + ſit A ad B. </s> + <s>His peractis ſequitur vim A pondus B mo<lb></lb> turam ex + ſecundo ſuperiùs demonſtrato theorema<lb></lb> te. </s> + <s>Quod erat propoſitum. </s> + </p> + <p type="head"> + + <s>LEMMA I.</s> + </p> + <p type="main"> + + <s>Duas vires ita connectere, vt ſi moueantur, <expan abbr="earũ">earum</expan> + <lb></lb> motus, in data ratione alter ad alterum ſe habeant <lb></lb> vires + contrariæ, aut medio aliquo, aut per ſe abſque <lb></lb> vllo + medio committuntur. </s> + <s>Si abſque medio com<lb></lb> mittant, eodem motu mouebuntur, maior + enim mi<lb></lb> norem eodem motu, quo ipſa moueri poterit, mo<lb></lb> + uebit: aut æquilibrium <expan abbr="faciẽt">facient</expan>, + ſi æquales ſint: vt ſi le<lb></lb> ue graui + committatur, ſiquidem leuitas grauitate <lb></lb> maior ſit, + attolletur graue: ſin verò grauitas maior <lb></lb> ſit, leue + deprimetur: ſi æqualia ſint, non mouebun<lb></lb> tur. + </s> + </p> + <p type="main"> + + <s>Si verò medio aliquo connectantur mediorum <lb></lb> varia ſunt + genera. </s> + <s>Aut enim medium eſt flexibile & <lb></lb> + <foreign lang="grc">ὁμοιομερὲς</foreign>, vt funis, catena, &c. </s> + <s>aut eſt inflexibile, il<lb></lb> lúdque aut rectum, aut curuum, + vt recta linea vel <lb></lb> curua vel angulus. </s> + <s>Atque hæc omnia aut continua <pb xlink:href="044/01/034.jpg" pagenum="26"></pb>ſunt aut diuiſa, ſimplicia aut + compoſita. </s> + </p> + <p type="main"> + + <s>Horum autem mediorum opera fit vt vires illis <lb></lb> applicatæ, quarum + iidem ſunt nutus, contrariis mo<lb></lb> tibus moueantur. </s> + <s>Id <expan abbr="autẽ">autem</expan> fit cùm in mediis illis inter <lb></lb> + <expan abbr="eorũ">eorum</expan> extrema interiacet quies vna vel + plures. </s> + <s><expan abbr="Exẽpli">Exempli</expan> + <lb></lb> gratia, ſi duo pondera funis extremitatibus alligata <lb></lb> + ſint, & funis clauo fixo & immobili incumbat pro<lb></lb> + pter illam quietem inter <expan abbr="vtrumq;">vtrumque</expan> pondus + <expan abbr="poſitã">poſitam</expan> + <expan abbr="nõ">non</expan> + <lb></lb> poterit <expan abbr="alterũ">alterum</expan> deorſum + moueri, quin <expan abbr="alterũ">alterum</expan> ſurſum + <lb></lb> moueatur. </s> + <s><expan abbr="Idẽ">Idem</expan> fiet in linea recta, ſi enim + illius extre<lb></lb> mitatibus pondera duo annexa ſint, & inter + ea ſit in <lb></lb> illa <expan abbr="punctũ">punctum</expan> + aliquod quieſcens, dum alterum ex illis <lb></lb> ponderibus deorſum + feretur, alterum aſcendet. </s> + <s><expan abbr="Pũ-ctum">Pun<lb></lb> ctum</expan> autem illud + quieſcens in linea illa recta, Gręcis <lb></lb> hypomochlium + dicitur, eò quòd vecti, quem <foreign lang="grc">μόχλον</foreign><lb></lb> vocant, ſubiiciatur. </s> + <s>Huius autem hypomochlij, in <lb></lb> recta linea ſe vecte collocatio + faciet, vt lineæ extre<lb></lb> ma ſecundum datam rationem + moueantur. </s> + <s>Si enim <lb></lb> recta linea in datam rationem diuiſa fuerit, hoc + eſt, <lb></lb> vt pars altera ad <expan abbr="alterã">alteram</expan> + eam habeat <expan abbr="rationẽ">rationem</expan>, <expan abbr="quã">quam</expan> quis <lb></lb> voluerit. (Quod <expan abbr="quidẽ">quidem</expan> quo modo fiat docet Euc.) & <lb></lb> + in puncto diuiſionis collocetur <expan abbr="hypomochliũ">hypomochlium</expan>, illius <lb></lb> lineæ extrema ſecundum + <expan abbr="illã">illam</expan> + <expan abbr="rationẽ">rationem</expan> mouebuntur: <lb></lb> ſiue + enim conſideretur <expan abbr="circulorũ">circulorum</expan>, + quos illa extrema <lb></lb> deſcribent, proportio ſiue ſpatium + quod illa in linea <lb></lb> perpendiculari notabunt vtroque modo illi motus, + <pb xlink:href="044/01/035.jpg" pagenum="27"></pb><expan abbr="partiũ">partium</expan> + <expan abbr="illarũ">illarum</expan> + <expan abbr="proportionẽ">proportionem</expan> + <expan abbr="ſeruabũt">ſeruabunt</expan>. </s> + <s>Sit <expan abbr="exẽpli">exempli</expan> gratia <lb></lb> linea AB, + quæ in puncto C in datam rationem ſecta <lb></lb> ſit, puta vt + pars AC quadrupla ſit ad partem CB, mo<lb></lb> ueatúrque circa + centrum C, punctum A deſcribet cir<lb></lb> culum <expan abbr="quadruplũ">quadruplum</expan> ad illum quem B <lb></lb> + deſcribet. </s> + <s>Eſt enim <expan abbr="eadẽ">eadem</expan> ratio in + cir<lb></lb> culo diametri ad <expan abbr="diametrũ">diametrum</expan>, quæ eſt <lb></lb> circunferentiæ ad + <expan abbr="circunferentiã">circunferentiam</expan> (vt <lb></lb> + alibi demonſtrauimus.) Hac igitur <lb></lb> ratione A puncti motus + quadruplus <lb></lb> + <figure id="id.044.01.035.1.jpg" xlink:href="044/01/035/1.jpg"></figure><lb></lb> erit + ad puncti B <expan abbr="motũ">motum</expan>. </s> + <s>Si verò ponamus AD <expan abbr="perpen-dicularẽ">perpen<lb></lb> dicularem</expan> eſſe, & linea AB + illi primùm <expan abbr="coincidẽs">coincidens</expan> circa + <lb></lb> punctum C, moueatur donec A ad D perueniat: tum <lb></lb> + <expan abbr="eodẽ">eodem</expan> momento B perueniet ad E: motum + igitur erit <lb></lb> A in linea <expan abbr="perpẽdiculari">perpendiculari</expan> tota circuli maioris diame<lb></lb> tro, + quæ eſt AD:B verò in <expan abbr="eadẽ">eadem</expan> + linea, minoris <expan abbr="tãtùm">tantùm</expan> + <lb></lb> circuli diametro <expan abbr="motũ">motum</expan> erit, + quę