GVIDIVBALDI
E MARCHIONIBVS
MONTIS
MECHANICORVM
LIBER.
PISAVRI
Apud Hieronymum Concordiam.
M. D. LXXVII.
Cum Licentia Superiorum.
PRAESENTI OPERE
CONTENTA.
De Libra.
De Vecte.
De Trochlea.
De Axe in peritrochio.
De Cuneo.
De Cochlea.
AD FRANCISCVM
MARIAM II
VRBINATVM
AMPLISSIMVM DVCEM
GVIDIVBALDI
E MARCHIONIBVS
MONTIS
PRAEFATIO.
DVAE res (AMPLISSIME PRIN
CEPS) quæ ad conciliandas homi
nibus facultates, vtilitas nempè, &
nobilitas, plurimùm valere con&longs;ue
uerunt. illæ ad exornandam mecha
nicam facultatem, & eam præ om
nibus alijs appetibilem reddendam con&longs;pira&longs;&longs;e
mihi videntur: nam &longs;i nobilitatem (quod pleriq;
modò faciunt) ortu ip&longs;o metimur, occurret hinc
Geometria, illinc verò Phi&longs;ica; quorum gemina
to complexu nobili&longs;&longs;ima artium prodit mechani
ca. &longs;i enim nobilitatem magis, tùm &longs;tratæ materiæ,
tùm argumentorum nece&longs;&longs;itati (quod Ari&longs;tote
les fatetur aliquandò) relatam volumus, omnium
procul dubiò nobili&longs;&longs;imam per&longs;piciemus. quæ
tur) ab&longs;oluit, & perficit; verùm etiam & phi&longs;ica
rum rerum imperium habet: quandoquidem
quodcunq; Fabris, Architectis, Baiulis, Agricolis,
Nautis, & quàm plurimis alijs (repugnantibus na
turæ legibus) opitulatur; id omne mechanicum
e&longs;t imperium. quippè quod aduer&longs;us naturam
vel eiu&longs;dem emulata leges exercet; &longs;umma id
certè admiratione dignum; veri&longs;&longs;imum tamen,
& à quocunque liberaliter admi&longs;&longs;um, qui pri
us ab Ari&longs;totele didicerit, omnia mechanica,
tùm problemata, tùm theoremata ad rotundam
machinam reduci, atq; ideo illo niti principio, Rotunda ma
china e&longs;t mouenti&longs;&longs;ima, & quò maior, eò mouen
tior. Verùm huic nobilitati adnexa e&longs;t &longs;umma re
rum ad vitam pertinentium vtilitas, quæ propte
rea omnes alias à diuer&longs;is artibus propagatas an
tecellit; quòd aliæ facultates po&longs;t mundi gene&longs;im
longa temporis intercapedine &longs;uos explicarunt
v&longs;us; i&longs;ta verò & in ip&longs;is mundi primordijs ita fuit
hominibus nece&longs;&longs;aria, vt ea &longs;ublata Sol de mun
do &longs;ublatus videretur. nam quacunq; nece&longs;&longs;ita
te Adæ vita degeretur; & quamuis etiam ca&longs;is
contectis &longs;tramine, & angu&longs;tis tugurijs, ac gurgu
&longs;tijs cœli de fenderet iniurias; &longs;ic & in corporis ve
&longs;titu, licet ip&longs;e nihil aliud &longs;pectaret, ni&longs;i vt imbres,
quodcunque tamen id fuit, omne mechanicum
fuit. neq; tamen huic facultati contingit, quod
ventis &longs;olet, qui cùm vndè oriuntur, ibi vehe
menti&longs;&longs;imi &longs;int, ad longinqua tamen fracti, de
bilitatiquè perueniunt: &longs;ed quod magnis flumini
bus crebriu&longs; accidit, quæ cùm in ip&longs;o ortu parua
&longs;int, perpetuò tamen aucta, eò ampliori ferun
tur alueo, quò à fontibus &longs;uis longius rece&longs;&longs;e
runt. Nam & temporis progre&longs;&longs;u mechanica fa
cultas &longs;ub iugo æquum arationis laborem di
&longs;pen&longs;are, atque aratrum agris circumagere cæ
pit. deinceps bigis, & quadrigis docuit comea
tus, merces, onera quælibet vehere, è finibus
no&longs;tri&longs; ad finitimos populos exportare, & ex il
lis contra importare ad nos. præterea cùm iam
res non tantùm nece&longs;&longs;itate, verùm etiam orna
tu, & commoditate metirentur, mechanicæ
fuit &longs;ubtilitatis, quòd nauigia remo impellere
mus; quòd gubernaculo exiguo in extrema pup
pi collocato ingentes triremium moles inflecte
remus; quòd vnius &longs;æpè manu pro multis fabro
rum manibus modò pondera lapidum, & tra
bium Fabris, & Architectis &longs;ubleuaremus; mo
dò tollenonis &longs;pecie aquas è puteis olitoribus e
xhauriremus. hinc etiam è liquidorum prælis vi
na, olea, vnguenta expre&longs;&longs;a, & quicquid liquohinc
magnas
contrarias partes
&longs;imus; hinc militiæ in aggeribus extruendis, in
con&longs;erenda manu, in opugnando, propugnan
doq; loca infinitæ ferè redundarunt vtilitates;
hinc demum Lignatores, Lapicidæ, Marmorarij
Vinitores, Olearij, Vnguentarij, Ferrarij, Auri
fices, Metallici, Chirurgi, Ton&longs;ores, Pi&longs;tores, Sar
tores, omnes deniq; opifices beneficiarij, tot, tan
taq; vitæ humanæ &longs;uppeditarunt commoda. Eant
nunc noui logodedali quidam mechanicorum
contemptores, perfricent frontem, &longs;i quam ha
bent, & ignobilitatem, atquè inutilitatem fal&longs;ò
criminari de&longs;inant: quòd &longs;i & adhuc id minimè
velint, eos quæ&longs;o in in&longs;citia &longs;ua relinquamus:
Ari&longs;totelemquè potius philo&longs;ophorum cory
phæum imitemur, cuius mechanici amoris ardo
rem acuti&longs;&longs;imæ illæ mechanicæ quæ&longs;tiones po&longs;te
ris traditæ &longs;atis declarant: qua quidem laude
Platonem magnificè &longs;uperauit; qui (vt te&longs;tatur
Plutarcus) Architam, & Eudoxum mechanicæ
vtilitatem impen&longs;ius colentes ab in&longs;tituto deter
ruit; quòd nobili&longs;&longs;imam philo&longs;ophorum po&longs;&longs;e&longs;
&longs;ionem in vulgus indicarent, ac publicarent; &
velut arcana philo&longs;ophiæ my&longs;teria proderent.
res &longs;anè meo quidem iudicio pro&longs;us vituperan
templationem quidem ocio&longs;am laudare; fructum
verò, & v&longs;um, arti&longs;q; finem improbare. &longs;ed præ
omnibus mathematicis vnus Archimedes ore
laudandus e&longs;t pleniore, quem voluit Deus in me
chanicis velut ideam &longs;ingularem e&longs;&longs;e, quam om
nes earum &longs;tudio&longs;i ad imitandum &longs;ibi propone
rent. is enim Cœle&longs;tem globum exiguo admo
dum, fragili què vitreo orbe conclu&longs;um ita efin
xit, &longs;imulatis a&longs;tris viuum naturæ opus, ac iura
poli motibus certis adeò præ &longs;e ferentibus; vt
æmula naturæ manus tale de &longs;e encomium &longs;it
promerita: &longs;ic manus naturam, vt natura ma
num ip&longs;a immitata putetur. is poli&longs;pa&longs;tu manu
leua, & &longs;ola, quinquies millenum modiorum
pondus attraxit. nauem in &longs;iccum litus eductam,
ac grauius oneratam &longs;olus machinis &longs;uis ad &longs;e
perindè pertraxit, ac &longs;i in mari remis, veli&longs;uè
impul&longs;a moueretur,
omnes Siciliæ vires non potuerunt) in mare de
duxit. ab i&longs;to etiam ea extiterunt bellica tor
menta, quibus Syracu&longs;æ aduer&longs;us Marcellum
ita defen&longs;æ &longs;unt, vt pa&longs;&longs;im eorum machinator
Briareus, & centimanus à Romanis appellare
tur. demum hac arte confi&longs;us eò proce&longs;&longs;it au
daciæ, vt eam vocem naturæ legibus adeò re
pugnantem protulerit. Da mihi, vbi &longs;i&longs;tam, ter
quod tamen non modò nos
vecte tantùm fieri potui&longs;&longs;e in præ&longs;enti libro doce
mus; verùm etiam, & omnis antiquitas (quod
multis forta&longs;&longs;è mirabile videbitur) id penitus
credidi&longs;&longs;e mihi videtur; quæ Neptuno tri
dentem tanquam vectem attribuit; cuius ope
terræ concu&longs;&longs;or vbiq; nuncupatur à poetis. ad
quod etiam a&longs;piciens celeberrimus no&longs;ter poeta
Neptunum inducit i&longs;ta machina &longs;yrtes, quò ma
gis apparerent Troianis, &longs;ubleuantem.
“Leuat ip&longs;e tridenti
& va&longs;tas aperit &longs;yrtes.”
Mechanici præterea fuerunt Heron, Cte&longs;ibius,
& Pappus, qui licet ad mechanicæ apicem, perin
de atq; Archimedes, euecti forta&longs;&longs;è minimè &longs;int;
mechanicam tamen facultatem egregiè percal
luerunt; tale&longs;q; fuerunt, & præ&longs;ertim Pappus, vt
eum me ducem &longs;equentem nemo (vt opinor) cul
pauerit. quod & propterea libentius feci, quòd
nè latum quidem vnguem ab Archimedeis prin
cipijs Pappus recedat. ego enim in hac præ&longs;ertim
facultate Archimedis ve&longs;tigijs hærere &longs;emper vo
lui: & licet eius lucubrationes ad
ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de æ
queponderantibus præ manibus
ver&longs;atur, in quò tanquam in copio&longs;i&longs;&longs;ima pœnu
omnia ferè mechanica dogmata repo&longs;ita mihi vi
dentur; quem &longs;anè libellum, &longs;i ætatis no&longs;træ mathe
matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;
&longs;ent &longs;anè
& ratas e&longs;&longs;e docent; &longs;ubtili&longs;&longs;imè, atquè veri&longs;
&longs;imè conuul&longs;as, & labefactatas. &longs;ed hoc vi
derint ip&longs;i. ego enim ad Pappum redeo, qui
ad v&longs;um mathematicarum vberiorem, emulu
mentorumquè acce&longs;&longs;iones amplificandas peni
tus conuer&longs;us, de quinque principibus machi
nis, Vecte nempè, Trochlea, Axe in peri
trochio, Cuneo, & Cochlea, multa egre
giè philo&longs;ophatus e&longs;t; demon&longs;trauit què quicquid
in machinis, aut cogitari peritè, aut acutè
definiri, aut certò &longs;tatui pote&longs;t, id omne quin
què illis infinita vi præditis machinis referen
dum e&longs;&longs;e. atquè vtinam iniuria temporis ni
hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam
den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra
rum orbem obtexi&longs;&longs;et, neque tanta mechani
cæ facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma
thematicarum proceres exi&longs;timarentur illi, qui
modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi
duas, & ob&longs;curas è medio tollunt. reperiun
tur enim aliqui, no&longs;traq; ætate emunctæ naris
mathematici, qui mechanicam, tùm mathe
maticè &longs;eor&longs;um, tùm phi&longs;icè con&longs;iderari po&longs;
&longs;e affirmant; ac &longs;i aliquando, vel &longs;ine demon
&longs;trationibus geometricis, vel &longs;ine vero motu
res mechanicæ con&longs;iderari po&longs;&longs;int: qua &longs;anè di
&longs;tinctione (vt leuius cum illis agam) nihil aliud mi
hi commini&longs;ci videntur, quàm vt dum &longs;e, tùm
phi&longs;icos, tùm mathematicos proferant, vtra
que (quod aiunt) &longs;ella excludantur. nequè
enim amplius mechanica, &longs;i à machinis ab&longs;tra
hatur, & &longs;eiungatur, mechanica pote&longs;t appel
lari. Emicuit tamen inter i&longs;tas tenebras (quam
uis alij quoquè nonnulli fuerint præclari&longs;&longs;imi)
Solis in&longs;tar Federicus Commandinus, qui multis
docti&longs;&longs;imis elucubrationibus ami&longs;&longs;um mathema
ticarum patrimonium non modò re&longs;taurauit,
verùm etiam auctiùs, & locupletiùs effecit.
erat enim &longs;ummus i&longs;te vir omnibus adeò facul
tatibus mathematicis ornatus, vt in eo Archi
tas, Eudoxus, Heron, Euclides, Theon, Ari
&longs;tarcus, Diophantus, Theodo&longs;ius, Ptolemæus
Apollonius, Serenus, Pappus, quin & ip
&longs;emet Archimedes (&longs;iquidem ip&longs;ius in Archi
medem &longs;cripta Archimedis olent lucernam) re & ecce repentè è tenebris (vt
confidimus) ac vinculis corporis in lucem, li
bertatem què productus mathematicas alieni&longs;
&longs;imo tempore optimo, & præ&longs;tanti&longs;&longs;imo patre
orbatas, nos verò ita con&longs;ternatos reliquit, vt e
ius de&longs;iderium vix longo &longs;ermone mitigare
po&longs;&longs;e videamur. Ille tamen perpetuò in alia
rum mathematicarum explicationem ver&longs;ans,
mechanicam facultatem, aut penitus præter
mi&longs;it, aut modicè attigit. Quapropter in hoc
&longs;tudium ardentiùs ego incumbere cæpi, nec me
vnquam per omne mathematum genus vagan
tem ea &longs;olicitudo de&longs;eruit; ecquid ex vno
quoquè decerpi, ac delibari po&longs;&longs;it; quo ad me
chanicam expoliendam, & exornandam acco
modatior e&longs;&longs;e po&longs;&longs;em. Nunc verò cùm mihi
videar, noni ea quidem omnia, quæ ad mecha
nicam pertinent, perfeci&longs;&longs;e; &longs;ed eò v&longs;q; tamen
progre&longs;&longs;us, vt ijs, qui ex Pappo, ex Vitruuio,
& ex alijs didicerint, quid &longs;it Vectis, quid Tro
chlea, quid Axis in peritrochio, quid Cuneus,
quid Cochlea; quomodoq; vt pondera moueri
po&longs;&longs;int, aptari debeant; adhuc tamen acciden
tia permulta, quæ inter potentiam, & pondus
vectis virtute illis in&longs;unt in&longs;trumentis, perdi&longs;ce
re cupiunt, opis aliquid adferre po&longs;&longs;im; putaui
tempus iam po&longs;tulare, vt prodirem; & nauatæ
Verùm quò facilius totius operis &longs;ub&longs;tructio
ad fa&longs;tigium &longs;uum per duceretur, nonnulla quo
què de libra fuerunt pertractanda, & præ&longs;er
tim dum vnico pondere alterum &longs;olum ip&longs;ius
brachium penitus deprimitur: que in re mi
rum e&longs;t quantas fecerint ruinas Iordanus (qui
inter recentiores maximæ fuit auctoritatis) &
alij; qui hanc rem &longs;ibi di&longs;cutiendam propo&longs;ue
runt. opus &longs;anè arduum, & for&longs;an viribus no
&longs;tris impar aggre&longs;si &longs;umus; in eo tamen digni, vt
no&longs;tros conatus, & indu&longs;triam ad præclara ten
dentem bonorum omnium perpetuus applau
&longs;us, approbatioq; comitetur; quòd ad &longs;tudium
tàm illu&longs;tre, tam magnificum, tam laudabile
contulimus quicquid habuimus virium. quod
&longs;anè qualecunq; &longs;it, tibi celeberrime PRINCEPS
nuncupandum cen&longs;uimus; cuius &longs;anè con&longs;ilij,
atq; in&longs;tituti no&longs;tri rationes multas reddere in
promptu e&longs;t: & primùm hæreditaria tibi in fa
miliam no&longs;tram promerita, quibus nos ita de
uictos habes; vt facilè intelligamus ad fortunas
non modò no&longs;tras, verùm & ad &longs;anguinem, &
vitam quoq; pro tua dignitate propendendam
parati&longs;&longs;imos e&longs;&longs;e debere. Præterea illud non
parui quoq; ponderis accedit, quòd à pueri
tia literarum omnium, &longs;ed præcipuè mathe
&longs;i illis adeptis vitam tibi acerbam, atq; in&longs;ua
uem &longs;tatueres. proinde in earum &longs;tudio infi
xus primam ætatis partem in illis percipiendis
exegi&longs;ti, eamquè &longs;æpius verè principe dignam
vocem protuli&longs;ti, te propterea mathematicis
præ&longs;ertim delectari, quòd i&longs;tæ maximè ex do
me&longs;tico illo, & vmbratili vitæ genere in Solem
(quod dicitur) & puluerem prodire po&longs;sint: cu
ius &longs;anè rei tuum flagranti&longs;simum ab ineunte æta
te peritiæ militaris de&longs;iderium, exploratum in
dicium poterat e&longs;&longs;e, ni&longs;i nimis emendicatæ men
tis e&longs;&longs;et ea proponere, quæ à te &longs;perari po&longs;&longs;ent;
quando tu penitus adole&longs;cens, egregia multa fa
cinora proficere matura&longs;ti. Tu enim cùm iam
à &longs;ancti&longs;&longs;imo Pontifice Pio V &longs;aluberrimæ Prin
cipum Chri&longs;tianorum coniunctionis fundamen
ta iacta e&longs;&longs;ent, alacer admodum ad debellan
dos Chri&longs;ti ho&longs;tes profectus, &longs;olidi&longs;&longs;imam, ac ve
ri&longs;&longs;imam gloriam tibi compara&longs;ti. Tu quoties de
&longs;umma rerum deliberatum e&longs;t, eas &longs;ententias
dixi&longs;ti, quæ &longs;ummam prudentiam cùm &longs;umma
animi excel&longs;itate coniunctam indicarent. ommit
tam interim pleraq; alia illis temporibus egre
giè, viriliter què à te ge&longs;ta, ne tibi ip&longs;i ea, quæ
omnibus &longs;unt manife&longs;ta, palàm facere videar:
tò tamen à te maiora, & præclara expectant
adhuc homines. Vale interim præ&longs;tanti&longs;&longs;imum
orbis decus, & &longs;i quando aliquid otij nactus
fueris has meas vigiliolas a&longs;picere ne dedi
gneris.
GVIDIVBALDI
E MARCHIONIBVS
MONTIS.
MECHANICORVM
LIBER.
DEFINITIONES.
Centrvm grauitatis vniu&longs;cu
iu&longs;q; corporis e&longs;t punctum quod
dam intra po&longs;itum, à quo &longs;i gra
ue appen&longs;um mente concipiatur,
dum fertur, quie&longs;cit; & &longs;eruat eam,
quam in principio habebat po&longs;i
tionem: neq; in ip&longs;a latione circumuertitur.
Hanc centri grauitatis definitionem Pappus Alexandrinus in
octauo Mathematicarum collectionum libro tradidit. Federicus
verò Commandinus in libro de centro grauitatis &longs;olidorum idem
centrum de&longs;cribendo ita explicauit.
Centrum grauitatis vniu&longs;cuiu&longs;q; &longs;olidæ figu
ræ e&longs;t punctum illud intra po&longs;itum, circa quod
vndiq; partes æqualium momentorum con&longs;i
&longs;tunt. &longs;i enim per tale centrum ducatur planum
figuram quomodocunq; &longs;ecans &longs;emper in par
tes æqueponderantes ip&longs;am diuidet.
COMMVNES NOTIONES.
I
Si ab æqueponderantibus æqueponderantia au
ferantur, reliqua æqueponderabunt.
II
Si æqueponderantibus æqueponderantia adii
ciantur, tota &longs;imul æqueponderabunt.
III
Quæ eidem æqueponderant, inter &longs;e æquè &longs;unt
grauia.
SVPPOSITIONES.
I
Vnius corporis vnum tantùm e&longs;t centrum gra
uitatis.
II
Vnius corporis centrum grauitatis &longs;emper in
eodem e&longs;t &longs;itu re&longs;pectu &longs;ui corporis.
III
Secundùm grauitatis centrum pondera deor
&longs;um feruntur.
DE LIBRA.
Anteqvam de libra &longs;ermo ha
beatur, vtres clarior eluce&longs;cat, &longs;it
libra AB recta linea; CD verò
trutina, quæ &longs;ecundum commu
nem con&longs;uetudinem horizonti
&longs;emper e&longs;t perpendicularis. pun
ctum autem C immobile, circa quod vertitur li
bra, centrum libræ
vocetur. itidemque
(quamuis tamen im
proprie) &longs;iue &longs;upra,
&longs;iue infra libram fue
rit con&longs;titutum. CA
verò, & CB, tum di
&longs;tantiæ, tum libræ
brachia nuncupen
tur. & &longs;i à centro li
bræ &longs;upra, vel infra
libram con&longs;tituto ip&longs;i AB perpendicularis duca
tur, hæc perpendiculum vocetur, quæ libram AB
&longs;ub&longs;tinebit; & quocunque modo moueatur libra,
ip&longs;i &longs;emper perpendicularis exi&longs;tet.
LEMMA.
Sit linea AB horizonti perpendicularis, & dia
metro AB circulus de&longs;cribatur AEBD, cuius
centrum C. Dico punctum B infimum e&longs;&longs;e lo
cum circumferentiæ circuli AEBD; punctum
verò A &longs;ublimiorem; & quælibet puncta, vt DE
æqualiter à puncto A di&longs;tantia æqualiter e&longs;&longs;e
deor&longs;um; quæ verò propius &longs;unt ip&longs;i A eis, quæ
magis di&longs;tant, &longs;ublimiora e&longs;&longs;e.
Producatur AB v&longs;q; ad mundi cen
trum, quod &longs;it F; deinde in circuli circum
connectanturq; FG FD FE. Quoniam
n. BF minima e&longs;t omnium, quæ à puncto
F ad circumferentiam AEBD ducun
tur; erit BF ip&longs;a FG minor. quare punctum
B propius erit puncto F, quàm G. hacq;
ratione o&longs;tendetur punctum B quouis alio
puncto circumferentiæ circuli AEDB
mundi centro propius e&longs;&longs;e. erit igitur pun
ctum B circumferentiæ circuli AEBD
infimus locus. Deinde quoniam AF per
centrum ducta maior e&longs;t ip&longs;a GF; erit
punctum A non
quouis alio puncto circumferentiæ circuli
AEBD &longs;ublimius. Præterea quoniam DF
FE &longs;unt æquales; puncta DE æqualiter
mundi centro di&longs;tabunt. & cum DF maior &longs;it FG; erit pun
ctum D ip&longs;i A propius puncto G &longs;ublimius. quæ omnia demon
&longs;trare oportebat.
PROPOSITIO I.
Si Pondus in eius centro grauitatis a recta &longs;u
&longs;tineatur linea, nunquam manebit, ni&longs;i eadem li
nea horizonti fuerit perpendicularis.
Sit pondus A, cuius centrum gra
uitatis B, quod à linea CE &longs;u&longs;ti
neatur. Dico pondus nunquam
perman&longs;urum, ni&longs;i CB horizonti
perpendicularis exi&longs;tat. &longs;it pun
ctum C immobile, quod vt pon
dus &longs;u&longs;tineatur, nece&longs;&longs;e e&longs;t. & cum
punctum C &longs;it immobile, &longs;i pon
dus A mouebitur, punctum B cir
culi circumferentiam de&longs;cribet,
cuius &longs;emidiameter erit CB. qua
re centro C, &longs;patio verò BC, cir
culus de&longs;cribatur BFDE. &longs;itq;
primum BC horizonti perpendicularís, quæ v&longs;q; ad D produca
tur; atq; punctum C &longs;it infra punctum B. Quoniam enim pondus
A &longs;ecundum grauitatis centrum B deor&longs;um mouetur; punctum
B deor&longs;um in centrum mundi, quò naturaliter tendit, per re
ctam lineam BD mouebitur: totum ergo pondus A eius cen
tro grauitatis B &longs;uper rectam lineam BC graue&longs;cet. cum au
tem pondus à linea CB &longs;u&longs;tineatur, linea CB totum &longs;u&longs;ti
nebit pondus A; &longs;uper quam deor&longs;um moueri non pote&longs;t, cum
ab ip&longs;a prohibeatur: per definitionem igitur centri grauitatis pun
ctum B, pondu&longs;q; A in hoc &longs;itu manebunt. & quamquam B quo
cunq; alio puncto circuli &longs;it &longs;ublimius, ab hoc tamen &longs;itu deor&longs;um
per circuli circumferentiam nequaquam mouebitur non enim ver
&longs;us F magis, quàm ver&longs;us E inclinabitur, cum ex vtraq; parte æqua
lis &longs;it de&longs;cen&longs;us; neq; pondus A in vnam magis, quàm in alteram
partem propen&longs;ionem habeat: quod non accidit in quouis alio
puncto circumferentiæ circuli (præter D) &longs;it ponderis eiu&longs;dem
puncto F ver&longs;us D &longs;it de&longs;cen&longs;us, at
verò ver&longs;us B a&longs;cen&longs;us. quare pun
ctum F deor&longs;um mouebitur. & quo
niam per rectam lineam in centrum
mundi moueri non pote&longs;t, cum à
puncto C immobili propter lineam
CF prohibeatur; deor&longs;um tamen
&longs;icuti eius natura po&longs;tulat, &longs;emper
mouebitur. & cum infimus locus &longs;it
D, per
tur, donec in D perueniat, in quo
&longs;itu manebit,
&longs;tet. tum quia deor&longs;um amplius moueri non pote&longs;t, cum ex pun
cto C &longs;it appen&longs;um; tum etiam, quia in eius centro grauitatis &longs;u&longs;ti
netur. Quando autem F erit in D, erit quoq; linea FC in DC,
&longs;imulq; horizonti perpendicularis. pondus ergo nunquam mane
bit, donec linea CF horizonti perpendicularis non exi&longs;tat. quod
o&longs;tendere oportebat. quod
o&longs;tendere oportebat.
Ex hoc elici pote&longs;t, pondus quocunq; modo
in dato puncto &longs;u&longs;tineatur, nunquam manere; ni
&longs;i quando a centro grauitatis ponderis ad id pun
ctum ducta linea horizonti &longs;it perpendicularis.
Vt ii&longs;dem po&longs;itis, &longs;u&longs;tineatur
pondus à lineis CG CH. Dico
&longs;i ducta BC horizonti &longs;it perpen
dicularis, pondus A manere. &longs;i verò
ducta CF non &longs;it horizonti per
pendicularis, punctum F deor&longs;um
v&longs;q; ad D moueri; in quo &longs;itu pon
dus manebit, ductaq; CD horizon
ti perpendicularis exi&longs;tet. quæ om
nia eadem ratione o&longs;tendentur. PROPOSITIO II.
Libra horizonti æquidi&longs;tans, cuius centrum
&longs;it &longs;upra libram, æqualia in extremitatibus, æqua
literq; à perpendiculo di&longs;tantia habens pondera,
&longs;i ab eiu&longs;modi moueatur &longs;itu, in eundem rur&longs;us
relicta, redibit; ibíq; manebit.
Sit libra AB recta li
nea horizonti æquidi
&longs;tans, cuius centrum C
&longs;it &longs;upra libram; &longs;itq; CD
rizonti perpendiculare
erit: atq; di&longs;tantia DA &longs;it
di&longs;tantiæ DB æqualis;
&longs;intq; in AB pondera æ
qualia,
centra &longs;int in AB
Moueatur AB libra ab
hoc &longs;itu, putá in EF, deinde relinquatur. dico libram EF in AB ho
rizonti æquidi&longs;tantem redire, ibíq; manere. Quoniam autem pun
ctum C e&longs;t immobile, dum libra mouetur, punctum D circuli cir
cumferentiam de&longs;cribet, cuius &longs;emidiameter erit CD. quare cen
tro C, &longs;patio verò CD, circulus de&longs;cribatur DGH. Quoniam
enim CD ip&longs;i libræ &longs;emper e&longs;t perpendicularis, dum libra erit in
EF, linea CD erit in CG, ita vt CG &longs;it ip&longs;i EF perpendicula
ris. Cùm autem AB bifariam à puncto D diuidatur, & pondera
in AB &longs;int æqualia; erit magnitudinis ex ip&longs;is AB compo&longs;itæ cen
trum grauitatis in medio, hoc e&longs;t in D. &
deribus erit in EF; erit magnitudinis ex vtri&longs;q; EF compo&longs;itæ cen
trum grauitatis G. & quoniam CG horizonti non e&longs;t perpendi
cularis;
nimè per&longs;i&longs;tet, &longs;ed deor&longs;um
circumferentiam GD mouebitur; donec CG horizonti fiat per
nec CG in CD redeat.
Quando autem CG erit
in CD, linea EF, cùm
ip&longs;i CG &longs;emper ad rectos
&longs;it angulos, erit in AB; in li
bra ergo EF in AB hori
zonti
bit, ibíq; manebit. quod
demon&longs;trare oportebat.
medis de
æqueponde
rantibus.
PROPOSITIO III.
Libra horizonti æquidi&longs;tans æqualia in extre
mitatibus, æqualiterq; à perpendiculo di&longs;tan
tia habens pondera, centro infernè collocato, in
hoc &longs;itu manebit. &longs;i verò inde moueatur, deor
&longs;um relicta, &longs;ecundùm partem decliuiorem mo
uebitur.
Sit libra AB rectá li
nea horizonti æquidi
&longs;tans, cuius centrum C
&longs;it infra libram; perpen
diculumq; &longs;it CD, quod
horizonti perpendiculare
erit; & di&longs;tantia AD &longs;it
di&longs;tantiæ DB æqualis;
&longs;intq; in AB pondera
æqualia, quorum grauita
tis centra &longs;int in punctis
AB. Dico primùm libram AB in hoc &longs;itu manere.
Quoniam
enim AB bifariam diuiditur à puncto D, & pondera in AB &longs;unt
æqualia; erit punctum D centrum grauitatis magnitudinis ex & CD libram &longs;u&longs;tinens ho
rizonti
moueatur autem libra AB ab hoc &longs;itu, putà in EF, deinde relinqua
tur. dico libram EF ex parte F moueri.
Quoniam igitur CD
ip&longs;i libræ &longs;emper e&longs;t perpendicularis, dum libra erit in EF, erit
CD in CG ip&longs;i EF perpendicularis. & punctum G magnitudi
nis ex EF compo&longs;itæ centrum grauitatis erit; quod dum moue
tur, circuli circumferentiam de&longs;cribet DGH, cuius &longs;emidiameter
CD, & centrum C. Quoniam autem CG horizonti non e&longs;t per
pendicularis, magnitudo ex EF ponderibus compo&longs;ita in hoc &longs;i
tu minimè manebit; &longs;ed &longs;ecundùm eius grauitatis centrum G deor
&longs;um per circumferentiam GH mouebitur. libra ergo EF ex par
te F deor&longs;um mouebitur, quod demon&longs;trare oportebat.
PROPOSITIO IIII.
Libra horizonti æquidi&longs;tans æqualia in ex
tremitatibus, æqualiterq; à centro in ip&longs;a libra
collocato, di&longs;tantia habens pondera; &longs;iue inde
moueatur, &longs;iue minus; vbicunq; relicta, manebit.
Sit libra recta linea A
B horizonti æquidi&longs;tans,
cuius centrum C in ea
dem &longs;it linea AB; di&longs;tan
tia verò CA &longs;it di&longs;tantiæ
CB æqualis: &longs;intq; pon
dera in AB æqualia, quo
rum centra grauitatis &longs;int
in Moueatur
libra, vt in DE, ibiquè
relinquatur. Dico primùm libram DE non moueri, in eoquè &longs;itu
manere. Quoniam enim pondera AB &longs;unt æqualia; erit magni
tudinis ex vtroq; pondere, videlicet A, & B compo&longs;itæ centrum
grauitatis C. quare idem punctum C, & centrum libræ, &
grauitatis totius ponderis erit. Quoniam autem centrum libræ
cum ponderibus in DE
mouetur, immobile re
manet, centrum quoq;
grauitatis, quod e&longs;t idem
C, non mouebitur. nec
igitur libra DE mouebi
tur, per definitionem
centri grauitatis, cum in
ip&longs;o &longs;u&longs;pendatur. Idip
&longs;um quoq; contingit libra in AB horizonti æquidi&longs;tante, vel in
quocunq; alio &longs;itu exi&longs;tente. Manebit ergo libra, vbi relinque
tur. quod demon&longs;trare oportebat.
Cum verò in iis, quæ dicta &longs;unt, grauitatis tantùm magnitudi
num, quæ in extremitatibus libræ po&longs;itæ &longs;unt æquales, ab&longs;q; lí
bræ grauitate con&longs;iderauerimus; quoniam tamen adhuc libræ bra
chia &longs;unt æqualia, idcirco idem libræ, eius grauitate con&longs;iderata,
vnà cum ponderibus, vel &longs;ine ponderibus eueniet. idem enim cen
trum grauitatis fine ponderibus libræ tantùm grauitatis centrum
erit. Similiter &longs;i pondera in libræ extremitatibus appendantur, vt
fieri &longs;olet, idem eueniet; dummodo ex &longs;u&longs;pen&longs;ionum punctis ad
centra grauitatum ponderum ductæ lineæ (quocunq; modo mo
ueatur libra) &longs;i protrahantur, in centrum mundi concurrant. vbi
enim pondera hoc modo &longs;unt appen&longs;a, ibi graue&longs;cunt, ac &longs;i in ii&longs;
dem punctis centra grauitatum haberent. præterea, quæ &longs;equun
tur, eodem pror&longs;us modo con&longs;iderare poterimus.
aliter &longs;entientibus dicta officere videntur; idcirco in hac parte ali
priam &longs;ententiam, &longs;ed Archimedem ip&longs;um, qui in hac eadem e&longs;&longs;e
Ii&longs;dem po&longs;itis, duca
tur FCG ip&longs;i AB, &
horizonti perpendicula
ris; & centro C, &longs;patio
què CA, circulus de&longs;cri
batur ADFBEG. erunt
puncta ADBE in circu
li circumferentia; cum li
bræ brachia &longs;int æqualia.
& quoniam in vnam con
ueniunt &longs;ententiam, a&longs;&longs;e
rentes &longs;cilicet libram DE
neq; in FG moueri, ne
que in DE manere, &longs;ed in AB horizonti æquidi&longs;tantem rediré.
hanc eorum &longs;ententiam nullo modo con&longs;i&longs;tere po&longs;&longs;e o&longs;tendam.
Non enim, &longs;ed &longs;i quod aiunt, euenerit, vel ideo erit, quia pondus
D pondere E grauius fuerit, vel &longs;i pondera &longs;unt æqualia, di&longs;tantiæ,
quibus &longs;unt po&longs;ita, non erunt æquales, hoc e&longs;t CD ip&longs;i CE non erit
æqualis, &longs;ed maior. Quòd autem pondera in DE &longs;int æqualia, &
di&longs;tantia CD &longs;it æqualis di&longs;tantiæ CE: hæc ex &longs;uppo&longs;itione pa
tent. Sed quoniam dicunt pondus in D in eo &longs;itu pondere in E
grauius e&longs;&longs;e in altero &longs;itu deor&longs;um: dum pondera &longs;unt in DE, pun
ctum C non erit amplius centrum grauitatis, nam non manent, &longs;i
ex C &longs;u&longs;pendantur; &longs;ed erit in linea CD, ex tertia primi Archi
medis de æqueponderantibus. non autem erit in linea CE, cum pon
dus D grauius &longs;it pondere E. &longs;it igitur in H, in quo &longs;i &longs;u&longs;pendan
tur, manebunt. Quoniam autem centrum grauitatis ponderum
in AB connexorum e&longs;t punctum C; ponderum verò in DE e&longs;t
punctum H: dum igitur pondera AB mouentur in DE, centrum
grauitatis C ver&longs;us D mouebitur, & ad D propius accedet; quod
e&longs;t impo&longs;sibile: cum pondera eandem inter &longs;e &longs;e &longs;eruent di&longs;tantiam.
Vniu&longs;cuiu&longs;q; enim corporis centrum grauitatis in eodem &longs;emper
e&longs;t &longs;itu re&longs;pectu &longs;ui corporis. & quamquam punctum C &longs;it duo
rum corporum AB centrum grauitatis, quia tamen inter &longs;e &longs;e ita à
libra connexa &longs;unt, vt &longs;emper eodem modo &longs;e &longs;e habeant; Ideo
punctum C ita eorum erit centrum grauitatis, ac &longs;i vna tantum libra
enim vna cum ponderi
bus vnum tantum conti
nuum efficit, cuius cen
trum grauitatis erit &longs;em
per in medio. non igitur
pondus in D pondere in
E e&longs;t grauius. Si autem
dicerent centrum graui
tatis non in linea CD,
&longs;ed in CE e&longs;&longs;e debere;
idem eueniet ab&longs;urdum.
Amplius &longs;i pondus D
deor&longs;um mouebitur, pondus E &longs;ur&longs;um mouebit. pondus igitur gra
uius, quàm &longs;it E, in eodemmet &longs;itu ponderi D æqueponderabit, &
grauia inæqualia æquali di&longs;tantia po&longs;ita æqueponderabunt. Adii
ciatur ergo ponderi E aliquod graue, ita vt ip&longs;i D contraponde
ret, &longs;i ex C &longs;u&longs;pendantur. &longs;ed cum &longs;upra o&longs;ten&longs;um &longs;it punctum C
centrum e&longs;&longs;e grauitatis æqualium ponderum in DE; &longs;i igitur pon
CE. &longs;itq; hoc centrum K. at per definitionem centri grauitatis, &longs;i
pondera &longs;u&longs;pendantur ex K, manebunt. ergo &longs;i &longs;u&longs;pendantur ex
C, non manebunt, quod e&longs;t contra hypote&longs;im: &longs;ed pondus E deor
&longs;um mouebitur. quòd &longs;i ex C quoque &longs;u&longs;pen&longs;a æqueponderarent;
bile. Non igitur pondus in E grauius eo, quod e&longs;t in D, ip&longs;i D æque
ponderabit, cum ex puncto C fiat &longs;u&longs;pen&longs;io. Pondera ergo in DE
æqualia ex eorum grauitatis centro C &longs;u&longs;pen&longs;a, æqueponderabunt,
manebuntquè. quod demon&longs;trare fuerat propo&longs;itum.
po&longs;sibile e&longs;&longs;e addere ip&longs;i E pondus adeo minimum, quin adhuc &longs;i
ex C &longs;u&longs;pendantur, pondus E &longs;emper deor&longs;um ver&longs;us G moueatur.
quod nos fieri po&longs;&longs;e &longs;uppo&longs;uimus, atque fieri po&longs;&longs;e credebamus. ex
ce&longs;&longs;um enim ponderis D &longs;upra pondus E, cum quantitatis ratio
nem habeat, non &longs;olum minimum e&longs;&longs;e, verum in infinitum diuidi
po&longs;&longs;e immaginabamur, quod quidem ip&longs;i, non &longs;olum minimum,
do demon&longs;trare nituntur.
Exponantur eadem.
à puncti&longs;què DE hori
zonti
&longs;it circulus LDM, cu
ius
in puncto D contingat,
ip&longs;iq; FDG &longs;it æqualis:
erit NC recta linea. &
quoniam angulus KEC
angulo HDN e&longs;t æqua
lis, angulusq; CEG an
gulo NDM e&longs;t etiam
æqualis; cum à &longs;emidiametris, æqualibusq; circumferentiis conti
neatur; erit reliquus mixtu&longs;què angulus KEG reliquo mixtoquè
HDM æqualis. & quia &longs;upponunt, quò minor e&longs;t angulus linea
horizonti perpendiculari, & circumferentia contentus, eò pondus
in eo &longs;itu grauius e&longs;&longs;e. vt quò minor e&longs;t angulus HD, & circumfe
rentia DG contentus angulo KEG, hoc e&longs;t angulo HDM; ita &longs;e
cundum hanc proportionem pondus in D grauius e&longs;&longs;e pondere in
E. Proportio autem anguli MDH ad angulum HDG minor e&longs;t
qualibet proportione, quæ &longs;it inter maiorem, & minorem quanti
tatem: ergo proportio ponderum DE omnium proportionum mi
nima erit. immo neq; erit ferè proportio, cum &longs;it omnium pro
portionum minima. quòd autem proportio MDH ad HDG &longs;it
omnium minima, ex hac nece&longs;sitate o&longs;tendunt; quia MDH exce
dit HDG angulo curuilineo MDG, qui quidem angulus omnium
angulorum rectilineorum minimus exi&longs;tit: ergo cum non po&longs;sit da
ri angulus minor MDG, erit proportio MDH ad HDG
proportionum minima. quæ ratio inutilis valde videtur e&longs;&longs;e; quia
quamquam angulus MDG &longs;it omnibus rectilineis angulis minor,
non idcirco &longs;equitur, ab&longs;olutè, &longs;impliciterq; omnium e&longs;&longs;e
minimum: nam ducatur à puncto D linea DO ip&longs;i NC perpendicu
laris, hæc vtra&longs;q; tanget circumferentias LDM FDG in puncto quia verò circumfe
rentiæ &longs;unt æquales, erit
angulus MDO mixtus
angulo ODG mixto
æqualis; alter ergo an
gulus, vt ODG minor
erit MDG, hoc e&longs;t mi
nor minimo. angulus
deinde OGH minor
erit angulo MDH; qua
re ODH ad angulum
bit
MDH ad eundem HDG. dabitur ergo quoquè proportio mi
nor minima, quam in infinitum adhuc minorem ita o&longs;tende
mus. De&longs;cribatur circulus DR, cuius centrum E, & &longs;emidiame
lus RDG angulo ODG. &longs;imiliter & angulus RDH angulo
ODH. minorem igitur proportionem habebit RDH ad HDG,
quàm ODH ad HDG. Accipiatur deinde inter EC vtcun
que punctum P, ex quo in di&longs;tantia PD alia de&longs;cribatur circum
ferentia DQ, quæ circumferentiam DR, circumferentiamquè
DG in puncto D continget; & angulus QDH minor erit
angulo RDH: ergo QDH ad HDG minorem habebit propor
tionem, quàm RDH ad HDG. eodemquè pror&longs;us modo, &longs;i
inter PC aliud accipiatur punctum, & inter hoc &C aliud, & &longs;ic
deinceps, infinitæ de&longs;cribentur circumferentiæ inter DO, & cir
cumferentiam DG; ex quibus proportionem in infinitum &longs;emper
minorem inueniemus. atque ideo proportionem ponderis in D
ad pondus in E non adeo minorem e&longs;&longs;e &longs;equitur, quin ad infini
tum ip&longs;a &longs;emper minorem reperiri po&longs;sit. & quia angulus MDG
in infinitum diuidi pote&longs;t; exce&longs;&longs;us quoque grauitatis D &longs;upra E
diuidi ad infinitum poterit.
Sed neque prætereundum
e&longs;t, ip&longs;os in demon&longs;tratio
ne angulum KEG maiorem
e&longs;&longs;e angulo HDG, tanquam
notum accepi&longs;&longs;e. quod e&longs;t
quidem verum, &longs;i DHEK
inter &longs;e &longs;e &longs;int æquidi&longs;tan
tes. Quoniam autem (vt
ip&longs;i quoque &longs;upponunt) li
neæ DHEK in centrum
mundi conueniunt; lineæ
DHEK æquidi&longs;tantes nun
quam erunt, & angulus KEG
angulo HDG non &longs;olum
maior erit, &longs;ed minor. vt
exempli gratia, producatur
FG v&longs;que ad centrum mun
di, quod &longs;it S; connectan
;turqué DSES. o&longs;tenden
dum e&longs;t angulum SEG mi
norem e&longs;&longs;e angulo SDG. du
catur à puncto E linea ET circulum DGEF contingens, ab eo
demqué puncto ip&longs;i DS æquidi&longs;tans ducatur EV. Quoniam igi
tur EVDS inter &longs;e &longs;e &longs;unt æquidi&longs;tantes: &longs;imiliter ETDO æqui
di&longs;tantes: erit angulus VET angulo SDO æqualis. & angulus
TEG angulo ODM e&longs;t æqualis; cum à lineis contingentibus,
circumferentii&longs;qué æqualibus contineatur: totus ergo angulus
VEG angulo SDM æqualis erit. Auferatur ab angulo SDM
angulus curuilineus MDG; ab angulo autem VEG angulus au
feratur VES; & angulus VES rectilineus maior e&longs;t curuilineo
MDG; erit reliquus angulus SEG minor angulo SDG.
Quare ex ip&longs;orum &longs;uppo&longs;itionibus non &longs;olum pondus in D gra
uius erit pondere in E; verùm è conuer&longs;o, pondus in E ip&longs;o D
grauius exi&longs;tet.
Rationes tamen af
ferunt, quibus demon
&longs;trare nituntur, libram
DE in AB horizon
ti æquidi&longs;tantem ex
nece&longs;sitate redire.
mum
dunt, idem pondus
grauius e&longs;&longs;e in A,
quàm in alio &longs;itu, quem
æqualitatis &longs;itum no
minant, cum linea
AB &longs;it horizonti æ
quidi&longs;tans. deinde quò propius e&longs;t ip&longs;i A, quouis alio remotiori
grauius e&longs;&longs;e. Vt pondus in A grauius e&longs;&longs;e, quàm in D; & in D,
quàm in L. &longs;imiliter in A grauius, quam in N; & in N grauius,
quàm in M. Vnum tantùm con&longs;iderando pondus in altero libræ
Quia (inquiunt) po&longs;ita trutina
in CF, pondus in A longius e&longs;t à trutina, quàm in D: & in D
longius, quàm in L. ductis enim DO LP ip&longs;i CF perpendicula
quod
deinde ex quo loco (aiunt) pon
dus velocius mouetur, ibi grauius e&longs;t; velocius autem ex A, quàm
ab alio &longs;itu mouetur; ergo in A grauius e&longs;t. &longs;imili modo, quò
propius e&longs;t ip&longs;i A, velocius quoque mouetur; ergo in D graAltera deinde cau&longs;a, quam ex rectiori, & obli
ctius de&longs;cendit, grauius e&longs;&longs;e videtur; cum pondus liberum, atq;
dit; ergo in A grauius erit. hocq; o&longs;tendunt accipiendo arcum
AN arcui LD æqualem; à puncti&longs;q; NL lineæ FG (quam
etiam directionis vocant) æquidi&longs;tantes ducantur NRLQ, quæ
lineas AB DO &longs;ecent in QR; & à puncto N ip&longs;i FG perpen
dicularis ducatur NT. rectèq; demon&longs;trant LQ ip&longs;i PO æqua
lem e&longs;&longs;e, & NR ip&longs;i CT; lineamq; NR ip&longs;a LQ maiorem e&longs;&longs;e.
