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