eſt BE. </s> + <s>Atqui diame<lb></lb> ter AD quadrupla eſt ad BE, quia ex hypotheſi + <expan abbr="ſemi-diametrorũ">ſemi<lb></lb> + diametrorum</expan> + <expan abbr="illorũ">illorum</expan> + <expan abbr="circulorũ">circulorum</expan> proportio eſt, vt 4 + ad 1. <lb></lb> Motus igitur <expan abbr="pũcti">puncti</expan> in linea A + <expan abbr="perpẽdiculari">perpendiculari</expan> ad <expan abbr="motũ">motum</expan> + <lb></lb> + <expan abbr="pũcti">puncti</expan> B quadruplus erit: <expan abbr="Idẽ">Idem</expan> dicetur ſi in data aliqua <lb></lb> + alia ratione ſecta ſit linea AB. <expan abbr="Demõſtratũ">Demonſtratum</expan> igitur + eſt <lb></lb> quomodo fieri poſſit vt rectæ lineę + extrema <expan abbr="ſecũdũ">ſecundum</expan> + <lb></lb> + <expan abbr="datã">datam</expan> + <expan abbr="rationẽ">rationem</expan> moueantur. </s> + <s>Si igitur illis extremis duæ <lb></lb> vires applicentur, <expan abbr="mouebũtur">mouebuntur</expan> + <expan abbr="eodẽ">eodem</expan> ipſo motu: ergo <lb></lb> + ſecundum datam vel propoſitam rationem. </s> + <s>Quod <lb></lb> aſſumptum erat. </s> + </p> + + + <pb xlink:href="044/01/036.jpg" pagenum="28"></pb> + <p type="main"> + + <s>Quod autem in vecte demonſtratum eſt, illud e<lb></lb> tiam in + reliquis mediis demonſtrandum erit, etſi <lb></lb> lemmati + ſatisfactum eſt, dum in vno exemplo id <lb></lb> probatum eſt. </s> + <s>Ante igitur ſecundum lemma de<lb></lb> monſtrabimus. </s> + </p> + <p type="head"> + + <s>LEMMA II.</s> + </p> + <p type="main"> + + <s>Proportionem proximè maiorem vel minorem, <lb></lb> quàm ſit + datæ vis ad datum pondus proportio, de<lb></lb> terminare vis + cuiuslibet quantitas ex motu ciere po|| <lb></lb> teſt, metienda + eſt. </s> + <s>Motum autem ciere poteſt vel in <lb></lb> ſubiecto ſuo, vel in + alieno. </s> + <s>Vis autem menſura non <lb></lb> ſumitur ex eo motu quem in + ſubiecto ſuo ciere po<lb></lb> teſt, eo quòd licet + vis quælibet certum motum ha<lb></lb> beat & <expan abbr="determinatũ">determinatum</expan> quo ſubiectum + ſuum mouet, <lb></lb> illius tamen quantitas, vt ſuprà diximus, + nondum <lb></lb> demonſtrata eſt. </s> + <s>Supereſt igitur vt vires motu illo <lb></lb> metiamur, quem in alieno + ſubiecto ciere poſſunt, vt <lb></lb> id fiat, quærenda + nobis ſunt ſubiecta quæ in homi<lb></lb> num + poteſtate ſint, & cum vi qualibet committi <lb></lb> + poſſint. </s> + <s>Omnium autem mobilium ſubiectorum, <lb></lb> maximè in hominum + poteſtate ſunt grauia: leuia e<lb></lb> nim coercere vix + poſſumus. </s> + <s>Ideo vires illis metiri <lb></lb> ſolemus, ſed grauibus quæ vel + vel figuris ſuis & com<lb></lb> page vel vaſe aliquo coercentur. </s> + <s>Itaque vſus homi<lb></lb> num certas quaſdam ponderum + menſuras ſibi ſta <lb></lb> tui, ponderibus ſcilicet + quibuſdam certa <expan abbr="quãtitate">quantitate</expan> + <pb xlink:href="044/01/037.jpg" pagenum="29"></pb>conſtantibus inditis + nominibus, vt eſſent <expan abbr="ponderũ">ponderum</expan> + <lb></lb> omnium communes menſuræ, vt ſunt libræ, vncia, <lb></lb> + <expan abbr="drachmę">drachmæ</expan>, &c. </s> + <s>quas famoſas menſuras <expan abbr="vocãt">vocant</expan>. </s> + <s>Quem<lb></lb> admodum igitur numeros numeris, ſic pondera <expan abbr="põ-deribus">pon<lb></lb> deribus</expan> metimur. </s> + <s><expan abbr="Tãtum">Tantum</expan> enim pondus eſſe dicitur, + <lb></lb> quot libras vncias drachmas æqualiter mouere po<lb></lb> + teſt dempto momento. </s> + <s>Nec tantum pondera hoc <lb></lb> modo metimur, ſed etiam alias omnes vires + motum <lb></lb> ad locum cientes. </s> + <s>Quot enim libras vir aut aliud a<lb></lb> nimal vel <expan abbr="vẽtus">ventus</expan> aut ignis, aut aliqua alia vis dempto + <lb></lb> momento mouere poterit, tot libris illam æqualem <lb></lb> + eſſe dicemus. </s> + <s>Si igitur data vel propoſita vis metien<lb></lb> da ſit, + ſiquidem naturalis ſit, quoniam docuimus <lb></lb> vim naturalem per + totum ſubiectum diffuſam eſſe <lb></lb> in rebus + homogeneis: id eſt, vt quæ eſt proportio <lb></lb> molis ad + molem, eadem ſit ponderis ad pondus: ſu<lb></lb> memus partem + aliquam illi homogeneam, aut ex i<lb></lb> pſo ſubiecto, aut + ex alio ipſi homogeneo, eámque <lb></lb> famoſa aliqua + menſura metiemur, vtramque ſcili<lb></lb> cet committendo + & obſeruando, quem motum al<lb></lb> tera in altera ciere + poſſit, vbi enim æquilibrium fa<lb></lb> cient motibus + extremorum, quibus affixæ fuerint, <lb></lb> proportionales erunt per 1 + theorema: motus autem <lb></lb> illi linearum dimenſione quam Geometria + docet, <lb></lb> noti erunt, & eorum proportio, nota igitur erit & <lb></lb> + virium proportio. </s> + <s>Atqui menſuræ famoſæ nota, per <pb xlink:href="044/01/038.jpg" pagenum="30"></pb>ſe eſt quantitas: + duorum autem proportione cogni<lb></lb> ta & alterius quantitate, + ſtatim & reliqui quantitas <lb></lb> innoteſcit per + ſecundam da. Eu. nota igitur vis + <expan abbr="quã-titate">quan<lb></lb> titate</expan> quæ + parti illi ineſt, noſcetur & vis <expan abbr="quãtitas">quantitas</expan> + <expan abbr="quę">quæ</expan><lb></lb> toti inerit: quæ enim + erit ratio molis ſubiecti vis da<lb></lb> tæ ad molem + particulę ſumptę, eadem erit vis totius <lb></lb> ad vim + partis. </s> + + + + + + <s>Hîc igitur erunt quatuor proportio<lb></lb> nalia, ſcilicet