Quoniam autem de&longs;cen&longs;u; ponderis ex A v&longs;q; ad N per circum
ip&longs;i vocant capere de directo) quàm de&longs;cen&longs;us ex L in D per cir
cumferentiam LD; cùm de&longs;cen&longs;us AN lineam CT pertran&longs;eat,
de&longs;cen&longs;us verò LD lineam PO; & CT maior e&longs;t PO; rectior erit
de&longs;cen&longs;us AN, quám de&longs;cen&longs;us LD. grauius ergo erit pondus
in A, quàm in L, & in quouis alio &longs;itu. eodemq; pror&longs;us
modo o&longs;tendunt, quò propius e&longs;t ip&longs;i A, grauius e&longs;&longs;e.
Vt &longs;int circumferentiæ LD DA inter &longs;e &longs;e æquales, & à puncto
D ip&longs;i AB perpendicularis ducatur DR; erit DR ip&longs;i CO æqua
lis. lineam deinde DR ip&longs;a LQ maiorem e&longs;&longs;e demon&longs;trant.
di
cuntq; de&longs;cen&longs;um DA magis capere de directo de&longs;cen&longs;u LD, ma
ior enim e&longs;t linea CO, quàm OP; quare pondus grauius erit
in D, quàm in L. quod ip&longs;um euenit in punctis NM. Suppo
&longs;itionem itaq;, qua libram DE in AB redire demon&longs;trant, vt
notam, manife&longs;tamq; proferunt. Nempè Secundùm &longs;itum pon
dus grauius e&longs;&longs;e, quanto in eodem &longs;itu minus obliquus e&longs;t de&longs;cen
&longs;us. huiu&longs;q; reditus cau&longs;am eam e&longs;&longs;e dicunt; Quoniam &longs;cilicet
de&longs;cen&longs;us ponderis in D rectior e&longs;t de&longs;cen&longs;u ponderis in E, cùm
minus capiat de directo pondus in E de&longs;cendendo, quàm pon
dus in D &longs;im liter de&longs;cendendo. Vt &longs;i arcus EV &longs;it ip&longs;i DA
æqualis, ducanturq; VH ET ip&longs;i FG perpendiculares; maior
erit DR, quàm TH. quare per &longs;uppo&longs;itionem pondus in D ra
tione &longs;itus grauius erit pondere in E. pondus ergo in D, cùm &longs;it
grauius, deor&longs;um mouebitur; pondus verò in E &longs;ur&longs;um, donec li
bra DE in AB redeat.
Altera huius quoq; reditus ratio e&longs;t, cùm trutina &longs;upra libram
e&longs;t in CF; linea CG e&longs;t meta. & quoniam angulus GCD ma
ior e&longs;t angulo GCE, & maior à meta angulus grauius reddit
pondus; trutina igitur &longs;uperius exi&longs;tente, grauius erit pondus in
D, quàm in E. idcirco D in A, & E in B redibit.
His itaq; rationibus conantur o&longs;tendere libram DE in AB re
dire; quæ meo quidem iuditio facile &longs;olui po&longs;&longs;unt.
Primùm itaq; quan
tum attinet ad ratio
nes pondus in A gra
uius e&longs;&longs;e, quàm in a
lio &longs;itu o&longs;tendentes,
quas ex longiori, &
propinquiori
linea FG, & ex velo
ciori, & rectiori mo
tu à puncto A dedu
cunt; primùm quidem
non demon&longs;trant, cur
pondus ex A velocius
moueatur, quàm ex alio &longs;itu. nec quia CA e&longs;t DO maior,
& DO ip&longs;a LP, propterea &longs;equitur tanquam ex vera cau&longs;a, pon
dus in A grauius e&longs;&longs;e, quàm in D; & in D, quàm in L. neq;
enim intellectus quie&longs;cit, ni&longs;i alia huius o&longs;tendatur cau&longs;a; cùm po
tius &longs;ignum, quàm vera cau&longs;a e&longs;&longs;e videatur. id ip&longs;um quoq; al
teri rationi contintingit, quam ex rectiori & obliquiori motu de
ducunt. Præterea quæcunq; ex velociori, & rectiori motu per
&longs;uadent pondus in A grauius e&longs;&longs;e, quàm in D; non ideo de
mon&longs;trant pondus in A, quatenus e&longs;t in A, grauius e&longs;&longs;e pon
dere in D, quatenus e&longs;t in D; &longs;ed quatenus à punctis DA rece
dit. Idcirco antequàm vlterius progrediar, o&longs;tendam primùm
pondus, quò propius e&longs;t ip&longs;is FG, minus grauitare; tum qua
tenus in eo &longs;itu, in quo reperitur, manet: tum quatenus ab eo
recedit. &longs;imulq; fal&longs;um e&longs;&longs;e, pondus in A grauius e&longs;&longs;e, quàm in
alio &longs;itu.
Producatur FG v&longs;q; ad mundi cen
trum, quod &longs;it S. & à puncto S circu
lum AFBG contingens ducatur. neq;
enim linea à puncto S circulum con
tingere pote&longs;t in A; nam ducta AS
triangulum ACS duos haberet angu
los rectos, nempè SAC ACS, quod
e&longs;t impo&longs;sibile. neq; &longs;upra punctum A
in circumferentia AF continget; cir
culum enim &longs;ecaret. tanget igitur in
fra, &longs;itq; SO. connectantur deinde SD
SL, quæ circumferentiam AOG in
punctis KH &longs;ecent. & Ck CH con
iungantur. Et quoniam pondus, quanto
propius e&longs;t ip&longs;i F, magis quoque inni
titur centro; vt pondus in D magis ver
&longs;ionis puncto C innititur tanquam
centro; hoc e&longs;t in D magis &longs;upra li
neam CD grauitat, quàm &longs;i e&longs;&longs;et in A
&longs;upra lineam CA; & adhuc magis in
L &longs;upra lineam CL; Nam cùm tres
anguli cuiu&longs;cunq; trianguli duobus re
ctis &longs;int æquales, & trianguli DCk æquicruris angulus DCk
minor &longs;it angulo LCH æquicruris trianguli LCH: erunt reli
qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH
CHL maiores. & horum dimidii; hoc e&longs;t angulus CDS angu
lo CLS maior erit. cùm itaq; CLS &longs;it minor, linea CL ma
gis adhærebit motui naturali ponderis in L pror&longs;us &longs;oluti. hoc
e&longs;t lineæ LS, quàm CD motui DS. pondus enim in L
berum, atq; &longs;olutum in centrum mundi per LS moueretur, pon
dusq; in D per DS. quoniam verò pondus in L totum &longs;uper LS
grauitat, in D verò &longs;uper DS: pondus in L magis &longs;upra lineam
CL grauitabit, quàm exi&longs;tens in D &longs;upra lineam DC. ergo
linea CL pondus magis &longs;u&longs;tentabit, quàm linea CD. Eodem
qué modo, quò pondus propius fuerit ip&longs;i F, magis ob hanc cau
&longs;am à linea CL &longs;u&longs;tineri o&longs;tendetur; &longs;emper enim angulus CLS quod etiam patet; quia &longs;i
lineæ CL, & LS in vnam coinciderent
lineam, quod euenit in FCS; tunc linea
CF totum &longs;u&longs;tineret pondus in F, im
mobilemq; redderet: neq; vllam pror
&longs;us grauitatem in circumferentia circu
li haberet. Idem ergo pondus propter
&longs;ituum diuer&longs;itatem grauius, leuiu&longs;q; erit.
non autem quia ratione &longs;itus interdum
maiorem re vera acquirat grauitatem,
interdum verò amittat, cùm eiu&longs;dem &longs;it
&longs;emper grauitatis, vbicunque reperiatur;
&longs;ed quia magis, minu&longs;uè in circumferen
tia grauitat, vt in D magis &longs;upra circum
ferentiam DA grauitat, quàm in L &longs;upra
circumferentiam LD. hoc e&longs;t, &longs;i pon
dus à circumferentiis, recti&longs;q; lineis &longs;u
&longs;tineatur; circumferentia AD magis &longs;u
&longs;tinebit pondus in D, quàm circumfe
rentia DL pondere exi&longs;tente in
nus enim coadiuuat CD, quàm CL.
Præterea quando pondus e&longs;t in L, &longs;i e&longs;
&longs;et omnino liberum, penitu&longs;q; &longs;olutum, deor&longs;um per LS moueretur;
ni&longs;i à linea CL prohiberetur, quæ pondus in L vltra lineam LS per
impellendoq; pondus partim &longs;u&longs;tentabit. ni&longs;i enim &longs;u&longs;tineret, ip&longs;iq;
reniteretur, deor&longs;um per lineam LS moueretur, non autem per
circumferentiam LD. &longs;imiliter CD ponderi in D renititur, cùm
illud per circumferentiam DA moueri cogat. eodemq; modo
exi&longs;tente pondere in A, linea CA pondus vltra lineam AS per
circumferentiam AO moueri compellet. e&longs;t enim angulus CAS
acutus; cùm angulus ACS &longs;it rectus. lineæ igitur CA CD ali
qua ex parte, non tamen ex æquo ponderi renituntur. & quotie&longs;
cunque angulus in circumferentia circuli à lineis à centro
mundi S, & centro C prodeuntibus, fuerit acutus; idem eue
nire &longs;imiliter o&longs;tendemus. Quoniam autem mixtus angulus CLD
rentia contineantur; & angulus C
erit reliquus quare circumferentia
DA, hoc e&longs;t de&longs;cen&longs;us ponderis in D propior erit motui natu
rali ponderis in D &longs;oluti, lineæ &longs;cilicet DS, quàm circumferen
tia LD lineæ LS. minus igitur linea CD ponderi in D reniti
tur, quàm linea CL ponderi in L. linea ideo CD minus &longs;u&longs;tinet,
quàm CL; pondu&longs;q; magis liberum erit in D, quàm in L:
cùm pondus naturaliter magis per DA moueatur, quàm per LD. quare grauius erit in D, quàm in L.
&longs;imiliter o&longs;tendemus CA
minus &longs;u&longs;tinere, quàm CD: pondu&longs;q; magis in A, quàm in D li
berum, grauiu&longs;q, e&longs;&longs;e. Ex parte deinde inferiori ob ea&longs;dem cau&longs;as,
quò pondus propius fuerit ip&longs;i G, magis detinebitur, vt in H ma
gis à linea CH, quàm in K à linea CK. nam cùm angulus CHS
maior &longs;it angulo CkS, ad rectitudinem magis appropinquabunt
&longs;e &longs;e lineæ CH HS, quàm Ck kS; atq; ob id pondus magis deti
nebitur à CH, quàm à Ck &longs;i enim CH HS in vnam conuenirent
lineam vt euenit pondere exi&longs;tente in G; tunc linea CG totum &longs;u
&longs;tineret' pondus in G, ita vt immobilis per&longs;i&longs;teret. quò igitur
minor erit angulus linea CH, & de&longs;cen&longs;u ponderis &longs;oluti, &longs;cilicet
HS contentus, eò minus quoq; eiu&longs;modi linea pondus detinebit.
& vbi minus detinebitur, ibi magis liberum, grauiu&longs;q; exi&longs;tet.
Præterea &longs;i pondus in k liberum e&longs;&longs;et, atq; &longs;olutum, per lineam
k S moueretur; à linea verò Ck prohibetur, quæ cogit pondus
citrà lineam k S per circumferentiam k H moueri. ip&longs;um enim
quodammodo retrahit, retrahendoq; &longs;u&longs;tinet. ni&longs;i enim &longs;u&longs;tineret.
pondus deor&longs;um per rectam k S moueretur, non autem per cir
cumferentiam k H. &longs;imiliter CH pondus retinet, cùm per circum
ior e&longs;t angulo CKS,
reliquus SHG reliquo SKH maior. circumferentia igitur k H, hoc
e&longs;t de&longs;cen&longs;us ponderis in k, propior erit motui naturali ponderis in
k &longs;oluti, hoc e&longs;t lineæ k S, quàm circumferentia HG lineæ HS. mi
nus idcirco detinet linea Ck, quàm CH: cùm pondus naturali
ter magis moueatur per k H, quàm per HG. &longs;imili ratione o&longs;ten
detur, quò minor erit angulus SkH, lineam Ck minus &longs;u&longs;tinere.
lus SOC non &longs;olum minor e&longs;t angulo
CKS, verùm etiam omnium angulorum
à punctis CS prodeuntium, verticemq;
in circumferuntia OkG habentium mi
nimus; erit
& eiu&longs;modi omnium minimus. ergo de
&longs;cen&longs;us ponderis in O propior erit motui
naturali ip&longs;ius in O &longs;oluti, quàm in alio
&longs;itu circumferentiæ OkG. lineaq; CO
minus pondus &longs;u&longs;tinebit, quàm &longs;i pon
dus in quouis alio fuerit &longs;itu eiu&longs;dem cir
cumferentiæ OG. &longs;imiliter quoniam con
tingentiæ angulus SOk, & angulo SDA,
& SAO, ac quibu&longs;cunq; &longs;imilibus e&longs;t mi
nor; erit de&longs;cen&longs;us ponderis in O motui
naturali ip&longs;ius ponderis in O &longs;oluti pro
pior, quàm in alio &longs;itu circumferentiæ
ODF. Præterea quoniam linea GO pon
dus in O dum deor&longs;um mouetur, impelle
re non pote&longs;t, ita vt vltra lineam OS mo
ueatur; cùm linea OS circulum non &longs;ecet,
&longs;ed contingat; angulu&longs;q; SOC &longs;it rectus, & non acutus; pondus
in O nihil &longs;upra lineam CO grauitabit. neq; centro innitetur.
quem
admodum in quouis alio puncto &longs;upra O accideret. erit igitur pon
dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu,
quàm in quouis alio circumferentiæ FOG. ac idcirco in hoc
grauius erit, hoc e&longs;t magis grauitabit, quàm in alio &longs;itu. & quò
propius fuerit ip&longs;i O remotiori grauius erit. lineaq; CO horizonti
æquidi&longs;tans erit. non tamen puncti C horizonti (vt ip&longs;i exi&longs;ti
mant) &longs;ed ponderis in O con&longs;tituti, cùm ex centro grauitatis
ponderis &longs;ummendus &longs;it horizon. quæ omnia demon&longs;trare opor
tebat.
Si autem libræ brachium ip&longs;o CO
fuerit maius, putá quantitate CD; erit
quoq; pondus in O grauius. circulus de
&longs;cribatur OH, cuius centrum &longs;it D, &longs;e
midiameterq; DO. tanget circulus OH
circulum FOG in puncto O, lineamq;
OS, quæ ponderis in O rectus, natura
li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con
tinget. & quoniam angulus SOH mi
nor e&longs;t angulo SOG, erit de&longs;cen&longs;us
ponderis in O per circumferentiam OH
motui naturali OS propior, quàm per
circumferentiam OG. magis ergo li
berum, atq; &longs;olutum, ac per con&longs;equens
grauius erit in O, centro libræ exi&longs;ten
te in D, quàm in C. &longs;imiliter o&longs;ten
detur, quò maius fuerit brachium DO,
pondus in O adhuc grauius e&longs;&longs;e.
Si verò idem circulus AFBG,
cuius centrum &longs;it R, propius fuerit
mundi centro S; circulumqué à pun
cto S ducatur contingens ST; punctum
T (vbi grauius e&longs;t pondus) magis
à puncto A di&longs;tabit, quàm punctum
O. ducantur enim à punctis OT ip&longs;i
CS perpendiculares OMTN; conne
ctanturq; RT; &longs;itq; centrum R in li
nea CS; lineaq; ARB ip&longs;i ACB æqui Quoniam igitur triangula COS
RTS &longs;unt rectangula; erit SC ad CO,
vt CO ad CM. &longs;imiliter SR ad RT,
vt RT ad RN. cùm itaq; &longs;it RT ip
maiorem habebit proportionem SC
ad CO, quàm SR ad RT. quare ma
iorem quoq; proportionem habebit
CO ad CM, quàm RT ad RN. mi
&longs;ecetur
igitur RN in P, ita vt RP &longs;it ip&longs;i
CM æqualis; & à puncto P ip&longs;is MONT æquidi&longs;tans ducatur
PQ, quæ circumferentiam AT &longs;ecet in Q: deniq; connectatur
RQ. quoniam enim duæ CO CM duabus RQRP &longs;unt æqua
lus MCO angulo PRQ æqualis. angulus autem MCA rectus
æqualis, & circumferentia OA circumferentiæ QA æqualis quo
que erit. punctum idcirco T, quia magis à puncto A di&longs;tat,
quàm Q; magis quoq; à puncto A di&longs;tabit, quàm punctum O. &longs;imiliter o&longs;tendetur, quò propius fuerit circulus mundi centro, eun
dem magis di&longs;tare. atq; ita vt prius demon&longs;trabitur pondus in cir
cumferentia TAF centro R inniti, in circumferentia verò TG
à linea detineri; atq; in puncto T grauius e&longs;&longs;e.
Si autem punctum G e&longs;&longs;et
in centro mundi; tunc quò
pondus propius fuerit ip&longs;i G,
grauius erit: & vbicunq; po
natur pondus præterquàm in
ip&longs;o G, &longs;emper centro C inni
tetur, vt in K. nam ducta
G k, efficiet hæc (&longs;ecun
dùm quam fit ponderis natu
ralis motus) vná cum libræ
brachio k C angulum acu
tum. æquicruris enim trian
guli CkG ad ba&longs;im anguli
ad k, & G &longs;unt &longs;emper acuti.
Conferantur autem inuicem hæc duo, pondus videlicet in k, &
pondus in D: erit pondus in k grauius, quàm in D. nam iuncta
DG, cùm tres anguli cuiu&longs;cunque trianguli duobus &longs;int rectis
æquales, & trianguli CDG æquicruris angulus DCG maior &longs;it
angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba&longs;im an
guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis
minores. horumq; dimidii; angulus &longs;cilicet CDG angulo CKG
minor erit. quare cùm pondus in k &longs;olutum naturaliter per
KG moueatur, pondusq; in D per DG, tanquam per &longs;patia,
quibus in centrum mundi feruntur; linea CD, hoc e&longs;t libræ
brachium magis adhærebit motui naturali ponderis in D pror
&longs;us &longs;oluti, lineæ &longs;cilicet DG; quàm Ck motui &longs;ecundùm kG
effecto. magis igitur &longs;u&longs;tinebit linea CD, quàm Ck.
ac pro
pterea pondus in k ex &longs;uperius dictis grauius erit, quàm in D.
Præterea quoniam pondus in K &longs;i e&longs;&longs;et omnino liberum, pror&longs;u&longs;q;
&longs;olutum, deor&longs;um per k G moueretur; ni&longs;i à linea C k prohibere
tur, quæ pondus vltra lineam KG per circumferentiam KH mo
ueri cogit; linea C k pondus partim &longs;u&longs;tinebit, ip&longs;iq; renitetur;
cùm illud per circumferentiam k H moueri compellat. &
quoniam angulus CDG minor e&longs;t angulo CkG, & angulus CDk
angulo CkH e&longs;t æqualis; erit reliquus GDk reliquo G k H maior.
circumferentia igitur k H motui naturali ponderis in k &longs;oluti, li
quàm circumferentia Dk li
neæ DG. quare linea CD
ponderi in D magis renititur,
quàm linea C k ip&longs;i ponde
ri in K. ergo pondus in k
grauius erit, quàm in D.
Similiter o&longs;tendetur pondus,
quò fuerit ip&longs;i F propius, vt
in L, minus grauitare: pro
pius verò ip&longs;i G, vt in H,
grauius e&longs;&longs;e.
Si verò centrum mundi
S e&longs;&longs;et inter puncta CG;
primùm quidem &longs;imili
ter o&longs;tendetur pondus vbi
cunq; po&longs;itum centro C
initi, vt in H. ductis enim
HG HS, angulus ad
ba&longs;im GHC æquicruris tri
anguli CHG e&longs;t &longs;emper
acutus: quare & SHC ip
&longs;o minor erit quoq; &longs;em
per acutus. ducatur au
tem à puncto S ip&longs;i CS
perpendicularis Sk. di
co pondus grauius e&longs;&longs;e in k, quàm in alio &longs;itu circumferentiæ FKG.
& quò propius fuerit ip&longs;i F, vel G, minus grauitare. Accipiantur
ver&longs;us F puncta DL, connectanturq; LC LS DC DS, produ
canturq; LS DS k SHS v&longs;q; ad circuli circumferentiam in EM
NO; connectanturq; CE, CM, CN, CO. Quoniam enim
quare vt LS ad DS ita erit SM
maior autem e&longs;t LS, quàm DS; & SM ip&longs;a SE.
eademq;
ratione kN minorem e&longs;&longs;e DM o&longs;tendetur. rur&longs;us quoniam re
ctangulum OSH æquale e&longs;t rectangulo kSN; ob eandem cau&longs;am
HO maior erit kN. eodemq; pror&longs;us modo kN omnibus a
liis per punctum S tran&longs;euntibus minorem e&longs;&longs;e demon&longs;trabitur.
& quoniam æquicrurium triangulorum CLE DCM latera LC
CE lateribus DC CM &longs;unt æqualia; ba&longs;is verò LE maior e&longs;t
DM: erit angulus LCE angulo DCM maior. quare ad ba&longs;im
anguli C
nores erunt. & horum dimidii, angulus &longs;cilicet CLS angulo CDS
minor erit. ergo pondus in
in D &longs;upra DC grauitabit. magisqué centro innitetur in L, quàm
in D. &longs;imiliter o&longs;tendetur in D magis
ergo
eademq; pror
&longs;us ratione quoniam kN minor e&longs;t HO, erit angulus CKS an
gulo CHS maior. quare pondus in H magis centro C innite
tur, quàm in k. & hoc modo o&longs;tendetur, vbicunq; in circum
ferentia FDG fuerit pondus, minus in K centro C inniti, quàm
in alio &longs;itu: & quò propius fuerit ip&longs;i F, vel G, magis inniti. dein
de quoniam angulus CkS maior e&longs;t CDS, & CDk æqualis
e&longs;t CkH: erit reliquus SkH reliquo SDk minor. quare cir
cumferentia k H propior erit motui naturali recto ponderis in K
&longs;oluti, lineæ &longs;cilicet k S, quàm circumferentia D k motui DS. &
ideo linea CD magis ip&longs;i ponderi in D renititur, quàm CK
ponderi in k con&longs;tituto. hacq; ratione o&longs;tendetur angulum
SHG maiorem e&longs;&longs;e SkH: & per con&longs;equens lineam CH magis
ponderi in H reniti, quàm CK ponderi in K. &longs;imiliter demon
&longs;trabitur lineam C
ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck
grauitare, quàm in quouis alio &longs;itu fuerit circumferentiæ FDG. & quò propius fuerit ip&longs;i F, vel G, minus grauitare.
grauius ergo
erit in k, quàm in alio &longs;itu: minu&longs;q; graue erit, quò propius fue
rit ip&longs;i F, vel G.
Si deniq; centrum C
e&longs;&longs;et in centro mundi,
pondus vbicunque con
&longs;titutum manere mani
fe&longs;tum e&longs;t. vt po&longs;ito pon
dere in D, linea CD to
tum &longs;u&longs;tinebit pondus;
cùm ip&longs;ius ponderis in D
horizonti &longs;it perpendicupondus ergo ma
nebit.
Quoniam autem in his hactenus demon&longs;tratis, nullam de gra
uitate brachii libræ mentionem fecimus, idcirco &longs;i brachii quoq;
grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu
dinis ex pondere, brachioq; compo&longs;itæ inueniri poterit, circulo
rumq; circumferentiæ &longs;ecundum di&longs;tantiam à centro libræ ad
hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue
ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libræ bra
chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i
derata grauitate reperiemus.
Ex dictis igitur, con&longs;iderando li
bram, vt longè à mundi centro a
be&longs;t, quemadmodum ip&longs;i fecere, &longs;i
cuti etiam actu e&longs;t, apparet fal&longs;itas
dicentium pondus in A grauius e&longs;&longs;e,
quàm in alio &longs;itu. &longs;imulq; fal&longs;um e&longs;&longs;e,
quò pondus à linea FG magis di&longs;tat nam punctum O pro
pius e&longs;t ip&longs;i FG, quàm punctum A. e&longs;t enim linea à puncto O ip&longs;i FG
perpendicularis ip&longs;a CA minor. de
inde ex puncto A pondus velocius mo
ueri, quàm ab alio &longs;itu, e&longs;t quoque
fal&longs;um. ex puncto enim O pondus ve
locius mouebitur, quàm ex puncto
A; cùm in O &longs;it magis liberum, atq;
&longs;olutum, quàm in alio &longs;itu: de&longs;cen&longs;us
qué ex puncto O propior &longs;it motui na
turali recto, quàm quilibet alius de
&longs;cen&longs;us.
Præterea cùm ex re
ctiori, & obliquiori
&longs;u
A
D; & in D, quàm in
L; primùm quidem fal
&longs;um exi&longs;timant, &longs;i pon
dus aliquod collocatum
fuerit in quocunq; &longs;itu
circunferentiæ, vt in D,
rectum eius de&longs;cen&longs;um
per rectam lineam DR
ip&longs;i FG parallelam, tam
quàm &longs;ecundùm mo
bere; &longs;icuti prius dictum
e&longs;t. In quocunq; enim
&longs;itu pondus aliquod con
&longs;tituatur, &longs;i naturalem
eius ad propium locum
motionem &longs;pectemus,
cùm rectá ad eum &longs;ua
ptè natura moueatur, &longs;up
po&longs;ita totius vniuer&longs;i figu
ra, eiu&longs;modi erit; vt
&longs;emper
naturaliter mouetur, ra
tionem habere videatur
lineæ à circumferentia ad centrum productæ. non igitur natura
les de&longs;cen&longs;us recti cuiuslibet &longs;oluti ponderis per lineas fieri po&longs;
&longs;unt inter &longs;e &longs;e parallelas; cùm omnes in centrum mundi conue
niant. &longs;upponunt deinde ponderis ex D in A per rectam lineam
ver&longs;us centrum mundi motum eiu&longs;dem e&longs;&longs;e quantitatis, ac &longs;i fui&longs;
&longs;et ex O in C: ita vt punctum A æqualiter à centro mundi &longs;it
di&longs;tans, vt C. quod e&longs;t etiam fal&longs;um; nam punctum A magis
à centro mundi di&longs;tat, quàm C: maior enim e&longs;t linea à cen
nea à centro mundi v&longs;q; ad A rectum &longs;ubtendat angulum à li
neis AC, & à puncto C ad centrum mundi contentum. ex qui
bus non &longs;olum &longs;uppo&longs;itio illa, qua libram DE in AB redire demon
&longs;trant, verùm etiam omnes ferè ip&longs;orum demon&longs;trationes ruunt.
ni&longs;i forta&longs;&longs;e dixerint, hæc omnia propter maximam à centro mun
di v&longs;q; ad nos di&longs;tantiam adeo in&longs;en&longs;ibilia e&longs;&longs;e, vt propter in&longs;en
&longs;ibilitatem tanquam vera &longs;upponi po&longs;sint: cùm omnes
hæc tractauerunt, tanquam nota &longs;uppo&longs;uerint. præ&longs;ertim quia
&longs;en&longs;ibilitas illa non efficit, quin de&longs;cen&longs;us ponderis ex L in D
(vt eorum verbis vtar) minus capiat de directo, quàm de&longs;cen
&longs;us DA. &longs;imiliter arcus DA magis de directo capiet, quàm cir
cumferentia EV. quocirca vera erit &longs;uppo&longs;itio; aliæq; demon
&longs;trationes in &longs;uo robore permanebunt. Concedamus etiam pon
&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; &
quælibet puncta in lineis horizonti æquidi&longs;tantibus accepta æ
qualiter à centro mundi di&longs;tare: non tamen propterea &longs;equetur,
veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius
e&longs;&longs;e, quàm in alio &longs;itu, vt in L. &longs;i enim verum e&longs;&longs;et, quò pon
dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam,
quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende
ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal
&longs;um e&longs;&longs;e ita demon&longs;tratur.
Sint circumferentiæ AL AM inter &longs;e &longs;e æquales; & conne
ctatur LM, quæ AB &longs;ecet in X: erit LM ip&longs;i FG æquidi&longs;tans,
ip&longs;iq; AB perpendicularis. & XM ip&longs;i XL æqualis erit.
&longs;i igi
tur pondus ex L moueatur in A per circumferentiam LA, rectus
eius motus erit &longs;ecundùm lineam LX. &longs;i verò moueatur ex A
in M per circumferentiam AM, &longs;ecundùm rectam eius motus
erit XM. quare de&longs;cen&longs;us ex L in A æqualis erit de&longs;cen&longs;ui ex A
in M; tum ob circumferentias æquales, tum propter rectas li
neas ip&longs;i AB perpendiculares æquales. ergo idem pondus in L
æquè graue erit, vt in A, quod e&longs;t fal&longs;um. cum longé grauius &longs;it
in A, quàm in L.
Quamuis autem AMLA æqualiter &longs;ecundùm ip&longs;os de directo
capiant; dicent forta&longs;&longs;e, quia tamen principium de&longs;cen&longs;us ex L
&longs;cilicet LD minus de directo capit, quàm principium de&longs;cen&longs;us
ex A, &longs;cilicet AN; pondus in A grauius erit, quàm in L. nam
cùm circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t)
æqualis, quæ &longs;ecundùm ip&longs;os de directo capit CT; LD verò
de directo capit PO. ideo pondus grauius erit in A, quàm in L.
quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer
&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum,
tum grauius, tum leuius e&longs;&longs;e. quod e&longs;t impo&longs;sibile.
hoc e&longs;t, &longs;i
de&longs;cen&longs;um con&longs;ideremus ponderis in L, quatenus ex L in A de
&longs;cendit, grauius erit, quàm &longs;i eiu&longs;dem ponderis de&longs;cen&longs;um con
&longs;ideremus ex L in D tantùm. neq; enim negare po&longs;&longs;unt ex ei&longs;
demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca
piat LX, &longs;iue PC. de&longs;cen&longs;us verò AM, quin &longs;imiliter de directo
quoq; hoc modo acci
piant, atq; ita accipe
re &longs;it nece&longs;&longs;e. &longs;i enim li
bram DE in AB redire
demon&longs;trare volunt, com
parando de&longs;cen&longs;us pon
deris in D cum de&longs;cen
&longs;u ponderis in E, nece&longs;&longs;e
e&longs;t, vt o&longs;tendant rectum
de&longs;cen&longs;um OC corre
&longs;pondentem circumferen
tiæ DA maiorem e&longs;&longs;e re
cto de&longs;cen&longs;u TH circum
ferentiæ EV corre&longs;pondente. &longs;i enim partem tantùm totius de
&longs;cen&longs;us ex D in A acciperent, vt D k; o&longs;tenderentq; magis cape
re de directo de&longs;cen&longs;um Dk, quàm æqualis portio de&longs;cen&longs;us ex
puncto E. &longs;equetur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon
dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo
ta &longs;it in kI. &longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo
lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S;
o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ
qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k
grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. & &longs;i rur&longs;us
o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re
ctiorem e&longs;&longs;e æquali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;equetur
libram SI ad AB propius accedere, nunquam tamen in AB per
uenire demon&longs;trabunt. &longs;i igitur libram DE in AB redire demon
&longs;trare volunt, nece&longs;&longs;e e&longs;t, vt de&longs;cen&longs;um ponderis ex D in A de di
recro capere quantitatem lineæ ex puncto D ip&longs;i AB ad rectos
angulos ductæ accipiant. atq; ita, &longs;i æquales de&longs;cen&longs;us DA AN
inuicem comparemus, qui æqualiter de directo capient OC CT,
eueniet idem pondus in D æquè graue e&longs;&longs;e, vt in A. &longs;i verò por
tiones tantum ex D A accipiamus; grauius erit in A, quàm
in D. ergo ex diuer&longs;itate tantùm modi con&longs;iderandi, idem pon
dus, & grauius, & leuius e&longs;&longs;e continget. non autem ex ip&longs;a na
In&longs;uper ip&longs;orum &longs;uppo&longs;itio non a&longs;&longs;erit, pondus &longs;ecun
dùm &longs;itum grauius e&longs;&longs;e, quantò in eodem &longs;itu minus obliquum
e&longs;t principium ip&longs;ius de&longs;cen&longs;us. Suppo&longs;itio igitur &longs;uperius alla
ta, hoc e&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo
dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; non &longs;olum ex his, quæ
diximus, vllo modo concedi pote&longs;t; &longs;ed quoniam huius oppo&longs;i
tum o&longs;tendere quoq; non e&longs;t difficile: &longs;cilicet idem pondus in
æqualibus circumferentiis, quò minus obliquus e&longs;t de&longs;cen&longs;us, ibi
minus grauitare.
Sint enim vt prius cir
inter &longs;e &longs;e æquales; &longs;itq;
punctum L propè F. &
connectatur LM, quæ
ip&longs;i AB perpendicularis
erit. & LX ip&longs;i XM
æqualis. deinde propè
M inter MG quoduis
accipiatur punctum P.
fiatq; circumferentia PO
circumferentiæ AM æ
qualis. erit punctum O
propè A. connectanturq; CL, CO, CM, CP, OP.
& à
puncto P ip&longs;i OC perpendicularis ducatur PN. & quoniam cir
cumferentia AM circumferentiæ OP e&longs;t æqualis: erit angu
lus
cto CNP e&longs;t æqualis: erit quoq; reliquus XMC trianguli MCX
reliquo NPC trianguli PCN æqualis. &longs;ed & latus CM lateri
CP e&longs;t æquale: ergo triangulum MCX triangulo PCN æquale
erit. latu&longs;q; MX lateri NP æquale.
quare linea PN ip&longs;i LX æqua
lis erit. ducatur præterea à puncto O linea OT ip&longs;i AC æqui
di&longs;tans, quæ NP &longs;ecet in V. atq; ip&longs;i OT à puncto P perpendi
cularis ducatur, quæ quidem inter OV cadere non pote&longs;t; nam
cùm angulus ONV &longs;it rectus; erit OVN acutus. quare OVP
obtu&longs;us erit. non igitur linea à puncto P ip&longs;i OT intra OV
duo enim anguli vnius
trianguli, vnus quidem
rectus, alter verò ob
tu&longs;us e&longs;&longs;et. quod e&longs;t im
po&longs;sibile. cadet ergo in
linea OT in parte VT. &longs;itq; PT. erit PT &longs;ecun
dùm ip&longs;os rectus circum
ferentiæ OP de&longs;cen&longs;us.
Quoniam igitur angulus
ONV e&longs;t rectus; erit
ior. quare OT ip&longs;a
quoq; ON maior exi&longs;tet. Cùm itaq; linèa OP angulos &longs;ubten
dat rectos ONP OTP; erit quadratum ex OP quadratis ex &longs;imiliter quadratis ex OT TP
&longs;imul æquale. quare quadrata &longs;imul ex ON NP quadratis ex
OT TP &longs;imul æqualia erunt. quadratum autem ex OT maius
e&longs;t quadrato ex ON; cum linea OT &longs;it ip&longs;a ON maior. ergo qua
dratum ex NP maius erit quadrato ex TP. ac propterea linea
TP minor erit linea PN, & linea LX. minus obliquus igitur e&longs;t
de&longs;cen&longs;us arcus LA, quàm arcus OP. ergo pondus in L, ex ip
&longs;orum dictis, grauius erit, quàm in O. quod ex iis, quæ &longs;upra di
ximus e&longs;t manife&longs;tè fal&longs;um, cùm pondus in O grauius &longs;it, quàm
in L. non igitur ex rectiori, & obliquiori motu ita accepto col
ligi pote&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo
dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us. Atq; hinc oritur omnis
fermé ip&longs;orum error in hac re, atq; deceptio: nam quamuis per
accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal
&longs;is fal&longs;um &longs;equitur, quemadmodum ex veris &longs;emper verum, nil
idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si
mis innituntur; fal&longs;i&longs;sima omninò colligunt, atq; concludunt.
decipiuntur quinetiam, dùm libræ contemplationem mathemati
cè &longs;impliciter a&longs;&longs;ummunt; cùm eius con&longs;ideratio &longs;it pror&longs;us me
chanica: nec vllo modo ab&longs;q; vero motu, ac ponderibus (en
bus eorum, quæ libræ accidunt, veræ caulæ reperiri nullo mo
do po&longs;sint.
Præterea &longs;i adhuc &longs;up
po&longs;itionem conceda
mus; à con&longs;ideratione
libræ longè recedunt;
dum eo pacto, vt libra
DE in AB redire de
beat, di&longs;currunt. &longs;emper
enim alterum pondus
&longs;eor&longs;um accipiunt, putá
D, vel E; ac &longs;i modò
modò alterum in libra
con&longs;titutum e&longs;&longs;et, nec
vllo modo ambo con
nexa; cuius tamen oppo&longs;itum omninò fieri oportet; neq; alterum
&longs;ine altero rectè con&longs;iderari pote&longs;t; cùm de ip&longs;is in libra con&longs;ti
tutis &longs;ermo habeatur. cùm enim dicunt, de&longs;cen&longs;um ponderis in
D minus obliquum e&longs;&longs;e de&longs;cen&longs;u ponderis in E; erit pondus in
D per &longs;uppo&longs;itionem grauius pondere in E: quare cùm &longs;it graui
us, nece&longs;&longs;e e&longs;t deor&longs;um moueri, libramq; DE in AB redire: di
&longs;cur&longs;us i&longs;te nullius pror&longs;us momenti e&longs;t. Primùm quidem &longs;em
per argumentantur, ac &longs;i pondera in DE de&longs;cendere debeant,
vnius tantùm &longs;ine alterius connexione con&longs;iderando de&longs;cen&longs;um.
po&longs;tremò tamen ob ponderum de&longs;cen&longs;uum comparationem colli
gentes inferunt, pondus in D deor&longs;um moueri, & pondus in E
&longs;ur&longs;um, vtraq; &longs;imul in libra inuicem connexa accipientes. ve
rùm ex ii&longs;demmet, quibus vtuntur, principiis, ac demon&longs;tratio
nibus, oppo&longs;itum eius, quod defendere conantur, facillimè col
ligi pote&longs;t. Nam &longs;i comparetur de&longs;cen&longs;us ponderis in D cum a
&longs;cen&longs;u ponderis in E, vt ductis EK DH ip&longs;i AB perpendicula
ribus; cùm angulus DCH &longs;it æqualis angulo ECk; & angulus
DHC rectus æqualis e&longs;t recto E k C; & latus DC lateri CE æqua
le: erit triangulum CDH triangulo CEk æquale, & latus DH lacùm
autem angulus DCA
&longs;it angulo ECB æqua
lis: erit quoq; circum
ferentia DA
tiæ BE æqualis. dum
itaq; pondus in D de
&longs;cendit per circumfe
rentiam DA, pondus
in E per circumferen
tiam EB ip&longs;i DA æ
qualem a&longs;cendit. & de
&longs;cen&longs;us
directo (more
capiet DH; a&longs;cen&longs;us verò ponderis in E de directo capiet Ek ip
&longs;i DH æqualem: erit itaq; de&longs;cen&longs;us ponderis in D a&longs;cen&longs;ui pon
deris in E æqualis, & qualis erit propen&longs;io vnius ad motum deor
sum, talis etiam erit re&longs;i&longs;tentia alterius ad motum &longs;ur&longs;um. re
&longs;i&longs;tentia &longs;cilicet violentiæ ponderis in E in a&longs;cen&longs;u naturali po
tentiæ ponderis in D in de&longs;cen&longs;u contrà nitendo apponitur; cùm
&longs;it ip&longs;i æqualis. quò enim pondus in D naturali potentia deor
&longs;um velocius de&longs;cendit, eò tardius pondus in E violenter a&longs;cendit.
quare neutrum ip&longs;orum alteri præponderabit, cùm ab æquali non
proueniat actio. Non igitur pondus in D pondus in E &longs;ur&longs;um
mouebit. &longs;i enim moueret; nece&longs;&longs;e e&longs;&longs;et, pondus in D maiorem
habere virtutem de&longs;cendendo, quàm pondus in E a&longs;cendendo;
&longs;ed hæc &longs;unt æqualia: ergo pondera manebunt. & grauitas pon
deris in D grauitati ponderis in E æqualis erit. Præterea quoniam
&longs;upponunt, quò pondus à linea directionis FG magis di&longs;tat, eò
grauius e&longs;&longs;e: Idcirco ductis quoq; à punctis DE ip&longs;i FG perpen
dicularibus DO EI; &longs;imili modo demon&longs;trabitur, triangulum
CDO triangulo CEI æqualem e&longs;&longs;e: & lineam DO ip&longs;i EI æqua
lem. tam igitur di&longs;tat à linea FG pondus in D, quàm pondus in
E. ex ip&longs;orum igitur rationibus, atq; &longs;uppo&longs;itionibus, pondera
in DE æquè grauia erunt. Amplius quid prohibet, quin libram
DE ex nece&longs;sitate in FG moueri &longs;imili ratione o&longs;tendatur? Pri
&longs;cen&longs;um ponderis in E ver&longs;us B rectiorem e&longs;&longs;e a&longs;cen&longs;u ponderis
in D ver&longs;us F; hoc e&longs;t minus capere de directo a&longs;cen&longs;um pon
deris in D in arcubus æqualibus a&longs;cen&longs;u ponderis in E. &longs;uppona
tur ergo &longs;ecundùm &longs;itum pondus leuius e&longs;&longs;e, quantò in eodem &longs;i
tu minus rectus e&longs;t a&longs;cen&longs;us: quæ quidem &longs;uppo&longs;itio, adeò ma
nife&longs;ta e&longs;&longs;e videtur, veluti ip&longs;orum altera. Quoniam igitur a&longs;cen
&longs;us ponderis in E rectior e&longs;t a&longs;cen&longs;u ponderis in D; per &longs;uppo&longs;i
tionem pondus in D leuius erit pondere in E. ergo pondus in
D &longs;ur&longs;um à pondere in E mouebitur, ita vt libra in FG perue
niat. atq; ita demon&longs;trari poterit, libram DE in FG moueri.
quæ quidem demon&longs;tratio inutilis e&longs;t pror&longs;us, ea&longs;demq; patitur
difficultates. licet enim tanquàm verum admittatur pondus in E
a&longs;cendendo grauius e&longs;&longs;e pondere in D &longs;imiliter a&longs;cendendo,
non tamen ex hoc &longs;equitur, pondus in E de&longs;cendendo grauius
e&longs;&longs;e pondere in D a&longs;cendendo. Neutra igitur harum demon
&longs;trationum libram DE, vel in AB redire, vel in FG moue
ri, o&longs;tendentium, vera e&longs;t.
Præterea &longs;i ip&longs;orum &longs;uppo&longs;itionem, eorumq; verborum vim
rectè perpendamus; alium certè habere &longs;en&longs;um con&longs;piciemus. nam
cùm &longs;emper &longs;patium, per quod naturaliter pondus mouetur, à cen
tro grauitatis ip&longs;ius ponderis ad centrum mundi, in&longs;tar rectæ li
neæ à centro grauitatis ad centrum mundi productæ, &longs;it &longs;umendum;
tantò huiusmodi ponderis de&longs;cen&longs;us, magis, minusuè obliquus
dicetur; quantò &longs;ecundùm &longs;patium in&longs;tar prædictæ lineæ de&longs;igna
tum, magis, aut minus (naturalem tamen locum petens, &longs;emperq;
magis ip&longs;i appropinquans) mouebitur; ita vt tantò obliquior de
&longs;cen&longs;us dicatur, quantò recedit ab eiu&longs;modi &longs;patio: rectior verò,
quantò ad idem accedit. & in hoc &longs;en&longs;u &longs;uppo&longs;itio illa nemini
difficultatem parere debet, adeò enim veritas eius con&longs;picua e&longs;t;
rationiq; con&longs;entanea: vt nulla pro&longs;us manife&longs;tatione egere vi
deatur.