vt + moles ad molem: ſic vis ad vim ex <lb></lb> quibus tria nota erunt: moles + enim metiri Geome<lb></lb> tria nos docuit, præterea vis partis, vt + demonſtraui<lb></lb> mus, nota eſt, ergo & vis totius + quantitas per deci<lb></lb> mam ſeptimam ſeptimi Elem. Eucl. + </s> + </p> + <p type="main"> + + <s>Si verò ſubiectum non ſit homogeneum, vi ta<lb></lb> men + naturali ſit præditum, ſi quidem data vis quam <lb></lb> metiri + volumus, ea ſitque motum ciere volumus, <lb></lb> tum conſiderabimus + quæ pars in illa, vis minimum <lb></lb> habeat: & ex ea totam + ipſius molem æſtimabimus: <lb></lb> ſi verò + ſit vis mouenda, ſtatuemus quaſi tota ſit + ho<lb></lb> mogenea ipſius parti, quæ vis plurimum in ſe + habe<lb></lb> bit, & ex ea totam ipſius molem + æſtimabimus. </s> + <s>Ita<lb></lb> que tum illius quanta minima, tum huius quanta <lb></lb> maxima + eſſe poteſt, vis quantitas nobis nota erit per <lb></lb> + proximè demonſtratam rationem. </s> + <s>Notis autem vi<lb></lb> rium quantitatibus, nota erit & earum + proportio: <lb></lb> ergo & proportio ipſa proximè maior vel + mi<lb></lb> nor: addita enim vel detracta ipſius denomi- <pb xlink:href="044/01/039.jpg" pagenum="31"></pb>nationi, vnitate erit + proximè maior vel mi<lb></lb> nor. </s> + </p> + <p type="main"> + + <s>Si verò vis data non ſit naturalis, <expan abbr="voluntariã">voluntariam</expan> qui<lb></lb> dem ita æſtimare + poſſumus, qualis vt plurimum <lb></lb> eſt, & ſi + quidem ea ſit, qua mouere volumus, eam <lb></lb> ſtatuemus, quanta + minima in eiuſdem generis ſub<lb></lb> iecto eſſe + poteſt, vt ſi vim hominis quinquaginta li<lb></lb> bris + æqualem ponamus, vim equi centum: ſi verò <lb></lb> ea ſit + quam mouere volumus, ſtatuemus eam <expan abbr="quã-ta">quan<lb></lb> ta</expan> maxima eſſe poteſt, vt vim + hominis 300 libra<lb></lb> rum, vim equi 500 librarum, & ſic + dé cæteris: ita vt <lb></lb> nullum ſit dubium quin illa minor + ſit, quam ſtatue<lb></lb> rimus, hæc verò maior. + </s> + </p> + <p type="main"> + + <s>Vis verò fortuitæ quantitatem nulla certa con<lb></lb> iectura + aſſequi poſſumus: ita vt quaſcunque + ma<lb></lb> chinas ei aptemus, modò moueat, modò non + mo<lb></lb> ueat, neque ad noſtrum <expan abbr="inſtitutũ">inſtitutum</expan> magnoperè + per<lb></lb> tinet illa inquiſitio: cum fortuitorum nulla ſit + diſci<lb></lb> plina. </s> + </p> + <p type="main"> + + <s>His igitur modis virium duarum datarum pro<lb></lb> portio proximè + maior nota fiet, quod in ſecundo <lb></lb> lemmate demonſtrandum + ſumpſeramus. </s> + </p> + <p type="main"> + + <s>Iam redeamus ad mediorum, quibus vi<lb></lb> res annectuntur, + conſide<lb></lb> rationem. </s> + </p> + + + <pb xlink:href="044/01/040.jpg" pagenum="32"></pb> + <figure id="id.044.01.040.1.jpg" xlink:href="044/01/040/1.jpg"></figure> + <p type="main"> + + <s>Docuimus quis ſit ſimplicis ve<lb></lb> ctis effectus, + ſimplex autem ve<lb></lb> ctis ſemicirculi conuerſione + ſuam <lb></lb> operationem abſouit, ita vt ſi <lb></lb> + vlteriùs F moueatur in in alio ſe<lb></lb> micirculo motus + prioribus contra<lb></lb> rios cieat: vt exempli gratia, ſit <lb></lb> + vectis AB, cuius hypomochium <lb></lb> ſit C, dum A punctum + deſcribet ſemicirculum AFD, <lb></lb> motus ille deorſum erit: + interea verò B punctum <lb></lb> deſcribet ſemicirculum BGE + aſcendendo: ſi verò A <lb></lb> tranſcendat, D incipiet + aſcendere: B verò tranſcen<lb></lb> dens, E <expan abbr="deſcẽdet">deſcendet</expan>: ideo excogitata + eſt vectis ratio per<lb></lb> petua ex plurium vectium + ſucceſſione circa idem <lb></lb> hypomochlium: eſt autem + illa tum in ergatis aut ſu<lb></lb> culis, tum in duorum tympanorum + homocentrico<lb></lb> rum, ſeu eadem axe transfixorum in planis + parallelis <lb></lb> aptatione, quorum ſemidiametri ſint in eadem + pro<lb></lb> portione quæ in vecte ad propoſitum motum + cien<lb></lb> dum neceſſaria eſt: centrum verò + eorum ſeu axis fi<lb></lb> xa ſit, ac ita vires aptentur, vt + maior minorem, mi<lb></lb> nor verò maiorem tympanum impellat. </s> + <s>Quemad<lb></lb> modum autem horum tympanorum homocentri<lb></lb> corum + opera vectis perpetui ratio <expan abbr="inuẽta">inuenta</expan> + eſt, ita eo<lb></lb> rum multiplicatione motus, & mouentis & + <expan abbr="mouẽ-di">mouen<lb></lb> di</expan>, proportio in + infinitum augeri poteſt. </s> + <s>Cuius rei <pb xlink:href="044/01/041.jpg" pagenum="33"></pb>maximus eſt + vſus: nec enim materia ad vectem, cu<lb></lb> ius longitudo + ſtadij vnius requireretur, idonea in<lb></lb> ueniri + poſſet: plurium autem tympanorum propor<lb></lb> tionalium + aptatione fiet machina tractabilis, cuius <lb></lb> vis maior erit quàm + vectis, cuius longitudo ſtadij v<lb></lb> nius eſſet. </s> + <s>Si enim duo tympani homocentrici a<lb></lb> ptentur, quorum proportio + ſit alterius ad alterum, <lb></lb> decupla, vis quæ libram vnam + æquabit, vim decem <lb></lb> libris æqualem dempto momento mouebit, + ſi con<lb></lb> gruè illis tympanis aptentur: ſi + verò adhuc duo alij <lb></lb> tympani fiant, quorum alterius ad alterum + decupla <lb></lb> ſit proportio, ac minor <expan abbr="illorũ">illorum</expan> ita aptetur, vt <expan abbr="maiorẽ">maiorem</expan> + <lb></lb> ex prioribus moueat, appendatur deinde minori ex <lb></lb> prioribus + tympanis vis