Si itaq; pondus &longs;olutum in &longs;itu D
collocatum ad propium locum mo
ueri debeat; proculdubio po&longs;ito cen
tro mundi S, per lineam DS moue
bitur. &longs;imiliter pondus in E &longs;olutum
per lineam ES mouebitur. quare &longs;i
(vt rei veritas e&longs;t) ponderis de&longs;cen
&longs;us magis, minu&longs;uè obliquus dicetur
&longs;ecundùm rece&longs;&longs;um, & acce&longs;&longs;um ad
&longs;patia per lineas DSES de&longs;ignata,
iuxta naturales ip&longs;orum ad propria lo
ca lationes; con&longs;picuum e&longs;t, minus
obliquum e&longs;&longs;e de&longs;cen&longs;um ip&longs;ius E
per EG, quàm ip&longs;ius D per DA:
cùm angulum SEG angulo SDA
minorem e&longs;&longs;e &longs;upra o&longs;ten&longs;um &longs;it. qua
re in E pondus magis grauitabit,
quàm in D. quod e&longs;t penitus oppo
&longs;itum eius, quod ip&longs;i o&longs;tendere cona
ti &longs;unt. In&longs;urgent autem forta&longs;&longs;e
contrarios, &longs;i igitur (dicent) pondus
in E grauius e&longs;t pondere in D, libra
DE in hoc &longs;itu minimè per&longs;i&longs;tet, quod
&longs;ed in FG mouebitur. quibus re&longs;pondemus, plurimum referre, &longs;iue
con&longs;ideremus pondera, quatenus &longs;unt inuicem di&longs;iuncta, &longs;iue quate
nus &longs;unt &longs;ibi inuicem connexa. alia e&longs;t enim ratio ponderis in E &longs;ine
connexione ponderis in D, alia verò eiu&longs;dem alteri ponderi con
nexi; ita vt alterum &longs;ine altero moueri non po&longs;sit. nam ponde
ris in E, quatenus e&longs;t &longs;ine alterius ponderis connexione, rectus
naturalis de&longs;cen&longs;us e&longs;t per lineam ES; quatenus verò connexum
e&longs;t ponderi in D, eius naturalis de&longs;cen&longs;us non erit amplius per
lineam ES, &longs;ed per lineam ip&longs;i CS parallelam. magnitudo enim
ex ponderibus ED, & libra DE compo&longs;ita, cuius grauitatis cen
trum e&longs;t C, &longs;i nullibi &longs;u&longs;tineatur, deor&longs;um eo modo, quo reperi
tur, &longs;ecundùm grauitatis centrum per rectam à centro grauita
tis C ad centrum mundi S ductam naturaliter mouebitur, donec libra igitur DE vná cum pon
deribus eo modo, quo reperitur, deor&longs;um mouebitur, ita vt pun
ctum C per lineam CS moueatur, donec C in S, libraq; DE in
Hk perueniat; habeatq; libra in Hk eandem, quam prius habe
bat po&longs;itionem; hoc e&longs;t Hk &longs;it ip&longs;i DE æquidi&longs;tans. connectantur
igitur DH Ek. manife&longs;tum e&longs;t, dum libra DE in Hk mouetur pun
cta DE per lineas DH Ek moueri, quippe exi&longs;tentibus inter &longs;e
&longs;e, ip&longs;iq; CS æqualibus, & æquidi&longs;tantibus. Quare pondera in
DE, quatenus &longs;unt &longs;ibi inuicem connexa, &longs;i ip&longs;orum naturalem mo
tum &longs;pectemus, non &longs;ecundùm lineas DS ES, &longs;ed &longs;ecundùm
LDH MEk ip&longs;i CS æquidi&longs;tantes mouebuntur. ponderis ve
rò in E liberi, ac &longs;oluti, naturalis propen&longs;io erit per ES: ponderis
autem in D &longs;imiliter &longs;oluti erit per DS. ac propterea non e&longs;t incon
ueniens idem pondus modò in E, modò in D, grauius e&longs;&longs;e in E,
quàm in D. &longs;i verò pondera in ED &longs;ibi inuicem connexa, quate
nusq; &longs;unt connexa con&longs;iderauerimus; erit ponderis in E natura
lis propen&longs;io per lineam MEK: grauitas enim alterius ponde
ris in D efficit, nè pondus in E per lineam ES grauitet, &longs;ed per
Ek. quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet
pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra
que enim &longs;e impediunt, nè ad propria loca Cùm igi
tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it &longs;ecundùm
LDH MEK: erit
dem lineas HDL KEM. atq; a&longs;cen&longs;us ponderis in E magis, mi
nu&longs;uè obliquus dicetur; quantò &longs;ecundùm &longs;patium magis, mi
nu&longs;uè iuxta lineam Mk mouebitur. hocq; pror&longs;us modo iuxta li
neam LH &longs;ummendus e&longs;t, tùm de&longs;cen&longs;us, tùm a&longs;cen&longs;us ponde
ris in D. &longs;i itaq; pondus in E deor&longs;um per EG moueretur; pon
dus in D &longs;ur&longs;um per DF moueret. & quoniam angulus CEK
æqualis e&longs;t angulo CDL, & angulus CEG angulo CDF æqua
lis; erit reliquus GEK reliquo LDF æqualis. cùm autem &longs;up
po&longs;itio illa, quæ ait, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, quan
tò in eodem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; tanquam clara,
atq; con&longs;picua admittatur; proculdubio hæc quoq; accipienda
erit; nempè, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo
dem &longs;itu minus obliquus e&longs;t a&longs;cen&longs;us. cùm non minus manife&longs;ta,
æqualis
igitur erit de&longs;cen&longs;us ponderis in E
a&longs;cen&longs;ui ponderis in D. eandem
enim obliquitatem habet de&longs;cen&longs;us
ponderis in E, quam habet a&longs;cen
&longs;us ponderis in D; & qualis erit
propen&longs;io vnius ad motum deor&longs;um,
talis quoq; erit re&longs;i&longs;tentia alterius ad
motum &longs;ur&longs;um.
pondus in D &longs;ur&longs;um mouebit. neq;
pondus in D deor&longs;um mouebitur, ita
vt &longs;ur&longs;um moueat pondus in E. nam
CDH æqualis; erit reliquus MEB
reliquo HDA æqualis. de&longs;cen&longs;us
igitur ponderis in D a&longs;cen&longs;ui ponde
ris in E æqualis erit. non ergo pon
dus in D pondus in E &longs;ur&longs;um moue
bit. ex quibus &longs;equitur pondera in
DE, quatenus &longs;unt &longs;ibi inuicem con
nexa, æquè grauia e&longs;&longs;e.
Alia deinde ratio, li
bram &longs;imiliter DE in AB
redire o&longs;tendens, cùm in
quiunt, exi&longs;tente trutina in
CF meta e&longs;t CG. & quo
niam angulus DCG maior
e&longs;t angulo ECG; pondus
in D grauius erit pondere
in E; ergo libra DE in AB
redibit: nihil meo iudicio
concludit. figmentumq;
hoc de trutina, & meta po
tius omittendum, ac &longs;ilen
dum; cùm &longs;it pror&longs;us voluntarium. nece&longs;sitas enim cur pondus
in D ex maiore angulo &longs;it grauius; curq; maior angulus maioris
&longs;it cau&longs;a grauitatis; nu&longs;quam apparet. &longs;i autem comparentur in
uicem anguli, cùm angulus GCD &longs;it æqualis angulo FCE; &longs;i angu
lus GCD e&longs;t cau&longs;a grauitatis; quare angulus FCE &longs;imiliter gra
uitatis non e&longs;t cau&longs;a? Huius autem rei eam in medium rationem
afferre videntur, quoniam CG e&longs;t meta, & CF trutina. &longs;i (inquiunt)
CG e&longs;&longs;et trutina, & CF meta, tunc angulus FCE grauitatis e&longs;&longs;et
cau&longs;a; non autem DCG ip&longs;i æqualis. quæ quidem ratio imma
ginaria pror&longs;us, ac voluntaria e&longs;&longs;e videtur. quid enim refert, &longs;iue tru
tina &longs;it in CF, &longs;iue in CG, cùm libra DE in eodem &longs;emper pun
cto C &longs;u&longs;tineatur? Vt autem eorum deceptio clarius appa
reat.
Sit eadem libra AB, cu
ius medium C. &longs;it deinde
tota FG trutina. eaq; im
mobilis exi&longs;tat; quæ libram
AB in puncto C &longs;u&longs;tineat.
moueaturq; libra in DE. &
quoniam trutina e&longs;t, & &longs;u
pra, & infra libram, quis
nam angulus erit cau&longs;a gra
uitatis, cùm libra DE in
eodemdicent for&longs;an, &longs;i trutina à potentia
in F &longs;u&longs;titeneatur, tunc CG erit tanquam meta, & angulus
DCG grauitatis erit cau&longs;a. &longs;i verò &longs;u&longs;tineatur in G, tunc FCE
erit cau&longs;a grauitatis, CF verò tanquam meta erit. cuius quidem
rei nulla videtur e&longs;&longs;e cau&longs;a, ni&longs;i immaginaria. meta enim (quod
aiunt) nullam pror&longs;us vim attractiuam, quandoq; ex maioris an
guli parte, quandoq; ex parte minoris habere videtur. Verùm à dua
bus potentiis &longs;u&longs;tineatur trutina, in F &longs;cilicet, & in G, quod præ ne
ce&longs;sitate fieri pote&longs;t, veluti &longs;i potentia in F &longs;it adeò debilis, vt ex &longs;e
ip&longs;a medietatem tantùm ponderis &longs;u&longs;tinere quæat: &longs;itq; potentia in
G ip&longs;i potentiæ in F æqualis, vtræq;
deribus &longs;u&longs;tineant. tunc quis nam angulus erit cau&longs;a grauitatis?
non
CF, & in F &longs;u&longs;tinetur. neq;
DCG, cùm trutina &longs;it in
CG, & in G quoq; &longs;u&longs;ti
neatur; non igitur anguli
grauitatis cau&longs;a erunt. ergo
neq; libra DE ab hoc &longs;itu
ob hanc cau&longs;am mouebiHanc autem eorum
&longs;ententiam dupliciter con
firmare videntur. primùm quidem a&longs;&longs;erunt Ari&longs;totelem in quæ&longs;tio
nibus mechanicis has duas tantùm quæ&longs;tiones propo&longs;ui&longs;&longs;e; eiu&longs;q;
demon&longs;trationes, tum maiori, & minori angulo, tùm trutinæ po&longs;i
tioni inniti. Affirmant deinde experientiam hoc idem docere;
hoc e&longs;t libram DE trutina exi&longs;tente in CF, in AB horizonti
æquidi&longs;tantem redire. quando autem trutina e&longs;t in CG, in FG
moueri. Verùm neq; Ari&longs;toteles, neq; experientia huic eorum
opinioni fauent, quin potius aduer&longs;antur. quantùm enim atti
net ad experientiam decipiuntur, ip&longs;a quidem experientia ma
nife&longs;tum e&longs;t hoc accidere, quando libræ quoq; centrum, vel &longs;u
pra, vel infra libram fuerit collocatum: non autem trutina dun
taxat &longs;upra, vel infra exi&longs;tente, id contingere.
Nam &longs;i libra AB habeat
centrum C &longs;upra libram;
&longs;itq; trutina CD infra li
bram; moueaturq; libra in
EF; tunc EF rur&longs;us in AB
horizonti æquidi&longs;tantem
redibit. &longs;imiliter &longs;i libra
centrum C habeat infra li
bram, &longs;itq; trutina CD &longs;u
pra libram, & moueatur
libra in EF; patet libram
ex parte F deor&longs;um moue
ri, trutina &longs;upra libram e
xi&longs;tente. & in quocunq; a
lio &longs;itu fuerit trutina, idem
&longs;emper eueniet. non igitur
trutina, &longs;ed centrum libræ
harum diuer&longs;itatum cau
&longs;a erit.
Animaduertendum e&longs;t
itaq; in hac parte difficulter materialem libram con&longs;titui po&longs;&longs;e,
quæ in vno tantùm puncto &longs;u&longs;tineatur; quemadmodum mente
concipimus. brachiaq; ab eiu&longs;modi centro adeò æqualia habeat,
non &longs;olum in longitudine, verùm etiam in latitudine, & profun
ditate, vt omnes partes hinc indé ad vnguem æqueponderent.
hoc enim materia difficilimè patitur. quocirca &longs;i centrum in ip&longs;a
libra e&longs;&longs;e con&longs;iderauerimus, ad &longs;en&longs;um confugiendum non e&longs;t:
cùm artificilia ad &longs;ummum illud perfectionis gradum ab artifice
deduci minimè po&longs;sint. In aliis verò experientia quidem appa
rentia docere poterit; propterea quod, quamquam centrum libræ
&longs;it &longs;emper punctum, quando tamen &longs;upra libram fuerit, parùm re
fert, &longs;i libra in eo puncto adamu&longs;&longs;im minimè &longs;u&longs;tineatur; quia cùm
&longs;it &longs;emper &longs;upra libram, idem &longs;emper eueniet. &longs;imili quoq; modo
quando e&longs;t infra libram: quod tamen non accidit centro in ip&longs;a li
bra exi&longs;tente. &longs;i enim ad vnguem &longs;emper in illo medio non &longs;u
&longs;tineatur, diuer&longs;itatem efficiet; cùm facillimum &longs;it, centrum il
Quòd autem Ari&longs;toteles duas tantùm quæ&longs;tiones propo
&longs;uerit, cur &longs;cilicet trutina &longs;uperius exi&longs;tente, &longs;i libra non &longs;it
horizonti æquidi&longs;tans in æquilibrium, hoc e&longs;t horizonti æqui
di&longs;tans redit: &longs;i autem trutina deor&longs;um fuerit con&longs;tituta, non
redit; &longs;ed adhuc &longs;ecundùm partem depre&longs;&longs;am mouetur: verum
quidem e&longs;t. non tamen eius demon&longs;trationes maiori, & mino
ri angulo, po&longs;itioniqué trutinæ (vt ip&longs;i dicunt) innituntur. In
hoc enim mentem philo&longs;ophi a&longs;ignantis rationem diuer&longs;itatis
motuum libræ minimè attingunt. tantùm enim abe&longs;t philo&longs;o
phum has diuer&longs;itates in angulos referre, vt potius in cau&longs;a e&longs;&longs;e
dicat magnitudinis alterius brachii libræ exce&longs;&longs;um à perpendiculo,
modò ex vna, modò ex altera parte contingentem.
Vt trutina &longs;uperius in
CF exi&longs;tente, perpendicu
lum erit FCG, quod &longs;e
cundùm ip&longs;um in centrum
mundi &longs;emper vergit;
quod quidem libram mo
tam in DE in partes di
uidit inæquales; & maior
pars e&longs;t ver&longs;us D: id au
tem, quod plus e&longs;t, deor
&longs;um fertur; ergo ex par
te D deor&longs;um libra moue
bitur, donec in AB re
deat. &longs;i verò trutina &longs;it
in CG deor&longs;um, erit GCF perpendiculum, quod libram DE
in partes inæquales &longs;imiliter diuidit: maior autem pars erit ver&longs;us
E; quare ex parte E deor&longs;um libra mouebitur. quod vt rectè in
telligatur, cùm trutina e&longs;t &longs;upra libram, libræ quoq; centrum &longs;u
pra libram e&longs;&longs;e intelligendum e&longs;t; & &longs;i deor&longs;um, centrum quoque
deor&longs;um: vt infra patebit. Aliter ip&longs;a Ari&longs;totelis demon&longs;tratio
nihil concluderet. exi&longs;tente enim centro in ip&longs;a libra, vt in C; quo
cunq; modo moueatur libra, nunquam perpendiculum FG libram, quare Ari&longs;totelis
&longs;ententia ip&longs;is non &longs;olum non fauet, verùm etiam maximè aduer
&longs;atur. quòd non &longs;olum ex &longs;ecunda, & tertia huius liquet; verùm
quia exi&longs;tente centro &longs;upra libram pondus eleuatum maiorem
propter &longs;itum acquirit grauitatem. ex quò contingit redditus li
bræ ad æqualem horizonti di&longs;tantiam. è contra verò, quando
centrum e&longs;t infra libram. Quæ omnia hoc modo o&longs;tendentur;
&longs;upponendo ea, quæ &longs;upra declarata &longs;unt. &longs;cilicet pondus ex quò
loco rectius de&longs;cendit, grauius fieri. & ex quo rectius a&longs;cendit, gra
uius quoq; reddi.
Sit libra AB horizonti
æquidi&longs;tans, cuius centrum
C &longs;it &longs;upra libram, perpen
diculumq; &longs;it CD. &longs;intq; in
AB ponderum æqualium
centra grauitatis po&longs;ita: mo
taq; &longs;it libra in EF. Dico
pondus in E maiorem ha
bere grauitatem, quàm pon
dus in F. & ob id libram
EF in AB redire. Produ
catur primùm CD v&longs;q; ad
mundi
inde AC CB EC CF HS
ip&longs;i HS æquidi&longs;tantes du
cantur Ek GFL. Quoniam
igitur naturalis de&longs;cen&longs;us re
ctus totius magnitudinis,
libræ &longs;cilicet EF &longs;ic con&longs;ti
tutæ vná cum ponderibus,
e&longs;t
trum H per rectam HS; erit
quoq; ponderum in EF ita po&longs;sitorum de&longs;cen&longs;us &longs;ecundùm re
ctas Ek FL ip&longs;i HS parallelas; &longs;icuti &longs;upra demon&longs;trauimus.
&longs;us ponderum in EF ma
gis, minu&longs;uè obliquus di
cetur &longs;ecundùm acce&longs;&longs;um,
& rece&longs;&longs;um iuxta lineas Ek
FL de&longs;ignatum.
tem
bus lateribus BD DE &longs;unt
æqualia; anguliq; ad D &longs;unt
CB æquale. & cùm pun
ctum C &longs;it immobile; dum
puncta AB mouentur, cir
culi circumferentiam de&longs;cri
bent, cuius &longs;emidiameter
erit AC. quare centro C,
circulus de&longs;cribatur AEBF.
puncta AB EF in circuli
circumferentia erunt. &longs;ed
cùm EF &longs;it ip&longs;i AB æqua
EAF circumferentiæ AFB
æqualis. quare dempta
communi AF, erit circumferentia EA circumferentiæ FB æqua
lis. Quoniam autem mixtus angulus CEA e&longs;t æqualis mixto
CFB; & HFB ip&longs;o CFB e&longs;t maior; angulus verò HEA ip&longs;o
CEA minor; erit angulus HFB angulo HEA maior. à quibus
gulo kEA maior. ergo de&longs;cen&longs;us ponderis in E minus obliquus
erit a&longs;cen&longs;u ponderis in F. & quamquam pondus in E de&longs;cen
dendo, & pondus in F a&longs;cendendo per circumferentias mouean
tur æquales; quia tamen pondus in E ex hoc loco rectius de&longs;cen
dit, quàm pondus in F a&longs;cendit: idcirco naturalis potentia pon
deris in E re&longs;i&longs;tentiam violentiæ ponderis F &longs;uperabit. quare
maiorem grauitatem habebit pondus in E, quàm pondus in F.
ergo pondus in E deor&longs;um, pondus verò in F &longs;ur&longs;um mouebitur: quod demon&longs;trare oportebat.
Huius autem effectus ratio ab Ari&longs;totele po&longs;ita, hic manife&longs;ta in
tueri pote&longs;t. &longs;it enim punctum N vbi CS EF &longs;e inuicem &longs;ecant.
& quoniam HE e&longs;t ip&longs;i HF æqualis; erit NE maior NF. li
nea ergo CS, quam perpendiculum vocat, libram EF in partes di
uidet inæquales. cùm itaq; pars libræ NE &longs;it maior NF; atq; id,
quod plus e&longs;t, nece&longs;&longs;e e&longs;t, deor&longs;um ferri: libra ergo EF ex parte E
deor&longs;um mouebitur, donec in AB redeat.
Ex iis præterea, quæ ha
ctenus dicta &longs;unt inferre li
cet, libram EF velocius ab
eo &longs;itu in AB moueri; vndè
linea EF in directum pro
tracta in centrum mundi
perueniat. vt &longs;it EFS recta
linea. & quoniam CD
CH, &longs;unt inter &longs;e &longs;e æqua
les. &longs;i igitur centro C, &longs;pa
tioq; CD, circulus de&longs;cri
batur DHM; erunt pun
cta DH in circuli circum
ferentia. Quoniam au
tem CH ip&longs;i EF e&longs;t per
pendicularis; continget li
nea EHS circulum DHM
in puncto H. pondus igi
tur in H (&longs;icuti &longs;upra de
mon&longs;trauimus) grauius
erit, quàm in alio &longs;itu circuli DHM. ergo magnitudo ex EF
ponderibus, & libra EF compo&longs;ita, cuius centrum grauitatis e&longs;t
in H, in hoc &longs;itu magis grauitabit, quàm in quocunq; alio &longs;itu ab hoc igitur &longs;itu velo
cius, quàm à quocunq;
alio mouebitur. & &longs;i H
propius fuerit ip&longs;i D mi
nus grauitabit, minu&longs;q;
ab eo &longs;itu mouebitur.
&longs;emper enim de&longs;cen&longs;us
obliquior e&longs;t, & minus re
ctus. libra ergo EF velo
cius ab hoc &longs;itu mouebi
tur, quàm ab alio &longs;itu. &
&longs;i propius ad AB acce
det, inde minus mouebi
tur. Deinde quò longius
punctum H à puncto C
di&longs;tabit, velocius moue
bitur; quod
&longs;totele in principio quæ&longs;t
io num mechanicarum, &
ex &longs;uperius dictis patet; verùm etiam ex iis, quæ infra in &longs;exta
propo&longs;itione dicemus, manife&longs;tum erit. libra igitur EF, quò ma
gis ab eius centro di&longs;tabit, adhuc velocius mouebitur.
Sit deinde libra AB,
cuius centrum C &longs;it infra li
bram; &longs;intq; in AB pon
dera æqualia; libraq; &longs;it
mota in EF. Dico maio
rem habere grauitatem
pondus in F, quàm pondus
in E. atq; ideo libram EF
deor&longs;um ex parte F moue
ri. Producatur DC ex
vtraq; parte v&longs;q; ad mun
di centrum S, & v&longs;q; ad
O, lineaq; HS ducatur,
cui à punctis EF æquidi
&longs;tantes ducantur GEk FL;
connectanturq; CE CF:
atq; centro C, &longs;patioq; CE
circulus de&longs;cribatur AEO
BF. &longs;imiliter demon&longs;tra
bitur puncta ABEF in
circuli circumferentia e&longs;&longs;e;
de&longs;cen&longs;umq; libræ EF vná
cum ponderibus rectum &longs;e
cundùm lineam HS fieri;
ponderumq; in EF &longs;ecun
lineas GK FL ip&longs;i HS æquidi&longs;tantes. Quoniam autem an
gulus CFP æqualis e&longs;t angulo CEO: erit angulus HFP angulo
HEO maior. angulus verò HFL æqualis e&longs;t angulo HEG. à
quibus igitur &longs;i demantur anguli HFP HEO, erit angulus
LFP angulo GEO minor. quare de&longs;cen&longs;us ponderis in F rectior
erit a&longs;cen&longs;u ponderis in E. ergo naturalis potentia ponderis in
F re&longs;i&longs;tentiam violentiæ ponderis in E &longs;uperabit. & ideo ma
iorem habebit grauitatem pondus in F, quàm pondus in E.
Pondus igitur in F deor&longs;um, pondus verò in E &longs;ur&longs;um mo
uebitur.
Ari&longs;totelis quoq; ratio hic per&longs;picua erit.
&longs;it enim punctum
&longs;ecant; erit NF maior
NE. & quoniam CO per
pendiculum (&longs;ecundùm
ip&longs;um) libram EF in par
tes inæquales diuidit, &
maior pars e&longs;t ver&longs;us F, hoc
e&longs;t NF; libra EF ex par
te F deor&longs;um mouebitur:
cùm id, quod plus e&longs;t, deor
&longs;um feratur.
Similiter, éx dictis
quoq; eliciemus libram EF
centrum habens infra li
bram, quò magis à &longs;itu
AB di&longs;tabit, velocius mo
ueri. centrum enim graui
tatis H, quò magis á pun
cto D di&longs;tat, eò volecius
pondus ex EF ponderibus,
libraq; EF compo&longs;itum
mouebitur, donec angulus
CHS rectus euadat. ad
huc in&longs;uper velocius moue
bitur, quò libram à centro
C magis di&longs;tabit.
Ex ip&longs;orum quinetiam rationibus, ac fal&longs;is &longs;upo&longs;itionibus iam
declaratos libræ effectus, ac motus deducere, ac manife&longs;tare libet;
vt quanta &longs;it veritatis efficacia appareat, quippè ex fal&longs;is etiam
eluce&longs;cere contendit.
Exponantur eadem, &longs;ci
licet &longs;it circulus AEBF;
libraqué AB, cuius cen
trum C &longs;it &longs;upra libram,
moueatur in EF. dico
pondus in E maiorem ibi
habere grauitatem, quàm
pondus in F; libramq; EF
in AB redire. Ducantur
à punctis EF ip&longs;i AB
perpendiculares EL FM,
quæ inter &longs;e æquidi&longs;tan
tes
Quoniam igitur angulus FNM e&longs;t æqualis angulo ENL, & an
gulus
quo
lo NMF &longs;imile. vt igitur NE ad EL, ita NF ad FM; & per
mutando vt EN ad NF, ita EL ad FM. &longs;ed cùm &longs;it HE ip&longs;i
HF æqualis, erit EN maior NF; quare & EL maior erit FM.
& quoniam dum pondus in E per
pondus in F per circumferentiam FB ip&longs;i circumferentiæ EA
æqualem a&longs;cendit; de&longs;cen&longs;u&longs;q; ponderis in E de directo (vt ip
&longs;i dicunt) capit EL: a&longs;cen&longs;us verò ponderis in F de directo ca
pit FM; minus de directo capiet a&longs;cen&longs;us ponderis in F, quàm
de&longs;cen&longs;us ponderis in E. maiorem igitur grauitatem habebit pon
dus in E, quàm pondus in F.
Producatur CD ex vtraq; parte in OP, quæ lineam EF in
puncto S &longs;ecet. & quoniam (vt aiunt) quò magis pondus à li
nea directionis OP di&longs;tat, eò fit grauius; idcirco hoc quoq; me
dio pondus in E maiorem habere
&longs;tendetur. Ducantur à punctis EF ip&longs;i OP perpendiculares EQ
FR. &longs;imili ratione o&longs;tendetur, triangulum QES triangulo RFS
&longs;imile e&longs;&longs;e; lineamq; EQ ip&longs;a RF maiorem e&longs;&longs;e. pondus itaq;
in E magis à linea OP di&longs;tabit, quàm pondus in F; ac propterea
pondus in E maiorem habebit grauitatem pondere in F. ex quibus
reditus libræ EF in AB manife&longs;tus apparet.
Si autem centrum libræ
&longs;it infra libram, tunc pon
dus depre&longs;&longs;um maiorem
habere grauitatem eleuato
ii&longs;dem mediis o&longs;tendetur.
ducantur à punctis EF ip
&longs;i AB perpendiculares EL
FM. &longs;imiliter demon&longs;tra
bitur EL maiorem e&longs;&longs;e
FM; & ob id de&longs;cen&longs;us
ponderis in F minus de di
recto capiet, quàm a&longs;cen
&longs;us ponderis in E: quocirca re&longs;i&longs;tentia violentiæ ponderis in E &longs;u
perabit naturalem propen&longs;ionem ponderis in F. ergo pondus in E
pondere in F grauius erit.
Producatur etiam CD ex vtraq; parte in OP; ip&longs;iq; à punctis
EF perpendiculares ducantur EQ FR. eodem pror&longs;us modo
o&longs;tendetur, lineam EQ maiorem e&longs;&longs;e FR. pondus ideò in E ma
gis à linea directionis OP di&longs;tabit, quàm pondus in F. maio
rem igitur grauitatem habebit pondus in E, quàm pondus in F. ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri.
Ari&longs;toteles itaq; has duas tantùm quæ&longs;tiones propo&longs;uit, ter
tiamq; reliquit; &longs;cilicet cùm centrum libræ in ip&longs;a e&longs;t libra: hanc
autem ommi&longs;sit, vt notam, quemadmodum res valde notas præ
termittere &longs;olet. nam cui dubium, &longs;i pondus in eius centro gra
uitatis &longs;u&longs;tineatur, quin maneat? Ea verò, quæ ex ip&longs;ius &longs;enten
tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten
tiam minimè protuli&longs;&longs;e nam cùm in &longs;ecunda parte &longs;e
cundæ quæ&longs;tionis proponit, cur libra, trutina deor&longs;um con&longs;tituta,
quando deor&longs;um lato pondere qui&longs;piam id amouet, non a&longs;cen
dit, &longs;ed manet? non a&longs;&longs;erit adhuc libram deor&longs;um moueri; &longs;ed
manere. quod in vltima quoq; conclu&longs;ione colligi&longs;&longs;e videtur.
Ve
rùm hoc non &longs;olum nobis non repugnat, &longs;ed &longs;i rectè intelligitur,
maximè &longs;uffragatur.
Sit enim libra AB
horizonti æquidi&longs;tans,
cuius centrum E &longs;it
infra libram. quia ve
rò Ari&longs;toteles libram,
&longs;icuti actu e&longs;t, con&longs;ide
rat; ideò nece&longs;&longs;e e&longs;t
trutinam, vel aliquid
aliud infra centrum E
collocare, vt EF
(quod quidem truti
na erit) ita vt centrum
E &longs;u&longs;tineat. &longs;itq; per
pendiculum ECD. & vt libra AB ab hoc moueatur &longs;itu; dicit
Ari&longs;toteles, ponatur pondus in B, quod cùm &longs;it graue, libram ex
parte B deor&longs;um mouebit; putá in G. ita vt propter impedimen
tum deor&longs;um amplius moueri non poterit. non enim dicit Ari
&longs;toteles, moueatur libra ex parte B deor&longs;um, quou&longs;q; libuerit; dein
de relinquatur, vt nos diximus: &longs;ed præcipit, vt in ip&longs;o B po
natur pondus, quod ex ip&longs;ius natura deor&longs;um &longs;emper mouebi
tur; donec libra trutinæ, &longs;iue alicui alii adhæreat. & quando B erit
in G, erit libra in GH; in quo &longs;itu, ablato pondere, manebit:
cùm maior pars libræ à perpendiculo &longs;it ver&longs;us G, quæ e&longs;t DG,
quàm DH. nec deor&longs;um amplius mouebitur; nam libra, vel
trutinæ, vel alteri cuipiam, quod centrum libræ &longs;u&longs;tineat, incum
bet. &longs;i enim huic non adhæreret, libra ex parte G deor&longs;um ex
ip&longs;ius &longs;ententia moueretur; cùm id, quod plus e&longs;t, &longs;cilicet DG,
deor&longs;um ferri &longs;it nece&longs;&longs;e.
Cæterum quis adhuc dicere poterit, &longs;i paruum imponatur pon
dus in B, mouebitur quidem libra deor&longs;um, non autem v&longs;q; ad
G. in quò &longs;itu &longs;ecundùm Ari&longs;totelem, ablato pondere, mane
re deberet. quod experimento patet; cùm in vna tantùm libræ
extremitate, impo&longs;ito onere, hocq; vel maiore, vel minore, libra
plus, minu&longs;uè inclinetur. Quod e&longs;t quidem veri&longs;&longs;imum, centro &longs;upra
libram, non autem infra, neq; in ip&longs;a libra collocato. Vt exempli
gratia.
Sit libra horizonti æ
quidi&longs;tans AB, cuius cen
trum C &longs;it &longs;upra libram,
perpendiculumq; CD ho
rizonti perpendiculare,
quod ex parte D produca
tur in H. Quoniam enim
con&longs;iderata libræ grauita
te, erit punctum D libræ
centrum grauitatis. &longs;i ergo
in B paruum imponatur
pondus, cuius centrum
grauitatis &longs;it in puncto B; magnitudinis ex libra AB, & pondere
in B compo&longs;itæ non erit amplius centrum grauitatis D; &longs;ed erit in
uitatem libræ AB. Connectatur CE.
Quoniam autem pun
ctum C e&longs;t immobile, dum libra mouetur, punctum E circuli cir
cumferentiam EFG de&longs;cribet, cuius &longs;emidiameter CE, & cen
trum C. quia verò CD horizonti e&longs;t perpendicularis, linea CE
horizonti perpendicularis nequaquam erit. quare magnitudo ex
AB, & pondere in B compo&longs;ita minimè in hoc &longs;itu manebit; &longs;ed
tiam EFG mouebitur; donec CE horizonti perpendicularis eua
dat; hoc e&longs;t, donec CE in CDF perueniat. atq; tunc libra AB
mota erit in kL, in quo &longs;itu libra vná cum pondere manebit. nec
deor&longs;um amplius mouebitur. Si verò in B ponatur pondus graui
us; centrum grauitatis totius magnitudinis erit ip&longs;i B propius, vt in
M. & tunc libra deor&longs;um, donec iuncta CM in linea CDH per
ueniat, mouebitur. Ex maiore igitur, & minore pondere in B po
&longs;ito, libra plus, minu&longs;uè inclinabitur. ex quo &longs;equitur pondus B
quarta circuli parte minorem &longs;emper circumferentiam de&longs;cribe
re, cùm angulus FCE &longs;it &longs;emper acutus. nunquam enim punctum
B v&longs;q; ad lineam CH perueniet, cùm centrum grauitatis ponde
ris, & libræ &longs;imul &longs;emper inter DB exi&longs;tat. quò tamen pondus
in B grauius fuerit, maiorem quoq; circumferentiam de&longs;cribet.
eò enim magis punctum B ad lineam CH accedet.
Habeat autem libra AB
centrum C in ip&longs;a libra, atq;
in eius medio: erit C libræ
centrum quoq; grauitatis;
à quo ip&longs;i AB, horizontiq;
perpendicularis ducatur FC
G. ponatur deinde in B
quoduis pondus; erit totius
magnitudinis centrum gra
uitatis putá in E; ita vt CE
ad EB &longs;it, vt pondus in B ad libræ grauitatem. & quoniam CE
non e&longs;t horizonti perpendicularis, libra AB, atq; pondus in B
in hoc &longs;itu nunquam manebunt; &longs;ed deor&longs;um ex parte B mouebun
tur, donec CE horizonti fiat perpendicularis. hoc e&longs;t donec li
bra AB in FG perueniat. ex quo patet, quolibet pondus in B
circuli quartam &longs;emper de&longs;cribere.
Sit autem centrum C in
fra libram AB. &longs;itq; DCE
perpendiculum. &longs;imiliter
po&longs;ito in B pondere, cen
trum grauitatis magnitudi
nis ex AB libra, & ponde
re in B compo&longs;itæ in linea
DB erit; vt in F; ita vt DF
ad FB &longs;it, vt pondus in B
ad libræ pondus. Iungatur CF.
& quoniam CD horizonti e&longs;t
perpendicularis; linea CF horizonti nequaquam perpendicula
ris exi&longs;tet. quare magnitudo ex AB libra, ac pondere in B com
po&longs;ita in hoc &longs;itu nunquam per&longs;i&longs;tet; &longs;ed deor&longs;um, ni&longs;i aliquid
impediat, mouebitur; donec CF in DCE perueniat: in quo &longs;itu
libra vná cum pondere manebit. & punctum B erit vt in G, atq;
punctum A in H, libraq; GH non amplius centrum infra, &longs;ed &longs;u
pra ip&longs;am habebit. quod idem &longs;emper eueniet; quamuis mini
mum imponatur pondus in B. ergo priu&longs;quam B perueniat ad
G; nece&longs;&longs;e e&longs;t libram, &longs;iue trutinæ deor&longs;um po&longs;itæ, vel alicui
&longs;tineat, occurrere; ibiq; ad
hærere. ex hoc &longs;equitur, pon
dus in B vltra lineam Dk
&longs;emper moueri; ac circuli
quarta maiorem &longs;emper cir
enim angulus FCE &longs;emper
obtu&longs;us, cùm angulus DCF
&longs;emper &longs;it acutus. quò au
tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe
rentiam de&longs;cribet. nam quò pondus in G leuius fuerit, eò ma
gis pondus in G eleuabitur; libraq; GH ad &longs;itum horizonti æqui
di&longs;tantem propius accedet. quæ omnia ex iis, quæ &longs;upra dixi
mus, manife&longs;ta &longs;unt.
His demon&longs;tratis.
Manife&longs;tum e&longs;t, centrum libræ cau&longs;am e&longs;&longs;e
diuer&longs;itatis effectuum in libra. atq; patet omnes Archimedis de
æqueponderantibus propo&longs;itiones ad hoc pertinentes in omni &longs;itu
veras e&longs;&longs;e. hoc e&longs;t &longs;iue libra &longs;it horizonti æquidi&longs;tans, &longs;iue non:
dummodo centrum libræ in ip&longs;a &longs;it libra; quemadmodum ip&longs;e
con&longs;iderat. & quamquam libra brachia habeat inæqualia, idem eue
niet; eodemq; pro&longs;us modo o&longs;tendetur, centrum libræ diuer&longs;imo
dè collocatum varios producere effectus.
Sit enim libra AB hori
zonti æquidi&longs;tans; & in AB
&longs;int pondera inæqualia, quo
rum grauitatis centrum &longs;it
C: &longs;u&longs;pendaturq; libra in
eodem puncto C. & mo
ueatur libra in DE. mani
lum in DE, &longs;ed in quouis
alio &longs;itu manere.
Sit autem centrum libræ
AB &longs;upra C in F; &longs;itq;
FC ip&longs;i AB, & horizonti
perpendicularis: & &longs;i mo
ueatur libra in DE, linea
CF mota erit in FG; quæ
cùm non &longs;it horizonti per
pendicularis, libra DE
deor&longs;um ex parte D moue
bitur, donec FG in FC
redeat: atq; tunc libra DE
in AB erit, in quò &longs;itu
quoq; manebit.
Et &longs;i centrum libræ F
&longs;it infra libram; &longs;itq; mota
libra in DE; primùm qui
dem manife&longs;tum e&longs;t li
bram in AB manere; in
DE verò deor&longs;um ex par
te E moueri: cùm linea
FG non &longs;it horizonti per
pendicularis.
Ex his determinatis &longs;i libra &longs;it
arcuata, vel libræ brachia angulum
con&longs;tituant; centrumq; diuer&longs;imo
dè collocetur (quamquam hæc pro
priè non &longs;it libra) varios tamen
huius quoq; effectus o&longs;tendere pote
rimus. Vt &longs;it libra ACB, cuius
centrum, circa quod vertitur, &longs;it C. ductaq; AB, &longs;it arcus &longs;iue angulus
ACB &longs;upra lineam AB; & in AB grauitatis centra ponderum
ponantur, quæ in hoc &longs;itu maneant. moueatur deinde libra ab
Dico li
bram ECF in ACB redire. to
tius magnitudinis centrum grauita
tis inueniatur D. & CD iunga
tur. Quoniam enim pondera AB
pendicularis erit. quando igitur
libra erit in ECF, linea CD erit
putá in CG; quæ cùm non &longs;it ho
rizonti perpendicularis; libra ECF in ACB redibit. quod idem
eueniet, &longs;i centrum C &longs;upra libram con&longs;tituatur, vt in H.
Si verò arcus, &longs;iue angulus
ACB, &longs;it infra lineam AB; eo
dem modo libram ECF, cuius
centrum, &longs;iue &longs;it in C, &longs;iue in H,
deor&longs;um ex parte F moueri o
&longs;tendemus.
Sit autem angulus ACB &longs;upra lineam AB; ac libræ centrum
&longs;it H; lineaq; CH libram &longs;u&longs;tineat; & moueatur libra in EKF:
libra EkF in ACB redibit.
Si verò centrum libræ &longs;it D, quocunq; modo moueatur libra;
vbi relinquetur, manebit.
Si deinde punctum H &longs;it infra lineam AB; tunc libra EkF
deor&longs;um ex parte F mouebitur.
Similiq; pror&longs;us ratione, &longs;i an
gulus ACB &longs;it infra lineam AB;
&longs;itq; libræ centrum H; &longs;u&longs;tineaturq;
libra linea CH; &longs;i libra ab hoc mo
ueatur &longs;itu, deor&longs;um ex parte pon
deris inferioris mouebitur. & &longs;i cen
trum libræ &longs;it D; vbi relinquetur,
manebit. &longs;i verò &longs;it in K; &longs;i ab eiu&longs;
modi moueatur &longs;itu, in eundem pro&longs;us redibit. quæ omnia ex iis,
quæ in principio diximus, &longs;unt manife&longs;ta. &longs;imiliter &longs;i centrum li
bræ, vel in altero brachiorum, vel intra, vel extra vtcunq; po
natur; eadem inueniemus.
PROPOSITIO. V.
Duo pondera in libra appen&longs;a, &longs;i libra inter
hæc ita diuidatur, vt partes ponderibus per
mutatim re&longs;pondeant; tàm in punctis appen&longs;is
ponderabunt, quàm &longs;i vtraq; ex diui&longs;ionis pun
cto &longs;u&longs;pendantur.
Sit AB libra, cuius centrum C; &longs;intq; duo pondera EF ex pun
ctis BG &longs;u&longs;pen&longs;a: diuidaturq; BG in H, ita vt BH ad HG
eandem habeat proportionem, quam pondus E ad pondus F.
Dico pondera EF tàm in BG ponderare, quàm &longs;i vtraq; ex pun
cto H &longs;u&longs;pendantur. fiat AC ip&longs;i CH æqualis.
& vt AC ad
CG, ita fiat pondus E ad pondus L. &longs;imiliter vt AC ad CB,
ita fiat pondus F ad pondus M. ponderaq; LM ex puncto A &longs;u
&longs;pendantur. Quoniam enim AC e&longs;t æqualis CH, erit BC ad
CH vt pondus M ad pondus F. & quoniam maior e&longs;t BC,
quàm CH; erit & pondus M ip&longs;o F maius. diuidatur igitur pon
dus M in duas partes QR, &longs;itq; pars Q ip&longs;i F æqualis; erit BC Præterea quo
niam CH e&longs;t æqualis ip&longs;i CA, erit HC ad CG, vt pondus
E ad pondus L: maior autem e&longs;t HC, quàm CG; erit & pondiuidatur itaq; pondus E in duas partes
NO ita, vt pars O &longs;it ip&longs;i L æqualis, erit HC ad CG, vt to
tum NO ad O; & diuidendo, vt HG ad GC, ita N ad O:
conuertendoq; vt CG ad GH, ita O ad N. & iterum com
ponendo, vt CH ad HG, ita ON ad N. vt autem GH
ad HB, ita e&longs;t F ad ON. quare ex æquali, vt CH ad HB, ita F
ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt
F ad N; & permutando, vt Q ad F, ita R ad N. e&longs;t autem pars
Q ip&longs;i F æqualis; quare & pars R ip&longs;i N æqualis erit. Itaq; cùm
pondus L &longs;it ip&longs;i O æquale, & pondus F ip&longs;i Q etiam æquale, atq;
pars R ip&longs;i N æqualis; erunt pondera LM ip&longs;is EF ponderibus
æqualia. & quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon
dus L; pondera EL æqueponderabunt. &longs;imiliter quoniam e&longs;t, vt
AC ad CB, ita
æqueponderabunt. Pondera igitur LM ponderibus EF in BG
appen&longs;is æqueponderabunt. cùm autem di&longs;tantia CA æqualis &longs;it
di&longs;tantiæ CH; &longs;i igitur vtraq; pondera EF in H appendantur,
pondera LM ip&longs;is EF ponderibus in H appen&longs;is æquepondera
bunt. &longs;ed LM ip&longs;is EF in GB quoq; æqueponderant: æquè
igitur grauia erunt pondera EF in GB, vt in H appen&longs;a. tàm igi
tur ponderabunt in BG, quàm in H appen&longs;a.
Sint autem pondera EF in CB appen&longs;a; &longs;itq; C libræ centrum;
& diuidatur CB in H, ita vt CH ad HB &longs;it, vt pondus in F ad
E. Dico pondera EF tàm in CB ponderare, quàm in puncto H.
fiat CA ip&longs;i CH æqualis, & vt CA ad CB, ita fiat pondus F ad
aliud D, quod appendatur in A. Quoniam enim CH e&longs;t æqua
lis CA, erit CH ad CB, vt F ad D; & maior quidem e&longs;t CB,
quàm CH; idcirco D pondere F maius erit. Diuidatur ergo D
in duas partes Gk, &longs;itq; G ip&longs;i F æqualis; erit vt BC ad CH,
vt Gk ad G; & diuidendo, vt BH ad HC, ita K ad G; & conuerVt autem CH ad HB, ita e&longs;t
vt igitur G ad k, ita e&longs;t F ad E; & permutando vt G
&longs;unt autem GF æqualia; erunt & kE inter &longs;e
&longs;e æqualia. cùm itaq; pars G &longs;it ip&longs;i F æqualis, & K ip&longs;i E; erit
totum C k ip&longs;is EF ponderibus æquale. & quoniam AC e&longs;t ip
&longs;i CH æqualis; &longs;i igitur pondera EF ex puncto H &longs;u&longs;pendantur,
pondus D ip&longs;is EF in H appen&longs;is æqueponderabit. &longs;ed & ip&longs;is
æqueponderat in CB, hoc e&longs;t F in B, & E in C; cùm &longs;it vt AC
ad CB, ita F ad. D. pondus enim E ex centro libræ C &longs;u&longs;pen
&longs;um non efficit, vt libra in alterutram moueatur partem. tàm igi
tur grauia erunt pondera EF in CB, quàm in H appen&longs;a.
Sit deniq; libra AB, & ex punctis AB &longs;u&longs;pen&longs;a &longs;int pondera
EF; &longs;itq; centrum libræ C intra pondera; diuidaturq; AB in
D, ita vt AD ad DB &longs;it, vt pondus F ad pondus E. Dico pon
dera EF tàm in AB ponderare, quám &longs;i vtraq; ex puncto D &longs;u&longs;pen
dantur. fiat CG æqualis ip&longs;i CD; & vt DC ad CA, ita fiat
pondus E ad aliud H; quod appendatur in D. vt autem GC ad
CB, ita fiat pondus F ad aliud K; appendaturq; k in G.
e&longs;t, vt BC ad CG, hoc e&longs;t ad CD, ita pondus k ad F; erit K ma
ius pondere F. quare diuidatur pondus k in L, & MN; fiatq;
pars L ip&longs;i F æqualis; erit vt BC ad CD, vt totum LMN ad
L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. vt
igitur BD ad DC, ita pars MN ad F. vt autem AD ad DB,
ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. cùm
verò AD &longs;it ip&longs;a CD maior; erit & pars MN pondere E
maior: diuidatur ergo MN in duas partes MN, &longs;itq; M æqua
lis ip&longs;i E. erit vt AD ad DC, vt NM ad M; & diuidendo, vt
AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M
ad N. vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N
vt E ad H; & permutando, vt M ad E, ita N ad H. &longs;ed ME
&longs;unt inter &longs;e æqualia, erunt NH inter &longs;e&longs;e quoq; æqualia. & quo
niam ita e&longs;t AC ad CD, vt H ad E: pondera HE æqueponde
rabunt. &longs;imiliter quoniam e&longs;t vt GC ad CB, ita F ad k, ponde
pondera igitur Ek HF in li
bra AB, cuius centrum C, æqueponderabunt. cùm autem GC
ip&longs;i CD &longs;it æqualis, & pondus H &longs;it ip&longs;i N æquale; pondera NH
æqueponderabunt. & quoniam omnia æqueponderant, demptis
hoc e&longs;t pondera EF & pondus LM ex centro libræ C &longs;u&longs;pen&longs;a.
quia verò pars L ip&longs;i F e&longs;t æqualis, & pars M ip&longs;i E æqualis; erit
totum LM ip&longs;is FE ponderibus &longs;imul &longs;umptis æquale. & cùm
&longs;it CG ip&longs;i CD æqualis, &longs;i igitur pondera EF ex puncto D &longs;u&longs;pen
dantur, pondera EF in D appen&longs;a ip&longs;i LM æqueponderabunt. quare
LM tàm ip&longs;is EF in AB appen&longs;is æqueponderat, quàm in pun
cto D appen&longs;is. libra enim &longs;emper eodem modo manet.
Ponde
quod
Cor.
Cor.
Cor.
Hæc autem omnia (mechanicè tamen ma
gis) aliter o&longs;tendemus.
Sit libra AB, cuius centrum C; &longs;intq; vt in primo ca&longs;u duo pon
dera EF ex punctis BG &longs;u&longs;pen&longs;a: &longs;itq; GH ad HB, vt pondus
F ad pondus E. Dico pondera EF tàm in GB ponderare, quàm
&longs;i vtraq; ex diui&longs;ionis puncto H &longs;u&longs;pendantur. Con&longs;truantur ea
dem, hoc e&longs;t fiat AC ip&longs;i CH æqualis, & ex puncto A duo ap
pendantur pondera LM, ita vt pondus E ad pondus L, &longs;it vt
CA ad CG; vt autem CB ad CA, ita &longs;it pondus M ad pondus
F. pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t)
æqueponderabunt. Sint deinde puncta NO centra grauitatis pon
derum EF; connectanturq; GN BO; iungaturq; NO, quæ tan
quam libra erit; quæ etiam efficiat lineas GN BO inter &longs;e &longs;e æqui
di&longs;tantes e&longs;&longs;e; à punctoq; H horizonti perpendicularis ducatur
HP, quæ NO &longs;ecet in P, atq; ip&longs;is GN BO &longs;it æquidi&longs;tans. deniq; connectatur GO, quæ HP &longs;ecet in R.
Quoniam igitur
HR e&longs;t lateri BO trianguli GBO æquidi&longs;tans; erit GH ad HB,
vt GR ad RO. &longs;imiliter quoniam RP e&longs;t lateri GN trianguli
OGN æquidi&longs;tans; erit GR ad RO, vt NP ad PO. quare
vt GH ad HB, ita e&longs;t NP ad PO. vt autem GH ad HB, ita
e&longs;t pondus F ad pondus E; vt igitur NP ad PO, ita e&longs;t pondus
F ad pondus E. punctum ergo P centrum erit grauitatis magni
tudinis ex vtri&longs;q; EF ponderibus compo&longs;itæ. Intelligantur itaq;
pondera EF ita e&longs;&longs;e à libra NO connexa, ac &longs;i vna tantùm e&longs;&longs;et
magnitudo ex vtri&longs;q; EF compo&longs;ita, in puncti&longs;q; BG appen&longs;a. &longs;i
igitur ponderum &longs;u&longs;pen&longs;iones BG &longs;oluantur, manebunt pondera
EF ex HP &longs;u&longs;pen&longs;a; &longs;icuti in GB prius manebant. pondera verò EF
in GB appen&longs;a ip&longs;is LM ponderibus æqueponderant, & pondera
EF ex puncto H &longs;u&longs;pen&longs;a, eandem habent con&longs;titutionem ad li
bram AB, quam in BG appen&longs;a: eadem ergo pondera EF ex
H &longs;u&longs;pen&longs;a ei&longs;dem ponderibus LM æqueponderabunt. æquè igi
tur &longs;unt grauia pondera EF in GB, vt in H appen&longs;a.
Similiter demon&longs;trabitur, pondera EF in quibu&longs;cunq; aliis pun
ctis appen&longs;a tàm
&longs;pendantur. &longs;i enim (vt &longs;upra docuimus) in libra pondera inue
niantur, quibus pondera EF æqueponderent; eadem pondera EF
ex H &longs;u&longs;pen&longs;a ei&longs;dem inuentis ponderibus æqueponderabunt; cùm
punctum P &longs;it &longs;emper eorum centrum grauitatis; & HP horizon
ri perpendicularis.
PROPOSITIO. VI.
Pondera æqualia in libra appen&longs;a eam in gra
uitate proportionem habent; quam di&longs;tantiæ, ex
quibus appenduntur.
Sit libra BAC &longs;u&longs;pen&longs;a ex puncto A; & &longs;ecetur AC vtcunq;
in D: ex punctis autem DC appendantur æqualia pondera EF. Dico pondus F ad pondus E eam in grauitate proportionem ha
bere, quam habet di&longs;tantia CA ad di&longs;tantiam AD. fiat enim vt
CA ad AD, ita pondus F ad aliud pondus, quod &longs;it G. Dico pri
múm pondera GF ex puncto C &longs;u&longs;pen&longs;a tantùm ponderare, quan
tùm pondera EF ex punctis DC. Secetur DC bifariam in H, &
ex H appendantur vtraq; pondera EF. ponderabunt EF &longs;imul
&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA
æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD:
deinde ex puncto B appendatur pondus L duplum ponderis F,
hoc e&longs;t æquale duobus ponderibus EF, quod quidem æqueponde
rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC.
igitur, vt CA ad AD, ita e&longs;t pondus F ad pondus G; erit compo
nendo vt CA AD ad AD, hoc e&longs;t vt Ck ad AD, ita ponde
ra &longs;ed cùm &longs;it, vt CA ad AD, ita F pon
dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus
G ad pondus F; & con&longs;equentium dupla, vt DA ad duplam ip&longs;ius
AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus
L. Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt
vt Ck ad &longs;ed vt Ck
ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad
AC. Vt igitur BA ad AC, ita FG pondera ad pondus L.
Qua
re ex &longs;exta eiu&longs;dem primi Archimedis, duo pondera FG ex pun
cto C &longs;u&longs;pen&longs;a tantùm ponderabunt, quantùm pondus L ex B;
hoc e&longs;t quantùm pondera EF ex punctis DC &longs;u&longs;pen&longs;a. Itaq; quo
niam pondera FG tantùm ponderant, quantum pondera EF; &longs;u
blato communi pondere F, tàm ponderabit pondus G in C ap
pen&longs;um, quàm pondus E in D. ac propterea pondus F ad pon
dus G. &longs;ed pondus F ad G erat, vt CA ad AD: ergo & F pon
dus ad pondus E eam in grauitate proportionem habebit, quam ha
bet CA ad AD. quod demon&longs;trare oportebat.
Si verò in libra
BAC pondera EF
æqualia ex punctis
BC &longs;u&longs;pendantur; &longs;i
militer dico pondus
E ad pondus F eam
in grauitate proportionem habere, quàm habet di&longs;tantia CA ad di
&longs;tantiam AB. fiat AD ip&longs;i AB æqualis, & ex puncto D &longs;u&longs;pen
datur pondus G æquale ponderi F; quod etiam ip&longs;i E erit æquale.
& quoniam AD e&longs;t æqualis ip&longs;i AB; pondera FG æqueponde
rabunt, eandemq; habebunt grauitatem. cùm autem grauitas pon
deris E ad grauitatem ponderis G &longs;it, vt CA ad AD; erit graui
tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc e&longs;t
CA ad AB. quod erat quoq; o&longs;tendendum.
ALITER.
Sit libra BAC, cu
ius centrum A; in pun
ctis verò BC pondera
appendantur æqualia G
F: &longs;itq; primùm cen
trum A vtcunque inter
BC. Dico pondus F ad
pondus G eam in graui
tate proportionem habere, quam habet di&longs;tantia CA ad di&longs;tan
tiam AB. fiat vt BA ad AC, ita pondus F ad aliud H, quod ap
pendatur in B: pondera HF ex A æqueponderabunt. &longs;ed cùm
pondera FG &longs;int æqualia, habebit pondus H ad pondus G ean
dem proportionem, quam habet ad F. vt igitur CA ad AB, ita
e&longs;t H ad G. vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui
tatem ip&longs;ius G; cùm in eodem puncto B &longs;int appen&longs;a. quare vt CA
ad AB, ita grauitas ponderis H ad grauitatem ponderis G. cùm au
tem grauitas ponderis F in C appen&longs;i &longs;it æqualis grauitati ponderis
H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA
ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. quod demon&longs;trare
oportebat.
Si verò libra B
AC &longs;ecetur vtcunq;
in D, & in DC ap
pendantur pondera
æqualia EF. Dico
&longs;imiliter ita e&longs;&longs;e gra
uitatem ponderis F ad grauitatem ponderis E, vt di&longs;tantia CA ad
di&longs;tantiam AD. fiat AB æqualis ip&longs;i AD, & in B appendatur
pondus G æquale ponderi E, & ponderi F. Quoniam enim AB e&longs;t
æqualis AD; pondera GE æqueponderabunt. &longs;ed cùm grauitas
ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, & graui
tas ponderis E &longs;it æqualis grauitati ponderis G; erit grauitas pon
deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA
ad AD. quod demon&longs;trare oportebat.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, quò pondus à centro
libræ magis di&longs;tat, eò grauius e&longs;&longs;e; & per con&longs;e
quens velocius moueri.
detur.
Sit enim &longs;tate
ræ &longs;capus AB, cu
ius trutina &longs;it in
C; &longs;itq; &longs;tateræ
appendiculum E. appendatur in A
pondus D, quod
æqueponderet ap
pendiculo E in F
appen&longs;o. aliud quoq; appendatur pondus G in A, quod etiam
appendiculo E in B appen&longs;o æqueponderet. Dico grauitatem
ponderis D ad grauitatem ponderis G ita e&longs;&longs;e, vt CF ad CB.
Quoniam enim grauitas ponderis D e&longs;t æqualis grauitati ponde
ris E in F appen&longs;i, & grauitas ponderis G e&longs;t æqualis grauitati pon
deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in
F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu
tas ip&longs;ius E in F, ad grauitatem ip&longs;ius E in B; grauitas autem pon
igitur grauitas ponderis D ad grauitatem ponderis G, ita e&longs;t CF
ad CB &longs;i ergo pars &longs;capi CB in partes diuidatur æquales, &longs;olo
pondere E, & propius, & longius à puncto C po&longs;ito; ponderum
grauitates, quæ ex puncto A &longs;u&longs;penduntur inter &longs;e &longs;e notæ erunt.
&longs;ius G grauitatis ip&longs;ius D tripla, quod demon&longs;trare oportebat.
Alio quoq; modo &longs;tatera vti po&longs;&longs;umus, vt
ponderum grauitates notæ reddantur.
Sit &longs;capus AB, cuius tru
tina &longs;it in C; &longs;itq; &longs;tateræ ap
pendiculum E, quod appen
datur in A; &longs;intqué pon
dera DG inæqualia, quorum
inter &longs;e &longs;e grauitatum propor
tiones quærimus: appenda
tur pondus D in B, ita vt ip&longs;i
E æqueponderet. &longs;imiliter pondus G appendatur in F, quod ei
dem ponderi E æqueponderet. dico D ad G ita e&longs;&longs;e, vt CF ad
CB. Quoniam enim pondera DE æqueponderant, erit D ad E,
vt CA ad CB. cùm autem pondera quoque GE æquepon
derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua
li pondus D ad pondus G ita erit, vt CF ad CB. quod o&longs;tende
re quoq; oportebat.
PROPOSITIO VII.
PROBLEMA.
Quotcunque datis in libra ponderibus
vbicunque appen&longs;is, centrum libræ inuenire,
ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma
neant.
Sit libra AB, &longs;intq; data quotcunque pondera CDEFG.
accipiantur in libra vtcunque puncta AHkLB, ex quibus
data pondera Centrum libræ inuenire oportet,
ex quo &longs;i fiat &longs;u&longs;pen&longs;io, data pondera maneant. Diuidatur
AH in M, ita vt HM ad MA, &longs;it vt grauitas ponderis
C ad grauitatem ponderis D. deinde diuidatur BL in N, ita
vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon
deris F. diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt
grauitas ponderum FG ad grauitatem ponderum CD. tandem
què diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon
derum CDFG ad grauitatem ponderis E. Quoniam igitur pon
dera CDFG tàm ponderant in O, quàm CD in M, & FG in N;
æqueponderabunt pondera CD in M, & FG in N, & pondus E
in K, &longs;i ex puncto P &longs;u&longs;pendantur. cùm verò pondera CD tan
tùm ponderent in M, quantùm in AH, & FG in N, quantùm
in LB; pondera CDFG ex AHLB punctis &longs;u&longs;pen&longs;a, & pon
dus E ex k, &longs;i ex P &longs;u&longs;pendantur, æqueponderabunt, atq; mane
bunt. Inuentum e&longs;t ergo centrum libræ P, ex quo data pondera
manent. quod facere oportebat.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, &longs;i ponderum CDEFG
centra grauitatis e&longs;&longs;ent in AHKLB punctis; e&longs;
&longs;et punctum P magnitudinis ex omnibus CD
EFG ponderibus compo&longs;itæ centrum graui
tatis.
Hoc enim ex definitione centri grauitatis patet, cùm ponde
ra, &longs;i ex puncto P &longs;u&longs;pendantur, maneant.
DE VECTE.
LEMMA.
Sint quatuor magnitudines A
BCD; &longs;itq; A maior B, & C ma
ior D. Dico A ad D maiorem
habere proportionem; quàm
habet B ad C.
Quoniam enim A ad C maiorem habet pro
portionem, quàm B ad C; & A ad D maio
rem
ad C: A igitur ad D maiorem habebit, quam B
ad C. quod demon&longs;trare oportebat.
PROPOSITIO I.
Potentia &longs;u&longs;tinens pondus vecti appen&longs;um;
eandem ad ip&longs;um pondus proportionem habe
bit, quam vectis di&longs;tantia inter fulcimentum, ac
ponderis &longs;u&longs;pen&longs;ionem ad di&longs;tantiam à fulcimen
to ad potentiam interiectam.
Sit vectis AB, cuius fulcimentum C; &longs;itq; pondus D ex A &longs;u
&longs;pen&longs;um AH, ita vt AH &longs;it &longs;emper horizonti perpendicularis:
&longs;itq; potentia &longs;u&longs;tinens pondus in B. Dico potentiam in B ad pon
dus D ita e&longs;&longs;e, vt CA ad CB. fiat vt BC ad CA, ita pondus D
ponderabit, exi&longs;tente C amborum grauitatis centro. quare poten
tia æqualis ip&longs;i E ibidem con&longs;tituta ip&longs;i D æqueponderabit, vecte
AB, eius fulcimento in C collocato, hoc e&longs;t prohibebit, ne pon
dus D deor&longs;um vergat, quemadmodum prohibet pondus E. Po
pondus E ad idem pondus D: ergo potentia in B ad pondus D
erit, vt CA ad CB; hoc e&longs;t vectis di&longs;tantia à fulcimento ad pon
deris &longs;u&longs;pendium ad di&longs;tantiam à fulcimento ad potentiam. quod
demon&longs;trare oportebat.
Hinc facilè o&longs;tendi pote&longs;t, fulcimentum quò
ponderi fuerit propius, minorem ad idem pon
dus &longs;u&longs;tinendum requiri potentiam.
Ii&longs;dem po&longs;i
tis, &longs;it fulcimen
tum in F ip&longs;i A
propius, quàm
C; fiatq; vt BF
ad FA, ita pon
dus D ad aliud
G, quod &longs;i appendatur in B, pondera DG ex fulcimento E quoniam autem BF maior e&longs;t BC, & CA
quàm idem D ad E: pondus igitur G minus erit pondere E. cùm
autem potentia in B ip&longs;i G æqualis ponderi D æqueponderet, mi
nor potentia, quàm ea, quæ ponderi E e&longs;t æqualis, pondus D &longs;u
&longs;tinebit; exi&longs;tente vecte AB, eius verò fulcimento vbi F, quàm &longs;i
fuerit vbi C. &longs;imiliter quoq; o&longs;tendetur, quò propius erit fulci
mentum ponderi D, adhuc &longs;emper minorem requiri potentiam
ad &longs;u&longs;tinendum pondus D.
COROLLARIVM.
Vnde palàm colligere licet, exi&longs;tente AF ip&longs;a
FB minore, minorem quoq; requiri potentiam
in ip&longs;o B pondere D &longs;u&longs;tinendo. æquali verò
æqualem. maiore verò maiorem.
PROPOSITIO II.
Alio modo vecte vti po&longs;sumus.
Sit vectis AB, cuius
fulcimentum &longs;it B, &
pondus C vtcunq; in
D inter AB appen
&longs;um; &longs;itq; potentia in
A &longs;u&longs;tinens pondus C.
Dico vt BD ad BA,
ita e&longs;&longs;e potentiam in A ad pondus C. appendatur in A pondus
E æquale ip&longs;i C; & vt AB ad BD, ita fiat pondus E ad aliud F.
& quoniam pondera CE &longs;unt inter &longs;e &longs;e æqualia, erit pondus C
ad pondus F, vt AB ad BD. appendatur quoq; pondus F in A.
& quoniam pondus E ad pondus F e&longs;t, vt grauitas ip&longs;ius E ad gra
uitatem
grauitas ponderis E ad grauitatem ponderis F, ita e&longs;t AB ab BD. vt autem AB ad BD, ita e&longs;t grauitas ponderis E ad grauitatem
uitas ponderis E ad
grauitatem ponderis
F ita erit, vt grauitas
ponderis E ad gra
uitatem ponderis C.
Pondera igitur CF Ponatur itaq; potentia in A &longs;u&longs;tinens
pondus F; erit potentia in A æqualis ip&longs;i ponderi F. & quoniam
pondus F in A appen&longs;um æquè graue e&longs;t, vt pondus C in D ap
pen&longs;um; eandem proportionem habebit potentia in A ad grauita
ris C in D appen&longs;i. Potentia verò in A ip&longs;i F æqualis &longs;u&longs;tinet
pondus F, ergo potentia in A pondus quoq; C &longs;u&longs;tinebit. Itaq;
cùm potentia in A &longs;it æqualis ponderi F, & pondus C ad pon
dus F &longs;it, vt AB ad BD; erit pondus C ad potentiam in A, vt
pondus C. potentia ergo ad pondus ita erit, vt di&longs;tantia fulci
mento, ac ponderis &longs;u&longs;pen&longs;ioni intercepta ad di&longs;tantiam à fulci
mento ad potentiam. quod oportebat demon&longs;trare.
ALITER.
Sit vectis AB, cuius fulcimentum &longs;it B, & pondus E ex puncto
C &longs;u&longs;pen&longs;um; &longs;itq; vis in A &longs;u&longs;tinens pondus E. Dico vt BC ad BA,
ita e&longs;&longs;e potentiam in A ad pondus E. Producatur AB in C, &
fiat BD æqualis BC; & ex puncto D appendatur pondus F æqua
le ponderi E; itemq; ex puncto A &longs;u&longs;pendatur pondus G ita, vt
pondus F ad pondus G eandem habeat proportionem, quam AB pondera FG æqueponderabunt.
cùm autem &longs;it CB æqua
lis BD, pondera quoq; FE æqualia æqueponderabunt. pondera
verò FEG in libra, &longs;eu vecte DBA appen&longs;a, cuius fulcimentum
e&longs;t B, non æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. po
natur itaq; in A tanta vis, vt pondera FEG æqueponderent; erit
potentia in A æqualis ponderi G. pondera enim FE
& vis in A nihil aliud efficere debet, ni&longs;i &longs;u&longs;tinere
dat. & quoniam pondera FEG, & potentia in A æqueponderant,
demptis igitur FG ponderibus, quæ æqueponderant, reliqua æque
ponderabunt; &longs;cilicet potentia in A ponderi E, hoc e&longs;t potentia
in A pondus E &longs;u&longs;tinebit, ita vt vectis AB maneat, vt prius erat.
Cùm autem potentia in A &longs;it æqualis ponderi G, & pondus E pon
deri F æquale; habebit potentia in A ad pondus E eandem pro
portionem, quam habet BD, hoc e&longs;t BC ad BA. quod demon
&longs;trare oportebat.
COROLLARIVM I.
Ex hoc etiam (vt prius) manife&longs;tum e&longs;&longs;e po
te&longs;t, &longs;i ponatur pondus E propius fulcimento B,
vt in H; minorem potentiam in A &longs;u&longs;tinere po&longs;
&longs;e ip&longs;um pondus.
Minorem enim proportionem habet HB ad BA, quam CB ad
BA. & quò propius pondus erit fulcimento, adhuc &longs;emper mino
rem po&longs;&longs;e potentiam &longs;u&longs;tinere pondus E &longs;imiliter o&longs;tendetur.
COROLLARIVM II.
Sequitur etiam potentiam in A &longs;emper mino
rem e&longs;&longs;e pondere E.
Sumatur enim inter AB quoduis punctum C, &longs;emper BC
minor erit BA.
COROLLARIVM III.
Ex hoc quoq; elici pote&longs;t, &longs;i duæ fuerint poten
tiæ, vna in A, altera in B, & vtraq; &longs;u&longs;tentet
pondus E; potentiam in A ad potentiam in B e&longs;
&longs;e, vt BC ad CA.
Vectis enim BA fungi
tur officio duorum
& AB &longs;unt tanquam duo
fulcimenta, hoc e&longs;t quan
do AB e&longs;t vectis, & poten
tia &longs;u&longs;tinens in A; erit eius
fulcimentum B. Quando verò BA e&longs;t vectis, & potentia in B;
erit A fulcimentum: & pondus &longs;emper ex puncto C remanet &longs;u
&longs;pen&longs;um. & quoniam potentia in A ad pondus E e&longs;t, vt BC ad
BA; vt autem pondus E ad potentiam, quæ e&longs;t in B, ita e&longs;t
BC ad CA. & hoc modo facilè etiam proportionem, quæ in
Quæ&longs;tionibus Mechanicis quæ&longs;tione vige&longs;ima nona ab Ari&longs;totele
ponitur, noui&longs;&longs;e poterimus.
COROLLARIVM IIII.
E&longs;t etiam manife&longs;tum, vtra&longs;q; potentias in A,
& B &longs;imul &longs;umptas æquales e&longs;&longs;e ponderi E.
Pondus enim E ad potentiam in A e&longs;t, vt BA ad BC; & idem
pondus E ad potentiam in B e&longs;t, vt BA ad AC; quare pondus
E ad vtra&longs;q; potentias in A, & B &longs;imul &longs;umptas e&longs;t, vt AB ad BC
CA &longs;imul, hoc e&longs;t ad BA. pondus igitur E vtri&longs;q; potentiis &longs;imul
&longs;umptis æquale erit.
PROPOSITIO III.
Alio quoq; modo vecte vti po&longs;sumus.
Sit Vectis AB,
cuius fulcimentum
B; &longs;itq; ex puncto
A pondus C appen
&longs;um; &longs;itq; potentia
in D vtcunq; inter
AB &longs;u&longs;tinens pon
dus C. Dico vt AB
ad BD, ita e&longs;&longs;e potentiam in D ad pondus C. Appendatur ex
puncto D pondus E æquale ip&longs;i C; & vt BD ad BA, ita fiat pon
dus E ad aliud F. & cùm pondera CE &longs;int inter &longs;e &longs;e æqualia; erit
pondus C ad pondus F, vt BD ad BA. appendatur pondus
F quoq; in D. & quoniam pondus E ad ip&longs;um F e&longs;t, vt grauitas
ponderis E ad grauitatem ponderis F; & pondus E ad pondus F
e&longs;t, vt BD ad BA: vt igitur grauitas ponderis E ad grauitatem
ponderis F, ita e&longs;t BD ad BA. vt autem BD ad BA, ita e&longs;t gra
uitas ponderis E ad grauitatem ponderis C; quare grauitas ponde
ris E ad grauitatem ponderis F eandem habet proportionem,
quam habet ad grauitatem ponderis C. pondera ergo CF eandem
habent grauitatem. &longs;it igitur potentia in D &longs;u&longs;tinens pondus F,
erit potentia in D ip&longs;i ponderi F æqualis. & quoniam pondus F
in D æquè graue e&longs;t, vt pondus C in A; habebit potentia in D
eandem proportionem ad grauitatem ponderis F, quam habet ad
grauitatem ponderis C. &longs;ed potentia in D pondus F &longs;u&longs;tinet; po
tentia igitur in D pondus quoq; C &longs;u&longs;tinebit: & pondus C ad po
tentiam in D ita erit, vt pondus C ad pondus F; & C ad F e&longs;t, vt
BD ad BA; erit igitur pondus C ad potentiam in D, vt BD ad
BA: & conuertendo, vt AB ad BD, ita potentia in D ad pondus
C. potentia ergo ad pondus e&longs;t, vt di&longs;tantia à fulcimento ad pon
deris &longs;u&longs;pendium ad di&longs;tantiam à fulcimento ad potentiam. quod
demon&longs;trare oportebat.
ALITER.
Sit vectis AB, cuius fulcimentum B; & ex puncto A &longs;it pon
dus C &longs;u&longs;pen&longs;um; &longs;itq; potentia in D &longs;u&longs;tinens pondus C. Dico
vt AB ad BD, ita e&longs;&longs;e potentiam in D ad pondus C. Produca
tur AB in E, fiatq; BE æqualis ip&longs;i BA; & ex puncto E appen
datur pondus F æquale ponderi C; & vt BD ad BE, ita fiat pon
dus F ad aliud G, quod ex puncto D &longs;u&longs;pendatur. pondera FG
æqueponderabunt. & quoniam AB e&longs;t æqualis BE, & pondera
FC æqualia; &longs;imiliter pondera FC æqueponderabunt. Pondera
verò FGC &longs;u&longs;pen&longs;a in vecte EBA, cuius fulcimentum e&longs;t B, non
æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. Ponatur igi
tur in D tanta vis, vt pondera FGC æqueponderent; erit po
tentia in D æqualis ponderi G: pondera enim FC æqueponde
rant, & potentia in D nil aliud efficere debet, ni&longs;i &longs;u&longs;tinere pon
dus G ne de&longs;cendat. & quoniam pondera FGC, & potentia in
D æqueponderant, demptis igitur FG ponderibus, quæ æquepon
derant; reliqua æqueponderabunt, &longs;cilicet potentia in D ponderi C. hoc e&longs;t potentia in D pondus C &longs;u&longs;tinebit, ita vt vectis AB ma
neat, vt prius. & cùm potentia in D &longs;it æqualis ponderi G, & pon
dus C æquale ponderi F; habebit potentia in D ad pondus C ean
dem proportionem, quam EB, hoc e&longs;t AB ad BD. quod de
mon&longs;trare oportebat.
COROLLARIVM I.
Ex hoc etiam pàtet, vt prius, &longs;i coftituatur pon
dus fulcimento B propius, vt in H; à minori po
tentia pondus ip&longs;um &longs;ub&longs;tineri debere.
Minorem enim proportionem habet HB ad BD, quàm AB ad
BD. & quò propius erit fulcimento, adhuc &longs;emper minorem re
quiri potentiam.
COROLLARIVM II.
Manife&longs;tum quoq; e&longs;t, potentiam in D &longs;emper
maiorem e&longs;&longs;e pondere C.
Si enim inter AB &longs;umatur quoduis punctum D, &longs;emper AB
maior erit BD.
Et aduertendum e&longs;t ha&longs;ce, quas attulimus demon&longs;trationes
non &longs;olum vectibus horizonti æquidi&longs;tantibus, verùm etiam ve
ctibus horizonti inclinatis ad hæc omnia o&longs;tendenda commodè
aptari po&longs;&longs;e. quod ex iis, quæ de libra diximus, patet.
PROPOSITIO IIII.
Si potentia pondus in vecte appen&longs;um mo
ueat; erit &longs;patium potentiæ motæ ad &longs;patium
moti ponderis, vt di&longs;tantia à fulcimento ad po
tentiam ad di&longs;tantiam ab eodem ad ponderis &longs;u
&longs;pen&longs;ionem.
Sit vectis AB, cuius ful
cimentum C; & ex puncto B
&longs;it pondus D &longs;u&longs;pen&longs;um; &longs;itq;
potentia in A mouens pon
dus D vecte AB. Dico &longs;pa
tium potentiæ in A ad &longs;pa
tium ponderis ita e&longs;&longs;e, vt CA
ad CB. Moueatur vectis AB,
& vt pondus D &longs;ur&longs;um mo
ueatur, oportet B &longs;ur&longs;um mo
ueri, A verò deor&longs;um. & quo
niam C e&longs;t punctum immobi
le; idcirco dum A, & B mo
uentur,
tias de&longs;cribent. Moueatur igi
tur AB in EF; erunt AE
BF circulorum circumferentiæ, quorum &longs;emidiametri &longs;unt CA
CB. tota compleatur circumferentia AGE, & tota BHF; &longs;intq;
KH puncta, vbi AB, & EF circulum BHF &longs;ecant. Quoniam e
cùm autem circumferentiæ AE
kH &longs;int &longs;ub eodem angulo ACE, & circumferentia AE ad to
tam circumferentiam AGE &longs;it, vt angulus ACE ad quatuor re
ctos; vt autem idem angulus HCk ad quatuor rectos, ita quoq;
e&longs;t circumferentia HK ad totam circumferentiam HBK; erit cir
cumferentia AE ad totam circumferentiam AGE, vt circumfe
AE ad circumferentiam kH, hoc e&longs;t BF, ita tota circumferen
tia AGE ad totam circumferentiam BHF. tota verò circumfe
rentia AGE ita &longs;e habet ad totam BHF, vt diameter circuli AEG Vt igitur circumferentia AE ad cir
culi BHF: vt autem diameter ad diametrum, ita &longs;emidiameter
ad &longs;emidiametrum, hoc e&longs;t CA ad CB: quare vt circumferen
tia AE ad circumferentiam BF, ita CA ad CF. circumferentia
verò AE &longs;patium e&longs;t potentiæ motæ, & circumferentia BF e&longs;t &longs;patium enim motus ponderis
D &longs;emper æquale e&longs;t &longs;patio motus puncti B, cùm in B &longs;it appen
&longs;um: &longs;patium ergo potentiæ motæ ad &longs;patium moti ponderis e&longs;t,
vt CA ad CB; hoc e&longs;t vt di&longs;tantia à fulcimento ad potentiam
ad di&longs;tantiam ab eodem ad ponderis &longs;u&longs;pen&longs;ionem. quod demon
&longs;trare oportebat.
Sit autem vectis AB, cu
ius fulcimentum B; potentia
qué mouens in A; & pondus
in C. dico &longs;patium potentiæ
translatæ ad &longs;patium transla
ti ponderis ita e&longs;&longs;e, vt BA ad
BC. Moueatur vectis, & vt
pondus sursum attollatur, ne
ce&longs;&longs;e e&longs;t puncta C A &longs;ur&longs;um
moueri. Moueatur igitur A
&longs;ur&longs;um v&longs;q; ad D; &longs;itq; ve
ctis motus BD. eodemq;
modo (vt prius dictum e&longs;t)
o&longs;tendemus puncta CA cir
culorum circumferentias de
&longs;cribere,
ita e&longs;&longs;e AD ad CE, vt &longs;emidiameter AB ad &longs;emidiametrum BC.
Eademq; ratione, &longs;i potentia e&longs;&longs;et in C, & pondus in A,
o&longs;tendetur ita e&longs;&longs;e CE ad AD, vt BC ad BA; hoc e&longs;t di&longs;tan
tia à fulcimento ad potentiam ad di&longs;tantiam ab eodem ad ponde
ris &longs;u&longs;pen&longs;ionem. quod oportebat demon&longs;trare.
COROLLARIVM.
Ex his manife&longs;tum e&longs;t maiorem habere pro
portionem &longs;patium potentiæ mouentis ad &longs;pa
tium ponderis moti, quàm pondus ad eandem
potentiam.
Spatium enim potentiæ ad &longs;patium ponderis eandem habet,
rò &longs;u&longs;tinens minor e&longs;t potentia mouente, quare minorem habebit
potentiam ip&longs;um &longs;u&longs;tinentem. &longs;patium igitur potentiæ mouentis
ad &longs;patium ponderis maiorem habebit proportionem, quàm pon
dus ad eandem potentiam.
PROPOSITIO V.
Potentia quomodocunq; vecte pondus &longs;u&longs;ti
nens ad ip&longs;um pondus eandem habebit propor
tionem, quam di&longs;tantia à fulcimento ad punctum,
vbi à centro grauitatis ponderis horizonti ducta
perpendicularis vectem &longs;ecat, intercepta, ad
di&longs;tantiam inter fulcimentum, & potentiam.
Sit vectis AB
horizonti æqui
di&longs;tans, cuius ful
cimentum N; &longs;it
deinde pondus
AC, cuius cen
trum grauitatis
&longs;it D, quod pri
mùm &longs;it infra ve
ctem; pondus ve
rò &longs;it ex punctis
AO &longs;u&longs;pen&longs;um;
& à puncto D horizonti, & ip&longs;i AB perpendicularis ducatur DE.
&longs;i verò alii &longs;int quoq; vectes AF AG, quorum fulcimenta &longs;int
HK; pondu&longs;q; AC in vecte AG ex punctis AQ &longs;it appen&longs;um;
in vecte autem AF in punctis AP: lineaq; DE producta &longs;ecet
AF in L, & AG in M. dico potentiam in F pondus AC &longs;u&longs;tinen
tem ad ip&longs;um pondus eam habere proportionem, quam habet kL
NB; & potentiam in G ad pondus eam, quam HM ad HG.
Quoniam enim DL horizonti e&longs;t perpendicularis, pondus AC
vbicunq; in linea DL fuerit appen&longs;um, eodem modo, quo reperi
tur, manebit. quare in vecte AB &longs;i &longs;u&longs;pen&longs;iones, quæ &longs;unt ad AO
&longs;oluantur, pondus AC in E appen&longs;um eodem modo manebit, &longs;i
cuti nunc manet; hoc e&longs;t &longs;ublato puncto A, & linea QO, codem
modo pondus in E appen&longs;um manebit, vt ab ip&longs;is AO pun
ctis &longs;u&longs;tinebatur; ex commentario Federici Commandini in &longs;extam
Archimedis
de libra. Itaq; quoniam pondus AC eandem ad libram habet con&longs;ti
tutionem, &longs;iue in AO &longs;u&longs;tineatur, &longs;iue ex puncto E &longs;it appen&longs;um;
eadem potentia in B idem pondus AC, &longs;iue in E, &longs;iue in AO
&longs;u&longs;pen&longs;um &longs;u&longs;tinebit. potentia verò in B &longs;u&longs;tinens pondus AC
in E appen&longs;um ad ip&longs;um pondus ita &longs;e habet, vt NE ad NB; po
tentia
&longs;um ad ip&longs;um pondus ita erit, vt NE ad NB. Non aliter o&longs;ten
detur pondus AC ex puncto L &longs;u&longs;pen&longs;um manere, &longs;icuti à pun
ctis AP &longs;u&longs;tinetur; potentiamq; in F ad ip&longs;um pondus ita e&longs;&longs;e, vt kL
ad KF. In vecte verò AG pondus AC in M appen&longs;um ita mane
re, vt à punctis AQ &longs;u&longs;tinetur; potentiamq; in G ad pondus
AC ita e&longs;&longs;e, vt HM ad HG; hoc e&longs;t vt di&longs;tantia à fulcimento
ad punctum, vbi à centro grauitatis ponderis horizonti ducta
perpendicularis vectem &longs;ecat, ad di&longs;tantiam à fulcimento ad poten
tiam. quod demon&longs;trare oportebat.
Si autem FBG e&longs;&longs;ent vectium fulcimenta, potentiæq; e&longs;&longs;ent
in KNH pondus &longs;u&longs;tinentes, &longs;imili modo o&longs;tendetur ita e&longs;&longs;e po
tentiam in H ad pondus, vt GM ad GH; & potentiam in N ad
pondus, vt BE ad BN; ac potentiam in k ad pondus, vt FL
ad Fk.
Et &longs;i vectes AB
AF AG habeant
fulcimenta in A,
& pondus &longs;it NO;
deinde ab eius
centro grauitatis
D ducatur ip&longs;i A
B, & horizonti
MEL; &longs;intq; po
tentiæ in FBG:
&longs;imiliter o&longs;tende
tur ita e&longs;&longs;e poten
tiam in G pondus NO &longs;u&longs;tinentem ad ip&longs;um pondus, vt AM
ad AG; ac potentiam in B, vt AE ad AB; & potentiam in F,
vt AL ad AF.
Sit deinde
vectis AB ho
rizonti æqui
di&longs;tans, cuius
fulcimentum
D; & &longs;it BE
pondus, cuius
centrum
tatis &longs;it F &longs;u
pra vectem: à
punctoq; F ho
rizonti, & ip&longs;i
AB ducatur
FH; pondu&longs;q; à puncto B, & PQ &longs;u&longs;tineatur. Sint deinde alii ve
ctes BL BM, quorum fulcimenta &longs;int NO; lineaq; FH producta &longs;e
cet BM in k, & BL in G; pondus autem in vecte BL in pun
ctis BP &longs;u&longs;tineatur; in vecte autem BM à puncto B, & PR. Di
co potentiam in L pondus BE vecte BL &longs;u&longs;tinentem ad ip&longs;um
pondus eam habere proportionem, quam NG ad NL; & po
tiamq; in M ad pondus eam, quam Ok ad OM. Quoniam e
nim à centro grauitatis F ducta e&longs;t kF horizonti perpendicularis,
ex quocunq; puncto lineæ kF &longs;u&longs;tineatur pondus, manebit; vt
nunc &longs;e habet. &longs;i igitur &longs;u&longs;tineatur in H, manebit vt prius; &longs;cili
cet &longs;ublato puncto B, & PQ, quæ pondus &longs;u&longs;tinent, pondus BE
manebit, &longs;icuti ab ip&longs;is &longs;u&longs;tinebatur. quare in vecte AB graue&longs;cet
in H, & ad vectem eandem habebit con&longs;titutionem, quam prius;
idcirco erit, ac &longs;i in H e&longs;&longs;et appen&longs;um. eadem igitur potentia ìdem
pondus BE, &longs;iue in H, &longs;iue in B, & Q &longs;uffultum, &longs;u&longs;tinebit. Potentia ve
rò in A &longs;u&longs;tinens pondus BE vecte AB in H appen&longs;um ad ip&longs;um
pondus eandem habet proportionem, quam DH ad DA; eadem
ergo potentia in A &longs;u&longs;tinens pondus BE in punctis BQ &longs;u&longs;tenta
tum ad ip&longs;um pondus erit, vt DH ad DA. Similiter o&longs;tende
tur pondus BE &longs;i in G &longs;u&longs;tineatur, manere; &longs;icuti à punctis BP
&longs;u&longs;tinebatur: & in puncto k, vt à punctis BR.
quare potentia in
L &longs;u&longs;tinens pondus BE ad ip&longs;um pondus ita erit, vt NG ad NL. potentia verò in M ad pondus, vt OK ad OM; hoc e&longs;t vt di&longs;tan
tia à fulcimento ad punctum, vbi à centro grauitatis ponderis ho
rizonti ducta perpendicularis vectem &longs;ecat, ad di&longs;tantiam à fulci
mento ad potentiam. quod demon&longs;trare quoq; oportebat.
Si verò LAM e&longs;&longs;ent fulcimenta, & potentiæ in NDO; &longs;imi
liter o&longs;tendetur ita e&longs;&longs;e potentiam in N ad pondus, vt LG ad L
N; & potentiam in D, vt AH ad AD; & potentiam in O, vt
Mk ad MO.
Et &longs;i vectes BA
BL BM habeant
fulcimenta in B, &
pondus &longs;upra
&longs;it NO; & ab eius
centro grauitatis F
ducatur ip&longs;i AB, &
horizonti perpendi
cularis FDEG; &longs;int
qué potentiæ in L
AM; &longs;imiliter o
&longs;tendetur ita e&longs;&longs;e po
tentiam in L pon
dus &longs;u&longs;tinentem ad ip&longs;um pondus, vt BD ad BL; & potentiam
in A ad pondus, vt BE ad BA, atq; potentiam in M, vt BG
ad BM.
Sit deniq;
vectis AB ho
rizonti æqui
di&longs;tans, cuius
fulcimentum
C, & pondus
DE habeat
trum grauita
tis F in ip&longs;o
vecte AB;
&longs;intq; deniq;
alii vectes G
H kL, quo
rum fulcimenta &longs;int MN; pondusq; in vecte GH &longs;u&longs;tineatur à
punctis GO; in vecte autem AB à punctis AP; & in uecte KL
à punctis KQ; & centrum grauitatis F &longs;it quoq; in utroq; uecte
GH kL; &longs;intq; potentiæ in HBL. Dico potentiam in H ad
pondus ita e&longs;&longs;e, ut NF ad NH; & potentiam in B ad pondus, ut
CF ad CB; ac potentiam in L ad pondus, ut MF ad ML. Quo
niam enim F centrum e&longs;t grauitatis ponderis DE, &longs;i igitur in F
tri grauitatis; eritq; ac &longs;i in F e&longs;&longs;et appen&longs;um; atq; in vecte eodem
modo manebit, &longs;iue à punctis AP, &longs;iue à puncto F &longs;u&longs;tineatur.
quod idem in vectibus GH kL eueniet; &longs;cilicet pondus eodem mo
do manere, &longs;iue in F, &longs;iue in GO, vel in kQ &longs;u&longs;tineatur. eadem
igitur potentia in B idem pondus DE, vel in F, vel in AP appen&longs;um
&longs;u&longs;tinebit: & quando appen&longs;um e&longs;t in F ad ip&longs;um pon
dus e&longs;t, vt CF ad CB, ergo potentia &longs;u&longs;tinens pondus DE in
AP appen&longs;um ad ip&longs;um pondus erit, vt CF ad CB. eodemq; mo
do potentia in H ad pondus in GO appen&longs;um ita erit, vt NF ad
NH. potentiaq; in L ad pondus in kQ appen&longs;um erit, vt MF
ad ML. quod o&longs;tendere quoq; oportebat.
Si verò HBL e&longs;&longs;ent fulcimenta, & potentiæ e&longs;&longs;ent in NCM; &longs;i
militer o&longs;tendetur potentiam in N ad pondus ita e&longs;&longs;e, vt HF ad
HN; & potentiam in C, vt BF ad BC, & potentiam in M, vt
LF ad LM.
Et &longs;i vectes BA
BC BD
cimenta in B, &longs;intq;
pondera in EF GH
kL, ita vt eorum
centra MNO gra
uitatis &longs;int in vecti
bus; &longs;intq; poten
tiæ in CAD: &longs;imi
liter o&longs;tendetur po
tentiam in C ad
pondus EF ita e&longs;&longs;e,
vt BM ad BC, & potentiam in A ad pondus GH, vt BN ad
BA, potentiamq; in D ad pondus KL, vt BO ad BD.
PROPOSITIO VI.
Sit AB recta linea, cui ad angulos &longs;it rectos
AD, quæ ex parte A producatur vtcunq; v&longs;q;
ad C; connectaturq; CB, quæ ex parte B quoq;
producatur v&longs;q; ad E. ducantur deinde à pun
cto B vtcunq; inter AB BE lineæ BF BG ip&longs;i
AB æquales; à puncti&longs;q; FG ip&longs;is perpendicula
res ducantur FH GK, quæ & inter &longs;e &longs;e, & ip&longs;i
AD con&longs;tituantur æ
quales, ac &longs;i BA AD
motæ &longs;int in BF FH,
& in BG GK; con
nectanturq; CH CK,
quæ lineas BF BG
in punctis MN &longs;e
cent. Dico BN mi
norem e&longs;&longs;e BM, &
BM ip&longs;a BA.
Connectantur BD BH
BK. & quoniam duæ lineæ
DA AB duabus HF FB
&longs;unt æquales, & angulus
DAB rectus recto HFB e&longs;t
anguli reliquis angulis æqua
les, & HB ip&longs;i DB æqualis.
&longs;imiliter o&longs;tendetur triangu
lum BkG triangulo BHF æqualem e&longs;&longs;e. quare centro B, inter
neas CH CK &longs;ecet in punctis OP; connectanturq; OB PB.
Quoniam igitur punctum k propius e&longs;t ip&longs;i E, quàm H; erit linea
Ck maior ip&longs;a CH, & CP ip&longs;a CO minor: ergo PK ip&longs;a OH
maior erit. Quoniam autem triangulum BkP æquicrure latera
Bk BP lateribus BH BO trianguli BHO æquicruris æqualia ha
bet, ba&longs;im verò KP ba&longs;i HO maiorem, erit angulus kBP an
gulo ergo reliqui ad ba&longs;im anguli, hoc e&longs;t kPB
PkB &longs;imul &longs;umpti, qui inter &longs;e &longs;unt æquales, reliquis ad ba&longs;im an
gulis, nempè OHB HOB, qui etiam inter &longs;e &longs;unt æquales, mino
res
æquales. quare & horum dimidii, &longs;cilicet NkB minor MHB.