centum libris æqualis, maio<lb></lb> ri verò ex + poſterioribus vis vni libræ æqualis, tum <lb></lb> hæc + illam mouebit, & ſic in infinitum motus extre<lb></lb> morum + proportio multiplicari poteſt. </s> + <s>Flexibili ve<lb></lb> rò medio quies ita aptari poteſt, vt duo + eius extre<lb></lb> ma diuerſis motibus moueantur, & quidem + ſecun<lb></lb> dum datam rationem. </s> + <s>Exemplum habemus in orga <lb></lb> nis polyſpactis, ſeu trochleis, in + quibus altero funis <lb></lb> extremo immobili manente, reliquum funis circa + <lb></lb> plures trochleas conuoluitur, quarum aliæ centris <lb></lb> + immobilibus fixæ ſunt, aliæ verò ipſis + contrariæ cen <lb></lb> tris mobilibus. </s> + <s>Atque ita circa illas conuoluitur fu<lb></lb> nis, vt inter eas ſit + ſpatium tantum, quanta eſt linea, <pb xlink:href="044/01/042.jpg" pagenum="34"></pb>per quam motum ciere volumus. </s> + <s>Quoties igitur fu<lb></lb> nis extremum quod moueri poteſt, trahitur, + <expan abbr="ſingulę">ſingulæ</expan><lb></lb> + reuolutiones æqualiter minuuntur, eáque ratione, <lb></lb> + <expan abbr="diſtãtia">diſtantia</expan> quæ eſt + inter trochleas contrarias minuitur, <lb></lb> ita ſcilicet vt quot + ſunt reuolutiones, in tot partes <lb></lb> diſtributum ſit + motus ſpatium. </s> + <s>Quot igitur reuolu<lb></lb> tiones erunt, totuplex erit motus extremitatis funis + <lb></lb> ad motum trochlearum mobilium verſus fixas. </s> + <s>Si i<lb></lb> gitur motus iſtius extrema conſtituantur, alterum <lb></lb> + <expan abbr="quidẽ">quidem</expan> funis illa extremitas quæ + mouetur, <expan abbr="alterũ">alterum</expan> ve<lb></lb> rò + terminus ſpatij illius, quod eſt à trochleis fixis ad <lb></lb> + mobiles contrarias: quæ erit proportio numeri con<lb></lb> uolutionum + funis ad vnitatem, eadem erit motus, <lb></lb> quo funis extremum mouebitur ad + motum quo al<lb></lb> terum extremum mouebitur: poteſt autem in + infi<lb></lb> nitum augeri conuolutionum numerus, ergo & mo<lb></lb> tus + illius proportio. </s> + </p> + <p type="main"> + + <s>Angulus autem ad motum ciendum ita ad<lb></lb> hibetur. </s> + <s>Diximus motus menſuram in nutus li<lb></lb> nea ſumi, quantum + igitur vis aliqua per eam verſus <lb></lb> locum naturalem mouetur, tanta + eſt, quantum verò <lb></lb> per eam à loco naturali reuellitur, + tanta eſt eius reſi<lb></lb> ſtentia. </s> + <s>Quod verò per lineam à loco naturali æquè <lb></lb> + diſtantem (id eſt, per eam quæ nutus lineas ſecat ad + <lb></lb> angulos rectos) mouetur, illud mouenti non reſi<lb></lb> + ſtit, omnium autem linearum inter illas intercepta- <pb xlink:href="044/01/043.jpg" pagenum="35"></pb>rum, ac cum illis in earum + interſectionis puncto <expan abbr="cõ-currentium">con<lb></lb> + currentium</expan>, quæ obliquè nutum verſus, aut contra + <lb></lb> nutum ferri dicuntur, quo propiùs quælibet ad nu<lb></lb> + tus lineam accedit, per illam rei motæ vis aut <expan abbr="reſiſtẽ-tia">reſiſten<lb></lb> + tia</expan> maior eſt: quò verò propiùs ad lineam + à loco na<lb></lb> turali æqui diſtantem accedit, eò + minor eſt. </s> + <s><expan abbr="Omniũ">Omnium</expan> + <lb></lb> autem maxima eſt in nutus linea, æquidiſtans + verò <lb></lb> à loco naturali motui per lineam nutus omnino op<lb></lb> + poſita eſt, obliquæ verò non ita quia ſecundum + illas <lb></lb> + <expan abbr="eodẽ">eodem</expan> ſpatio delata vis propiùs + ad <expan abbr="locũ">locum</expan> + <expan abbr="naturalẽ">naturalem</expan> acce<lb></lb> dit, aut ab + eo recedit, quàm eſſet, cùm moueri cœpit. </s> + </p> + <figure id="id.044.01.043.1.jpg" xlink:href="044/01/043/1.jpg"></figure> + <p type="main"> + <s>Sit exempli gratia AB linea <lb></lb> nutus, vis alicuius, puta ponde<lb></lb> + ris, ſitque A ſurſum & contra nu<lb></lb> tum: B verò + deorſum & nutum <lb></lb> verſus, deſcribatúrque + circulus, <lb></lb> cuius AB ſit diameter, quàm CD, <lb></lb> alia + diameter ſecet ad angulos <lb></lb> rectos in centro E: omnes igitur <lb></lb> + lineæ à centro E ad circunferentiam circuli ductæ, <lb></lb> + quæ <expan abbr="cadẽt">cadent</expan> intra <expan abbr="ſemicirculũ">ſemicirculum</expan> CAD, contra + <expan abbr="nutũ">nutum</expan> + <expan abbr="aſcẽ-dere">aſcen<lb></lb> dere</expan> + dicentur, quatenus circunferentiam ſpectant: <lb></lb> quatenus verò + centrum ſpectant, deſcendere dicen<lb></lb> tur: <expan abbr="cõtra">contra</expan> verò omnes in ſemicirculo + CBD à centro <lb></lb> ad circunferentiam ductæ deſcendere + circunferen<lb></lb> tiam verſus, & centrum verſus + aſcendere dicentur: <pb xlink:href="044/01/044.jpg" pagenum="36"></pb>illæ igitur erunt, quæ obliquè nutum verſus aut + con<lb></lb> tra nutum ferri <expan abbr="dicũtur">dicuntur</expan>: linea + verò CED, neque <expan abbr="aſcẽ-det">aſcen<lb></lb> det</expan>, neque deſcendet: lineæ + verò in ipſam ad angu<lb></lb> los rectos incidentes nutus + lineæ erunt, quoniam li<lb></lb> neæ AB parallelæ erunt: + ſi igitur à centro E <expan abbr="ducãtur">ducantur</expan> + <lb></lb> lineæ ad circunferentiam inter A & D, puta EF, EG, <lb></lb> EH, + quarum EF ſit proxima lineæ AB:EH verò pro<lb></lb> xima + lineæ CD, ac per illas moueantur contra <expan abbr="nutũ">nutum</expan> + <lb></lb> tres vires æquales eodem tempore, ita vt prima per <lb></lb> lineam + EF perueniat ad punctum F: ſecunda