Cùm autem angulus BkG æqualis &longs;it angulo BHF, erit NkG
ip&longs;o MHF maior. &longs;i igitur à puncto k con&longs;tituatur angulus GKQ
ip&longs;i FHM æqualis, fiet triangulum GkQ triangulo FHM æqua
le; nam duo anguli ad FH vnius duobus ad Gk alterius &longs;unt
æquales, & latus FH lateri Gk e&longs;t æquale, erit GQ ip&longs;i FM æ
quale. ergo GN maior erit ip&longs;a FM.
Cùm itaq; BG ip&longs;i BF &longs;it æqua
lis, erit BN minor ip&longs;a BM. Quòd autem BM &longs;it ip&longs;a BA minor,
e&longs;t manife&longs;tum; cùm BM ip&longs;a BF, quæ ip&longs;i BA e&longs;t æqualis, &longs;it
minor. quod demon&longs;trare oportebat.
In&longs;uper &longs;i intra BG BE alia vtcunq; ducatur linea ip&longs;i BG æ
qualis; fiatq; operatio, quemadmodum &longs;upra dictum e&longs;t; &longs;imili
ter o&longs;tendetur lineam BR minorem e&longs;&longs;e BN. & quò propius fue
rit ip&longs;i BE, adhuc minorem &longs;emper e&longs;&longs;e.
Si verò æqualia triangula BFH BGK &longs;int
deor&longs;um inter BC BA con&longs;tituta; connectan
turq; HC KC, quæ lineas BF BG ex parte
FG productas in punctis MN &longs;ecent erit BN
maior BM, & BM ip&longs;a BA.
Nam producatur CH
Ck v&longs;q; ad circumferentiam
in OP, Connectanturq; BO
BP; &longs;imili modo o&longs;tende
tur lineam Pk maiorem e&longs;
&longs;e OH, angulumq; PkB mi
norem e&longs;&longs;e angulo OHB. &
quoniam angulus BHF e&longs;t
æqualis angulo BkG; erit to
tus PKG angulus angulo
OHF minor: quare reliquus
GKN reliquo FHM maior
erit. &longs;i it aq; con&longs;tituatur angu
lus GkQ ip&longs;i FHM æqua
lis, linea KQ ip&longs;am GN ita
&longs;ecabit, vt GQ ip&longs;i FM æqua
lis euadat: quare maior. erit
GN, quàm FM; quibus &longs;i
æquales adiiciantur BF BG,
erit BN ip&longs;a BM maior. &
cùm BM &longs;it ip&longs;a FB maior,
erit quoq; ip&longs;a BA maior. &longs;i
militer o&longs;tendetur, quò pro
pius fuerit BG ip&longs;i BC, li
neam BN &longs;emper maiorem
e&longs;&longs;e. PROPOSITIO VII.
Sit recta linea AB, cuì perpendicularis exi
&longs;tat AD, quæ ex parte D producatur vtcunq; v&longs;q;
ad C; connectaturq; CB, quæ producatur e
tiam v&longs;q; ad E; & inter AB BE lineæ &longs;imiliter
vtcunq; ducantur BF BG ip&longs;i AB æquales; à
punctisq; FG lineæ FH GK ip&longs;i AB æquales,
ip&longs;is verò BF BG
pendiculares
ac &longs;i BA AD motæ
&longs;int in BF FH BG
GK: Connectanturq;
CH CK, quæ lineas
BF BG productas &longs;e
cent in punctis MN.
Dico BN maiorem e&longs;
&longs;e BM, & BM ip&longs;a BA.
Connectantur BD BH Bk,
& centro B, interuallo quidem
BD, circulus de&longs;cribatur. &longs;imi
liter vt in præcedenti demon
&longs;trabimus puncta kHDOP in
circuli circumferentia e&longs;&longs;e, trian
gulaq; ABD FBH GBk in
ter &longs;e &longs;e æqualia e&longs;&longs;e, atq; lineam
Pk maiorem OH, angulumq;
PKB minorem e&longs;&longs;e angulo O
HB. Quoniam igitur angulus BHF æqualis e&longs;t angulo BkG,
lo OHF minor: quare reliquus
GkN reliquo FHM maior
erit. &longs;i igitur fiat angulus GK
Q ip&longs;i FHM æqualis, erit trian
gulum GKQ triangulo FHM
æquale, & latus GQ lateri FM
æquale; ergo maior erit GN ip
&longs;a FM; ac propterea BN ma
ior erit BM. BM autem ma
ior erit BA; nam BM maior e&longs;t
ip&longs;a BF. quod demon&longs;trare
oportebat.
Eodemq; pror&longs;us modo, quo
propius fuerit BG ip&longs;i BE, li
neam BN &longs;emper maiorem e&longs;&longs;e
o&longs;tendetur.
Si autem triangula BFH BGK deor&longs;um in
ter AB BC con&longs;tituantur, ducanturq; CHO
CKP, quæ lineas BF BG &longs;ecent in punctis M
N; erit linea BN minor ip&longs;a BM, & BM
ip&longs;a BA.
Connectantur enim BO BP,
&longs;imiliter o&longs;tendetur angulum
PKB minorem e&longs;&longs;e OHB. &
quoniam angulus FHB æqua
lis e&longs;t angulo GkB; erit angu
lus GkN angulo FHM ma
ior: quare & linea GN ma
ior erit ip&longs;a FM. ideoq; linea
Cùm autem maior &longs;it BF ip&longs;a
BM; erit BM ip&longs;a BA minor. Si
miliq; modo o&longs;tendetur, quò
propius fuerit BG ip&longs;i BC, li
neam BN &longs;emper minorem
e&longs;&longs;e.
PROPOSITIO VIII.
Potentia pondus &longs;u&longs;tinens centrum grauitatis
&longs;upra vectem horizonti æquidi&longs;tantem habens,
quò magis pondus ab hoc &longs;itu vecte eleuabitur;
minori &longs;emper, vt &longs;u&longs;tineatur, egebit potentia:
&longs;i verò deprimetur, maiori.
Sit vectis AB horizonti æquidi&longs;tans, cuius fulcimentum C;
pondus autem BD, eiu&longs;dem verò grauitatis centrum &longs;it &longs;upra ve
ctem vbi H: &longs;itq; potentia &longs;u&longs;tinens in A. moueatur deinde ve
ctis AB in EF, &longs;itq; pondus motum in FG. Dico primùm mino
rem
A pondus BD vecte AB. &longs;it k centrum grauitatis ponderis FG;
deinde tùm ex H, tùm ex K ducantur HL kM ip&longs;orum horizon
tibus perpendiculares, quæ in
&longs;i quoq; AB perpendicularis. ducatur deinde kN ip&longs;i EF perpen
dicularis, quæ ip&longs;i HL æqualis erit, & CN ip&longs;i CL æqualis. Quo
&longs;tinens pondus BD ad ip&longs;um pondus eam habebit proportionem,
quam CL ad CA. rur&longs;us quoniam kM horizonti e&longs;t perpendicu
laris, potentia in E pondus FG &longs;u&longs;tinens ita erit ad pondus, vt
CM ad CE. Cùm autem CN NK ip&longs;is CL LH &longs;int æquales,
CM ad CE. quàm CL ad CA: & cùm pondera BD FG &longs;int
æqualia, e&longs;t enim idem pondus; ergo minor erit proportio po
tentiæ in E pondus FG &longs;u&longs;tinentis ad ip&longs;um pondus, quàm po
tentiæ in A pondus BD &longs;u&longs;tinentis ad ip&longs;um pondus. Quare
minor potentia in E &longs;u&longs;tinebit pondus FG, quàm potentia in A
pondus BD. & quò pondus magis eleuabitur; &longs;emper o&longs;tendetur
minorem adhuc potentiam pondus &longs;u&longs;tinere; cùm linea PC mi
nor &longs;it linea CM. &longs;it deinde vectis in QR, & pondus in QS,
cuius dico maiorem requiri potentiam in R
ad ducatur à cen
tro grauitatis O linea OT horizonti perpendicularis. & quo
niam HL OT, &longs;i ex parte L, atq; T producantur, in centrum
mundi conuenient; erit CT maior CL: e&longs;t autem CA ip&longs;i CR
æqualis, habebit ergo TC ad CR maiorem proportionem, quàm
LC ad CA. Maior igitur erit potentia in R &longs;u&longs;tinens pondus
QS, quàm in A &longs;u&longs;tinens BD. &longs;imiliter o&longs;tendetur; quò vectis
RQ magis à vecte AB di&longs;tabit deor&longs;um vergens, &longs;emper maio
rem potentiam requiri ad &longs;u&longs;tinendum pondus: di&longs;tantia enim CV
longior e&longs;t CT. Quò igitur pondus à &longs;itu horizonti æquidi&longs;tan
te magis eleuabitur à minori &longs;emper potentia pondus &longs;u&longs;tinebitur;
quò verò magis deprimetur, maiori, vt &longs;u&longs;tineatur, egebit potentia.
quod demon&longs;trare oportebat.
Ex
Hinc facile elicitur potentiam in A ad poten
tiam in E ita e&longs;&longs;e, vt CL ad CM.
Nam ita e&longs;t LC ad CA, vt potentia in A ad pondus; vt au
tem CA, hoc e&longs;t CE ad CM, ita e&longs;t pondus ad potentiam in E;
quare ex æquali potentia in A ad potentiam in E ita erit, vt CL
ad CM.
Similiq; ratione non &longs;olum o&longs;tendetur, potentiam in A ad po
tentiam in R ita e&longs;&longs;e, vt CL ad CT; &longs;ed & potentiam quoq; in E
ad potentiam in R ita e&longs;&longs;e, vt CM ad CT. & ita in reliquis.
Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimen
tum B; & centrum grauitatis H ponderis CD &longs;it &longs;upra vectem;
moueaturq; vectis in BE, pondu&longs;q; in FG. dico minorem po
tentiam in E &longs;u&longs;tinere pondus FG vecte EB, quàm potentia in
A pondus CD vecte AB. &longs;it k centrum grauitatis ponderis FG,
& à centris grauitatum Hk ip&longs;orum horizontibus perpendicuQuoniam enim (ex &longs;upra demon&longs;tratis)
&longs;ed vt BM ad
BE, ita potentia in E &longs;u&longs;tinens pondus FG ad ip&longs;um pondus; &
vt BL ad BA, ita potentia in A ad pondus CD; minorem
habebit proportionem potentia in E ad pondus FG, quàm potenErgo potentia in E minor erit poten
tia in A. &longs;imiliter o&longs;tendetur, quò magis pondus eleuabitur, &longs;em
per minorem potentiam pondus &longs;u&longs;tinere. Sit autem vectis in
BO, & pondus in PQ, cuius centrum grauitatis &longs;it R. dico maio
rem potentiam in O requiri ad &longs;u&longs;tinendum pondus PQ vecte BO,
quàm pondus CD vecte BA. ducatur à puncto R horizonti per
& quoniam BS maior e&longs;t BL, habebit BS ad
BO maiorem proportionem, quàm BL ad BA; quare maior erit
potentia in O &longs;u&longs;tinens pondus PQ, quàm potentia in A &longs;u&longs;ti
nens pondus CD. & hoc modo o&longs;tendetur' quò vectis BO ma
gis à vecte AB deor&longs;um tendens di&longs;tabit, &longs;emper maiorem ponderi
Hinc quoq; vt &longs;upra patet pontentiam in A ad potentiam in E e&longs;
&longs;e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL
ad BS. atque potentiam in E ad potentiam in O, vt BM
ad BS.
Præterea &longs;i in B alia intelligatur potentia, ita vt duæ &longs;int poten
tiæ pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon
dus PQ vecte BO, quàm pondus CD vecte BA aduer&longs;o au
tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve
cte BE, quàm pondus CD vecte AB. ducta enim kN ip&longs;i EB
perpendicularis, erit EN ip&longs;i AL æqualis: quare EM ip&longs;a LA
maior erit. ergo maiorem habebit proportionem EM ad E
quàm LA ad A
quæ &longs;unt proportiones potentiæ ad pondus.
Similiter o&longs;tendetur potentiam in
nentem ad potentiam in eodem puncto
e&longs;&longs;e, vt LA ad EM; ad potentiam autem in B pondus vecte O
&longs;u&longs;tinentem ita e&longs;&longs;e, vt AL ad OS. quæ verò vectibus E
&longs;u&longs;tinent inter &longs;e &longs;e e&longs;&longs;e, vt EM ad OS.
Deinde vt in iis, quæ &longs;uperius dicta &longs;unt, demon&longs;trabimus po
tentiam in
EM ad M
L
Sit autem vectis A
horizonti æquidi&longs;tans,
cuius fulcimentum
grauitati&longs;q; centrum H
ponderis AC &longs;it &longs;upra
vectem: moueaturq; ve
ctis in
in EF, potentiaq; in G. &longs;imiliter vt &longs;upra o&longs;ten
detur potentiam in G
pondus EF
tem minorem e&longs;&longs;e potentia in D pondus AC &longs;u&longs;tinente. cùm
BL, minorem habebit
proportionem MB ad
BG, quàm LB ad BD. atq; hoc modo o&longs;ten
detur, quò pondus ve
cte magis eleuabitur, mi
norem &longs;emper. ad pon
dus &longs;u&longs;tinendum requi
ri potentiam. Simili
ter &longs;i moueatur vectis
in BO, potentiaq; &longs;u
&longs;tinens in N, o&longs;tendetur potentiam in N maiorem e&longs;&longs;e potentia in
D. maiorem enim habet proportionem SB ad BN, quàm LB
ad BD. o&longs;tendetur etiam, quò magis pondus deprimetur; ma
iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. quod demon
&longs;trare oportebat. quod demon
&longs;trare oportebat.
Hinc quoq; liquet potentias in GDN inter &longs;e &longs;e ita e&longs;&longs;e, vt
BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS.
COROLLARIVM.
Ex his manife&longs;tum e&longs;t; &longs;i potentia vecte &longs;ur
&longs;um moueat pondus, cuius centrum grauitatis
&longs;it &longs;upra vectem, quò magis pondus eleuabitur;
&longs;emper minorem potentiam requiri vt pondus
moueatur.
Vbi enim potentia pondus &longs;u&longs;tinens e&longs;t &longs;emper minor, erit
quoq; potentia ip&longs;um mouens &longs;emper minor.
Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon
deris, &longs;iue propinquius, &longs;iue remotius fuerit à vecte AB horizon
ti æquidi&longs;tante, eandem potentiam in A pondus nihilominus
&longs;u&longs;tinere: vt &longs;i centrum grauitatis H ponderis BD longius ab&longs;it
à vecte BA, quàm centrum grauitatis N ponderis PV, dum
modo ducta à puncto H perpendicularis HL horizonti, vectiq;
AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD æquale;
erit tùm pondus BD, tùm pondus PV, ac &longs;i ambo in L e&longs;
&longs;ent appen&longs;a; atque &longs;unt æqualia, cùm loco vnius ponderis ac
cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD,
pondus quoq; PV &longs;u&longs;tinebit. Vecte autem EF, quò centrum
grauitatis longius fuerit à vecte, eò facilius potentia idem pon
dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius
&longs;it à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta
men vt ducta à puncto k vecti FE perpendicularis tran&longs;eat per
X; &longs;itq; pondus FG ponderi YZ æquale; & à punctis kX ip
&longs;orum horizontibus perpendiculares ducantur KM X9; erit C9
maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs;
&longs;et appen&longs;um, & pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um. quo
CM ad CE, maior potentia in E &longs;u&longs;tinebit pondus YZ, quàm
FG. In vecte autem QR è conuer&longs;o demon&longs;trabitur, &longs;cilicet
quò centrum grauitatis eiu&longs;dem ponderis &longs;it longius à vecte, eò
maiorem e&longs;&longs;e potentiam pondus &longs;u&longs;tinentem. maior enim e&longs;t
CT, quàm CI; & ob id maiorem habebit proportionem CT
ad CR, quàm CI ad CR. Similiter demon&longs;trabitur, &longs;i pondus
intra potentiam, & fulcimentum fuerit collocatum; vel poten
tia intra fulcimentum, & pondus. Quod idem etiam potentiæ
eueniet mouenti. vbi enim minor potentia &longs;u&longs;tinet pondus, ibi
minor potentia mouebit; & vbi maior in &longs;u&longs;tinendo, ibi maior
quoq; in mouendo requiretur.
PROPOSITIO VIIII.
Potentia pondus &longs;u&longs;tinens infra vectem ho
rizonti æquidi&longs;tantem ip&longs;ius centrum grauitatis
uabitur maiori &longs;emper potentia, vt &longs;u&longs;tineatur,
egebit. &longs;i verò deprimetur, minori.
Sit vectis AB horizonti æquidi&longs;tans, cuius fulcimentum C;
&longs;itq; pondus AD, cuius centrum grauitatis L &longs;it infra vectem;
&longs;itq; potentia in B &longs;u&longs;tinens pondus AD: moueatur deinde ve
ctis in FG, & pondus in FH. Dico primum maiorem requiri
potentiam in G ad &longs;u&longs;tinendum pondus FH vecte FG, quàm
&longs;it potentia in B pondere exi&longs;tente AD vecte autem AB. &longs;it M
grauitatis centrum ponderis FH, & à punctis LM ip&longs;orum ho
rizontibus perpendiculares ducantur Lk MN: ip&longs;i verò FG per
pendicularis ducatur MS, quæ æqualis erit LK, & CK ip&longs;i CS
erit etiam æqualis. Quoniam igitur CN maior e&longs;t Ck, habe
bit
tentia uerò in B ad pondus AD eandem habet, quam kC ad CB:
& vt potentia in G ad pondus FH, ita e&longs;t NC ad CG; ergo
maiorem habebit proportionem potentia in G ad pondus FH,
quàm potentia in B ad pondus AD. maior igitur e&longs;t potentia
in G ip&longs;a potentia in B. &longs;i verò vectis &longs;it in OP, & pondus in
OQ; erit potentia in B maior, quàm in P. eodem enim mo
do o&longs;tendetur CR minorem e&longs;&longs;e Ck, & CR ad CP minorem
habere proportionem, quàm Ck ad CB; & ob id potentiam in
B maiorem e&longs;&longs;e potentia in P. & hoc modo o&longs;tendetur, quò ma
gis à &longs;itu AB pondus eleuabitur, &longs;emper maiorem potentiam ad
pondus &longs;u&longs;tinendum requiri. è è contra verò &longs;i deprimetur.
quod
demon&longs;trare oportebat.
Hinc quoq; facilè elici pote&longs;t potentias in PBG inter &longs;e &longs;e ita
e&longs;&longs;e, vt CR ad Ck; & vt Ck ad CN; atq; vt CN ad CR.
Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimentum
B; pondu&longs;q; CD habeat centrum grauitatis O infra vectem; &longs;itq;
potentia in A &longs;u&longs;tinens pondus CD. Moueatur deinde vectis in
Dico maiorem re
quiri potentiam in E, vt pondus &longs;u&longs;tineatur, quàm in A; & ma
iorem in A, quàm in F. ducantur à centris grauitatum horizon
tibus perpendiculares NM OP QR, quæ ex parte NOQ
protractæ in centrum mundi conuenient. &longs;imiliter vt &longs;upra o&longs;ten
detur BM
iorem
iorem, quàm BR ad BF: & propter hoc potentiam in E maio
rem e&longs;&longs;e potentia in A; & potentiam in A maiorem potentia in
F. & quò vectis magis à &longs;itu AB eleuabitur, &longs;emper o&longs;tendetur,
maiorem requiri potentiam ponderi &longs;u&longs;tinendo. &longs;i verò depri
metur, minorem.
Hinc patet etiam potentias in EAF inter &longs;e &longs;e ita e&longs;&longs;e, vt BM ad
BP; & vt BP ad BR; ac vt BM ad BR.
In&longs;uper &longs;i in B altera &longs;it potentia, ita vt duæ &longs;int potentiæ pondus
&longs;u&longs;tinentes, maiore opus e&longs;t potentia in B pondus kL &longs;u&longs;tinente
vecte BF, quàm pondus CD vecte AB. & adhuc maiore vecte
AB, quàm vecte BE. maiorem enim habet proportionem RF
ad FB, quàm PA ad AB; & PA ad AB maiorem habet, quàm
EM ad EB.
Similiterq; o&longs;tendetur potentias in B pondus vectibus &longs;u&longs;tinen
tes inter &longs;e &longs;e ita e&longs;&longs;e, vt EM ad AP; & ut
AP ad FR; atque ut
EM ad FR.
Præterea potentia in B ad potentiam in F ita erit, ut RF ad
RB; & potentia in B ad potentiam in A, ut PA ad PB, & po
tentia
Sit autem vectis
AB horizonti æqui
di&longs;tans, cuius fulci
mentum B; & pon
dus AC, cuius cen
trum grauitatis &longs;it in
fra vectem: &longs;itq; po
tentia in D pondus
vectis in BE BF, &
potentia in GH: &longs;i
militer o&longs;tendetur po
tentiam in G maiorem e&longs;&longs;e debere potentia in D; & potentiam in
D maiorem potentia in H. maiorem enim proportionem habet
KB ad BG, quàm BL ad BD; & BL ad BD maiorem, quàm
MB ad BH. & hoc modo o&longs;tendetur, quò vectis magis à &longs;itu
AB eleuabitur, adhuc &longs;emper maiorem e&longs;&longs;e debere potentiam pon
dus &longs;u&longs;tinentem. quò autem magis deprimetur; minorem.
quod
demon&longs;trare oportebat.
Similiter in his potentiæ in GDH inter &longs;e &longs;e ita. erunt, vt BK
ad BL; & vt BL ad BM; deniq; vt Bk ad BM.
COROLLARIVM.
Ex his patet etiam, &longs;i potentia vecte &longs;ur&longs;um
moueat pondus, cuius centrum grauitatis &longs;it in
fra vectem; quò magis pondus eleuabitur, &longs;em
per maiorem requiri potentiam, vt pondus mo
ueatur.
Nam &longs;i potentia pondus &longs;u&longs;tinens &longs;emper e&longs;t maior: erit quoq;
potentia mouens &longs;emper maior.
Et his etiam facilè elicietur, &longs;i centrum grauitatis eiu&longs;dem pon
deris, &longs;iue propius, &longs;iue remotius fuerit à vecte AB horizonti æ
quidi&longs;tante; eandem potentiam in B pondus &longs;u&longs;tinere. vt &longs;i cen
trum grauitatis L ponderis AD &longs;it remotius à vecte BA, quàm
centrum grauitatis N ponderis PV; dummodo ducta à puncto L
perpendicularis LK horizonti, vectiq; AB tran&longs;eat per N: &longs;imili
ter vt in præcedenti o&longs;tendetur, eandem potentiam in B, & pondus
AD, & pondus PV &longs;u&longs;tinere. In vecte auté EF, quò
longius aberit à vecte, eò maiori opus erit potentia ponderi &longs;u&longs;ti
nendo. vt centrum grauitatis M ponderis FH remotius &longs;it à ue
cte EF, quàm S centrum grauitatis ponderis XZ; ducantur à pun
ctis MS horizontibus perpendiculares MI SG; erit CI maior
CG: ac propterea maior e&longs;&longs;e debet potentia in E pondus FH &longs;u
&longs;tinens, quàm pondus XZ. Contra uerò in uecte OR o&longs;tende
tur, quò &longs;cilicet centrum grauitatis eiu&longs;dem ponderis longius ab
&longs;it à uecte, à minori potentia pondus &longs;u&longs;tineri. minor enim e&longs;t
CY, quàm CT. Simili quoq; modo demon&longs;trabitur, &longs;i pondus
&longs;it intra potentiam, & fulcimentum; uel potentia intra fulci
mentum, & pondus. Quod idem potentiæ eueniet mouenti:
uebit. & vbi maior potentia in &longs;u&longs;tinendo; ibi quoq; maior in mo
uendo aderit.
PROPOSITIO X.
Potentia pondus &longs;u&longs;tinens in ip&longs;o vecte cen
trum grauitatis habens, quomodocunq; vecte
transferatur pondus; eadem &longs;emper, vt &longs;u&longs;tinea
tur, potentia opus erit.
Sit vectis AB horizonti æquidi&longs;tàns, cuius fulcimentum C.
E verò centrum grauitatis ponderis in ip&longs;o &longs;it vecte.
Moueatur
deinde uectis in FG, Hk; & centrum grauitatis in LM. dico ean
dem potentiam in kBG idemmet &longs;emper &longs;u&longs;tinere pondus.
Quoniam enim pondus in uecte AB perinde &longs;e habet, ac &longs;i e&longs;&longs;et
uecte Hk. ac &longs;i in M e&longs;&longs;et appen&longs;um; di&longs;tantiæ uerò CL CE
CM &longs;unt inter &longs;e &longs;e æquales; nec non CK CB CG inter &longs;e æ
quales; erit potentia in B ad pondus, ut CE ad CB; atque poten
vt CL ad CG. eadem igitur potentia in k
pondus &longs;u&longs;tinebit. quod demon&longs;trare oportebat.
Similiter o&longs;tendetur, &longs;i pondus e&longs;&longs;et intra potentiam, & fulci
mentum; vel potentia inter fulcimentum, & pondus. quod idem
potentiæ mouenti eueniet.
PROPOSITIO XI.
Si vectis di&longs;tantia inter fulcimentum, & poten
tiam ad di&longs;tantiam fulcimento, punctoq;, vbi
à centro grauitatis ponderis horizonti ducta
perpendicularis vectem &longs;ecat, interiectam ma
iorem habuerit proportionem, quàm pondus
ad potentiam; pondus vtiq; à potentia moue
bitur.
Sit véctis AB, ex
punctoq; A &longs;u&longs;penda
tur pondus C; hoc e&longs;t
punctum A &longs;emper &longs;it
punctum, vbi perpen
dicularis à grauitatis
centro ponderis du
cta vectem &longs;ecat; &longs;itq;
potentia in B, ac fulcimentum &longs;it D; & DB ad DA maiorem
habeat proportionem, quàm pondus C ad potentiam in B. Di
co pondus Cà potentia in B moueri. fiat vt BD ad DA, ita
pondus E ad potentiam in B; atq; pondus E quoq; appendatur
in A: patet potentiam in B æqueponderare ip&longs;i E; hoc e&longs;t pon
dus & quoniam BD ad DA maiorem habet pro
portionem, quàm Cad potentiam in B; & vt BD ad DA, ita
tentiam: igitur E ad
potentiam maiorem
habebit proportio
nem, quàm pondus
C ad eandem potenquare pondus
E maius erit ponde
re C. & cùm potentia ip&longs;a E æqueponderet, potentia igitur ip&longs;i
C non æqueponderabit, &longs;ed &longs;ua ui deor&longs;um verget. pondus igitur
C à potentia in B mouebitur vecte AB, cuius fulcimentum
e&longs;t D.
Si verò &longs;it vectis AB, &
fulcimentum A, pondu&longs;q; C
in D appen&longs;um, & potentia
in B; & BA ad AD maio
rem habeat proportionem,
quàm pondus C ad poten
tiam in B. dico pondus C à
potentia in B moueri. fiat vt BA ad AD; ita pondus E ad poten
nebit. &longs;ed cùm BA ad AD maiorem habeat proportionem,
quàm pondus C ad potentiam in B; & vt BA ad AD, ita e&longs;t
pondus E ad potentiam in B: pondus igitur E ad potentiam,
quæ e&longs;t in B, maiorem habebit proportionem, quàm pondus C & ideo pondus E maius erit pondere C.
potentia verò in B &longs;u&longs;tinet pondus E; ergo potentia in B pondus
C minus pondere E in D appen&longs;um mouebit vecte AB, cuius fulci
mentum e&longs;t A.
Sit rur&longs;us vectis
AB, cuius fulcimen
B &longs;it appen&longs;um; &longs;itq;
potentia in D: &
DA ad AB maio
rem habeat propor
tionem, quàm pon
dus C ad potentiam, quæ e&longs;t in D. dico pondus C à potentia
in D moueri. fiat vt DA ad AB, ita pondus E ad potentiam in
D; & &longs;it pondus E ex puncto B &longs;u&longs;pen&longs;um: potentia in D pondus
E &longs;u&longs;tinebit. &longs;ed DA ad AB maiorem habet proportionem,
quàm C ad potentiam in D; & vt DA ad AB, ita e&longs;t pondus E
ad potentiam in D; pondus igitur E ad potentiam, quæ e&longs;t in D,
maiorem habebit proportionem, quàm pondus C ad eandem po
tentiam. quare pondus E maius e&longs;t pondere C.
& cùm poten
tia in D pondus E &longs;u&longs;tineat, potentia igitur in D pondus C in B
appen&longs;um vecte AB, cuius fulcimentum e&longs;t A, mouebit. quod
demon&longs;trare oportebat.
ALITER.
Sit vectis AB, &
pondus C in A ap
pen&longs;um & poten
tia in B; &longs;itqué fulci
mentum D: & DB
ad DA maiorem habeat proportionem, quàm pondus C ad po
tentiam in B. dico pondus C à potentia in B moueri.
fiat BE ad
EA, vt pondus C ad potentiam, erit punctum E inter BD. opor
tet enim BE ad EA minorem habere proportionem, quàm DB
ad DA, & ideo BE minor erit BD. & quoniam potentia in B &longs;u
&longs;tinet pondus C in A appen&longs;um uecte AB, cuius
igitur potentia in B, quàm data, idem pondus &longs;u&longs;tinebit fulcimen
to D. data ergo potentia in B pondus C mouebit uecte AB, cuius
fulcimentum e&longs;t D.
Sit deinde vectis AB, & fulci
mentum A, & pondus C in D
appen&longs;um, &longs;itq; potentia in B; &
AB ad AD maiorem habeat pro
portionem, quàm pondus C ad
potentiam in B. dico pondus C
à potentia in B moueri. Fiat AB ad AE, vt pondus C ad poten
nece&longs;&longs;e e&longs;t enim AE
maiorem e&longs;&longs;e AD. & &longs;i pondus C e&longs;&longs;et in E appen&longs;um, potentia
minor autem potentia in B, quàm data, &longs;u&longs;ti
Sit rur&longs;us vectis AB, cu
ius fulcimentum A, & pon
dus C in B &longs;it appen&longs;um;
&longs;itq; potentia in D; & DA
ad AB maiorem habeat
proportionem, quàm pondus C ad potentiam in D. dico pon
dus C à potentia in D moueri. fiat vt pondus C ad potentiam,
DA ad AB, quàm DA ad AE. & &longs;i pondus C appendatur in
mi
cte AB, cuius fulcimentum e&longs;t A. quod oportebat demon
&longs;trare.
PROPOSITIO XII.
PROBLEMA.
Datum pondus à data potentia dato vecte
moueri.
Sit pondus A vt centum, potentia verò mouens &longs;it vt decem;
&longs;itq; datus vectis BC. oportet potentiam, quæ e&longs;t decem pondus
A centum vecte BC mouere. Diuidatur BC in D, ita vt CD
ad DB eandem habeat proportionem, quàm habet centum ad
decem, hoc e&longs;t decem ad vnum; etenim &longs;i D fieret fulcimentum,
con&longs;tat potentiam vt decem in C æqueponderare ponderi A in B
appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. accipiatur inter BD quod
uis punctum E, & fiat E fulcimentum. Quoniam enim maior
e&longs;t proportio CE ad EB, quàm CD ad DB; maiorem habebit
proportionem CE ad EB, quàm pondus A ad potentiam decem
in C: potentia igitur decem in C pondus A centum in B appen
&longs;um vecte BC, cuius fulcimentum &longs;it E, mouebit.
Si verò &longs;it vectis
BC, & fulcimen
tum B. diuidatur CB
in D, ita vt CB ad
BD eandem habeat
proportionem,
habet centum ad decem: & &longs;i pondus A in D &longs;u&longs;pendatur, & po
tentia in C, potentia vt decem in C pondus A in D appen&longs;um &longs;u
&longs;tinebit. accipiatur inter DB quoduis punctum E, ponaturq; pon
dus A in E; & cùm &longs;it maior proportio CB ad BE, quàm
BC ad BD; maiorem habebit proportionem CB ad BE, quàm
pondus A centum ad potentiam decem. potentia igitur decem
in C pondus A centum in E appen&longs;um mouebit vecte BC, cu
ius fulcimentum e&longs;t B. quod facere oportebat.
Hoc autem fieri non po
te&longs;t exi&longs;tente vecte BC, cuius
fulcimentum &longs;it B, & pondus
A centum in C appen&longs;um: po
natur enim potentia &longs;u&longs;tinens
pondus A vtcunq; inter BC, quare opor
tet datam potentiam maiorem e&longs;&longs;e pondere A. &longs;it igitur poten
tia data vt centum quinquaginta. diuidatur BC in D, ita vt CB
ad BD &longs;it, vt centum quinquaginta ad centum; hoc e&longs;t tria ad duo:
dus A in C accipiatur itaq; inter DC quoduis pun
portio EB ad BC, quàm DB ad BC; habebit EB ad BC maio
rem proportionem, quàm pondus A ad potentiam in E. poten
appen&longs;um vecte BC, cuius fulcimentum e&longs;t B, mouebit. quod
facere oportebat.
COROLLARIVM.
Hinc manife&longs;tum e&longs;t &longs;i data potentia &longs;it dato
pondere maior; hoc fieri po&longs;&longs;e, &longs;iue ita exi&longs;ten
te vecte, vt eius fulcimentum &longs;it inter pondus,
& potentiam; &longs;iue pondus inter fulcimentum,
& potentiam habente; &longs;iue demum potentia in
ter pondus, & fulcimentum con&longs;tituta.
Sin autem data potentia minor, vel æqualis
dato pondere fuerit; palam quoq; e&longs;t id ip&longs;um
dumtaxat a&longs;&longs;e qui po&longs;&longs;e vecte ita exi&longs;tente, vt eius
fulcimentum &longs;it inter pondus, & pontentiam;
habente.
PROPOSITIO XIII.
PROBLEMA.
Quotcunq; datis in vecte ponderibus vbicun
què appen&longs;is, cuius fulcimentum &longs;it quoq; da
tum, potentiam inuenire, quæ in dato puncto
data pondera &longs;u&longs;tineat.
Sint data pondera ABC in vecte DE, cuius fulcimentum F,
vbicunq; in punctis DGH appen&longs;a: collocandaq; &longs;it potentia in
puncto E. potentiam inuenire oportet, quæ in E data pondera
ABC vecte DE &longs;u&longs;tineat. diuidatur DG in k, ita vt Dk ad KG
&longs;it, vt pondus B ad pondus A; deinde diuidatur kH in L, ita vt kL
ad LH, &longs;it vt pondus C ad pondera BA; atq; vt FE ad FL, ita
fiant pondera ABC &longs;imul ad potentiam, quæ ponatur in E. di
co potentiam in E data pondera ABC in DGH appen&longs;a vecte
DE, cuius fulcimentum e&longs;t F, &longs;u&longs;tinere. Quoniam enim &longs;i ponde
ra ABC &longs;imul e&longs;&longs;ent in L appen&longs;a, potentia in E data pondera
in L appen&longs;a &longs;u&longs;tineret; pondera verò ABC tàm in L ponderant,
quam
ponderant, quàm &longs;i A in D, & B in G appen&longs;a e&longs;&longs;ent; ergo po
tentia in E data pondera ABC in DGH appen&longs;a vecte DE, cu
ius fulcimentum e&longs;t F, &longs;u&longs;tinebit. Si autem potentia in quouis
alio puncto vectis DE (præterquàm in F) con&longs;tituenda e&longs;&longs;et,
vt in k; fiat vt Fk ad FL, ita pondera ABC ad potentiam: &longs;i
ctis DGH appen&longs;a &longs;u&longs;tinere. quod facere oportebat.
Ex hac, & ex quinta huius, &longs;i pondera ABC &longs;int in vecte
DE quomodocunq; po&longs;ita; oporteatq; potentiam inuenire, quæ
in E data pondera &longs;u&longs;tinere debeat: ducantur à centris grauita
tum ponderum ABC horizontibus perpendiculares, quæ ve
ctem DE in DGH punctis &longs;ecent; cæteraq; eodem modo fiant:
Manife&longs;tum e&longs;t, potentiam in E, vel in K data pondera &longs;u&longs;tinere.
idem enim e&longs;t, ac &longs;i pondera in DGH e&longs;&longs;ent appen&longs;a.
PROPOSITIO XIIII.
PROBLEMA.
Data quotcunq; pondera in dato vecte vbi
cunq; & quomodocunq; po&longs;ita à data potentia
moueri.
Sit datus vectis DE, & &longs;int data pondera vt in præcedenti co
rollario; &longs;itq; A vt centum, B vt quinquaginta, C vt triginta;
dataq; potentia &longs;it vt triginta. exponantur eadem, inueniaturq;
punctum L; deinde diuidatur LE in F, ita vt FE ad FL &longs;it, vt
centum octoginta ad triginta, hoc e&longs;t &longs;ex ad vnum: & &longs;i F fieret
fulcimentum, potentia vt triginta in E &longs;u&longs;tineret pondera ABC.
accipiatur igitur inter LF quoduis punctum M, fiatq; M fulci
mentum: manife&longs;tum e&longs;t potentiam in E vt triginta pondera
ABC vt centum octoginta vecte DE mouere. quod facere
oportebat.
Hoc autem vniuersè a&longs;&longs;equi minimè poterimus, &longs;i in extremita
te vectis fulcimentum e&longs;&longs;et, vt in D; quia proportio DE, ad DL
hoc e&longs;t proportio ponderum ABC ad potentiam, quæ pondera
&longs;u&longs;tinere debeat, &longs;emper e&longs;t data. quod multo quoq; minus fieri
po&longs;&longs;et, &longs;i ponenda e&longs;&longs;et potentia inter DL.
PROPOSITIO XV.
PROBLEMA.
Quia verò dum pondera vecte mouentur,
vectis quoq; grauitatem habet, cuius nulla ha
ctenus mentio facta e&longs;t: idcirco primùm quo
modo inueniatur potentia, quæ in dato puncto
datum vectem, cuius fulcimentum &longs;it quoq; da
tum, &longs;u&longs;tineat, o&longs;tendamus.
Sit datus vectis AB, cuius fulcimentum &longs;it datum C; &longs;itq;
punctum D, in quo collocanda &longs;it potentia, quæ vectem AB &longs;u
&longs;tinere debeat, ita vt immobilis per&longs;i&longs;tat. ducatur à puncto C
linea CE horizonti perpendicularis, quæ vectem AB in duas di
uidat partes AE EF, &longs;itq; partis AE centrum grauitatis G, &
partis EF centrum grauitatis H; à punctisqué GH horizon
tibus perpendiculares ducantur Gk HL, quæ lineam AF
in punctis KL &longs;ecent. quoniam enim vectis AB à linea CE in duas
diuiditur partes AE EF; ideo vectis AB nihil aliud erit, ni&longs;i
duo pondera AE EF in vecte, &longs;iue libra AF po&longs;ita; cuius &longs;u
&longs;pen&longs;io, &longs;iue fulcimentum e&longs;t C. quare pondera AE EF ita erunt
po&longs;ita, ac &longs;i in kL e&longs;&longs;ent appen&longs;a. diuidatur ergo kL in M,
ita vt kM ad ML, &longs;it vt grauitas partis EF ad grauitatem par
tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad
potentiam, quæ &longs;i collocetur in D (dummodo DA horizonti
AB deor&longs;um premendo &longs;u&longs;tinebit. quod inuenire oportebat.
Si verò potentia in puncto B ponenda e&longs;&longs;et.
fiat vt CF ad CM
ita pondus AB ad potentiam. &longs;imili modo o&longs;tendetur poten
tiam in B vectem AB &longs;u&longs;tinere. &longs;imiliterq; demon&longs;trabitur in quo
cunq; alio &longs;itu (præterquàm in e) ponenda fuerit potentia, vt in
N. fiat enim vt CO ad CM, ita AB ad potentiam; quæ &longs;i pona
tur in N, vectem AB &longs;u&longs;tinebit.
Adiiciatur autem pondus in vecte appen&longs;um,
&longs;iue po&longs;itum; vt iisdem po&longs;itis &longs;it pondus P in
A appen&longs;um; potentiaq; &longs;it ponenda in B, ita
vt vectem AB vnà cum pondere P &longs;u&longs;tineat.
Diuidatur AM in Q, ita vt AQ ad QM &longs;it, ut grauitas ue
ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat
grauitas AB, & P &longs;imul ad potentiam, quæ ponatur in B: patet
potentiam in B uectem AB unà cum pondere P &longs;u&longs;tinere. Si ue
rò e&longs;&longs;et CA ad CM, vt AB ad P; e&longs;&longs;et punctum C eorum centrum
grauitatis, & ideo vectis AB vná cum pondere P ab&longs;q; potentia in
B manebit. &longs;ed &longs;i ponderum grauitatis centrum e&longs;&longs;et inter CF, vt
in O; fiat vt CF ad CO, ita AB&P &longs;imul ad potentiam, quæ
in B, & vectem AB, & pondus P &longs;u&longs;tinebit.
Similiter o&longs;tendetur, &longs;i plura e&longs;&longs;ent pondera in vecte AB ubi
cunq;, & quomodocunq; po&longs;ita.
In&longs;uper ex his non &longs;olum, ut in decimaquarta huius docuimus,
quomodo &longs;cilicet data pondera ubicunq; in uecte po&longs;ita data poten
tia dato uecte mouere po&longs;&longs;umus, eodem modo grauitate uectis
con&longs;iderata idem facere poterimus; uerùm etiam accidentia reli
qua, quæ &longs;upra ab&longs;q; uectis grauitatis con&longs;ideratione demon&longs;tra
ta &longs;unt; &longs;imili modo uectis grauitate con&longs;iderata vná cum ponde
ribus, uel &longs;ine ponderibus o&longs;tendentur.
DE TROCHLEA.
Trochleae in&longs;trumento pon
dus multipliciter moueri pote&longs;t;
quia verò in omnibus e&longs;t eadem
ratio: ideo (vt res euidentior ap
pareat) in iis, quæ dicenda &longs;unt,
intelligatur pondus &longs;ur&longs;um ad re
ctos horizontis plano angulos hoc modo &longs;em
per moueri.
Sit pondus A, quod ip&longs;i ho
rizontis plano &longs;ur&longs;um ad rectos
angulos &longs;it attollendum; & vt
fieri &longs;olet, trochlea duos habens
orbiculos, quorum axiculi &longs;int
in BC, &longs;upernè appendatur;
trochlea verò duos &longs;imiliter ha
bens orbiculos, quorum axicu
li &longs;int in DE, ponderi alligetur:
ac per omnes vtriu&longs;q; trochleæ
orbiculos circunducatur ducta
rius funis, quem in altero eius ex
tremo, putá in F, oportet e&longs;&longs;e
religatum. potentia autem mo
uens ponatur in G, quæ dum
de&longs;cendit, pondus A &longs;ur&longs;um ex
aduer&longs;o attolletur; quemadmo
dum Pappus in octauo libro Ma
thematicarum collectionum a&longs;
&longs;erit; nec non Vitruuius in deci
mo de Architectura, & alii.
Quomodo autem hoc trochleæ in&longs;trumen
tum reducatur ad vectem; cur magnum pondus
ab exigua virtute, & quomodo, quantoq; in tem
pore moueatur; cur funis in vno capite debeat
e&longs;&longs;e religatus; quodq; &longs;uperioris, inferioris&qacute;ue
trochleæ fuerit officium; & quomodo omnis in
dus inueniri po&longs;sit; dicamus.
LEMMA.
Sint rectæ lineæ AB CD parallelæ, quæ in
punctis AC circulum ACE contingant, cuius
centrum F: & FA FC connectantur. Dico
AFC rectam lineam e&longs;&longs;e.
Ducatur FE ip&longs;is AB CD æquidi&longs;tans.
& quoniam AB, & FE &longs;unt parallelæ, &
angulus BAF e&longs;t rectus; erit & AFE re
ctus. eodemq; modo CFE rectus erit.
li
nea igitur quod erat de
mon&longs;trandum.
PROPOSITIO I.
Si funis trochleæ &longs;upernè appen&longs;æ orbiculo
circunducatur, alterumq; eius extremum pon
deri alligetur, altero interim à potentia pondus
&longs;u&longs;tinente apprehen&longs;o: erit potentia ponderi
æqualis.
Sit pondus A,
cui alligatus &longs;it fu
nis in B; trochleaq;
habens orbiculum C
EF, cuius centrum
D, &longs;ur&longs;um appenda
tur; &longs;itq; D quoq;
centrum axiculi; &
circa orbiculum uo
luatur funis BC EF
G; &longs;itq; potentia
in G &longs;u&longs;tinens pon
dus A. dico poten
tiam in G ponderi A
æqualem e&longs;&longs;e. Sit FG
æquidi&longs;tans CB.
Quoniam igitur pon
CB horizonti plano perpendicularis <*> quare FG eidem plano perSint CF
in horizontis con
nectantur DC DF; erit CF recta linea, & anguli DCB DFG recti.
erit linea CF horizonti æquidi&longs;tans. cùm verò
CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t
D; nam in axiculo
centrum axiculi, & orbiculi, etiam vtri&longs;que circumuolutis
immobile remanet. Itaq; cùm di&longs;tantia DC &longs;it æqualis di&longs;tantiæ
DF, potentiaq; in F ponderi A in C appen&longs;o æqueponderet, cùm
(nam idem e&longs;t) con&longs;tituta ponderi A æqualis. Idem enim effi
cit potentia in G, ac &longs;i in G aliud e&longs;&longs;et appen&longs;um pondus æquale
ponderi A; quæ pondera in CF appen&longs;a æquæponderabunt. Præ
terea, cùm in neutram fiat motus partem, idem erit vnico exi
e&longs;&longs;ent funes BC FG alligati in vecte, &longs;iue libra CF.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;&longs;e pote&longs;t, idem pon
dus ab eadem potentia ab&longs;q; ullo huius tro
chleæ auxilio nihilominus &longs;u&longs;tineri po&longs;&longs;e.
Sit enim pondus H æquale
ponderi A, cui alligatus &longs;it funis
kL; &longs;itq; potentia in L &longs;u&longs;tinens
pondus H. cùm autem pondus
ab&longs;q; vllo adminiculo &longs;u&longs;tinere
volentes tanta vi opus &longs;it, quanta
ponderi e&longs;t æqualis; erit potentia
in L ponderi H æqualis; pondus
verò H ip&longs;i ponderi A e&longs;t æquale,
cui potentia in G e&longs;t æqualis; erit
igitur potentia in G potentiæ in L
æqualis. quod idem e&longs;t, ac &longs;i
potentia idem pondus &longs;u&longs;tineret.
Præterea &longs;i potentiæ in G, &
in L inuicem fuerint æquales, &longs;eor
&longs;um autem ponderibus minores;
patet potentias ponderibus &longs;u&longs;ti
nendis non &longs;ufficere. &longs;i verò maiores, manife&longs;tum e&longs;t pondera à
pontentiis moueri. & &longs;ic in eadem e&longs;&longs;e proportione potentiam in
L. ad pondus H, veluti potentia in G ad pondus A.
Sed quoniam in demon&longs;tratione a&longs;&longs;umptum fuit axiculum cir
cumuerti, qui vt plurimum immobilis manet; idcirco immobili
quoq; manente axiculo idem o&longs;tendatur.
Sit orbiculus trochleæ CEF, cu
ius centrum D; &longs;itq; axiculus GHk,
cuius idem &longs;it centrum D. Ducatur
CG DkF diameter horizonti æ
quidi&longs;tans. & quoniam dum orbi
culus circumuertitur, circumferen
tia circuli CEF &longs;emper e&longs;t æquidi
&longs;tans circumferentiæ axiculi GHk;
circa enim axiculum circumuerti
tur; & circulorum æquidi&longs;tantes cir
cumferentiæ idem habent centrum;
erit punctum D &longs;emper & orbiculi,
& axiculi centrum. Itaq; cùm DC &longs;it æqualis DF, & DG ip&longs;i
Dk; erit GC ip&longs;i kF æqualis. &longs;i igitur in vecte, &longs;iue libra CF
pondera appendantur æqualia, æqueponderabunt. di&longs;tantia enim
CG æqualis e&longs;t di&longs;tantiæ kF; axiculu&longs;q; GHK immobilis gerit
vicem centri, &longs;iue fulcimenti. immobili igitur manente axicu
lo, &longs;i ponatur in F potentia &longs;u&longs;tinens pondus in C appen&longs;um; erit
potentia in F ip&longs;i ponderi æqualis. quod erat o&longs;tendendum.
Et cùm idem pror&longs;us &longs;it, &longs;iue axiculus circumuertatur, &longs;iue mi
nus; liceat propterea in iis, quæ dicenda &longs;unt, loco axiculi cen
trum tantùm accipere.
PROPOSITIO II.
Si funis orbiculo trochleæ ponderi alligatæ
circumducatur, altero eius extremo alicubi reli
gato, altero uerò à potentia pondus &longs;u&longs;tinente
apprehen&longs;o; erit potentia ponderis &longs;ubdupla.
Si pondus A; &longs;it BCD
orbiculus trochleæ pon
deri A alligate, cuius cen
trum E; funis deinde FB
CDG circa orbiculum
voluatur, qui religetur in
F; &longs;itq; potentia in G &longs;u
&longs;tinens pondus A. dico
potentiam in G &longs;ubdu
plam e&longs;&longs;e ponderis A. &longs;int
funes FB GD puncti E
horizonti perpendicula
res, qui inter &longs;e &longs;e æqui
di&longs;tantes
funes FB GD circulum
BCD in BD punctis.
connectatur BD; erit BD
per centrum E ducta,
ip&longs;iu&longs;qué centri horizonti æquidi&longs;tans. Cùm autem potén
tia in G trochlea pondus A &longs;u&longs;tinere debeat, funem ex altero ex
tremo religatum e&longs;&longs;e oportet, puta in F; ita vt F æqualiter &longs;altem
potentiæ in G re&longs;i&longs;tat, alioquin potentia in G nullatenus pondus
&longs;u&longs;tinere po&longs;&longs;et. Et quoniam potentia fune &longs;u&longs;tinet orbiculum,
qui reliquam trochleæ partem, cui appen&longs;um e&longs;t pondus, &longs;u&longs;tinet
axiculo; grauitabit hæc trochleæ pars in axiculo, hoc e&longs;t in centro
E. quare pondus A in eodem quoq; centro E ponderabit, ac &longs;i
in E e&longs;&longs;et appen&longs;um. po&longs;ita igitur potentia, quæ in G, vbi D
(idem enim pror&longs;us e&longs;t) erit BD tanquam vectis, cuius fulci
mentum erit B, pondus in E appen&longs;um, & potentia in D. con
uenienter enim fulcimenti rationem ip&longs;um B &longs;ubire pote&longs;t, exi
&longs;tente fune FB immobili. cæterum hoc po&longs;terius magis eluce&longs;cet.
Quoniam autem potentia ad pondus eandem habet proportio
nem,
ad BD: potentia igitur in G ponderis A &longs;ubdupla erit. quod de
mon&longs;trare oportebat.
Hoc igitur ita &longs;e ha
bet vnico exi&longs;tente fune
FBC DG ip&longs;i orbiculo
circumducto, ac &longs;i duo e&longs;
&longs;ent funes BF GD ve
cti BD alligati, cuius ful
cimentum erit B, pon
dus in E appen&longs;um, &
potentia &longs;u&longs;tinens in D,
vel quod idem e&longs;t in G.
COROLLARIVM I.
Ex hoc itaq; manife&longs;tum e&longs;t, pondus hoc mo
do à minori in &longs;ubdupla proportione potentia
&longs;u&longs;tineri, quam &longs;ine vllo huiu&longs;modi trochleæ
auxilio.
Veluti &longs;it pondus H ponderi A
æquale, cui religatus &longs;it funis kL;
potentiaq; in L &longs;u&longs;tineat pondus H;
erit potentia in L &longs;eor&longs;um ponderi
H, & ponderi A æqualis; &longs;ed poten
tia in G &longs;ubdupla e&longs;t ponderis A,
quare potentia in G &longs;ubdupla erit po
tentiæ, quæ e&longs;t in L. & hoc modo in
huiu&longs;cemodi reliquis omnibus pro
portio inueniri poterit.
COROLLARIVM. II.
Manife&longs;tum e&longs;t etiam; &longs;i duæ fuerint poten
tiæ vna in G, altera in F, pondus A &longs;u&longs;tinentes;
vtra&longs;q; &longs;imul ponderi A æquales e&longs;&longs;e: & vnam
quamque &longs;u&longs;tinere dimidium ponderis A.
Hoc autem ex tertio, & quarto corollario &longs;ecundæ huius in
tractatu de vecte patet.
COROLLARIVM III.
Illud quoq; præterea innote&longs;cit, cur &longs;cilicet fu
nis ex altero religatus e&longs;&longs;e debeat extremo.
PROPOSITIO III.
Si vtri&longs;q; duarum trochlearum &longs;ingulis or
biculis, quarum altera &longs;upernè, altera verò in
fernè con&longs;tituta, ponderiq; alligata fuerit, cir
cunducatur funis; altero eius extremo alicubi
religato, altero verò à potentia pondus &longs;u&longs;ti
nente detento; erit potentia ponderis &longs;ub du
pla.
Sit pondus A; &longs;it BCD orbiculus trochleæ pon
deri A alligatæ, cuius centrum K; EFG verò
&longs;it trochleæ &longs;ur&longs;um appen&longs;æ, cuius centrum H. deinde LBC DME FGN funis circa orbicu
los ducatur, qui religetur in L; &longs;itq; potentia in
N &longs;u&longs;tinens pondus A. dico potentiam in N
&longs;ubduplam e&longs;&longs;e ponderis A. &longs;i enim potentia &longs;u
&longs;tinens pondus A vbi M collocata foret, e&longs;&longs;et
vtiq; potentia in M &longs;ubdupla ponderis A. po
e&longs;t e
A &longs;ine trochlea &longs;u&longs;tineret, cui æqueponderat
pondus in N ponderis A dimidio æquale.
quare vis in N æqualis dimidio ponderis A
ip&longs;um A &longs;u&longs;tinebit. Potentia igitur in N &longs;u&longs;ti
nens pondus A &longs;ubdupla e&longs;t ip&longs;ius A. quod
demon&longs;trare oportebat.
Si verò vt in &longs;ecunda figura &longs;it fu
nis BC DEF GHkL orbiculis cir
cum uolutus, & religatus in B; poten
tiaq; in L pondus A &longs;u&longs;tineat: erit
potentia in L &longs;imiliter ponderis &longs;ubdu
pla. orbiculus enim trochleæ &longs;upe
rioris, ip&longs;aqué trochlea penitus &longs;unt
inutiles: & idem e&longs;t, ac &longs;i funis reli
gatus e&longs;&longs;et in F, & potentia in L &longs;u
&longs;tineret pondus &longs;ola trochlea ponderi
alligata, quæ potentia ponderis A o&longs;ten
&longs;a e&longs;t &longs;ubdupla.
COROLLARIVM.
Ex his &longs;equitur, &longs;i duæ &longs;int potentiæ in BL;
vtra&longs;q; inter &longs;e &longs;e æquales e&longs;&longs;e.
Vtraq; enim &longs;eor&longs;um e&longs;t ip&longs;ius A &longs;ubdupla.
PROPOSITIO IIII.
Sit vectis AB, cuius fulcimentum &longs;it A; qui
bifariam diuidatur in D: &longs;itq; pondus C in D
appen&longs;um; duæq; &longs;int potentiæ æquales in BD
pondus C &longs;u&longs;tinentes. Dico unamquamq; poten
tiam in BD ponderis C &longs;ubtriplam e&longs;&longs;e.
Quoniam enim altera
potentia e&longs;t in D colloca
ta, & pondus C in eodem
puncto D e&longs;t appen&longs;um;
potentia in D partem
ponderis C &longs;u&longs;tinebit ip
&longs;i potentiæ D æqualem.
quare potentia in B partem &longs;u&longs;tinebit reliquam, quæ pars dupla erit
ip&longs;ius potentiæ in B; cùm pondus ad potentiam eandem habeat
proportionem, quam AB ad AD: & potentiæ in BD &longs;unt æqua
les; ergo potentia in B duplam &longs;u&longs;tinebit partem eius, quam &longs;u&longs;ti
net potentia in D. diuidatur ergo pondus C in duas partes, qua
rum vna &longs;it reliquæ dupla; quod fiet, &longs;i in tres partes æquales EFG
diui&longs;erimus: tunc enim FG dupla erit ip&longs;ius E. Itaq; potentia
in D partem E &longs;u&longs;tinebit, & potentiam in B reliquas FG. vtreq;
igitur inter &longs;e &longs;e æquales potentiæ in BD &longs;imul totum &longs;u&longs;tinebunt
pondus C. & quoniam potentia in D partem E &longs;u&longs;tinet, quæ ter
tia e&longs;t pars ponderis C, ip&longs;iq; e&longs;t æqualis; erit potentia in D &longs;ub
tripla ponderis C. & cùm potentia in B &longs;u&longs;tineat partes FG, qua
rum potentia in B e&longs;t &longs;ubdupla; erit in B potentia vni partium FG,
putà G æqualis. G verò tertia e&longs;t pars ponderis C; potentia
igitur in B &longs;ubtripla erit ponderis C. Vnaquæq; ergo potentia in
BD &longs;ubtripla e&longs;t ponderis C. quod demon&longs;trare oportebat.
Et &longs;i duo e&longs;&longs;ent vectes AB EF bifariam in GD diui&longs;i, quorum
fulcimenta e&longs;&longs;ent AF, & pondus C in DG vtriq; vecti appen
&longs;um, ita tamen vt in vtroq; æqualiter ponderet; duæq; e&longs;&longs;ent
æquales potentiæ in BG: eadem pror&longs;us ratione o&longs;tendetur,
vnamquamq; potentiam in B, & G ponderis C &longs;ubtriplam
e&longs;&longs;e.
PROPOSITIO V.
Si vtri&longs;q; duarum
quarum altera &longs;upernè, altera verò infernè con&longs;ti
tuta, ponderiq; alligata fuerit, circumducatur fu
nis; altero eius extremo inferiori trochleæ reli
gato, altero verò à potentia pondus &longs;u&longs;tinente
detento: erit potentia ponderis &longs;ubtripla.
Sit pondus A; &longs;it BCD orbiculus tro
chleæ ponderi A alligate, cuius centrum
E; & FGH trochleæ &longs;ur&longs;um appen&longs;æ, cu
ius centrum k; & LFGHBCDM funis
orbiculis circumducatur, qui religetur in L
trochleæ inferiori; &longs;itq; potentia in M &longs;u
&longs;tinens pondus A. dico potentiam in M
&longs;ubtriplam e&longs;&longs;e ponderis A. ducantur FH
BD per centra kE horizonti æquidi&longs;tan
tes, &longs;icut in præcedentibus dictum e&longs;t. Quo
niam enim funis FL trochleam &longs;u&longs;tinet in
feriorem, quæ &longs;u&longs;tinet orbiculum in eius
centro E; erit funis in L vt potentia &longs;u&longs;ti
nens orbiculum, ac &longs;i in ip&longs;o E centro e&longs;&longs;et;
potentia verò in M e&longs;t, ac &longs;i e&longs;&longs;et in D;
efficietur igitur DB tanquam vectis, cuius
pra o&longs;ten&longs;um e&longs;t) ex E &longs;u&longs;pen&longs;um à dua
bus potentiis altera in D, altera in E &longs;u&longs;ten
tatum. Cùm autem in pondere &longs;u&longs;tinendo
vectes FH BD immobiles maneant, &longs;i in
funibus FL HB appendantur pondera, e
beat fulcimentum in medio; alioquin ex al
tera parte deor&longs;um fieret motus, quod
non contingit. tam igitur &longs;u&longs;tinet funis FL,
quàm HB. deinde quoniam ex medio ve
cte BD pondus &longs;u&longs;penditur, idcirco &longs;i duæ fuerint potentiæ in BD & quamquam funis
tamen ex eodemmet puncto &longs;u&longs;tinet, vbi appen&longs;um e&longs;t pondus, non
efficiet propterea, quin potentiæ in BD &longs;int inter &longs;e &longs;e æquales;
opitulatur enim tàm vni, quàm alteri. potentiæ verò in BD eæ
dem &longs;unt, ac &longs;i e&longs;&longs;ent in HM; quare tàm &longs;u&longs;tinebit funis MD,
quàm HB. ita verò &longs;u&longs;tinet HB, atq; FL; funis igitur MD ita
&longs;u&longs;tinebit, &longs;icut FL, hoc e&longs;t, ac &longs;i in D, & L appen&longs;a e&longs;&longs;ent pon
dera æqualia. Cùm itaq; æqualia pondera à potentiis &longs;u&longs;tinean
tur æqualibus, potentiæ in ML æquales erunt; quarum eadem pror
&longs;us e&longs;t ratio, ac &longs;i e&longs;&longs;ent ambæ in DE. Itaq; cùm pondus A in
medio vectis BD &longs;it appen&longs;um, duæq; potentiæ &longs;int æquales in
DE pondus &longs;u&longs;tinentes; erit B fulcimentum, ac vnaquæq; potentia,
&longs;iue in DE, &longs;iue in ML &longs;ubtripla ponderis A. ergo potentia in M
&longs;u&longs;tinens pondus &longs;ubtripla erit ponderis A. quod o&longs;tendere o
portebat.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, vnumquemq; funem
MD FL HB tertiam &longs;u&longs;tinere partem pon
deris A.
Præterea, &longs;i funis ex M per a
lium adhuc deferatur orbiculum &longs;u
periorem in trochlea &longs;ur&longs;um &longs;imi
liter appen&longs;a con&longs;titutum, cuius
centrum N; ita vt perueniat in O;
ibiq; à potentia detineatur; erit po
tentia in O &longs;u&longs;tinens pondus A iti
dem &longs;ubtripla ip&longs;ius ponderis. fu
nis enim MD tantùm ponderis &longs;u
&longs;tinet, ac &longs;i in D appen&longs;um e&longs;&longs;et
pondus æquale tertiæ parti ponde
O ip&longs;i æqualis, hoc e&longs;t &longs;ubtripla
ponderis A. Potentia igitur in O
&longs;ubtripla e&longs;t ponderis A.
Et ne idem &longs;æpius repetatur, no
ui&longs;&longs;e oportet potentiam in O &longs;em
per æqualem e&longs;&longs;e ei, quæ e&longs;t in M;
hoc e&longs;t &longs;i potentia in M e&longs;&longs;et &longs;ub
quadrupla, &longs;ubquintupla, vel huiu&longs;
modi aliter ip&longs;ius ponderis; poten
tia quoq; in O erit itidem &longs;ubqua
drupla, &longs;ubquintupla, atq; ita dein
ceps eiu&longs;demmet ponderis, quem
madmodum &longs;e habet potentia
in M.
PROPOSITIO VI.
Sint duo vectes AB CD bifariam diui&longs;i in
EF, quorum fulcimenta &longs;int. in BD; &longs;itq; pon
dus G in EF vtriq; vecti appen&longs;um, ita ut ex
vtroq; æqualiter ponderet; duæq; &longs;int potentiæ
in AC æquales pondus &longs;u&longs;tinentes. Dico unam
quamq; potentiam in AC &longs;ubquadruplam e&longs;
&longs;e ponderis G.
Cùm enim potentiæ in
AC totum &longs;u&longs;tineant pon
dus G, potentiaq; in A ad
partem ponderis, quod &longs;u&longs;ti
net, &longs;it vt BE ad BA; po
tentia
ip&longs;ius G, quod &longs;u&longs;tinet, ita
&longs;it vt DF ad DC; & vt BE
ad BA, ita e&longs;t DF ad DC;
erit potentia in A ad partem ponderis, quod &longs;u&longs;tinet, vt poten
tia in C ad ip&longs;ius ponderis, quod &longs;u&longs;tinet, partem; & potentiæ
in AC &longs;unt æquales; æquales igitur erunt partes ponderis G,
quæ à potentiis &longs;u&longs;tinentur. quare vnaquæq; potentia in A C di
midium &longs;u&longs;tinebit ponderis G. Potentia verò in A &longs;ubdupla e&longs;t pon
deris, quod &longs;u&longs;tinet: ergo potentia in A dimidio dimidii, hoc
e&longs;t quartæ portioni ponderis G æqualis erit; ideoq; &longs;ubquadrupla
erit ponderis G. neq; aliter demon&longs;trabitur potentiam in C &longs;ub
quadruplam e&longs;&longs;e eiu&longs;dem ponderis G. quod demon&longs;trare opor
tebat.
Si verò tres &longs;int vectes
AB CD EF bifariam di
ui&longs;i in GHk, quorum fulci
menta &longs;int BDF; & pondus
L eodem modo in GHK
appen&longs;um; &longs;intq; tres poten
tiæ in ACE æquales pondus
&longs;u&longs;tinentes; &longs;imiliter o&longs;ten
detur vnamquamque po
tentiam &longs;ub&longs;excuplam e&longs;&longs;e
ponderis L. atq; hoc ordi
ne &longs;i quatuor e&longs;&longs;ent vectes,
& quatuor potentiæ; erit vnaquæq; potentia &longs;uboctupla ponderis.
atq; ita deinceps in infinitum.
PROPOSITIO VII.
Si tribus duarum trochlearum orbiculis,
altera &longs;upernè vnico duntaxat, altera verò infer
nè duobus autem in&longs;ignita orbiculis, ponderiq;
alligata con&longs;tituta fuerit, funis circumponatur; al
tero eius extremo alicubi religato, altero verò à
potentia pondus &longs;u&longs;tinente retento; erit potentia
ponderis &longs;ubquadrupla.
Sit pondus A; &longs;int tres orbiculi, quorum
centra BCD; orbiculu&longs;q;, cuius centrum D,
&longs;it trochleæ &longs;ur&longs;um appen&longs;æ; quorum verò
&longs;unt centra BC, &longs;int trochleæ ponderi A alli
gatæ; funi&longs;q; EFGHkLNOP per omnes
circumducatur orbiculos, qui religetur in E;
&longs;itq; vis in P &longs;u&longs;tinens pondus A. dico po
tentiam in P &longs;ubquadruplam e&longs;&longs;e ponderis
A. ducantur kL GF ON per rotularum
centra, & horizonti æquidi&longs;tantes, quæ (ex
iis, quæ dicta &longs;unt) tanquam vectes erunt.
& quoniam propter vectem, &longs;iue libram kL,
cuius fulcimentum, &longs;iue centrum e&longs;t in me
dio, tàm &longs;u&longs;tinet funis kG, quàm LN, cùm
in neutram partem fiat motus. nec non
propter vectem GF, è cuius medio veluti &longs;u
&longs;pen&longs;um dependet onus; &longs;i duæ e&longs;&longs;ent in GF
potentiæ, &longs;eu in HE (e&longs;t enim par vtriu&longs;q;
&longs;itus ratio, vt iam &longs;epius dictum e&longs;t) e&longs;&longs;ent
vtiq; huiu&longs;modi potentiæ inuicem æquales.
quare ita &longs;u&longs;tinet funis HG, vt EF. &longs;imiliter
o&longs;ten detur funem PO tàm &longs;u&longs;tinere, quàm
LN: quare funes PO kG EF LN æqua
liter &longs;u&longs;tinent. æqualiter igitur funis PO &longs;u
&longs;tinet, vt kG. &longs;i ergo duæ intelligantur e&longs;
&longs;e potentiæ in OG, &longs;eu in PH, quod idem e&longs;t, pondus nihilomi
nus &longs;u&longs;tinentes, quemadmodum funes &longs;u&longs;tinent, æquales vtiq; e&longs;
&longs;ent; & GF ON duorum vectium vires gerent; quorum fulci
menta erunt FN, & pondus A in BC medio vectium appen&longs;um.
& quoniam omnes funes æqualiter &longs;u&longs;tinent, tàm &longs;u&longs;tinebunt
duo PO LN, quàm duo KGEF; tàm igitur &longs;u&longs;tinebit vectis
ON, quàm vectis GF. quare in vtroq; vecte ON GF æquali
ter pondus erit ergo vnaquæq; potentia in PH &longs;ubquadru
pla ponderis A. & cùm funis KG potentiæ loco &longs;umatur, quippè
qui haud &longs;ecus &longs;u&longs;tinet, quàm PO; erit potentia in P &longs;u&longs;tinens pon
dus A ip&longs;ius ponderis &longs;ubquadrupla. quod demon&longs;trare oportebat.
COROLLARIVM I.
Hinc manife&longs;tum e&longs;t vnumquemq; funem EF
GK LN OP quartam &longs;u&longs;tinere partem pon
deris A.
COROLLARIVM II.
Patet etiam orbiculum, cuius centrum C,
non minus eo, cuius centrum e&longs;t B, &longs;u&longs;tinere.
ALITER.
Adhuc ii&longs;dem po&longs;itis, &longs;i duæ e&longs;&longs;ent poten
tiæ æquales pondus A &longs;u&longs;tinentes, vna in O
tiarum ponderis A &longs;ubtripla. &longs;ed quoniam
vectis GF, cuius fulcimentum e&longs;t F bifariam
diui&longs;us e&longs;t in C; &longs;i igitur ponatur in G poten
tia idem pondus &longs;u&longs;tinens, vt potentia in C;
erit potentia in G &longs;ubdupla potentiæ, quæ e&longs;
&longs;et in C; nam &longs;i potentia in C &longs;e ip&longs;a pon
dus in C appen&longs;um &longs;u&longs;tineret, e&longs;&longs;et vtiq; ip
&longs;i ponderi æqualis; & idem pondus, &longs;i à po
tiæ in G duplum; potentia veró in C &longs;ubtri
pla e&longs;&longs;et ponderis A; ergo potentia in G
&longs;ub&longs;excupla e&longs;&longs;et ponderis A. Cùm itaq;
potentia in O &longs;ubtripla &longs;it ponderis A, &
potentia in G &longs;ub&longs;excupla; erunt vtræq; &longs;i
mul potentiæ in OG ip&longs;ius ponderis A &longs;ub
duplæ. tertia enim pars cum &longs;exta dimi
dium efficit. quoniam autem potentiæ in
OG, &longs;iue in PH (vt prius dictum e&longs;t)
&longs;unt inter &longs;e æquales, ac vtræq; &longs;imul &longs;ubdu
plæ &longs;unt ponderis A. erit vnaquæq; potenPotentia igitur in P &longs;u&longs;tinens pon
dus A ip&longs;ius ponderis A &longs;ubquadrupla erit. quod erat o&longs;ten
dendum.
Si verò funis religetur in E,
& &longs;ecundùm quatuor adhuc
circumuoluatur orbiculos, per
ueniatq; ad P. &longs;imiliter o&longs;ten
detur potentiam in P &longs;ubqua
druplam e&longs;&longs;e ponderis A. idem enim e&longs;t, ac &longs;i funis re
ligatus e&longs;&longs;et in L, potentiaq;
&longs;u&longs;tineret pondus fune tribus
tantùm orbiculis circumdu
cto, quorum centra e&longs;&longs;ent B
CQ. orbiculus enim cuius
centrum D e&longs;t pœnitus inu
tilis. PROPOSITIO VIII.
Sint duo vetes AB CD bifariam diui&longs;i in EF,
quorum fulcimenta &longs;int AC, & pondus G in
punctis EF vtriq; vecti &longs;it appen&longs;um, ita vt ex
vtroq; æqualiter ponderet; tre&longs;q; &longs;int potentiæ
æquales in BDE pondus G &longs;u&longs;tinentes. Dico
vnamquamq; &longs;eor&longs;um ex dictis potentiis &longs;ub
quintuplam e&longs;&longs;e ponderis G.
Quoniam enim pondus G
appen&longs;um e&longs;t in EF, & tres
&longs;unt potentiæ in EBD æqua
les; ideo potentia in E partem
tantùm ponderis G &longs;u&longs;tinebit
ip&longs;i potentiæ in E æqualem;
potentiæ verò in BD partem
&longs;u&longs;tinebunt reliquam; & pars,
dupla; pars autem, quam &longs;u
&longs;tinet D, erit &longs;imiliter ip&longs;ius D dupla; propter proportionem
BA ad AE, & DC ad CF. Cùm itaq; potentiæ in BD &longs;int æqua
à potentiis BD &longs;u&longs;tinentur, inter &longs;e &longs;e æquales; & vnaquæq; du
pla eius partis, quæ à potentia in E &longs;u&longs;tinetur. diuidatur er
go pondus G in tres partes, quarum duæ &longs;int inter &longs;e &longs;e æquales,
nec non vnaquæq; &longs;eor&longs;um alterius tertiæ partis dupla. quod
fiet, &longs;i in quinq; partes æquales HKLMN diuidatur; pars
enim compo&longs;ita ex duabus partibus kL dupla e&longs;t partis H; pars
quoq; MN eiu&longs;dem partis H e&longs;t &longs;imiliter dupla. quare & pars
kL parti MN erit æqualis. Su&longs;tineat autem potentia in E par
tem H; & potentia in B partes KL; potentia verò in D partes
dus G; & vnaquæq; potentia in BD duplum &longs;u&longs;tinebit eius, quod
&longs;u&longs;tinet potentia in E. Cùm itaq; potentia in E partem H &longs;u&longs;ti
neat, quæ quinta e&longs;t pars ponderis G, ip&longs;iq; &longs;it æqualis; erit po
tentia in E &longs;ubquintupla ponderis G. & quoniam potentia in B
partes kL &longs;u&longs;tinet, quæ quidem duplæ &longs;unt potentiæ B, & partis H;
erit quoq; potentia in B ip&longs;i H æqualis: quare &longs;ubquintupla erit
ponderis G. Non aliter o&longs;tendetur potentiam in D &longs;ubquintu
plam e&longs;&longs;e ponderis G. vnaquæq; igitur potentia in BDE &longs;ubquin
tupla e&longs;t ponderis G. quod demon&longs;trare oportebat.
Si verò &longs;int tres vectes AB
CD EF bifariam diui&longs;i in
GHk, quorum fulcimenta
&longs;int ACE; & pondus L eo
dem modo in GHk &longs;it ap
pen&longs;um; quatuorq; &longs;int po
tentiæ æquales in BDFG
pondus L &longs;u&longs;tinentes; &longs;imili
modo o&longs;tendetur vnam
quamq; potentiam in BD
FG &longs;ub&longs;eptuplam e&longs;&longs;e ponde
ris L. & &longs;i quatuor e&longs;&longs;ent vectes, & quinq; potentiæ æquales pon
dus &longs;u&longs;tinentes; eodem quoq; modo o&longs;tendetur vnamquamq;
potentiam &longs;ubnonuplam e&longs;&longs;e ponderis. atq; ita deinceps.
PROPOSITIO VIIII.
Si quatuor duarum trochlearum binis orbi
culis, quarum altera &longs;upernè, altera vero in
fernè, ponderiq; alligata, di&longs;po&longs;ita fuerit, cir
cumducatur funis; altero eius extremo inferiori
dus &longs;u&longs;tinente retento: erit potentia ponderis
&longs;ubquintupla.
Sit pondus A, cui alligata &longs;it trochlea duos
habens orbiculos, quorum centra &longs;int BC;
&longs;itq; trochlea &longs;ur&longs;um appen&longs;a duos alios ha
bens orbiculos, quorum centra &longs;int DE; funi&longs;q;
per omnes circumducatur orbiculos, qui tro
chleæ inferiori religetur in F; &longs;itqué poten
tia in G &longs;u&longs;tinens pondus A. dico poten
tiam in G &longs;ubquintuplam e&longs;&longs;e ponderis A. ducantur Hk LM per centra BC horizon
ti æquidi&longs;tantes, quas eodem modo, quo &longs;u
pra dictum e&longs;t, e&longs;&longs;e tanquam vectes o&longs;tende
mus, quorum fulcimenta kM, & pondus A
ex medio vtriu&longs;q; vectis BC &longs;u&longs;pen&longs;um, & tres
potentiæ in LHC pondus &longs;u&longs;tinentes, quas
&longs;imili modo æquales e&longs;&longs;e demon&longs;trabimus; fu
nes enim idem efficiunt, ac &longs;i e&longs;&longs;ent potentiæ.
& quoniam pondus æqualiter ex vtroq; ve
cte HK LM ponderat, quod quidem o&longs;ten
detur quoque, vt in præcedentibus demon
L, &longs;eu in G, quod idem e&longs;t; tùm in H, atq;
in C, hoc e&longs;t in F, &longs;ubquintupla ponderis A.
Potentia ergo in G &longs;u&longs;tinens pondus A ip&longs;ius
A &longs;ubquintupla erit. quod o&longs;tendere opor
tebat.
Si verò funis in F adhuc de
feratur circa alium orbiculum,
cuius centrum N, qui religetur
in O; &longs;imiliter duplici medio
(vt in &longs;eptima huius) demon
&longs;trabitur potentiam in G pon
dus A &longs;u&longs;tinentem &longs;ub&longs;excu
plam e&longs;&longs;e ponderis A. Primùm
quidem ex tribus vectibus LM
Hk FP, quorum fulcimenta
&longs;unt MkP, & pondus in me
dio vectium appen&longs;um; & tres
potentiæ in LHF æquales pon
dus &longs;u&longs;tinéres. deinde ex poten
tiis in LHN, quarum vnaquæq;
&longs;ubquintupla e&longs;&longs;et ponderis A. e&longs;&longs;ent enim ambæ &longs;imul poten
tiæ in LH &longs;ubduplæ &longs;exquialte
ræ ip&longs;ius ponderis,
in F &longs;ubdecupla e&longs;&longs;et, cùm &longs;it ip
&longs;ius N &longs;ubdupla: &longs;ed duæ quin
tæ cùm decima dimidium ef
ficiunt, quòd &longs;i per terna diui
datur, &longs;exta pars ponderis re
&longs;pondebit vnicuiq; potentiæ in
LHF. ex quibus patet poten
tiam in G &longs;ub&longs;excuplam e&longs;&longs;e
ponderis A. &longs;imiliterq; demon
&longs;trabitur vnumquemque orbi
culum æqualem &longs;u&longs;tinere por
tionem.
Quòd &longs;i, vt in tertia figura
funis in O protrahatur; per
aliumq; circumducatur orbi
culum, cuius centrum Q; qui
deinde in R trochleæ relige
tur inferiori; erit potentia in atq;
ita in infinitum procedendo
proportio potentiæ ad pon
dus quotcunq; &longs;ubmulti
plex inueniri poterit. dein
de &longs;emper o&longs;tendetur vt in
præcedentibus; &longs;i potentia
pondus &longs;u&longs;tinens fuerit, vel
&longs;ubquadrupla, vel &longs;ubquitu
pla, vel quouis alio modo &longs;e
habebit ad pondus; &longs;imiliter
vnumquemque funem, vel
quartam, vel quintam, vel
quamuis aliam partem &longs;u&longs;ti
nere ponderis, quemadmo
dum potentia ip&longs;a; funes e
nim idem efficiunt, ac &longs;i tot
e&longs;&longs;ent potentiæ: orbiculi ve
rò, ac &longs;i tot e&longs;&longs;ent vectes.
COROLLARIVM
Ex his manife&longs;tum e&longs;t orbiculos trochleæ, cui
e&longs;t alligatum pondus, efficere, vt pondus mino
quod quidem trochleæ &longs;uperioris orbiculi non
efficiunt.
Noui&longs;&longs;e tamen oportet, quòd (vt fieri &longs;olet) inferioris tro
chleæ orbiculus, cuius centrum N, minor e&longs;&longs;e debet eo, cuius cen
trum C; hic autem minor adhuc eo, cuius centrum B; ac deniq;
&longs;i plures fuerint orbiculi in trochlea inferiori ponderi alligata, &longs;em
per cæteris maior e&longs;&longs;e debet, qui annexo ponderi e&longs;t propinquior.
oppo&longs;ito autem modo di&longs;ponendi &longs;unt in trochlea &longs;uperiori. quod
fieri con&longs;ueuit, ne funes inuicem complicentur; nam quantùm
ad orbiculos attinet, &longs;iue magni fuerint, &longs;iue parui, nihil refert;
cùm &longs;emper idem &longs;equatur.
Præterea notandum e&longs;t, quod etiam ex dictis facilè patet, &longs;i
funis, &longs;iue religetur in R trochleæ inferiori, &longs;iue in S, maximam
indè oriri differentiam inter potentiam, & pondus: nam &longs;i relige
tur in S, erit potentia in G ponderis &longs;ub&longs;excupla. &longs;i verò in R,
&longs;ub&longs;eptupla. quod trochleæ &longs;uperiori non contingit, quia &longs;iue
religetur funis (vt in præcedenti figura) in T, &longs;iue in O; &longs;em
per potentia in G &longs;ub&longs;excupla erit ip&longs;ius ponderis.
Po&longs;t hæc con&longs;iderandum e&longs;t, quonam modo vis moueat pon
dus; necnon potentiæ mouentis, ponderi&longs;q; moti &longs;patium, atque
tempus.
PROPOSITIO X.
Si funis orbiculo trochleæ &longs;ur&longs;um appen&longs;æ
fuerit circumuolutus, cuius altero extremo &longs;it al
ligatum pondus; alteri autem mouens collocata
&longs;it potentia: mouebit hæc vecte horizonti &longs;em
per æquidi&longs;tante.
Sit pondus A, &longs;it orbiculus trochleæ &longs;ur
&longs;um appen&longs;æ' cuius centrum K; &longs;it deinde
funis HBCDEF aligatus ponderi A in H,
orbiculoq; circumductus; &longs;itq; trochlea ita in
L appen&longs;a, & nullum alium habeat motum
præter liberam orbiculi circa axem ver&longs;ionem;
&longs;itq; potentia in F mouens pondus A. Dico
potentiam in F &longs;emper mouere pondus A
vecte horizonti æquidi&longs;tante. ducatur BKE
horizonti æquidi&longs;tans; &longs;intq; BE puncta, vbi
funes BH, & EF circulum tangunt; erit BkE
k. &longs;icut &longs;upra o&longs;ten&longs;um e&longs;t.
dum itaq; vis
in F deor&longs;um tendit ver&longs;us M, vectis EB
mouebitur, cùm totus orbiculus moueatur,
hoc e&longs;t circumuertatur. dum igitur F e&longs;t in M, &longs;it punctum E ve
ctis v&longs;q; ad I motum; B autem v&longs;q; ad C, ita vt vectis &longs;it in
CI. fiat deinde NM æqualis ip&longs;i FE: & quando punctum E
erit in I,
tem erat in B erit in C; ita vt ducta CI per centrum K tran&longs;eat.
dum autem B e&longs;t in C, &longs;it punctum H in G; eritq; BH ip&longs;i
CBG æqualis; cùm &longs;it idem funis. & quoniam dum EF tendit
in NM, adhuc &longs;emper remanet EFM horizonti perpendicularis,
circulumq; tangens in puncto E; ita vt ducta à puncto E per cen
trum k, &longs;it &longs;emper horizonti æquidi&longs;tans. quod idem euenit funi
BG, & puncto B. dum igitur circulus, &longs;iue orbiculus circumuer
titur, &longs;emper mouetur vectis EB, &longs;emperq; adhuc remanet alius
vectis in EB. &longs;iquidem ex ip&longs;ius rotulæ natura, in qua &longs;emper
dum mouetur, remanet diameter ex B in E (quæ vectis vicem ge
rit) euenit, vt recedente vna, &longs;emper altera &longs;uccedat; eiu&longs;modi
durante circumductione: atq; ita fit, vt potentia &longs;emper moueat
pondus vecte EB horizonti æquidi&longs;tante. quod demon&longs;trare opor
tebat.
Ii&longs;dem po&longs;itis, &longs;patium potentiæ pondus
mouentis e&longs;t æquale &longs;patio eiu&longs;dem ponderis
moti.
Quoniam enim o&longs;ten&longs;um e&longs;t, dum F e&longs;t in M, pondus A, hoc
e&longs;t punctum H e&longs;&longs;e in G; & cùm funis HBCDEF &longs;it æqualis
GBCDENFM, e&longs;t enim idem funis; dempto igitur communi
GBCDENF, erit HG ip&longs;i FM æqualis. &longs;imiliterq; o&longs;tende
tur, de&longs;cen&longs;um F &longs;emper æqualem e&longs;&longs;e a&longs;cen&longs;ui H. ergo &longs;patium
potentiæ æquale e&longs;t &longs;patio ponderis. quod erat demon&longs;tran
dum.
Præterea potentia idem pondus per æquale
&longs;patium in æquali tempore mouet, tàm fune
hoc modo orbiculo trochleæ &longs;ur&longs;um appen&longs;æ
circumuoluto, quàm &longs;ine trochlea: dummo
do ip&longs;ius potentiæ lationes in velocitate &longs;int æ
quales.
Ii&longs;dem po&longs;itis &longs;it aliud pondus P
æquale ponderi A, cui alligatus &longs;it
funis TQ
et &longs;it TQ ip&longs;i HB æqualis; moueat
qué
ad rectos angulos horizonti, quem
admodum mouetur pondus A. di
co per æquale &longs;patium in eodem
tempore potentiam in Q pondus
P, & potentiam in F pondus A
mouere. quod idem e&longs;t, ac &longs;i e&longs;&longs;et
idem pondus in æquali tempore
motum; &longs;icut propo&longs;uimus. Pro
ducatur EF in S, & TQ in R;
fiantq; QR FS non &longs;olum inter
&longs;e &longs;e, verùm etiam ip&longs;i BH æqua
les. Cùm autem TQ QR &longs;int
ip&longs;is HB FS æquales, & vis in Q
moueat pondus P per rectam T
QR; vis autem in F moueat A
per rectam HB, & velocitates
motuum vtriu&longs;q; potentiæ &longs;int æquales; tunc in eodem tempore
potentia in Q erit in R, & potentia in F erit in S; cùm &longs;patia &longs;int
æqualia. &longs;ed dum potentia in Q e&longs;t in R, pondus P, hoc e&longs;t
punctum T erit in Q; cùm TQ &longs;it ip&longs;i QR æqualis. & dum po
tentia in F e&longs;t in S, pondus A, hoc e&longs;t punctum H erit in B; &longs;ed
&longs;patium TQ æquale e&longs;t &longs;patio HB, potentiæ ergo in FQ æquali
ter motæ pondera PA æqualia per æqualia &longs;patia in eodem tempo
re mouebunt. quod erat demon&longs;trandum
PROPOSITIO XI.
Si funis orbiculo trochleæ ponderi alligatæ
fuerit circumuolutus, qui in altero eius extre
mouente pondus appræhen&longs;o; vecte &longs;emper ho
rizonti æqui&longs;tante potentia mouebit.
Sit pondus A; Sit orbiculus.
CED trochleæ ponderi A alli
gatæ ex kH; &longs;itq; KH ad rectos
angulos horizonti, ita vt pon
dus &longs;emper trochleæ motum, &longs;i
ue &longs;ur&longs;um, &longs;iue deor&longs;um factum
&longs;equatur; &longs;itq; orbiculi centrum
K; & funis orbiculo circumuo
lutus &longs;it BCDEF, qui relige
tur in B, ita vt in B immobilis
maneat; & &longs;it potentia in F mo
uens pondus A. dico potentiam
in F &longs;emper mouere
cte horizonti æquidi&longs;tante. &longs;int
BC EF inter &longs;e &longs;e, ip&longs;iq; kH æ
quidi&longs;tantes, & eiu&longs;dem kH ho
rizonti perpendiculares, tangen
te&longs;q;
et connectatur EC, quæ per cen
trum k tran&longs;ibit, horizontiq;
æquidi&longs;tans erit; &longs;icuti prius di
ctum e&longs;t. Quoniam enim or
biculus CED circa eius cen
trum K vertitur; ideo dum vis
in F trahit &longs;ur&longs;um punctum E,
deberet punctum C de&longs;cende
re, ac trahere deor&longs;um B; &longs;ed fu
nis in B e&longs;t immobilis, & BC
potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur;
ac per con&longs;equens tota trochlea, & pondus; & EkC erit tanquam
vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC
ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF,
quòd &longs;i punctum C omnino fue
rit immobile, moueaturq; ve
ctis EC in NC; & diuidatur
NC bifariam in L: erunt CL
LN ip&longs;is Ck KE æquales.
quare &longs;i vectis EC e&longs;&longs;et in CN,
punctum k e&longs;&longs;et in L; & &longs;i du
catur LM horizonti perpendi
cularis, quæ &longs;it etiam æqualis
kH; e&longs;&longs;et pondus A, hoc e&longs;t
punctum H in M. &longs;ed quoniam
potentia in F dum tendit &longs;ur
&longs;um mouendo orbiculum, &longs;em
per mouetur &longs;uper rectam EFG,
quæ &longs;emper e&longs;t quoq; æquidi
&longs;tans BC; nece&longs;&longs;e erit orbicu
lum trochleæ &longs;emper inter li
neas EG BC e&longs;&longs;e: & centrum
k, cum &longs;it in medio, &longs;uper
rectam lineam HkT &longs;emper
moueri. Itaq; ducatur per L li
nea PTLQ horizonti, & EC
æquidi&longs;tans, quæ &longs;ecet Hk pro
ductam in T; & centro T, &longs;pa
tio verò TQ, circulus de&longs;criba
tur QRPS, qui æqualis erit circulo CED; & puncta PQ tangent furectangulum enim e&longs;t PECQ, &
PT TQ ip&longs;is EK kC &longs;unt æquales. deinde per T ducatur R
TS diameter circuli PQS æquidi&longs;tans ip&longs;i NC; fiatqué TO æqua
lis kH. dum autem centrum k motum erit v&longs;q; ad lineam PQ,
tunc centrum k erit in T. o&longs;ten&longs;um e&longs;t enim centrum orbiculi &longs;u
per rectam HT &longs;emper moueri. idcirco vt centrum k &longs;it in li
nea PQ ip&longs;i EC æquidi&longs;tante, nece&longs;&longs;e e&longs;t vt &longs;it in T. & vt vectis
EC eleuetur in angulo ECN, nece&longs;&longs;e e&longs;t, vt &longs;it in RS, non aucùm totus orbiculus &longs;ur
&longs;um moueatur, toru&longs;q; mutet totum locum; habet tamen C ratio
nem fulcimenti, quia minus mouetur C, quàm k, & E: punctum
enim E mouetur v&longs;q; ad R, & K v&longs;q; ad T, punctum verò C v&longs;q;
ad S tantùm. quare dum centrum K e&longs;t in T, po&longs;itio orbiculi erit
QR PS: & pondus A. hoc e&longs;t punctum H erit in O; cùm TO
&longs;it æqualis kH; po&longs;itio verò EC, &longs;cilicet vectis moti, erit RS, po
tentiaq; in F mota erit &longs;ur&longs;um per rectam EFG. eodem autem
tempore, quo k erit in T, &longs;it potentia in G: dum autem vectis EC
hoc modo mouetur, adhuc &longs;emper remanent GP BQ inter &longs;e &longs;e æ
quidi&longs;tantes, atq; horizonti perpendiculares, ita vt vbi orbiculum
tangunt, vt in punctis PQ; &longs;emper linea PQ erit diameter orbi
culi, & tanquam vectis horizonti æquidi&longs;tans. dum igitur orbi
culus mouetur, & circumuertitur, &longs;emper etiam mouetur vectis
EC, & &longs;emper remanet alius vectis in orbiculo horizonti æqui&longs;tans,
vt PQ; ita vt potentia in F &longs;emper moueat pondus vecte hori
zonti æquidi&longs;tante, cuius fulcimentum erit &longs;emper in linea CB; &
pondus in medio vectis appen&longs;um; potentiaq; in linea EG. quod
erat o&longs;tendendum.
Ii&longs;dem po&longs;itis, &longs;patium potentiæ pondus
mouentis duplum e&longs;t &longs;patii eiu&longs;dem ponderis
moti.