verò <lb></lb> per EG perueniat ad + G, tertia per EH perueniat ad H, <lb></lb> dico vis motæ per EF maiorem + fore reſiſtentiam, <lb></lb> quàm illius quæ per EG aut + EH, mouebitur & illius <lb></lb> quæ per EG mouebitur, maiorem + quàm eius quæ <lb></lb> per EH mouebitur. </s> + <s>Ducantur enim à punctis FGH <lb></lb> in lineam ED perpendiculares FK, GL, + HM, illæ <expan abbr="erũt">erunt</expan> + <lb></lb> nutus lineæ: quanta igitur erit FK, tantum vis prima <lb></lb> mota + cenſebitur, quanta verò GH, tantum vis ſecun<lb></lb> da: + quanta verò HM, tantum vis tertia mota cenſebi<lb></lb> tur: at + qui quæ eſt ratio motus ad motum in viribus <lb></lb> æqualibus, + per quartam <expan abbr="cõcluſionem">concluſionem</expan> + huius tracta<lb></lb> tus, eadem eſt reſiſtentiæ ad + reſiſtentiam: quæ igitur <lb></lb> erit ratio linearum illarum + perpendicularium, inter <lb></lb> ſe eadem erit & <expan abbr="reſiſtẽtiarum">reſiſtentiarum</expan> virium per lineas ob<lb></lb> + liquas motarum, à quibus illæ perpendiculares du<lb></lb> + ctæ ſunt, atqui quo lineæ illæ perpendiculares pro- <pb xlink:href="044/01/045.jpg" pagenum="37"></pb>piùs ad AB circuli + diametrum accedunt, eò ſunt ma<lb></lb> iores per decimam quartam + tertij Elem. Eu. ergo & <lb></lb> vires per eas lineas delatę à + <expan abbr="quarũ">quarum</expan> extremitatibus du<lb></lb> + <expan abbr="cẽtur">centur</expan>, maiores <expan abbr="reſiſtẽtias">reſiſtentias</expan> + <expan abbr="habebũt">habebunt</expan>: at qui quò magis <lb></lb> ad + AB, accedunt eo magis ab ED recedunt: quò igi<lb></lb> tur magis ad + ED accedent, eò minores erunt <expan abbr="reſiſtẽ-tiæ">reſiſten<lb></lb> tiæ</expan>, hinc ſequitur + tanquam corollarium. </s> + <s>Si duæ vi<lb></lb> res perſe æquales triangulo rectangulo + aptentur, cu<lb></lb> ius latus alterum, angulum rectum <expan abbr="conſtituẽtium">conſtituentium</expan>: <lb></lb> + ſit earum nutus linea, ac per illud vis altera mouea<lb></lb> tur, + altera verò per latus angulo recto oppoſitum, <lb></lb> quæ erit + ratio huius lateris ad illud, eadem erit reſi<lb></lb> ſtentia + vis illius ad vis huius <expan abbr="reſiſtẽtiam">reſiſtentiam</expan>. </s> + <s>Atqui duo<lb></lb> rum illorum laterum proportio in infinitum + auge<lb></lb> ri vel minui poteſt, ergo & + reſiſtentia virium illis ap<lb></lb> plicatarum. </s> + <s>Hoc igitur modo poſſumus vti angulo <lb></lb> ſeu triangulo ad + motus ciendos, nempe eo immoto <lb></lb> vires per eius latera mouendo. </s> + </p> + <p type="main"> + + <s>Sed & alia ratione eo vti poſſumus, ipſum + ſcilicet <lb></lb> triangulum mouendo, qui tunc cuneus dicitur. </s> + <s>Vt <lb></lb> autem eo hac ratione vtamur, vires ita diſponere o<lb></lb> + portet, vt altera illarum vni ex lateribus angulum <lb></lb> rectum + conſtituentibus incumbat, altera verò lateri <lb></lb> ipſum + ſubtendenti. </s> + <s>Illa enim tantùm mouebitur, <lb></lb> quantum latus cui altera incumbit. </s> + <s>Sit exempli gra <pb xlink:href="044/01/046.jpg" pagenum="38"></pb><figure id="id.044.01.046.1.jpg" xlink:href="044/01/046/1.jpg"></figure><lb></lb> tia + triangulus ABC, cuius angulus B <lb></lb> rectus ſit, latus verò + illum ſubtendens <lb></lb> ſit AC, incumbátque vis D lateri AB, + <lb></lb> vis verò E lateri AC, ſitque vis D nutus <lb></lb> linea BC, + vis verò E nutus ſit linea AB, erigatúrque à <lb></lb> + puncto C linea CF <expan abbr="perpẽdicularis">perpendicularis</expan> ad BC æqualis AB, <lb></lb> à qua vis E + <expan abbr="nõ">non</expan> diſcedat. </s> + <s>Si triangulum illud in plano <lb></lb> fixo moueatur, donec AB perueniat ad CF, + mota e<lb></lb> rit vis D nutu ſuo tantùm, quanta eſt + linea BC, vis ve<lb></lb> rò E tantum, quanta eſt linea AB. + </s> + <s>Cùm autem poſſit <lb></lb> in infinitum augeri & minui, + laterum illorum pro<lb></lb> portio, poſſunt etiam duorum + illorum <expan abbr="extremorũ">extremorum</expan> + <lb></lb> motus in data ratione conſtitui. </s> + <s>Quanta enim erit <lb></lb> BC ad AB, tantus erit motus vis D ad motum vis E: + <lb></lb> ergo & in hoc medio primum lemma noſtrum de<lb></lb> + monſtratum eſt. </s> + </p> + <p type="main"> + + <s>In hoc autem medij genere hoc diuerſum ab <lb></lb> aliis mediis accidit, + quòd non tam facilè vtrin<lb></lb> que motus eo cietur, ac in + illis, in quibus ſi virium <lb></lb> proportio momento vel ſuperet, + vel minor ſit pro<lb></lb> portione motus <expan abbr="extremorũ">extremorum</expan> medij, tum motus hinc <lb></lb> + vel inde cietur. </s> + <s>Atqui in hoc propter ſuperficie<lb></lb> rum contactum, quarum pori + vel aſperitates ſe<lb></lb> ſe mutuò + ſubingrediuntur, & ita inuicem adhæ<lb></lb> rent, fit vt + ægriùs cieatur motus, faciliùs ve- <pb xlink:href="044/01/047.jpg" pagenum="39"></pb>rò mouebuntur, ſi + illæ ſuperficies leues fue<lb></lb> rint, vt ſi pinguibus + inungantur, vel ex materia <lb></lb> leui conſtent. </s> + <s>Ideò enim reliqua faciliùs mo<lb></lb> uentur, quòd circa + puncta veluti quædam mo<lb></lb> ueantur. </s> + </p> + <p type="main"> + + <s>Simplicis autem trianguli rectilinei aut cu<lb></lb> nei in diuturnis + motibus ciendis rarus eſt vſus, <lb></lb> tum ob illud quod + notauimus incommodum ex <lb></lb> vitio materiæ, quo fit vt in eo + quaſſatione opus <lb></lb> ſit, tum etiam quòd breui eius + operatio termine<lb></lb> tur. </s> + <s>Ideo illo vtimur aut cùm ſolutionem con<lb></lb> tinui molimur, + quæ breui tempore fit, vel <lb></lb> cùm aliquid diſtendere aut + aliquid figere volu<lb></lb> mus. </s> + </p> + <p type="main"> + + <s>Anguli verò curui linei magnus eſt vſus, præ<lb></lb> + ſertim helicis cylindricæ. </s> + <s>Nihil enim aliud eſt he<lb></lb> lix quàm triangulus curuus: + ſi enim alteram ex <lb></lb> lineis rectis angulum conſtituentibus, + cylindri <lb></lb> baſi ipſi lineæ æquali obuolueris, + reliquam ve<lb></lb> rò ſeruato eodem quem conſtituunt + angulo, ſu<lb></lb> per cylindri ſuperficiem curuaueris, + habebis he<lb></lb> licem cylindricam, quam ſi iterum ſeruato + eo<lb></lb> dem angulo in rectum extendas, habebis trian<lb></lb> gulum + rectilineum. </s> + <s>Hæc autem helix commo<lb></lb> diſſima eſt, tum + quòd in parua mole triangulum <pb xlink:href="044/01/048.jpg" pagenum="40"></pb>longiſſimum obuolutum contineat tum quod + par<lb></lb> tes eius omnes ſibi inuicem <expan abbr="cõgruant">congruant</expan>: omnes enim <lb></lb> partes helicis cylindricæ, aut + circa eundem, aut circa <lb></lb> æquales cylindros deſcriptæ, + ſeruato eodem angulo <lb></lb> ſibi inuicem + ſuppoſitæ congruunt. </s> + <s>Quo fit, vt ſi ca<lb></lb> ui cylindri interiori ſuperficiei + impreſſa ſit helix, a<lb></lb> lia verò cylindri + ſuperficiei connexę ipſi cauę æqua<lb></lb> + li, ſeruato eodem qui in illo eſt, angulo, ſibi + inui<lb></lb> cem & omnes vnius omnibus alterius partibus con<lb></lb> + gruant. </s> + </p> + <p type="main"> + + <s>Huius autem medij cùm plures ſint partes, con<lb></lb> ſtat enim + duabus ſuperficiebus, pluribus etiam mo<lb></lb> dis variari poteſt. </s> + <s>In ſumma autem eius affectus hic <lb></lb> eſt, vt cylindri + baſi in ſuperficie immobili circum a<lb></lb> xem + conuerſa, vis mouenda helicem premat: dum <lb></lb> enim cylindrus circum + axem conuertetur, vis mo<lb></lb> uenda qualibet reuolutione tantùm + ſecundum cy<lb></lb> lindri longitudinem mouebitur, quanta eſt + diſtan<lb></lb> tia duarum helicis ſpirarum. </s> + <s>Quæ igitur erit propor<lb></lb> tio circunferentiæ circuli baſim + cylindri <expan abbr="conſtituẽ-tis">conſtituen<lb></lb> tis</expan> ad illam <expan abbr="diſtãtiam">diſtantiam</expan>, eadem erit motus + orbicularis <lb></lb> cuiuſlibet puncti in cylindri ſuperficie + ſignati ad mo<lb></lb> tum rectum vis helicem prementis. </s> + </p> + <p type="main"> + + <s>Illud igitur medium ex duobus compoſitum eſt, <lb></lb> recto + ſcilicet & curuo: ita igitur vires illi aptandæ <lb></lb> + ſunt, vt eius quam mouere volumus, nutus linea ſit <pb xlink:href="044/01/049.jpg" pagenum="41"></pb>cylindri longitudo: illa + verò qua mouere volumus, <lb></lb> in orbem moueatur, aut ſi eius + nutus linea recta ſit, <lb></lb> circa cylindrum fluens illum moueat + contingendo. </s> + </p> + <p type="main"> + + <s>Et hæc quidem de mediis in ſuo genere <expan abbr="conſiſtẽ-tibus">conſiſten<lb></lb> tibus</expan>. </s> + <s>Poſſunt autem fieri eorum Syzygiæ vectis + ſci<lb></lb> licet cum trochleis, helicis cum vecte, aut cum + tro<lb></lb> chleis, aut <expan abbr="omniũ">omnium</expan> + ſimul. </s> + <s>Vectis cum trochleis ſi fu<lb></lb> nis illud extremum, quod mouetur + in trochleis, er<lb></lb> gatis aut ſuculis inuoluatur: helix cum + vecte <expan abbr="coniũ-getur">coniun<lb></lb> getur</expan>, + ſi tympanus circum axem moueatur, ſitque i<lb></lb> ta + denticulatus, vt dentes ipſius helicem in cylindro <lb></lb> excauatam + ingrediantur, quam machinam helicem <lb></lb> perpetuam vulgò vocant, + eò quòd eius conuerſio <lb></lb> perpetua eſſe + poſſit, cum helicis ſimplicis operatio <lb></lb> non excedat + ipſius longitudinem. </s> + <s>Quorum <expan abbr="omniũ">omnium</expan> + <lb></lb> tum inter ſe, tum ad ſubiecta mouenda accommo<lb></lb> + datio adeò varia eſt, vt ſcripto <expan abbr="cõprehẽdi">comprehendi</expan> vix poſſit. + <lb></lb> + </s> + <s>Ex his autem quæ dicta ſunt, mediocris ingenij me<lb></lb> + chanicus poterit ea prout ipſi neceſſe erit aptare. </s> + <s>In <lb></lb> machinis autem omnibus hæc cautio <expan abbr="adhibẽda">adhibenda</expan> eſt, <lb></lb> vt earum ſtructura firma + ſit, præſertim verò vbi cir<lb></lb> cum axes + aliquas fit motus, deinde vt vincula, qui<lb></lb> bus illis vires <expan abbr="affingũtur">affinguntur</expan>, valida ſint, illarum + enim o<lb></lb> mnium vis in ſuo ſtatu manendi vtrique vi + aptandæ <lb></lb> æqualis eſſe debet: aget enim in illas + vis <expan abbr="vtraq;">vtraque</expan>, quòd <lb></lb> ſi medium + firmum non ſit, motus in ipſa machina <pb xlink:href="044/01/050.jpg" pagenum="42"></pb>ciebitur. </s> + <s>Itaque diſſoluetur, ideò manca videtur pe<lb></lb> titio + illa Archimedis in hoc problemate <foreign lang="grc">δὸς ποῦ στῶ τὰν + <lb></lb> γὰν κινῶ</foreign>, quòd locum <expan abbr="tãtùm">tantùm</expan> in quo conſiſtat, + ſibi da<lb></lb> ri poſtulet, cùm pręterea vincula + quibus terra à loco <lb></lb> ſuo naturali remota + ſuſtineri poſſet, petere debue<lb></lb> rit: id + <expan abbr="autẽ">autem</expan> vt & ea quæ de motu in + infinitum augen<lb></lb> do vel minuendo diximus, ita intelligenda + ſunt, vt <lb></lb> ſciamus infinita hominum poteſtati, + quacunque ar<lb></lb> te iuuetur, non