Cùm enim o&longs;ten&longs;um &longs;it, dum k e&longs;t in T, pondus A, hoc e&longs;t
punctum H e&longs;&longs;e in O, & in eodem etiam tempore potentiam in
F e&longs;&longs;e in G: & quoniam funis BCDEF e&longs;t æqualis funi BQS
PG; funis enim e&longs;t idem; & funis circa &longs;emicirculum CDE e&longs;t
æqualis funi circa &longs;emicirculum QSP; demptis igitur communi
bus BQ, & FP; erit reliquus FG ip&longs;is CQ, & EP &longs;imul &longs;umptis
æqualis. &longs;ed EP ip&longs;i TK e&longs;t æqualis, & CQ ip&longs;i quoq; Tk æqualis,
&longs;unt enim Pk TC parallelogramma rectangula; quare lineæ EP
CQ &longs;imul ip&longs;ius Tk duplæ erunt. funis igitur FC ip&longs;ius TK du
plus erit. & quoniam kH e&longs;t æqualis TO, dempto communi kO,
erit kT ip&longs;i HO æqualis; quare funis FG ip&longs;ius HO duplus erit; quod erat
demon&longs;trandum.
Potentia deinde idem pondus in æquali tem
pore per dimidium &longs;patium mouebit fune circa
orbiculum trochleæ ponderi alligatæ reuoluto,
quàm &longs;ine trochlea; dummodo ip&longs;ius potentiæ
velocitates motuum &longs;int æquales.
Sit enim (ii&longs;dem po&longs;i
tis) aliud pondus V æqua
le ponderi A, cui alligatus
&longs;it funis 9X; &longs;itq; poten
tia in X mouens pondus
V. dico &longs;i vtriu&longs;q; poten
tiæ motuum velocitates
&longs;int æquales, in eodem
tempore potentiam in F
mouere pondus A per di
midium &longs;patium eius, per
quod à potentia in X mo
uetur pondus V; quod
idem e&longs;t, ac &longs;i e&longs;&longs;et idem
pondus in æquali tempo
re motum. Moueat po
tentia in X pondus V, po
tentiaq; perueniat in Y;
&longs;itq; XY æqualis ip&longs;i FG;
& fiat YZ æqualis X9, ita
vt quando potentia in X
erit in Y, &longs;it pondus V,
hoc e&longs;t punctum 9 in Z.
&longs;ed 9 Z e&longs;t æqualis FG, Itaq; dum poten
tiæ erunt in GY, pondera AV erunt in OZ. in eodem autem
tempore erunt potentiæ in GY, ip&longs;arum enim velocitates mo
tuum &longs;unt æquales; quare vis in F pondus A in eodem tempore
mouebit per dimidium &longs;patium eius, per quod mouetur à poten
tia in X pondus V: & pondera &longs;unt æqualia; Potentia ergo idem
pondus in æquali tempore per dimidium &longs;patium mouebit fune,
trochleaq; hoc modo ponderi alligata, quàm &longs;ine trochlea; dum
modo potentiæ motuum velocitates &longs;int æquales. quod erat de
mon&longs;trandum.
PROPOSITIO XII.
Si funis circa plures reuoluatur orbiculos, al
tero eius extremo alicubi religato, altero au
tem à potentia pondus mouente detento; poten
tia vectibus horizonti &longs;emper æquidi&longs;tantibus
mouebit.
Sit pondus A, &longs;it orbiculus CED tro
chleæ ponderi alligatæ ex kS ad rectos an
gulos horizonti; ita vt pondus &longs;emper eius
motum &longs;ur&longs;um, ac deor&longs;um factum &longs;equa
tur. &longs;it deinde orbiculus circa centrum L
trochleæ &longs;ur&longs;um appen&longs;æ &longs;itq; funis circa
orbiculos reuolutus BCDEHMNO,
qui religatus &longs;it in B; &longs;itq; vis in O mouens
pondus A mouendo &longs;e deor&longs;um per OP.
dico potentiam in O &longs;emper mouere pon
dus A vectibus horizonti &longs;emper æquidi
&longs;tantibus. ducatur NH per centrum L ho
culi, cuius centrum e&longs;t L. ducatur deinde
EC per centrum k &longs;imiliter horizonti æqui
ius centrum e&longs;t k. Moueatur potentia in
O deor&longs;um, quæ dum deor&longs;um mouetur, ve
ctem NH mouebit; & dum vectis moue
&longs;um, vti &longs;upra dictum e&longs;t. dum autem H
mouetur &longs;ur&longs;um, mouet etiam &longs;ur&longs;um E; &
vectem EC, cuius fulcimentum e&longs;t C, &longs;ed
fulcimentum C non pote&longs;t mouere deor
&longs;um B; ideo orbiculus, cuius centrum K, &longs;ur
&longs;um mouebitur, & per con&longs;equens trochlea, & pondus A; vt in
præcedenti dictum e&longs;t. & quoniam ob eandem cau&longs;am in præce
dentibus a&longs;signatam in HN, & EC &longs;emper remanent vectes hori
zonti æquidi&longs;tantes; potentia ergo mouens pondus A &longs;emper
eum mouebit vectibus horizonti æquidi&longs;tantibus. quod erat o
&longs;tendendum.
Et &longs;i funis circa plures &longs;it reuolutus orbiculos; &longs;imiliter o&longs;tende
tur, potentiam mouere pondus vectibus horizonti &longs;emper æqui
di&longs;tantibus: & vectes orbiculorum trochleæ &longs;uperioris &longs;emper
e&longs;&longs;e, vt HN, quorum fulcimenta erunt &longs;emper in medio: vectes au
tem orbiculorum trochleæ inferioris &longs;emper exi&longs;tere, vt EC; quo
Ii&longs;dem po&longs;itis, &longs;patium potentiæ duplum e&longs;t
&longs;patii ponderis.
Sit motum centrum K v&longs;q; ad centrum R; & orbiculus &longs;it FTG.
deinde per centrum R ducatur GF ip&longs;i EC æquidi&longs;tans: tangent
funes EH CB orbiculum in GF punctis. fiat deniq; RQ æqua
lis KS. dum igitur k erit in R; pondus A, &longs;cilicet punctum S erit
in q. & dum centrum orbiculi e&longs;t in R, &longs;it potentia in O mota
in P. & quoniam funis BCDEHMNO e&longs;t æqualis funi BFT
GHMNP; e&longs;t enim idem funis; & FTG æqualis e&longs;t CDE; dem
ptis igitur communibus BF, & GHMNO, erit reliquus OP ip
&longs;is FCEG &longs;imul &longs;umptis æqualis: & per con&longs;equens duplus kR,
& QS & cùm OP &longs;it &longs;patium potentiæ motæ, & SQ &longs;patium pon
deris moti; erit &longs;patium potentiæ duplum &longs;patii ponderis. quod
erat o&longs;tendendum.
Præterea potentia idem pondus in æquali
tempore per dimidium &longs;patium mouebit fune
circa duos orbiculos reuoluto, quorum vnus
&longs;it trochleæ &longs;uperioris, alter verò &longs;it trochleæ
ponderi alligatæ; quàm &longs;ine trochleis: dummo
do ip&longs;ius potentiæ lationes &longs;int æqualiter ve
loces.
Ii&longs;dem namq; po&longs;itis, &longs;it pon
dus V æquale ip&longs;i A, cui alliga
tus &longs;it funis X9; &longs;itq;
mouens
dus mouet, perueniat in Y: fiant
qué XY Z9 ip&longs;i OP æquales;
erit Z9 dupla QS. & &longs;i vtriu&longs;
que potentiæ velocitates mo
tuum &longs;int æquales; patet pon
dus V duplum pertran&longs;ire &longs;pa
tium in eodem tempore eìus,
quod pertran&longs;it pondus A. in eo
dem enim tempore potentia in
X peruenit ad Y, & potentia in
O ad P; ponderaq; &longs;imiliter in
Z Q. quod erat demon&longs;tran
dum.
PROPOSITIO XIII.
Fune circa &longs;ingulos duarum trochlearum
orbiculos, quarum altera &longs;upernè, altera verò
infernè, ponderiq; alligata fuerit, reuoluto;
altero etiam eius extremo inferiori trochleæ re
to: erit decur&longs;um trahentis potentiæ &longs;patium, mo
ti ponderis &longs;patii triplum.
Sit pondus A; &longs;it BCD orbiculus tro
chleæ ponderi A ex EQ &longs;u&longs;pen&longs;o alligatæ;
&longs;itq; orbiculi centrum E; &longs;it deinde FGH
orbiculus trochleæ &longs;ur&longs;um appen&longs;æ, cuius
centrum k; &longs;itq; funis LFGHDCBM
circa omnes reuolutus orbiculos, tro
chleæq; inferiori in L religatus: &longs;itq; in
M potentia mouens. dico &longs;patium de
cur&longs;um à potentia in M, dum mouet pon
dus, triplum e&longs;&longs;e &longs;patii moti ponderis A.
Moueatur potentia in M v&longs;q; ad N; &
centrum E &longs;it motum v&longs;q; ad O; & L v&longs;
que ad P; atq; pondus A, hoc e&longs;t pun
ctum Q v&longs;q; ad R; orbiculu&longs;q; motus, &longs;it
TSV. ducantur per EO lineæ ST BD
horizonti æquidi&longs;tantes, quæ inter &longs;e &longs;e
quoq; æquidi&longs;tantes erunt. quoniam au
tem dum E e&longs;t in O, punctum Q e&longs;t in
R; erit EQ æqualis OR, & EO ip&longs;i QR
æqualis; &longs;imiliter LQ æqualis erit PR,
& L P ip&longs;i QR æqualis. tres igitur QR
EO LP inter &longs;e &longs;e æquales erunt; quibus
etiam &longs;unt æquales BS DT. & quoniam fu
nis LFGHDCBM æqualis e&longs;t funi PF
GHTVSN, cùm &longs;it idem funis, & qui
circa &longs;emicirculum TVS e&longs;t æqualis funi
circa &longs;emicirculum BCD; demptis igi
tur communibus PFGHT' & SM; erit
reliquus MN tribus BS LP DT &longs;imul
&longs;umptis æqualis. BS verò LP DT &longs;imul
tripli &longs;unt EO, & ex con&longs;equenti QR.
ti triplum erit. quod erat demon&longs;trandum.
Tempus quoq; huius motus manife&longs;tum e&longs;t, eadem enim po
tentia in æquali tempore &longs;patio &longs;ecundùm triplum ampliori &longs;ine
huiu&longs;modi trochleis idem pondus mouebit, quàm cum ei&longs;dem
hoc modo accomodatis. &longs;patium ponderis &longs;ine trochleis moti
æquale e&longs;t &longs;patio potentiæ. & hoc modo in omnibus inueniemus
tempus.
PROPOSITIO XIIII.
Fune circa tres duarum trochlearum orbicu
los, quarum altera &longs;upernè vnico dumtaxat, al
tera verò in&longs;ernè, duobus autem in&longs;ignita or
biculis, ponderi&qacute;ue alligata fuerit, reuoluto;
altero eius e&longs;tremo alicubi religato, altero autem
à potentia pondus mouente detento: erit decur
&longs;um trahentis potentiæ &longs;patium moti ponderis
&longs;patii quadruplum.
Sit pondus A, &longs;int duo orbiculi,
tra
pondus motum trochleæ &longs;ur&longs;um, & deor&longs;um
&longs;emper &longs;equatur: &longs;it deinde orbiculus, cuius cen
trum L, trochleæ &longs;ur&longs;um appen&longs;æ in <35>; &longs;itq;
funis circa omnes orbiculos circumuolutus BC
DEFGHZMNO, religatu&longs;q; in B; &longs;itq; po
tentia in O mouens pondus A. dico &longs;patium,
quod mouendo pertran&longs;it potentia in O, qua
druplum e&longs;&longs;e &longs;patii moti ponderis A. mouean
tur orbiculi trochleæ ponderi alligatæ; & dum
centrum k e&longs;t in R, centrum I &longs;it in S, & pon
dus A, hoc e&longs;t punctum
rit æqualis. orbiculi enim inter &longs;e &longs;e eandem
&longs;emper &longs;eruant di&longs;tantiam; & k
qualis erit. ducantur per orbiculorum centra
lineæ FH QT EC VX NZ horizonti æqui
di&longs;tantes, quæ tangent funes in FHQTEC
VX NZ punctis, & inter &longs;e &longs;e quoq; æquidi
&longs;tantes erunt: & EQ CT VN XZ non &longs;o
lum inter &longs;e &longs;e, &longs;ed etiam ip&longs;is IS KR
les erunt. & dum centra kI &longs;unt in RS, po
tentia in O &longs;it mota in P. & quoniam funis
BCDEFGHZMNO e&longs;t æqualis funi BT9
QFGHXYVP, e&longs;t enim
ca T9Q XYV &longs;emicirculos &longs;unt æquales funibus, qui &longs;unt circa
CDE ZMN; Demptis igitur communibus BT, QF GHX,
& VO; erit OP æqualis ip&longs;is VN XZ CT QE &longs;imul &longs;umptis.
quatuor verò VN ZX CT QE &longs;unt inter &longs;e &longs;e æquales, & &longs;imul
quadruplæ kR, & &longs;pa
tium igitur potentiæ quadruplum e&longs;t &longs;patii ponderis. quod erat
o&longs;tendendum.
Et &longs;i funis in P circa alium adhuc reuoluatur orbiculum ver&longs;us
<35>, potentiaqué mouendo &longs;e deor&longs;um moueat &longs;ur&longs;um pondus; &longs;imi
liter o&longs;tendetur &longs;patium potentiæ quadruplum e&longs;&longs;e &longs;patii ponderis.
Si verò funis in B circumuoluatur al
teri orbiculo, qui deinde trochleæ in
&longs;u&longs;tinens pondus A &longs;ubquintupla pon
deris. & &longs;i in O &longs;it potentia mouens
pondus A; &longs;imiliter demon&longs;trabitur
&longs;patium potentiæ in O quintuplum e&longs;
&longs;e &longs;patii ponderis A.
Et &longs;i funis ita circa orbiculos apte
tur, vt potentia in O &longs;u&longs;tinens pon
dus &longs;it ponderis &longs;ub&longs;extupla; & loco
potentiæ &longs;u&longs;tinentis ponatur in O po
tentia mouens pondus: eodem modo
o&longs;tendetur &longs;patium potentiæ &longs;extu
plum e&longs;&longs;e &longs;patii ponderis moti. & &longs;ic
procedendo in infinitum proportiones
&longs;patii potentiæ ad &longs;patium ponderis
moti quotcunq; multiplices inuenien
tur.
COROLLARIVM I.
Ex his manife&longs;tum e&longs;t ita &longs;e habere pondus
ad potentiam ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium
potentiæ mouentis ad &longs;patium ponderis moti.
Vt &longs;i pondus A quintuplum &longs;it potentiæ in O pondus A &longs;u&longs;ti
nentis; erit & &longs;patium OP potentiæ pondus mouentis quintuplum
&longs;patii
COROLLARIVM II.
Patet etiam per ea, quæ dicta &longs;unt, orbiculos
trochleæ, quæ ponderi e&longs;t alligata, efficere; vt à
moto pondere minus, quàm à trahente poten
tia de&longs;cribatur &longs;patium; maioriq; tempore datum
æquale &longs;patium de&longs;cribi, quàm &longs;ine illis. quod
quidem orbiculi trochleæ &longs;uperioris non effi
ciunt.
Multiplici o&longs;ten&longs;a ponderis ad potentiam proportione, iam ex
aduer&longs;o potentiæ ad pondus proportio multiplex o&longs;tendatur.
PROPOSITIO XV.
Si funis orbiculo trochleæ à potentia &longs;ur&longs;um
detentæ fuerit circumuolutus; altero eius extre
mo alicubi religato, alteri verò pondere appen
&longs;o; dupla erit ponderis potentia.
Sit trochlea habens orbiculum, cuius
centrum A; & &longs;it pondus B alligatum fu
ni CDEFG, qui circa orbiculum &longs;it re
uolutus, ac tandem religatus in G: &longs;itq;
potentia in H &longs;u&longs;tinens pondus. dico po
tentiam in H duplam e&longs;&longs;e ponderis B. du
catur DF per
&longs;tans.
A, qui pondus &longs;u&longs;tinet; erit potentia &longs;u&longs;ti
nens
ergo in A exi&longs;tente, pondere verò in D
appen&longs;o, funiq; CD religato; erit DF
tanquam vectis, cuius fulcimentum erit
F, pondus in D, & potentia in A. po
ad FA, & DF dupla e&longs;t ip&longs;ius FA; Po
tentia igitur in A, &longs;iue in H, quod idem e&longs;t, ponderis B dupla erit.
quod demon&longs;trare oportebat.
Præterea con&longs;iderandum occurrit, cùm hæc omnia maneant,
idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo
uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al
ligati.
ALITER.
Ii&longs;dem po&longs;itis, &longs;i in G appen&longs;um e&longs;&longs;et pondus k æquale pon
deri B, pondera B k æqueponderabunt in libra DF, cuius centrum
A. potentia verò in H &longs;u&longs;tinens pondera Bk e&longs;t ip&longs;is &longs;imul &longs;um
ptis æqualis, & pondera BK ip&longs;ius B &longs;unt dupla; potentia ergo in
H ponderis B dupla erit. & quoniam funis religatus in G nihil a
liud efficit, ni&longs;i quòd pondus B &longs;u&longs;tinet, ne de&longs;cendat; quod idem
efficit pondus k in G appen&longs;um: potentia igitur in H &longs;u&longs;tinens
pondus B, fune religato in G, dupla e&longs;t ponderis B. quod de
mon&longs;trare oportebat.
PROPOSITIO XVI.
Ii&longs;dem po&longs;itis &longs;i in H &longs;it potentia mouens pon
dus, mouebit hæc eadem vecte horizonti &longs;em
per æquidi&longs;tante.
Hoc etiam (&longs;icut in &longs;uperioribus dictum
e&longs;t) o&longs;tendetur. moueatur enim orbiculus
&longs;ur&longs;um, po&longs;itionemq; habeat MNO, cuius
centrum L: & per L ducatur MLO ip&longs;i DF,
& horizonti æquidi&longs;tans. & quoniam funes
tangunt circulum MON in punctis MO;
ideo cùm potentia in A, &longs;eu in H, quod
idem e&longs;t, moueat pondus B in D appen&longs;um
vecte DF, cuius fulcimentum e&longs;t F; &longs;emper
adhuc remanebit alius vectis, vt MO hori
zonti æquidi&longs;tans, ita vt &longs;emper potentia
moueat pondus vecte horizonti æquidi&longs;tan
te, cuius fulcimentum e&longs;t &longs;emper in linea
OG, & pondus in MC, potentiaq; in cen
tro orbiculi.
Ii&longs;dem po&longs;itis, &longs;patium ponderis moti duplum
e&longs;t &longs;patii potentiæ mouentis.
Sit motus orbiculus à centro A
v&longs;q; ad centrum L; & pondus B,
hoc e&longs;t punctum C, in eodem tem
pore &longs;it motum in P; & potentia in
H v&longs;q; ad K; erit AH ip&longs;i LK æqua
lis, & AL ip&longs;i Hk. & quoniam fu
nis CDEFG e&longs;t æqualis funi PM
NOG, idem enim e&longs;t funis, & fu
nis circa &longs;emicirculum MNO æ
qualis e&longs;t funi circa &longs;emicirculum
DEF; demptis igitur communi
bus DP FG, erit PC æqualis
DM FO &longs;imul &longs;umptis, qui funes
&longs;unt dupli ip&longs;ius AL, & con&longs;equen
ter ip&longs;ius Hk. &longs;patium ergo pon
deris moti CP duplum e&longs;t &longs;patii
Hk potentiæ. quod oportebat de
mon&longs;trare.
COROLLARIVM
Ex hoc manife&longs;tum e&longs;t, idem pondus trahi
ab eadem potentia in æquali tempore per du
plum &longs;patium trochlea hoc modo accommoda
ta, quàm &longs;ine trochlea; dummodo ip&longs;ius poten
tiæ lationes in velocitate &longs;int æquales.
Spatium enim ponderis moti &longs;ine trochlea æquale e&longs;t &longs;patio
potentiæ.
Si autem funis in G circa alium reuoluatur
orbiculum, cuius centrum k; &longs;itq; huiu&longs;mo
di orbiculi trochlea deor&longs;um affixa, quæ nul
lum alium habeat motum, ni&longs;i liberam orbi
culi circa axem reuolutionem; funi&longs;q; relige
tur in M; erit potentia in H &longs;u&longs;tinens pondus
B &longs;imiliter ip&longs;ius ponderis dupla. quod qui
dem manife&longs;tum e&longs;t, cùm idem pror&longs;us &longs;it,
&longs;iue funis &longs;it religatus in M, &longs;iue in G. orbicu
lus enim, cuius centrum k, nihil efficit; penitu&longs;
qué inutilis e&longs;t.
Si verò &longs;it potentia in M &longs;u&longs;tinens pon
dus B, & trochlea &longs;uperior &longs;it &longs;ur&longs;um appen
&longs;a; erit potentia in M æqualis ponderi B.
Quoniam enim potentia in G &longs;u&longs;tinens
pondus B æqualis e&longs;t ponderi B, & ip&longs;i po
tentiæ in G æqualis e&longs;t potentia in L; e&longs;t
enim GL vectis, cuius fulcimentum e&longs;t k;
& di&longs;tantia Gk di&longs;tantiæ kL e&longs;t æqualis;
erit igitur potentia in L, &longs;iue (quod idem e&longs;t)
in M, ponderi B æqualis.
Huiu&longs;modi autem motus fit vectibus DF LG, quorum fulci
menta &longs;unt kA, & pondus in D, & potentia in F. &longs;ed in vecte
LG potentia e&longs;t in L, pondus verò, ac &longs;i e&longs;&longs;et in G.
Si deinde in M &longs;it potentia mouens pondus, transferaturq; po
tentia in N, pondus autem motum fuerit v&longs;q; ad O; erit MN
&longs;patium potentiæ æquale &longs;patio CO ponderis. Cùm enim funis
MLGFDC æqualis &longs;it funi NLGFDO. e&longs;t enim idem funis;
dempto communi MLGFDO; erit &longs;patium MN potentiæ æ
quale &longs;patio CO ponderis.
Et &longs;i funis in M circa plures reuoluatur orbiculos, &longs;emper erit
potentia altero eius extremo pondus &longs;u&longs;tinens æqualis ip&longs;i ponderi.
&longs;patiaq; ponderis, atq; potentiæ mouentis &longs;emper o&longs;tendentur
æqualia.
PROPOSITIO XVII.
Si vtri&longs;q; duarum trochlearum &longs;ingulis orbicu
lis, quarum vna &longs;upernè à potentia &longs;u&longs;tineatur,
altera verò infernè, ibiq; affixa, con&longs;tituta fue
rit, funis circumducatur; altero eius extremo &longs;u
periori trochleæ religato, alteri verò pondere
appen&longs;o; tripla erit ponderis potentia.
Sit orbiculus, cuius centrum A, tro
chleæ infernè affixæ; & &longs;it funis BCD
EFG non &longs;olum huic orbiculo circumuo
lutus, verùm etiam orbiculo trochleæ &longs;u
perioris, cuius centrum k; &longs;itq; funis in
B &longs;uperiori trochleæ religatus; & in G &longs;it ap
pen&longs;um pondus H; potentiaq; in L &longs;u&longs;ti
neat pondus H. dico potentiam in L tri
plam e&longs;&longs;e ponderis H. &longs;i enim duæ e&longs;&longs;ent
potentiæ pondus H
K, altera in B, erunt vtræq; &longs;imul triplæ
ponderis H, & potentia in B ip&longs;i ponderi
æqualis. & quoniam &longs;ola potentia in L
vtri&longs;q; &longs;cilicet potentiæ in KB e&longs;t æqua
lis. &longs;u&longs;tinet enim potentia in L; tùm po
tentiam in K, tùm potentiam in B; idem
qué efficit potentia in L, ac &longs;i duæ e&longs;&longs;ent
potentiæ, vna in k, altera in B: Tri
pla igitur erit potentia in L ponderis H. quod demon&longs;trare oportebat.
Si autem in L &longs;it potentia mouens pondus.
di
co &longs;patium ponderis moti triplum e&longs;&longs;e &longs;patii po
tentiæ motæ.
In præcedenti.
Moueatur centrum or
biculi K v&longs;q; ad M; cuius
quidem motus &longs;patium
motæ potentiæ &longs;patio e&longs;t
æquale, &longs;icuti &longs;upra dictum
e&longs;t: & quando k erit in M,
B erit in N; & NB æqualis
erit M k; & dum k e&longs;t in M,
&longs;it pondus H, hoc e&longs;t pun
ctum G motum in O; & per
MK ducantur EF PQ ho
rizonti æquidi&longs;tantes; erit
vnaquæq; EP BN FQ ip
&longs;i KM æqualis. & quoniam
funis BCDEFG æqualis
e&longs;t funi NCDPQO;
idem enim e&longs;t funis; & fu
nis circa &longs;emicirculum ER
F æqualis e&longs;t funi circa &longs;e
micirculum PSQ: dem
ptis igitur communibus
BCDE, & FO, erit OG
tribus QF NB PE &longs;imul
&longs;umptis æqualis. &longs;ed QF
NB PE &longs;imul triplæ &longs;unt
Mk, hoc e&longs;t &longs;patii poten
tiæ motæ; &longs;patium ergo
GO ponderis H moti tri
plum e&longs;t &longs;patii potentiæ motæ. quod o&longs;tendere oportebat.
PROPOSITIO XVIII.
Si vtriu&longs;q; duarum trochlearum binis orbicu
lis, quarum altera &longs;upernè à potentia &longs;u&longs;tineatur,
altera verò infernè, ibiq; annexa, collocata fue
rit, funis circumnectatur; altero eius extremo
alicubi, non autem &longs;uperiori trochleæ religato,
alteri verò pondere appen&longs;o; quadrupla erit
ponderis potentia.
Sit trochlea inferior, duos habens orbiculos,
quorum centra AB; &longs;it qué trochlea &longs;uperior
duos &longs;imiliter habens orbiculos, quorum cen
tra CD; funi&longs;q; EFGHKLMNOP &longs;it cir
ca omnes orbiculos reuolutus, qui &longs;it religatus
in E; & in P appendatur pondus Q; &longs;itq; po
tentia in R. dico potentiam in R quadruplam
e&longs;&longs;e ponderis q. Cùm enim &longs;i duæ intelligan
tur potentiæ, vna in k, altera in D, potentia
qualis erit ponderi; erunt duæ &longs;imul potentiæ,
vna in D, altera in k, pondus Q &longs;u&longs;tinentes,
triplæ eiu&longs;dem ponderis. Potentia verò in C
dupla e&longs;t potentiæ in k, & per con&longs;equens pon
deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs;
potentia in C; duæ igitur potentiæ in DC qua
druplæ &longs;unt ponderis q. & cùm potentia in R
orbiculis &longs;u&longs;tineat pondus Q, erit
ac &longs;i duæ e&longs;&longs;ent potentiæ, vna in D, altera in C,
& vtræq; &longs;imul pondus Q &longs;u&longs;tinerent. ergo po
tentia in R quadrupla e&longs;t ponderis q. quod
oportebat demon&longs;trare. COROLLARIVM
Ex quo patet, &longs;i funis fuerit religatus in G, &
circa orbiculos, quorum centra &longs;unt BCD reuo
lutus; potentiam in R pondus &longs;u&longs;tinentem &longs;imili
ter ponderis Q quadruplam e&longs;&longs;e. orbiculus enim,
cuius centrum A, nihil efficit.
Si autem in R &longs;it potentia mouens pondus.
dico
&longs;patium ponderis moti quadruplum e&longs;&longs;e &longs;patii
potentiæ.
Moueantur centra CD orbiculorum v&longs;q; ad
ST; erunt ex &longs;uperius dictis CS DT &longs;patio
potentiæ æqualia; & per CSDT ducantur Hk
VX NO YZ horizonti æquidi&longs;tantes; &
centra CD &longs;unt in ST, &longs;it pondus Q, hoc e&longs;t
punctum P motum in 9. & quoniam funis EF
GHKLMNOP æqualis e&longs;t funi EFGVX
LMYZ 9; cùm &longs;it idem funis: & funes circa
&longs;emicirculos NIO H
bus, qui &longs;unt circa &longs;emicirculos Y<35>Z V
demptis igitur communibus EFGH kLMN
& O9; erit P9 ip&longs;is NY ZO VH
mul &longs;umptis æqualis. quatuor autem NY ZO
VH Xk &longs;imul quadrupli &longs;unt DT, hoc e&longs;t
&longs;patii potentiæ; &longs;patium igitur P9 ponderis
quadruplum e&longs;t &longs;patii potentiæ quod demon
&longs;trandum fuerat.
Si autem funis &longs;it re
ligatus in E trochleæ &longs;u
periori, & potentia in R
&longs;u&longs;tineat pondus Q; e
rit potentia in R ponde
ris Q quintupla. & &longs;i in
R &longs;it potentia mouens
pondus; erit &longs;patium pon
deris moti quintuplum
&longs;patii potentiæ. quæ om
nia &longs;imili modo o&longs;ten
dentur, &longs;icut in præce
dentibus demon&longs;tra
tum e&longs;t.
Si verò potentia in R &longs;ub&longs;tineat pon
dus Q trochlea tres orbiculos habente,
quorum centra &longs;int ABC; & &longs;it alia tro
chlea infernè affixa duos, vel tres orbicu
los habens, quorum centra DEF; &longs;itq;
funis circa omnes orbiculos reuolutus, &longs;i
ue in G, &longs;iue in H religatus; &longs;imiliter
o&longs;tendetur potentiam in R &longs;excuplam
e&longs;&longs;e ponderis q. Et &longs;i in R &longs;it potentia
mouens pondus, o&longs;tendetur &longs;patium pon
deris moti &longs;excuplum e&longs;&longs;e &longs;patii poten
tiæ.
Et &longs;i funis &longs;it religatus in K trochleæ
&longs;uperiori, & in R &longs;it potentia pondus
&longs;u&longs;tinens; &longs;imili modo o&longs;tendetur poten
tiam in R &longs;eptuplam e&longs;&longs;e ponderis q.
Et &longs;i in R &longs;it potentia mouens, o&longs;ten
detur &longs;patium ponderis Q &longs;eptuplum e&longs;&longs;e
&longs;patii potentiæ. atq; ita in infinitum
omnis potentiæ ad pondus multiplex
proportio inueniri poterit. &longs;emperq; o
&longs;tendetur, ita e&longs;&longs;e pondus ad potentiam
ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium poten
tiæ pondus mouentis ad &longs;patium ponde
ris moti.
Vectium autem ip&longs;orum orbiculorum
motus in his fit hoc modo, videlicet vectes
orbiculorum trochleæ &longs;uperioris mouen
tur, vti dictum e&longs;t in decima &longs;exta huius;
hoc e&longs;t habent fulcimentum in extremita
te, potentiam in medio, pondus in altera extremitate appen&longs;um. ve
ctes verò trochleæ inferioris habent fulcimentum in medio, pon
dus, & potentiam in extremitatibus.
COROLLARIVM
Manife&longs;tum e&longs;t in his, orbiculos trochleæ &longs;u
perioris efficere, vt pondus moueatur maiori
potentia, quàm &longs;it ip&longs;um pondus, & per maius
&longs;patium potentiæ &longs;patio, & per æquale tempo
re minori; quod quidem orbiculi trochleæ in
ferioris non efficiunt.
Alio quoq; modo hanc potentiæ ad pondus multiplicem propor
tionem inuenire po&longs;&longs;umus.
PROPOSITIO XVIIII.
Si vtriu&longs;q; duarum trochlearum &longs;ingulis orbi
culis, quarum altera &longs;upernè appen&longs;a, altera ve
rò infernè à &longs;u&longs;tinente potentia
funis circumuoluatur; altero eius extremo alicu
bi religato, alteri autem pondere appen&longs;o; du
pla erit ponderis potentia.
Sit orbiculus trochleæ &longs;upernè appen&longs;æ, cu
ius centrum &longs;it A; & BCD &longs;it trochleæ infe
rioris; &longs;it deinde funis EBC DFGHL reli
gatus in E; & in L &longs;it appen&longs;um pondus M;
&longs;itq; potentia in N &longs;u&longs;tinens pondus M.
dico potentiam in N duplam e&longs;&longs;e ponderis
M. Cùm enim &longs;upra o&longs;ten&longs;um &longs;it potentiam
in L, quæ pondus, exempli gratia, O &longs;u&longs;ti
neat
ponderis; potentia igitur in N ponderi O æ
qualis pondus M potentiæ in L æquale &longs;u&longs;ti
nebit; ponderi&longs;q; M dupla erit. quod demon
&longs;trare oportebat.
ALITER.
Ii&longs;dem po&longs;itis.
Quoniam potentia in F,
&longs;eu in D, quod idem e&longs;t, æqualis e&longs;t ponde
ri M; & BD e&longs;t vectis, cuius fulcimentum
e&longs;t B, & potentia in N e&longs;t, ac &longs;i e&longs;&longs;et in me
dio vectis, & pondus æquale ip&longs;i M, ac &longs;i e&longs;
&longs;et in D propter funem FD; quod idem
e&longs;t, ac &longs;i BCD e&longs;&longs;et orbiculus trochleæ &longs;upe
rioris, pondusq; appen&longs;um e&longs;&longs;et in fune DF,
&longs;icut in decimaquinta, & decima&longs;exta dictum e&longs;t; ergo potentia in
N dupla e&longs;t ponderis M. quod erat o&longs;tendendum.
Si autem in N &longs;it potentia mouens pondus M, erit &longs;patium
ponderis M duplum &longs;patii potentiæ in N. quod ex duodecima
huius manife&longs;tum e&longs;t; &longs;patium enim puncti L deor&longs;um ten
dentis duplum e&longs;t &longs;patii N &longs;ur&longs;um; erit igitur è conuer&longs;o &longs;patium
potentiæ in N deor&longs;um tendentis dimidium
&longs;um moti.
Sicut autem ex tertia, quinta, &longs;eptima huius, &c.
colligi po&longs;&longs;unt
ponderis O rationes quotcunq; multiplices ip&longs;ius potentiæ in L,
tis ponderis M quotcunq; multiplices. Atq; ita ex decimatertia
dentur quotcunq; multiplices
&longs;patii ponderis M ad &longs;patium
potentiæ mouentis in N con&longs;ti
tutæ.
Poterit quoq; ex decima&longs;e
ptima decimaoctaua huius mul
tiplex inueniri proportio, quam
habet potentia pondus &longs;u&longs;ti
nens ad ip&longs;um pondus; &longs;icut
proportio potentiæ in N ad pon
dus M ex decimaquinta, & deci
ma&longs;exta o&longs;tendebatur: inuenie
turq; ita e&longs;&longs;e pondus ad poten
tiam pondus &longs;u&longs;tinentem, vt &longs;pa
tium potentiæ mouentis ad &longs;pa
tium ponderis.
Vectium motus in his fit
hoc modo, videlicet vectes or
biculorum trochleæ inferioris
mouentur, vt vectis BD, quæ
mouetur, ac &longs;i B e&longs;&longs;et fulcimen
tum, & pondus in D, & poten
tia in medio. Vectes verò or
biculorum trochleæ &longs;uperioris mouentur, vt FH, cuius fulcimen
tum e&longs;t in medio, pondus in H, & potentia in F.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, orbiculos trochleæ
inferioris in his efficere, vt pondus maiori po
per maius &longs;patium &longs;patio potentiæ, & minori
tempore per æquale. quod quidem orbiculi &longs;u
perioris trochleæ non efficiunt.
Cognitis proportionibus multiplicibus, iam ad &longs;uperparticu
lares accedendum e&longs;t.
PROPOSITIO XX.
Si vtriu&longs;q; duarum trochlearum &longs;ingulis or
biculis, quarum altera &longs;upernè à potentia &longs;u&longs;ti
neatur, altera verò infernè, ponderiq; alligata,
mo alicuibi, altero verò inferiori trochleæ reli
gato; pondus potentiæ &longs;e&longs;quialterum erit.
Sit ABC orbiculus
trochleæ &longs;uperioris, &
DEF trochleæ inferio
ris ponderi G alligatæ;
&longs;itq; funis HABCDE
Fk circa orbiculos re
uolutus, qui &longs;it religatus
in K, & in H trochleæ
inferiori; &longs;itq; potentia
in L &longs;u&longs;tinens pondus
G. dico pondus poten
tiæ &longs;e&longs;quialterum e&longs;&longs;e.
funis CD AH tertiam
&longs;u&longs;tinet partem ponde
ris G, erit vnaquæq; po
tentia in DH &longs;ubtripla
ponderis G; quibus &longs;i
mul a&longs;&longs;umptis e&longs;t æqua
eius, quæ e&longs;t in H. quare potentia in L &longs;ub&longs;e&longs;quialtera e&longs;t ponde
ris G. pondus ergo G ad pontentiam in L e&longs;t, vt tria ad duo;
hoc e&longs;t &longs;e&longs;quialterum. quod demon&longs;trare oportebat.
Si autem in L &longs;it potentia mouens pondus.
Dico &longs;patium potentiæ &longs;patii ponderis &longs;e&longs;quial
terum e&longs;&longs;e.
Ii&longs;dem po&longs;itis, perueniat orbi
culus ABC v&longs;q; ad MNO, &
DEF ad PQR; & H in S; &
pondus G v&longs;q; ad T. Et quoniam
funis HABCDEFK e&longs;t æqualis
funi SMNOPQRk, cùm &longs;it
idem funis; & funes circa &longs;emicir
culos ABC MNO &longs;unt inter &longs;e
&longs;e æquales; qui verò &longs;unt circa
DEF PQR &longs;imiliter inter &longs;e æ
quales; Demptis igitur AS CP
RK communibus, erunt duo CO
MA tribus DP HS FR æqua
les. &longs;ed vterq; CO AM &longs;eor&longs;um
e&longs;t æqualis &longs;patio potentiæ motæ.
quare duo CO MA, &longs;imul &longs;patii
potentiæ dupli erunt: tre&longs;q; DP
HS FR &longs;imul &longs;imili modo &longs;patii
ponderis moti tripli erunt. dimidia
verò pars, hoc e&longs;t &longs;patium poten
tiæ motæ ad tertiam, ad &longs;patium
&longs;cilicet ponderis moti ita &longs;e habet,
vt duplum dimidii ad duplum ter
tii; hoc e&longs;t, vt totum ad duas ter
tias, quod e&longs;t vt tria ad duo. &longs;patium ergo potentiæ in L &longs;pa
tii ponderis G moti &longs;e&longs;quialterum e&longs;t. quod o&longs;tendere opor
tebat.
PROPOSITIO XXI.
Si tribus duarum trochlearum orbiculis, qua
rum altera vnius tantùm orbiculi &longs;upernè à po
tentia &longs;u&longs;tineatur, altera verò duorum infernè,
ponderiq; alligata, collocata fuerit, funis cir
cumuoluatur; altero eius extremo alicubi, altero
autem &longs;uperiori trochleæ religato: pondus poten
tiæ &longs;e&longs;quitertium erit.
Sit pondus A trochleæ inferiori alliga
tum, quæ duos habeat orbiculos, quorum
centra &longs;int BC; &longs;uperiorq; trochlea orbicu
lum habeat, cuius centrum D; & &longs;it funis
EFGHkLMN circa omnes orbiculos re
uolutus, qui religatus &longs;it in N, & in E tro
chleæ &longs;uperiori; &longs;itqué potentia in O
&longs;u&longs;tinens pondus A. dico pondus po
Quoniam enim
vnu&longs;qui&longs;q; funis NM HG EF KL quar
tam &longs;u&longs;tinent partem ponderis A, & omnes
&longs;imul totum &longs;u&longs;tinent pondus; tres HG
EF kL &longs;imul tres &longs;u&longs;tinebunt partes pon
deris A. quare pondus A ad hos omnes
&longs;imul erit, vt quatuor ad tria: & cùm po
tentia in O idem efficiat, quod HG EF kL
&longs;imul efficiunt; omnes enim &longs;u&longs;tinet; erit po
tentia in O tribus &longs;imul HG EF kL æ
qualis; & ob id pondus A ad potentiam
in O erit, vt quatuor ad tria; hoc e&longs;t &longs;e&longs;qui
tertium. quod demon&longs;trare oportebat.
Si vero in O &longs;it potentia mouens pondus A.
Dico &longs;patium potentiæ in O decur&longs;um &longs;patii pon
deris A moti &longs;e&longs;quitertium e&longs;&longs;e.
Ii&longs;dem po&longs;itis, &longs;it centrum B motum
in P; &C v&longs;q; ad Q; & D in R; & E in
S eodem tempore: & per centra ducantur
ML 9Z FG TV Hk XY horizonti,
& inter &longs;e &longs;e æquidi&longs;tantes. Similiter, vt in
præcedente o&longs;tendetur tres
quatuor TG VF ZL 9M æquales e&longs;&longs;e. &
quoniam tres XH SE Yk &longs;imul triplæ
&longs;unt &longs;patii potentiæ, quatuor verò TG VF
ZL 9M &longs;imul quadruplæ &longs;unt &longs;patii pon
deris moti; erit &longs;patium potentiæ ad &longs;pa
tium ponderis, vt tertia pars ad quartam.
&longs;ed tertia pars ad quartam e&longs;t, vt tres ter
tiæ ad tres quartas, hoc e&longs;t, vt totum ad
tres quartas; quod e&longs;t, vt quatuor ad tria.
&longs;patium ergo potentiæ &longs;patii ponderis mo
ti &longs;e&longs;quitertium e&longs;t. quod erat demon
&longs;trandum.
Si verò funis in E per alium circumuol
uatur orbiculum, qui deinde trochleæ in
feriori religetur; &longs;imiliter o&longs;tendetur pro
portionem ponderis ad
dus &longs;u&longs;tinentem &longs;e&longs;quiquartam e&longs;&longs;e. quòd
&longs;i in O &longs;it potentia mouens pondus, o&longs;ten
detur &longs;patium potentiæ &longs;patii ponderis &longs;e&longs;
quiquartum e&longs;&longs;e. & &longs;ic in infinitum proce
dendo quamcunq; &longs;uperparticularem pro
portionem ponderis ad potentiam inuenie
mus; &longs;emperq; reperiemus, ita e&longs;&longs;e pondus
ad potentiam pondus &longs;u&longs;tinentem, vt &longs;pa
tium potentiæ mouentis ad &longs;patium ponde
ris moti.
Motus verò vectium fit hoc mo
do, videlicet vectis ML fulci
mentum e&longs;t M, cùm funis &longs;it re
ligatus in N, & pondus in me
dio, & potentia in L. ve
rò punctum L tendit &longs;ur&longs;um, quod
à fune KL mouetur, idcirco K &longs;ur
&longs;um mouebitur, & vectis HK ful
cimentum erit H, pondus ac &longs;i e&longs;
&longs;ent in k, & potentia in medio;
vectis autem FG fulcimentum
erit G, pondus in medio; & poten
tia in F. punctum enim F &longs;ur&longs;um
mouetur à fune EF. Præterea
G in orbiculo deor&longs;um tendit,
quia H quoque in eius orbiculo
deor&longs;um mouetur. PROPOSITIO XXII.
Si vtri&longs;que duarum trochlearum &longs;ingulis
orbiculis, quarum altera &longs;upernè à potentia
&longs;u&longs;tineatur, altera verò infernè, ponderiq; alli
gata, collocata fuerit, circumducatur funis; al
tero eius extremo alicubi, altero autem &longs;uperio
ri trochleæ religato. erit potentia ponderis &longs;e&longs;
quialtera.
Sit orbiculus ABC trochleæ ponderi D al
ligatæ; & EFG trochleæ &longs;uperioris, cuius
centrum H; &longs;it deinde funis k ABCEFGL
circa orbiculos reuolutus, & religatus in L, &
in k trochleæ &longs;uperiori; &longs;itq; potentia in M
&longs;u&longs;tinens pondus D. dico potentiam ponde
ris &longs;e&longs;quialteram e&longs;&longs;e. Quoniam enim poten
tia in E &longs;u&longs;tinens pondus D &longs;ubdupla e&longs;t pon
deris D, potentiæ verò in E dupla e&longs;t poten
tia in H; erit potentia in H ponderi D æqua
lis; & cùm potentia in K &longs;ubdupla &longs;it ponde
ris D; erunt vtræq; &longs;imul potentiæ in H k &longs;e&longs;
quialteræ ponderis D. Itaq; cùm potentia in
M duabus potentiis in Hk &longs;imul &longs;umptis &longs;it
æqualis, quemadmodum in &longs;uperioribus o
&longs;ten&longs;um e&longs;t; erit potentia in M &longs;e&longs;quialtera
ponderis D. quod oportebat demon&longs;trare.
Si verò in M &longs;it potentia mouens pondus,
&longs;imiliter vt in præcedentibus o&longs;tendetur, &longs;pa
tium ponderis &longs;patii potentiæ &longs;e&longs;quialterum
e&longs;&longs;e.
Et &longs;i funis in K per alium circumuoluatur
orbiculum, cuius centrum &longs;it N; qui dein
de trochleæ inferiori religetur in O; & po
tentia in M &longs;u&longs;tineat pondus D. dico pro
portionem potentiæ ad pondus &longs;e&longs;quiter
tiam e&longs;&longs;e.
Quoniam enim potentia in E &longs;u&longs;tinens
in H; erit potentia in H &longs;ub&longs;e&longs;quialtera pon
deris D. &longs;imili quoq; modo quoniam po
tentia in O, quæ e&longs;t, ac &longs;i e&longs;&longs;et in centro or
&longs;ius autem O dupla e&longs;t potentia in N; erit
quoq; potentia in N &longs;ub&longs;e&longs;quialtera ponde
ris D. quare duæ &longs;imul potentiæ in HN pon
dus D &longs;uperant tertia parte, &longs;e &longs;e habentq; ad
D in ratione &longs;e&longs;quitertia: & cùm potentia
in M duabus &longs;it potentiis in HN &longs;imul &longs;um
ptis æqualis, &longs;uperabit itidem potentia in
M pondus D tertia parte. ergo proportio
potentiæ in M ad pondus D &longs;e&longs;quitertia
e&longs;t. quod demon&longs;trare oportebat.