ſubeſſe: quamuis + enim Geometri<lb></lb> ca conſideratio in infinitum ſeſe + <expan abbr="extẽdat">extendat</expan>, ſunt ta<lb></lb> + men certi fines, vltra quos natura rerum nos pro<lb></lb> gredi non + patitur: ſunt præterea vitia materiæ, <lb></lb> quæ + Geometra non conſiderat, illa tamen non <lb></lb> obſtant quò + minùs id quod proponitur, <expan abbr="verũ">verum</expan> + ſit in <lb></lb> intellectu. </s> + <s>An verò id quod proponitur tale ſit, vt in <lb></lb> opus educi + poſſit, conſiderabit Geometer habita ra<lb></lb> tione + circunſtantiarum, præſertim verò temporis, <lb></lb> quod + ipſi præſcribetur, & ſumptuum quos facere <lb></lb> + poterit is qui aliquid faciendum proponet, quæ ſi <lb></lb> + abundè ſuppetant, nihil non fieri poterit. </s> + </p> + <p type="main"> + + <s>Vt igitur hunc tractatulum concludamus, ac ve<lb></lb> lut in ſummam + contrahamus: In motibus ciendis <lb></lb> tria ſunt conſideranda. </s> + <s>Vis qua motum ciere volu<lb></lb> mus, vis quam mouere volumus, & motum + quo <lb></lb> mouere volumus: duo enim quælibet ex illis ter<lb></lb> + tium determinant. </s> + <s>Si enim vi parua vim magnam <pb xlink:href="044/01/051.jpg" pagenum="43"></pb>mouere volumus, id nonniſi paruo motu facere <lb></lb> + poſſumus: ſi verò vim aliquam magno motu + mo<lb></lb> uere velimus, vi magna mouente ad id opus eſt. </s> + <s>Si <lb></lb> vi parua magnum motum ciere volumus, mini<lb></lb> mam vim + mouendam eſſe oportet: vt puta, ſi libra <lb></lb> vna centum + libras mouere velimus, oportet motum <lb></lb> illius, motu huius <expan abbr="cẽtuplo">centuplo</expan> maiorem eſſe. </s> + <s>Si verò ve<lb></lb> limus libra vna aliam vim ita mouere, vt ea + cen<lb></lb> tuplo citiùs moueatur, quàm libræ illius + pondus, il<lb></lb> lam centeſimam tantùm libræ vnius + partem eſſe ne<lb></lb> ceſſe eſt: ſi + verò libram vnam ita mouere velimus, vt <lb></lb> centuplo citius moueatur, + quàm vis quæ illam mo<lb></lb> uebit, vi centum libris maiore ad + id opus erit. </s> + <s><expan abbr="Neq;">Neque</expan> + <lb></lb> patitur natura ſibi in his vim fieri: ſi enim + eiuſmodi <lb></lb> proportio aliquo modo infringi poſſet, + ſtatim da<lb></lb> retur <foreign lang="grc">αὐτώμα + ἐνδέλεχες</foreign>, vel vt <expan abbr="vocãt">vocant</expan>, + motus perpetuus <lb></lb> in perpetua materia. </s> + </p> + <p type="main"> + + <s>Ex his igitur fundamentis inuentæ ſunt machinæ <lb></lb> omnes, + quotquot antehac ſunt excogitatę, & quot<lb></lb> quot + poſthac excogitari poterunt, ad ea referentur. </s> + </p> + <p type="main"> + + <s>Itaque ſi propoſitæ cuiuſcunque machinæ + effe<lb></lb> ctum ſcire velimus, conſideranda ſunt duo + eius <lb></lb> extrema, quibus vires applicantur: quæ enim erit <lb></lb> ratio + motus vnius ex illis extremis, ad motum <lb></lb> alterius eadem erit & + virium, quæ illis extre<lb></lb> mis ad motum ciendum applicari + poterunt, <pb xlink:href="044/01/052.jpg" pagenum="44"></pb>addito aut dempto + momento, vt ſi dum <expan abbr="alterũ">alterum</expan> + ma<lb></lb> chinæ extremum palmo vno mouetur, alterum cen<lb></lb> tum + palmis moueatur vis quælibet huic annexa, al<lb></lb> teram alteri + annexam centuplam momento minùs, <lb></lb> mouebit: ſed motu + centuplo: mouebit autem & re<lb></lb> liquas omnes vires, quæ + infra centuplam proportio<lb></lb> nem ad eam habebunt: ſi verò + non vim centuplam <lb></lb> mouere, ſed in data vi motum centuplum ciere + veli<lb></lb> mus, eam quidem in extremo quod centuplo citiùs <lb></lb> + mouetur, locare oportebit, alteram verò alteri extre<lb></lb> mo centuplo + maiorem adhibere neceſſe erit. </s> + </p> + <p type="main"> + + <s>Hinc oriuntur tria problemata, videlicet data vi <lb></lb> datum pondus mouere, + quod iam <expan abbr="demõſtrauimus">demonſtrauimus</expan>: <lb></lb> item data vi datum motum ciere, quod ex + <expan abbr="præcedẽ-tis">præceden<lb></lb> + tis</expan> demonſtratione abſoluitur: tertium, datam vim, + <lb></lb> dato motu mouere, quod quidem faciliùs demon<lb></lb> + ſtratur, quàm abſoluitur. </s> + <s>Scimus enim, vt id fiat, <lb></lb> vim aliam proportionalem (vt docuimus) + requiri, <lb></lb> quo modo autem illam habere poſſimus, explicare + <lb></lb> hoc opus, hic labor eſt. </s> + <s>Non enim quemadmodum <lb></lb> organa ad motus in data proportione ciendos + ha<lb></lb> bere poſſumus: ita & vires infinitæ + magnitudinis po<lb></lb> teſtati noſtræ ſubſunt. + </s> + </p> + <p type="main"> + + <s>Quod enim ad vires in grauitatis & leuitatis gene<lb></lb> re + conſiſtentes attinet, eas vt moueant antè moueri <lb></lb> + oportet à loco vel ſitu naturali, vi <expan abbr="autẽ">autem</expan> aliqua id fie- <pb xlink:href="044/01/053.jpg" pagenum="45"></pb>ri oportet, quæ iam in promptu ſit. </s> + <s>Quæ ſi tanta eſt, <lb></lb> vt poſſit motum + propoſitum ciere, fruſtrà fiat, ſi alij <lb></lb> à + loco naturali remouendæ, qua poſtea ad motum <lb></lb> vtendum + ſit, adhibeatur. </s> + </p> + <p type="main"> + + <s>Nullum igitur ex eiuſmodi viribus commodum <lb></lb> percipi poteſt, + niſi quis in futurum ſibi proſpiciens, <lb></lb> multa à + loco vel ſitu naturali ex otio remoueat, vt <lb></lb> iis, cùm opus + erit ad motus ciendos vtatur. </s> + <s>Hac ra<lb></lb> tione quantas vires in promptuario habebit, tantos <lb></lb> + motus ciere poterit. </s> + <s>Præcipuum igitur quod ad hu<lb></lb> ius problematis + conſtructionem pertinet, eſt, vt vi<lb></lb> res quàm + maximas poteſtati noſtræ ſubiiciamus, vo<lb></lb> + luntariis aut fortuitis viribus naturales præparando, <lb></lb> aut à + natura præparatas, quæ poteſtati noſtræ + ſub<lb></lb> ſunt, accipiendo. </s> + <s>Qualia multa ſi mortales aduerte<lb></lb> rent, fieri + poſſent, vt aliàs, Deo duce, docebimus. </s> + </p> + <p type="main"> + + <s>Earum autem virium quæ in raritatis & denſita<lb></lb> tis + proportione conſiſtunt, ſubiecta plurimùm in <lb></lb> + noſtra poteſtate ſunt: multa enim ſunt naturalia + ſub <lb></lb> iecta actu, denſa potentia verò rara. </s> + <s>Si quod igitur <lb></lb> eorum potentia proxima ſit rariſſimum, + ita vt nullo <lb></lb> negotio actus ille raritatis induci poſſit, + <expan abbr="concluda-túrq;">concluda<lb></lb> + túrque</expan> loco aliquo anguſto, poſtea inducatur + ille a<lb></lb> ctus, cùm rara <expan abbr="maiorẽ">maiorem</expan> locum occupent, quum den<lb></lb> ſa, fiet vt + locus in omnem partem diſtendatur, illius <lb></lb> autem partes + minùs cohærentes, tantum impellen <pb xlink:href="044/01/054.jpg" pagenum="46"></pb>tur, quanta eſt + proportio molis rei rarefactæ ad mo<lb></lb> lem illius cùm + denſa eſſet: illa autem raritatis po<lb></lb> tentia + proxima eſt in compoſitione ſulphuris <lb></lb> & nitri: + ea igitur & ſimilibus ſubiectis, <lb></lb> in data vi datus + motus <lb></lb> cieri poteſt. </s> + </p> + <p type="head"> + + <s>FINIS.</s> + </p> + <p type="head"> + + <s>Errata quæ inter imprimendum irrepſerunt, <lb></lb> ſic + corrigito.</s> + </p> + <p type="main"> + + <s><emph type="italics"></emph>Pagina<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>linea<emph.end type="italics"></emph.end> 10, <emph type="italics"></emph>munere, lege nuere. </s> + + <s>pag.<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 12, <emph type="italics"></emph>comparationibus, lege + comparati, omni<lb></lb> bus. </s> + + <s>pag.<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>finis, lege fini. </s> + + <s>pag. eadem. lin. penult. Altero, lege Alto.</s> + + <s>pag.<emph.end type="italics"></emph.end> 20. <lb></lb> + <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 16. <emph type="italics"></emph>quum<emph.end type="italics"></emph.end> C <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> L. <emph type="italics"></emph>lege quam. </s> + + <s>pag.<emph.end type="italics"></emph.end> 25. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>nunc ſit ratio, lege + minor ſit. </s> + + <s>linea<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>Vi<lb></lb> res + maiori litera<emph.end type="italics"></emph.end> (<emph type="italics"></emph>eſt + enim ſectionis initium. </s> + + <s>lin.<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>mittant, lege + mittantur. </s> + + <s>pag.<emph.end type="italics"></emph.end> 27. <emph type="italics"></emph>li.<emph.end type="italics"></emph.end> 20. <lb></lb> + <emph type="italics"></emph>puncti in linea<emph.end type="italics"></emph.end> A, <emph type="italics"></emph>lege puncti<emph.end type="italics"></emph.end> A <emph type="italics"></emph>in linea. </s> + + <s>pag.<emph.end type="italics"></emph.end> 28. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 9. <emph type="italics"></emph>vis, lege Vis, maiori + litera<emph.end type="italics"></emph.end> (<emph type="italics"></emph>eſt enim + <lb></lb> ſectionis initium. </s> + + <s>lin. e. motu ciere, leg. motu quem ciere. </s> + + <s>lin.<emph.end type="italics"></emph.end> 23. <emph type="italics"></emph>metiri + ſolemus, lege non <lb></lb> ſolemus. </s> + + <s>lin. vlt. ſtatui, lege ſtatuit.</s> + + <s>pag.<emph.end type="italics"></emph.end> 32. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>abſouit, lege + abſoluit. </s> + + <s>lin.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>dele<emph.end type="italics"></emph.end> F. </s> + + <s><emph type="italics"></emph>lin.<emph.end type="italics"></emph.end><lb></lb> 12. <emph type="italics"></emph>&<emph.end type="italics"></emph.end> 13, <emph type="italics"></emph>pone virgulam poſt<emph.end type="italics"></emph.end> D <emph type="italics"></emph>&<emph.end type="italics"></emph.end> E.</s> + + <s> + <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 8. <emph type="italics"></emph>hypomochium, lege hypomochlium. </s> + + <s>pag.<emph.end type="italics"></emph.end> 37. <emph type="italics"></emph>li.<emph.end type="italics"></emph.end><lb></lb> 13. <emph type="italics"></emph>reſiſtentia, + lege reſiſtentiæ. </s> + + <s>pag.<emph.end type="italics"></emph.end> 39. <emph type="italics"></emph>lin.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>leui, lege læui. </s> + + <s>pag.<emph.end type="italics"></emph.end> 40. <emph type="italics"></emph>linea<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>ſuppoſitæ, + lege <lb></lb> ſuperpoſitæ</s> + + <s>li.<emph.end type="italics"></emph.end> 7. <emph type="italics"></emph>connexæ, lege + conuexæ. </s> + + <s>lin.<emph.end type="italics"></emph.end> 13. <emph type="italics"></emph>affectus, lege + effectus.<emph.end type="italics"></emph.end></s> + </p> + </chap> + </body> + <back></back> + </text> +</archimedes>