Si autem in M &longs;it potentia mouens pon
dus, &longs;imili modo o&longs;tendetur &longs;patium ponderis D &longs;patii potentiæ in
M &longs;e&longs;quitertium e&longs;&longs;e.
Et &longs;i funis in O per alium circumuoluatur orbiculum, qui tro
chleæ &longs;uperiori deinde religetur; eodem modo demon&longs;trabimus
proportionem potentiæ in M pondus &longs;u&longs;tinentis ad pondus &longs;e&longs;
quiquartam e&longs;&longs;e. & &longs;i in M &longs;it potentia mouens, &longs;imiliter o&longs;ten
detur &longs;patium ponderis &longs;patii potentiæ &longs;e&longs;quiquartum e&longs;&longs;e. pro
cedendoq; hoc modo in infinitum quamcunq; proportionem
potentiæ ad pondus &longs;uperparticularem inueniemus; &longs;emperqué
vt &longs;patium ponderis ad &longs;patium potentiæ pondus mouentis.
Motus verò vectis EG e&longs;t, ac &longs;i G e&longs;&longs;et fulcimentum, cùm
funis &longs;it religatus in L; pondus ac &longs;i in E e&longs;&longs;et appen&longs;um, & po
tentia in medio. Vectis verò CA fulcimentum e&longs;t A pondus in
medio, & potentia in C. & K fulcimentum e&longs;t vectis Pk, pon
dus in P, & potentia in medio. quæ omnia &longs;icut in præceden
ti o&longs;tendentur.
PROPOSITIO XXIII.
Si vtri&longs;q; duarum trochlearum &longs;ingulis or
biculis, quarum altera &longs;upernè à potentia &longs;u&longs;ti
neatur, altera verò infernè, ponderiq; alligata,
extremo alicuibi, non autem trochleis religato;
æqualis erit ponderi potentia.
Sit orbiculus trochleæ &longs;uperioris
ABC, cuius centrum D; & EFG
trochleæ ponderi H alligatæ, cu
ius centrum k; & &longs;it funis LEF
GABCM circa orbiculos reuo
lutus, religatu&longs;q; in LM; &longs;itq;
potentia in N &longs;u&longs;tinens pondus
H. dico potentiam in N æqua
lem e&longs;&longs;e ponderi H. Accipiatur
quoduis punctum O in AG. &
quoniam &longs;i in O e&longs;&longs;et potentia &longs;u
dupla e&longs;t ea, quæ e&longs;t in D, &longs;iue
(quod idem e&longs;t) in N; erit po
tentia in N ponderi H æqualis.
quod demon&longs;trare oportebat.
Et &longs;i in N &longs;it potentia mouens pondus.
Dico
&longs;patium potentiæ in N æqualem e&longs;&longs;e &longs;patio pon
deris H moti.
Quoniam enim &longs;patium puncti O moti, duplum e&longs;t, tùm &longs;patii
ALITER.
Ii&longs;dem po&longs;itis, transfera
tur centrum orbiculi ABC
v&longs;q; ad P; orbiculu&longs;q; po&longs;i
tionem habeat QRS; dein
de eodem tempore orbiculus
EFG &longs;it in TVX, cuius cen
trum &longs;it Y; & pondus perue
nerit in Z. ducantur per or
biculorum centra lineæ GE
TX AC QS horizonti æqui
di&longs;tantes. & &longs;icut in aliis
demon&longs;tratum fuit, duo fu
nes AQ CS duobus XG
TE æquales erunt; &longs;ed AQ
CS &longs;imul dupli &longs;unt &longs;patii po
tentiæ motæ; & duo XG TE
&longs;imul &longs;unt &longs;imiliter dupli &longs;pa
tii ponderis; erit igitur
potentiæ &longs;patio ponderis æ
quale. quod demon&longs;trare o
portebat.
Quod etiam &longs;i vtraq; trochlea duos
habuerit orbiculos, quorum centra
&longs;int ABCD, funi&longs;q; per omnes cir
cumuoluatur, qui in LM religetur;
&longs;imiliter o&longs;tendetur potentiam in N
æqualem e&longs;&longs;e ponderi H. vnaquæq;
enim potentia in EF &longs;u&longs;tinens pon
dus &longs;ubquadrupla e&longs;t ponderis; & po
tentiæ in CD duplæ &longs;unt earum,
quæ &longs;unt in EF; erit vnaquæq; po
tentia in CD &longs;ubdupla ponderis H. quare potentiæ in CD &longs;imul &longs;umptæ
ponderi H erunt æquales. & quo
niam potentia in N duabus in CD
pontentiis e&longs;t æqualis; erit potentia
in N ponderi H, æqualis.
Et &longs;i in N &longs;it potentia mouens, &longs;i
mili modo o&longs;tendetur, &longs;patium po
tentiæ æquale e&longs;&longs;e &longs;patio ponderis.
Si autem vtraq; trochlea tres, vel
quatuor, vel quotcunq; habeat orbi
culos; &longs;emper o&longs;tendetur
N æqualem e&longs;&longs;e ponderi H; & &longs;pa
tium potentiæ pondus mouentis æ
quale e&longs;&longs;e &longs;patio ponderis moti.
Vectium autem motus hoc pacto &longs;e habent; orbiculorum qui
dem trochleæ &longs;uperioris, veluti AC in præcedenti figura fulcimen
tum e&longs;t C, pondus verò in A appen&longs;um, & potentia in D medio.
vectes autem orbiculorum trochleæ inferioris ita mouentur, vt ip
&longs;ius GE fulcimentum &longs;it E, pondus in medio appen&longs;um, & po
tentia in G.
PROPOSITIO XXIIII.
Si tribus duarum trochlearum orbiculis, qua
rum altera vnius dumtaxat orbiculi &longs;upernè à
potentia &longs;u&longs;tineatur, altera verò duorum infer
nè, ponderiq; alligata fuerit con&longs;tituta, cir
cundetur funis; vtroq; eius extremo alicubi, &longs;ed
non &longs;uperiori trochleæ religato: duplum erit
pondus potentiæ.
Sint AB centra orbiculorum
trochleæ ponderi C alligatæ; D ve
rò &longs;it centrum orbiculi trochleæ &longs;u
perioris; &longs;it deinde funis per om
nes orbiculos circumuolutus, reli
gatu&longs;q; in EF; & &longs;it potentia in
G &longs;u&longs;tinens pondus C. dico pon
dus C duplum e&longs;&longs;e potentiæ in G.
Quoniam enim &longs;i in H k duæ e&longs;
&longs;ent potentiæ pondus &longs;u&longs;tinentes
duobus funibus orbiculis trochleæ
inferioris tantùm circumuolutis, e&longs;
&longs;et vtiq; vtraq; potentia in k H &longs;ub
quadrupla ponderis C; &longs;ed poten
tia in G æqualis e&longs;t potentiis in Hk
&longs;imul &longs;umptis; vniu&longs;cuiu&longs;q; enim
potentiæ in H, & k dupla e&longs;t: erit
potentia in G &longs;ubdupla ponderis
C. pondus ergo potentiæ duplum
erit. quod demon&longs;trare opor
tebat.
Et &longs;i in G &longs;it potentia mouens pondus.
Dico
&longs;patium potentiæ duplum e&longs;&longs;e &longs;patii ponderis.
Ii&longs;dem po&longs;itis, &longs;int
moti orbiculi, &longs;imiliter
demon&longs;trabitur ambos
illos LM NO æquales
e&longs;&longs;e quatuor PQ RS
TV XY. &longs;ed LM NO
&longs;imul dupli &longs;unt &longs;patii po
tentiæ in G motæ; &
quatuor PQ RS TV
XY &longs;imul quadrupli &longs;unt
&longs;patii ponderis moti. &longs;pa
tium igitur potentiæ ad
&longs;patium ponderis e&longs;t tan
quam &longs;ubduplum ad &longs;ub
quadruplum. erit ergo
potentiæ &longs;patium pon
deris &longs;patii duplum.
Hinc autem con&longs;iderandum
e&longs;t quomodo fiat motus; quia,
cùm funis &longs;it religatur in F, vectis
NO in prima figura habebit ful
cimentum O, pondus in medio,
& potentia in N. &longs;imiliter quo
niam funis e&longs;t religatus in E, ve
ctis PQ habebit
pondus in medio, & potentia in
q. idcirco partes orbiculorum
in N, & Q &longs;ur&longs;um mouebuntur;
orbiculi ergo non in eandem, &longs;ed
in contrarias mouebuntur partes,
videlicet vnus
ni&longs;tror&longs;um. & quoniam potentiæ
in NQ eædem &longs;unt, quæ &longs;unt in
LM; potentiæ igitur in LM æ
quales &longs;ur&longs;um mouebuntur. ve
ctis igitur LM in neutram moue
bitur partem. quare neq; orbicu
lus circumuertetur. Itaq; LM
erit tanquam libra, cuius centrum
D, ponderaqué appen&longs;a in LM
æqualia quartæ parti ponderis C;
vnu&longs;qui&longs;q; enim funis LN MQ
quartam &longs;u&longs;tinet partem ponderis C. mouebitur ergo totus orbi
culus, cuius centrum D, &longs;ur&longs;um; &longs;ed non circumuertetur.
Et &longs;i funis in F circa alios duos
voluatur orbiculos, quorum cen
tra &longs;int HK, qui deinde religetur
in L; erit proportio ponderis ad
potentiam &longs;e&longs;quialtera.
Si enim quatuor e&longs;&longs;ent potentiæ
cupla ponderis C, quare quatuor
&longs;imul potentiæ in MNOI qua
tuor &longs;extæ erunt ponderis C. &
quoniam duæ &longs;imul potentiæ in
HD quatuor potentiis in MNOI
&longs;unt æquales; & potentia in G æ
qualis e&longs;t potentiis in DH: erit
potentia in G quatuor &longs;imul po
tentiis in MNOI æqualis; & ob
id quatuor &longs;extæ erit ponderis C.
proportio igitur ponderis C ad po
tentiam in G &longs;e&longs;quialtera e&longs;t.
Et &longs;i in G &longs;it potentia mouens,
&longs;imili modo o&longs;tendetur &longs;patium
potentiæ &longs;patii ponderis &longs;e&longs;quialte
rum e&longs;&longs;e.
Et &longs;i funis in L adhuc circa duos
alios orbiculos reuoluatur &longs;imi
liter o&longs;tendetur proportionem
ponderis ad potentiam &longs;e&longs;qui
tertiam e&longs;&longs;e. quòd &longs;i in G &longs;it
potentia mouens, o&longs;tende
tur &longs;patium potentiæ &longs;patii ponde
ris &longs;e&longs;quitertium e&longs;&longs;e, atq; ita dein
ceps in infinitum procedendo,
quamcunq; proportionem ponderis ad potentiam &longs;uperparticula
rem inueniemus &longs;emperq; reperiemus ita e&longs;&longs;e pondus ad poten
tiam pondus &longs;u&longs;tinentem, vt &longs;patium potentiæ mouentis ad &longs;pa
tium ponderis à potentia moti.
Motus vectium fit hoc modo, vectis YZ, cùm funis &longs;it religatus
in E, habet fulcimentum in Y, pondus in B medio appen&longs;um, &
potentia in Z. & vectis PQ habet fulcimentum in P potentia in
medio, & pondus in q. oportet enim orbiculos, quorum cen
tra &longs;unt BD in eandem partem moueri, videlicet vt QZ &longs;ur
&longs;um moueantur. & quoniam funis religatus e&longs;t in L, erit T fulci
mentum vectis ST, qui pondus habet in medio, & potentia in
S. & quia S mouetur &longs;ur&longs;um, nece&longs;&longs;e e&longs;t etiam R &longs;ur&longs;um moue
ri; & ideo F erit fulcimentum vectis FR, & pondus erit in R,
& potentia in medio. orbiculi igitur, quorum centra &longs;unt H k,
in contrariam mouentur partem eorum, quorum centra &longs;unt BD:
quare partes
ver&longs;us XV. vectis igitur VX in neutram partem mouebitur, cùm
P, & F deor&longs;um moueantur; & VX erit tanquam vectis, in cuius
medio erit pondus appen&longs;um, & in VX duæ potentiæ æquales
&longs;extæ parti ponderis C. potentiæ enim in MO hoc e&longs;t funes PV
FX &longs;extam &longs;u&longs;tinent partem ponderis C. totus igitur orbiculus,
cuius centrum A &longs;ur&longs;um vnà cum trochlea mouebitur; non au
tem circumuertetur.
PROPOSITIO XXV.
Si tribus duarum trochlearum orbiculis,
quarum altera binis in&longs;ignita rotulis à potentia
&longs;upernè detineatur; altera verò vnius tantùm
rotulæ infernè
rit, circumuoluatur funis; vtroq; eius extremo
alicuibi, non autem inferiori trochleæ religa
to: dupla erit ponderis potentia.
Sit pondus A trochleæ inferiori alligatum,
quæ orbiculum habeat, cuius centrum &longs;it B; tro
chlea verò &longs;uperior duos orbiculos habeat,
quorum centra &longs;int CD; &longs;itq; funis circa om
nes orbiculos reuolutus, qui in EF &longs;it religatus;
potentiaq; &longs;u&longs;tinens pondus &longs;it in G. dico po
tentiam in G ponderis A duplam e&longs;&longs;e. &longs;i enim
tia in C dupla potentiæ in K; quare duæ &longs;imul
potentiæ in CD vtriu&longs;q; &longs;imul potentiæ in H k
duplæ erunt. &longs;ed potentiæ in H k ponderi A &longs;unt
æquales, & potentiæ in CD ip&longs;i potentiæ in G
&longs;unt etiam æquales; potentia igitur in G ponde
ris A dupla erit. quod oportebat demon&longs;trare.
Si autem in G &longs;it potentia mouens pon
dus, &longs;imiliter vt in præcedenti o&longs;tendetur &longs;pa
tium ponderis &longs;patii potentiæ duplum e&longs;&longs;e.
Hinc quoq; con&longs;iderandum e&longs;t vectem PQ
non moueri, quia vectis LM habet fulcimen
tum in L, potentia in medio, & pondus in M.
vectis autem NO habet fulcimentum in O,
potentia in medio, & pondus in N. quare M, & N &longs;ur&longs;um mo
uebuntur. in contrarias igitur partes orbiculi, quorum centra
&longs;unt CD mouentur. idcirco vectis PQ in neutram partem mo
uebitur; eritq;, ac &longs;i in medio e&longs;&longs;et appen&longs;um pondus, & in PQ
duæ potentiæ æquales dimidio ponderis A. vtraq; enim potentia
in HK &longs;ubdupla e&longs;t ponderis A. totus igitur orbiculus, cuius
centrum B &longs;ur&longs;um mouebitur, &longs;ed non circumuertetur.
Et &longs;i funis in F duobus aliis adhuc circumuol
uatur orbiculis, quorum centra &longs;int HK, qui de
inde religetur in L; erit proportio potentiæ in G
ad pondus A &longs;e&longs;quialtera.
Si enim in MNOP quatuor e&longs;&longs;ent poten
tiæ pondus &longs;u&longs;tinentes, vnaquæq; &longs;ubquadru
pla e&longs;&longs;et ponderis A: &longs;ed cùm potentia in k
&longs;it dupla potentiæ in N; erit potentia in k
ponderis A &longs;ubdupla. & quoniam potentia
in D duabus in MO potentiis e&longs;t æqualis; erit
quoq; potentia in D ponderis A &longs;ubdupla.
cùm autem adhuc potentia in C potentiæ in P
&longs;it dupla, erit &longs;imiliter
&longs;ubdupla. tres igitur potentiæ in CD k tribus
medietatibus ponderis A &longs;unt æquales. quo
niam autem potentia in G potentiis in CDK
e&longs;t æqualis, erit potentia in G tribus medie
tatibus ponderis A æqualis. Proportio igi
tur potentiæ ad pondus &longs;e&longs;quialtera e&longs;t.
Si verò in G &longs;it potentia mouens, erit &longs;pa
tium ponderis &longs;patii potentiæ &longs;e&longs;quialterum.
Et &longs;i funis in L adhuc circa duos alios or
biculos reuoluatur, &longs;imiliter o&longs;tendetur pro
portionem potentiæ ad pondus &longs;e&longs;quitertiam
e&longs;&longs;e. & &longs;ic in infinitum omnes proportiones
potentiæ ad pondus &longs;uperparticulares inue
niemus. o&longs;tendemu&longs;q; potentiam pondus
&longs;u&longs;tinentem ad pondus ita e&longs;&longs;e, vt &longs;patium
ponderis moti ad &longs;patìum potentiæ pondus
mouentis.
Motus vectium fiet hoc
modo, videlicet Q erit ful
cimentum vectis QR, po
tentia in medio, pondus
in R; & vectis Z 9 fulci
mentum erit Z, pondus in
medio, potentiaq; in 9. &longs;i
militer X erit fulcimentum
vectis VX, potentia in me
dio, pondus in V. & quo
niam V &longs;ur&longs;um mouetur, Y
quoq; &longs;ur&longs;um mouebitur;
& vectis YF fulcimentum
erit F: quare F, & Z in orbi
culis deor&longs;um mouebun
tur. & ob id vectis ST in
neutram mouebitur par
tem; & ST erit tamquam
libra, cuius centrum D, &
pondera in ST æqualia
quartæ parti ponderis A. vnu&longs;qui&longs;q; enim funis SZ
TF quartam &longs;u&longs;tinet par
tem ponderis A. orbicu
lus ergo, cuius centrum D,
&longs;ur&longs;um mouebitur; non au
tem circumuertetur.
Hactenus proportiones ponderis ad potentiam multiplices,
& &longs;ubmultiplices; deinde &longs;uperparticulares, &longs;ub&longs;uperparticu
lare&longs;qué declaratæ fuerunt: nunc autem reliquum e&longs;t, vt propor
tiones inter pondus, & potentiam &longs;uperpartientes, & multi
plices &longs;uperparticulares, multiplicesqué &longs;uperpartientes mani
fe&longs;tentur.
PROPOSITIO XXVI.
PROBLEMA.
Si proportionem &longs;uperpartientem inuenire
volumus, quemadmodum &longs;i proportio, quam
habet pondus ad potentiam pondus &longs;u&longs;tinen
tem fuerit &longs;uperbipartiens, &longs;icut quinque ad
tria.
nens, proportionemq; habeat pondus B ad
potentiam in A, vt quinq; ad vnum; hoc e&longs;t,
&longs;it potentia in A &longs;ubquintupla ponderis B: de
inde eodem fune circa alios orbiculos reuo
potentiæ in A. & quoniam pondus B ad po
tentiam in A e&longs;t, vt quinq; ad vnum; &
potentia in A ad potentiam in C e&longs;t, vt vnum
ad tria; erit pondus B ad potentiam in C, vt
quinq; ad tria; hoc e&longs;t &longs;uperbipartiens.
Et hoc modo omnes proportiones ponde
ris ad potentiam &longs;uperpartientes inuenientur;
vt &longs;i &longs;upertripartientem quis inuenire volue
rit; eodem incedat ordine; fiat &longs;cilicet poten
tia in A &longs;u&longs;tinens pondus B &longs;ub&longs;eptupla ip
&longs;ius ponderis B; deinde fiat potentia in C ip
&longs;ius A quadrupla; erit pondus B ad poten
tiam in C, vt &longs;eptem ad quatuor: vídelicet
&longs;upertripartiens.
Si verò in C &longs;it potentia mo
uens pondus erit &longs;patium
&longs;patii ponderis &longs;uperbipartiens.
e&longs;t &longs;patii potentiæ in A, ita videlicet &longs;e habent,
vt quinq; ad quindecim; & &longs;patium potentiæ
e&longs;t, vt quindecim ad tria; erit igitur &longs;patium
potentiæ in C ad &longs;patium ponderis B, vt
quinq; ad tria; videlicet &longs;uperbipartiens. & &longs;emper o&longs;tendemus, ita
e&longs;&longs;e &longs;patium potentiæ mouentis ad &longs;patium ponderis; vt pondus
ad potentiam pondus &longs;u&longs;tinentem.
Similiq; pror&longs;us ratione proportionem potentiæ ad pondus &longs;u
&longs;i enim C e&longs;&longs;et inferius, & in ip&longs;o
appen&longs;um e&longs;&longs;et pondus; B verò &longs;uperius, in quo e&longs;&longs;et potentia pon
dus in C &longs;u&longs;tinens, e&longs;&longs;et potentia in B &longs;uperbipartiens ponderis
in C appen&longs;i: cùm B ad A &longs;it,
C, vt vnum ad tria.
Si autem multiplicem &longs;uperparticularem in
uenire voluerimus; vt proportio, quam habet
pondus ad potentiam pondus &longs;u&longs;tinentem, &longs;it
duplex &longs;e&longs;quialtera, vt quinq; ad duo.
Eodem modo, quo &longs;uperpartientes inuenimus, has quo
que omnes multiplices &longs;uperparticulares reperiemus. vt fiat
pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve
ro in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu
nis &longs;it religatus in D, non autem trochleæ &longs;uperiori, vel in F: erit
pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum
&longs;e&longs;quialterum.
Et è conuer&longs;o proportionem potentiæ ad pondus multiplicem
&longs;uperparticularem inueniemus; & vt in reliquis o&longs;tendetur, ita e&longs;
&longs;e &longs;patium potentiæ mouentis ad &longs;patium ponderis, vt pondus
ad potentiam pondus &longs;u&longs;tinentem.
Omnem quoq; multiplicem &longs;uperpartientem
eodem modo inueniemus; vt &longs;i proportio, quam
habet pondus ad potentiam, &longs;it duplex &longs;uperbi
partiens, vt octo ad tria.
Fiat potentia in A pondus B &longs;u&longs;tinens &longs;uboctupla ponderis B;
& potentia in C potentiæ in A &longs;it tripla; erit pondus B ad po
tentiam in C, vt octo ad tria. & è conuer&longs;o omnem potentiæ ad
& vt in cæteris reperiemus ita e&longs;&longs;e pondus ad potentiam pondus
&longs;u&longs;tinentem, vt &longs;patium potentiæ mouentis ad &longs;patium pon
deris.
Notandum autem e&longs;t, quòd cùm in præcedentibus
nibus &longs;æpius dictum fuerit, potentiam pondus &longs;u&longs;tinentem ip&longs;ius
ponderis duplam e&longs;&longs;e, vel triplam, & huiu&longs;modi; vt in decima
quinta huius o&longs;ten&longs;um e&longs;t; quia tamen potentia non &longs;olum pon
dus, verùm etiam trochleam &longs;u&longs;tinet; idcirco maioris longè vir
tutis, maiori&longs;q; ip&longs;i ponderi proportionis con&longs;tituenda videtur
ip&longs;a potentia. quod quidem verum e&longs;t, &longs;i etiam trochleæ graui
tatem con&longs;iderare voluerimus. &longs;ed quoniam inter potentiam, &
pondus proportionem quærimus: ideo hanc trochleæ grauitatem
ommi&longs;imus, quam &longs;iquis etiam con&longs;iderare voluerit, vim ip&longs;i po
tentiæ æqualem trochleæ addere poterit. Quod ip&longs;um etiam in
fune ob&longs;eruari poterit. & &longs;icut hoc in decimaquinta con&longs;ideraui
mus, idem quoq; in reliquis aliis con&longs;iderare poterimus.
Noui&longs;&longs;e etiam oportet, quòd &longs;icuti proportio
nes omnes inter potentiam, & pondus vnico
fune inuentæ fuerunt; ita etiam pluribus funi
bus, trochlei&longs;qué eædem inueniri poterunt. vt
&longs;i multiplicem &longs;uperparticularem proportionem
pluribus funibus inuenire voluerimus, veluti &longs;i
proportio, quam habet pondus ad potentiam
pondus &longs;u&longs;tinentem, fuerit duplex &longs;e&longs;quialtera, vt
quinq; ad duo; oportet hanc proportionem ex
pluribus componere. vt (exempli gratia) ex pro
portione &longs;e&longs;quiquarta, vt quinqué ad quatuor,
& ex dupla, vt quatuor ad duo. exponatur igitur po
tentia in A pondus B &longs;u&longs;tinens, ad quam pondus
quatuor: deinde alio fune inueniatur
cuius dupla &longs;it potentia in A. &
vt quinq; ad quatuor; & A ad C, vt quatuor ad
duo; erit pondus B ad potentiam in C, vt quin
que ad duo; hoc e&longs;t proportionem habebit du
plicem &longs;e&longs;quialteram.
Et notandum e&longs;t hanc quoq;
niri po&longs;&longs;e, &longs;i proportionem quinq; ad duo ex pluri
bus componamus, vt quinq; ad quindecim & quin
decim ad viginti & viginti ad duo. Et hoc modo
non &longs;olum omnem aliam proportionem inuenie
mus, &longs;ed quamcunq, multis, infinitisqué mo
dis comperiemus. omnis enim proportio ex infi
nitis proportionibus componi pote&longs;t. vt patet
in commentario Eutocii in quartam propo&longs;itio
nem &longs;ecundi libri Archimedis de &longs;phera, & cy
lindro.
Po&longs;&longs;umus quoq; pluribus funibus, trochleis
verò inferioribus tantùm, vel &longs;uperioribus vti.
Sit pondus A, cui alligata &longs;it trochlea
orbiculum habens, cuius centrum B;
religetur funis in C, qui circa orbiculum
reuoluatur, funi&longs;q; perueniat in D: erit
dupla ponderis A. deinde funis in D
alteri trochleæ religetur, & circa huius
trochleæ orbiculum alius reuoluatur fu
nis, qui religetur in E, & perueniat in
quod &longs;u&longs;tinet
D dimidium ponderis A &longs;u&longs;tineret &longs;i
ne trochlea; quare potentia in F &longs;ubqua
drupla erit ponderis A. & &longs;i adhuc fu
nis in F alteri trochleæ religetur, &
per eius orbiculum circumuoluatur a
lius funis, qui religetur in G, & per
ueniat in H; erit potentia in H &longs;ub
dupla potentiæ in F. ergo potentia in
H &longs;uboctupla erit ponderis A. & &longs;ic
in infinitum &longs;emper &longs;ubduplam poten
tiam
Et &longs;i in H &longs;it potentia mouens, erit
&longs;patium potentiæ &longs;patii ponderis octu&longs;patium enim D duplum e&longs;t &longs;pa
tii ponderis A, & &longs;patium F &longs;patii D
duplum; erit &longs;patium F &longs;patii ponde
ris A quadruplum. &longs;imiliter quoniam
&longs;patium potentiæ in H
F, erit &longs;patium potentiæ in H &longs;patii
ponderis A octuplum.
Sit deinde pondus A funi alliga
tum, qui orbiculo trochleæ &longs;uperio
ris &longs;it circumuolutus, & religatus in
B; &longs;itq; potentia in C &longs;u&longs;tinens pon
dus A: erit potentia in C ponderis A
dupla, deinde C alteri funi religetur,
qui per alterius trochleæ orbicu
lum circumuoluatur, & religetur
in D; erit potentia in E dupla po
tentiæ in C. Quare potentia in E
quadrupla erit ponderis A. & &longs;i ad
huc E alteri funi religetur, qui etiam
circa orbiculum alterius trochleæ re
uoluatur, & religetur in F; erit poten
tia in G dupla potentiæ in E. ergo
potentia in G octupla erit ponderis
A. & &longs;ic in infinitum &longs;emper præ
cedentis potentiæ potentiam du
plam inueniemus.
Si autem in G &longs;it potentia mo
uens,
plum &longs;patii potentiæ in G. &longs;patium
enim ponderis A duplum e&longs;t &longs;patii
potentiæ in C, & C duplum e&longs;t &longs;patii
ip&longs;ius E; quare &longs;patium ponderis
A &longs;patii potentiæ in E quadruplum
erit. &longs;imiliter quoniam &longs;patium E
duplum e&longs;t &longs;patii potentiæ in G; erit ergo &longs;patium ponderis A
octuplum &longs;patii potentiæ in G.
COROLLARIVM.
Ex his manife&longs;tum e&longs;t maiorem &longs;emper ha
bere proportionem &longs;patium potentiæ mouen
tis ad &longs;patium ponderis moti, quàm pondus
ad eandem potentiam.
Hoc autem ex iis, quæ in corollario quartæ huius de vecte dicta
&longs;unt, patet.
PROPOSITIO XXVII.
PROBLEMA.
Datum pondus à data potentia trochleis
moueri.
Data potentia, vel e&longs;t maior, vel æqualis, vel minor dato
pondere.
Et &longs;i e&longs;t maior, tunc poten
tia, vel ab&longs;q; alio in&longs;trumento,
vel fune circa orbiculum trochleæ
&longs;ur&longs;um appen&longs;æ reuoluto datum
pondus mouebit. Minor enim po
tentia; quàm data, ponderi æque
ponderat, data ergo mouebit.
Quod idem fieri pote&longs;t iuxta om
nes propo&longs;itiones, quibus poten
tia pondus &longs;u&longs;tinens, vel æqualis,
vel minor pondere o&longs;ten&longs;a e&longs;t.
Si autem æqualis,
pondus mouebit fune
per orbiculum trochleæ
ponderi alligatæ circum
uoluto. potentia enim
&longs;u&longs;tinens pondus &longs;ubdu
pla e&longs;t ponderis, poten
tia igitur ponderi æqua
lis datum pondus mo
uebit. Quod etiam
cundum
quibus potentiam pon
dere minorem e&longs;&longs;e o
&longs;ten&longs;um e&longs;t, fieri po
te&longs;t.
Si verò minor, &longs;it datum pondus
vt &longs;exaginta, potentia verò mouens inueniatur poten
tia in A &longs;u&longs;tinens pondus B, quæ pon
deris B &longs;it &longs;ubquintupla. & quoniam
potentia in A pondus &longs;u&longs;tinens e&longs;t
vt duodecim; maior igitur poten
tia, quàm duodecim in A pondus
B mouebit. Quare potentia vt tre
decim in A pondus B mouebit. quod
facere oportebat.
uendis ponderibus, potentiam ali
quando for&longs;itan melius mouere mo
uendo &longs;e deor&longs;um, quàm mouendo
&longs;e &longs;ur&longs;um. vt circumuoluatur adhuc
funis per alium trochleæ &longs;uperioris
orbiculum, cuius centrum C, funi&longs;q;
admodum erat in A. Ideo poten
tia vt tredecim in D pondus B mo
uebit. & quia mouet &longs;e deor&longs;um,
forta&longs;&longs;e trahet facilius, quàm in A;
atq; tempus e&longs;t idem, &longs;icut etiam
erat in A.
PROPOSITIO XXVIII.
PROBLEMA.
Propo&longs;itum &longs;it nobis efficere, potentiam pon
dus mouentem, & pondus per data &longs;patia &longs;ibi in
uicem longitudine commen&longs;urabilia moueri.
Sit datum &longs;patium potentiæ, vt tria,
ponderis verò, vt quatuor. inueniatur po
tentia in A pondus B &longs;u&longs;tinens, quæ pon
deris &longs;it &longs;e&longs;quitertia, vt quatuor ad trìa. &longs;i
igitur in A &longs;it potentia mouens pondus;
erit &longs;patium ponderis &longs;patii potentiæ &longs;e&longs;
quitertium, vt quatuor ad tria. quod face
re oportebat.
Hoc autem & ex iis, quæ dicta &longs;unt in
vige&longs;ima &longs;ecunda, & in vige&longs;imaquinta
huius efficere po&longs;&longs;umus &longs;olo fune. Quòd &longs;i
pluribus funibus id efficere voluerimus,
non &longs;olum multis, &longs;ed infinitis modis hoc
efficere poterimus, vt &longs;upra dictum e&longs;t.
Quare hoc affirmare po&longs;&longs;umus, quod qui
dem mirum e&longs;&longs;e videtur: videlicet.
COROLLARIVM. I.
Ex his manife&longs;tum e&longs;&longs;e, Quamlibet datam in
numeris proportionem inter pondus, & poten
tiam; & inter &longs;patium ponderis moti, & &longs;patium
potentiæ motæ; infinitis modis trochleis inueni
ri po&longs;&longs;e.
COROLLARIVM II.
Ex dictis etiam manife&longs;tum e&longs;t, quò pondus
facilius mouetur, eò quoq; tempus maius e&longs;&longs;e;
quò verò difficilius, eò minus e&longs;&longs;e. & è con
uer&longs;o.
DE AXE IN
PERITROCHIO.
Fabricam, &
ius in&longs;trumenti Pappus in octauo
mathematicarum collectionum
libro docet; axemq; vocat AB,
tympanum verò CD circa idem
centrum; & &longs;cytalas in foramini
bus tympani EF GH & c. ita vt potentia,
quæ &longs;emper in &longs;cytalis e&longs;t, vt in F, dum circum
uertit tympanum, & axem, &longs;ur&longs;um moueat pon
dus K axi appen&longs;um fune LM circa axem reuo
luto. Nobis igitur re&longs;tat, vt o&longs;tendamus, cur ma
gna pondera ab exigua virtute, quouè etiam mo
do hoc in&longs;trumento moueantur; temporis quin
etiam, &longs;patiiq; mouentis inuicem potentiæ, ac
moti ponderis rationem aperiamus; huiu&longs;modi
que in&longs;trumenti v&longs;um ad vectem reducamus.
PROPOSITIO I.
Potentia pondus &longs;u&longs;tinens axe in peritrochio
ad pondus eandem habet proportionem, quam
&longs;emidiameter axis ad &longs;emidiametrum tympani
vná cum &longs;cytala.
Sit diameter axis AB, cuius centrum C; &longs;it diameter tympani
DCE circa idem centrum; &longs;intq; AB DE in eadem recta linea;
&longs;int deinde &longs;cytalæ in foraminibus tympani DF GH & c inter &longs;e &longs;e
æquales, atq; æquè di&longs;tantes; &longs;itq; FE horizonti æquidi&longs;tans;
pondus autem K in fune BL circa axem volubili &longs;it appen&longs;um. &
potentia in F &longs;u&longs;tineat pondus K. Dico potentiam in F ad pondus
k ita &longs;e habere, vt CB ad CF. fiat vt CF ad CB, ita pondus
k ad aliud M, quod appendatur in F. & quoniam pondera M k
appen&longs;a &longs;unt in FB; erit FB tanquam vectis, &longs;iue libra; quia ve
rò C e&longs;t punctum immobile, circa quod axis, tympanusq; reuol
uuntur; erit C fulcimentum vectis FB; vellibræ centrum. cùm
rabunt. Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver
gat, ponderi K æqueponderabit; ip&longs;iq; M æqualis erit. idem enim
præ&longs;tat potentia, quod pondus M. pondus igitur K ad poten
pondus erit, vt CB ad CF, hoc e&longs;t, &longs;emidiameter axis ad &longs;emiSimiliter etiam o&longs;ten
detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in q. tunc enim &longs;u&longs;ti
neret vecte CQ; & ad pondus eam haberet proportionem, quam
habet CB ad Cq. Videlicet &longs;emidiameter axis ad &longs;emidiame
trum tympani vná cum &longs;cytala Eq. quod demon&longs;trare opor
tebat.
COROLLARIVM.
Manife&longs;tum e&longs;t potentiam &longs;emper minorem
e&longs;&longs;e pondere.
Semidiameter enim axis &longs;emper &longs;emidiametro tympani mi
nor e&longs;t. & potentia eò minor e&longs;t pondere, quò &longs;emidiameter axis
minor e&longs;t &longs;emidiametro tympani vná cum &longs;cytala. quare quò lon
gior e&longs;t CF, vel CQ; & quò breuior e&longs;t CB, minor adhuc &longs;em
per potentia in F, vel in Q pondus k &longs;u&longs;tinebit. quò enim minor
e&longs;t CB, eò minorem habebit proportionem &longs;emidiameter axis
ad &longs;emidiametrum tympani vná cum &longs;cytala.
Hoc autem loco con&longs;iderandum occurrit, quòd &longs;i in alia &longs;cyta
la appendatur pondus, vt in T, &longs;u&longs;tinens pondus k; it a nempè, vt
pondus in T appen&longs;um, pondusq; k circa axem con&longs;titutum
maneant; erit pondus in T grauius pondere M in F appen&longs;o.
iungatur enim TB, & à puncto C horizonti perpendicularis du
catur CI, quæ lineam TB &longs;ecet in I; tandemq; connectatur
TC, quæ æqualis erit CF. Quoniam autem pondera appen&longs;a
&longs;unt in TB, perindè &longs;e &longs;e habebunt, ac &longs;i in punctis TB ip&longs;orum
centra grauitatum haberent; vt antea dictum e&longs;t. & quia ma
nent, erit punctum I (ex prima huius de libra) amborum &longs;imul
grauitatis centrum; cùm &longs;it CI horizonti perpendicularis. &longs;ed
quoniam angulus BCI e&longs;t rectus, erit BIC acutus, lineaq; BI
ip&longs;a BC maior erit. quare angulus CIT erit obtu&longs;us; atq;
ideo linea CT ip&longs;a TI maior erit. Cùm autem CT maior &longs;it
TI, & IB maior BC; maiorem habebit proportionem TC ad
CB, quàm TI ad IB; & conuertendo, minorem habebit pro
portionem BC ad CT, hoc e&longs;t ad CF, quàm BI ad IT; vt ex
vige&longs;ima &longs;exta quinti elementorum (iuxta Commandini editio
nem) patet. Quoniam verò punctum I e&longs;t ponderum in TB
vt BI ad IT. pondus verò in F ad idem pondus in B e&longs;t, vt BC
ad CF; maiorem igitur proportionem habebit pondus in T ad
pondus in B, quàm pondus in F ad idem pondus in B. ergo
Si verò loco ponderis in T animata potentia &longs;u&longs;tinens pon
dus k con&longs;tituatur; quæ ita degrauet &longs;e, ac &longs;i in centrum mundi
tendere vellet; quemadmodum &longs;uapte natura efficit pondus in T
appen&longs;um; erit hæc eadem ponderi in T appen&longs;o æqualis; alio
quin non &longs;u&longs;tineret; quæ quidem ip&longs;a potentia in F collocata ma&longs;icuti enim &longs;e &longs;e habet pondus in T ad pondus in F, ita
& potentia in T ad potentiam in F; cùm potentiæ &longs;int ponderi
bus æquales. verùm &longs;i vnaquæq; potentia &longs;eor&longs;um &longs;umpta, tàm
in T, quàm in F &longs;u&longs;tinens pondus
moueri &longs;e vellet, veluti apprehen&longs;a manu &longs;cytala; tunc eademmet
potentia, vel in F, vel in T con&longs;tituta idem pondus k &longs;u&longs;tinere po
terit; cùm &longs;emper in cuiu&longs;cunq; extremitate &longs;cytalæ ponatur, ab
eodem centro C æquidi&longs;tans fuerit, ac &longs;ecundum eandem circum
ferentiam ab eodem centro æqualiter &longs;emper di&longs;tantem perpen&longs;io
nem habeat. neq; enim (&longs;icuti pondus) proprio nutu magis in
centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, &longs;eu
quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. pro
pterea non eodem modo res &longs;e &longs;e habet, &longs;iue pondera, &longs;iue anímatæ
potentiæ ii&longs;dem locis eodem munere abeundo fuerint con&longs;titutæ.
Potentia autem mouet pondus vecte FB, videlicet dum po
tentia in F circumuertit tympanum, circumuertit etiam axem; &
FB fit tamquam vectis, cuius fulcimentum C, potentia mouens
in F, & & dum punctum F peruenit in N;
punctum H erit in F, & punctum B erit in O; ita vt ducta NO
tran&longs;eat per C; eodemq; tempore pondus k motum erit in P, ita
vt OBP &longs;it æqualis ip&longs;i BL, cùm &longs;it idem funis.
Deinde ex quarta huius de vecte facilè eliciemus &longs;patium po
tentiæ mouentis ad &longs;patium ponderis moti ita e&longs;&longs;e, vt &longs;emidiame
ter tympani cùm &longs;cytala ad &longs;emidiametrum axis, hoc e&longs;t, vt CF
ad CB, cùm circumferentia FN ad BO, &longs;it vt CF ad CB. & quo
niam BL, e&longs;t æqualis OBP, dempta communi BP, erit OB ip
&longs;i PL æqualis. quare FN &longs;patium potentiæ ad PL &longs;patium pon
deris erit, vt CF ad CB, videlicet &longs;emidiameter tympani cùm
&longs;cytala ad &longs;emidiametrum axis. Quod idem o&longs;tendetur, poten
tia vel in Q, vel in qualibet alia &longs;cytala exi&longs;tente, vt in S. cùm
enim &longs;cytalæ &longs;int &longs;ibi inuicem æquales, atq; æqualiter di&longs;tantes;
vbicunq; &longs;it potentia æquali mota velocitate &longs;emper æquali tem
pore æquale &longs;patium pertran&longs;ibit, hoc e&longs;t ex Q in R, vel ex Sin T
eodem tempore mouebitur, quò ex F in N. &longs;ed quò tempore po
tentia ex F in N mouetur, eodemmet pror&longs;us pondus k ex L in
P quoq; mouetur; vbicunq; igitur &longs;it potentia, erit &longs;patium poten
tiæ ad &longs;patium ponderis moti, vt CF ad CB, hoc e&longs;t &longs;emidia
meter tympani cum &longs;cytala, ad &longs;emidiametrum axis.
COROLLARIVM. I.
Ex his manife&longs;tum e&longs;t, ita e&longs;&longs;e pondus ad po
tentiam pondus &longs;u&longs;tinentem, vt &longs;patium poten
tiæ mouentis ad &longs;patium ponderis moti.
COROLLARIVM II.
Manife&longs;tum e&longs;t etiam, maiorem &longs;emper ha
bere proportionem &longs;patium potentiæ mouentis
ad &longs;patium ponderis moti, quàm pondus ad ean
dem potentiam.
Præterea quò circulus FHN circa &longs;cytalas e&longs;t maior, eò quoq;
in pondere mouendo maius &longs;umetur tempus; dummodo potentia
æquali moueatur velocitate. tempu&longs;q; eò maius erit, quò diame
ter vnius diametro alterius e&longs;t maior. circulorum enim circumfe
rentiæ ita &longs;e habent, vt diametri. Cùm vero ex trige&longs;ima &longs;exta
quarti libri Pappi Mathematicarum collectionum, duorum inæ
qualium circulorum æquales circumferentias inuenire po&longs;simus;
ideo tempus quoq; portionum circulorum inæqualium hoc modo
inueniemus. è conuer&longs;o autem, quò maior erit axis circumferen
tia citius pondus &longs;ur&longs;um mouebitur. maior enim pars funis BL
in vna circumuer&longs;ione completa circa circulum ABO reuoluitur,
quàm &longs;i minor e&longs;&longs;et; cùm funis circumuolutus &longs;it circumferen
tiæ circuli æqualis, circa quem reuoluitur.
COROLLARIVM.
Ex his manife&longs;tum e&longs;t, quò facilius pondus mo
uetur, tempus quoq; eò maius e&longs;&longs;e; & quò dif
ficilius, eò tempus minus e&longs;&longs;e. & è conuer&longs;o.
PROPOSITIO II.
PROBLEMA.
Datum pondus à data potentia axe in peritro
chio moueri.
Sit datum pondus &longs;exagin
ta; potentia verò vt decem.
exponatur quædam recta li
nea AB, quæ diuidatur in C,
ita vt AC ad CB eandem
habeat proportionem, quam &longs;exaginta ad decem. & &longs;i CB axis
&longs;emidiameter e&longs;&longs;et, & CA &longs;emidiameter tympani cùm &longs;cytalis;
derare. Accipiatur autem inter BC quoduis punctum D; fiatq;
BD &longs;emidiameter axis, & DA &longs;emidiameter tympani cùm &longs;cy
talis; ponaturq; pondus &longs;exaginta in B fune circa axem, & potentia Quoniam enim AD ad DB maiorem habet proportio
nem, quam AC ad CB; maiorem habebit proportionem AD ad
DB, quam pondus &longs;exaginta in B appen&longs;um ad potentiam vt decemQuare potentia in A pondus &longs;exaginta axe in peritro
chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, & DA &longs;emidia
meter tympani cùm &longs;cytalis. quod erat faciendum.
ALITER.
Organicè verò melius erit hoc pacto.
Exponatur axis, cuius
diameter &longs;it BD, & cen
trum C, quem quidem
axem maiorem, vel mino
rem con&longs;tituemus, veluti
magnitudo, ponderi&longs;q; grauitas po&longs;tulat. producatur deinde BD
v&longs;q; ad A: fiatq; BC ad CA, vt decem ad &longs;exaginta. & &longs;i CA tym
pani cùm &longs;cytalis &longs;emidiameter e&longs;&longs;et, potentia decem in A ponde
ri &longs;exaginta in B æqueponderaret. producatur verò BA ex parte
A, & in hac producta linea quoduis accipiatur punctum E; fiatq;
CE &longs;emidiameter tympani cùm &longs;cytalis; ponaturq; potentia vt
decem in E; habebit EC ad CB maiorem proportionem, quàm
pondus &longs;exaginta in B ad potentiam vt decem in E. potentia igi
tur vt decem in E mouebit pondus &longs;exaginta in B appen&longs;um fune
circa axem, cuius &longs;emidiameter e&longs;t CB, & CE &longs;emidiameter tym
pani cùm &longs;cytalis. quod facere oportebat.
Sub hoc facultatis genere &longs;unt ergatæ, &longs;uccu
læ, terebræ, tympana cum &longs;uis axibus, &longs;iue dentata,
&longs;iue non; & &longs;imilia.
Terebra verò habet etiam ne&longs;cioquid cochleæ; dum enim mo
uet pondus, &longs;cilicet dum perforat, ex &longs;ua ferè natura &longs;emper vlte
rius progreditur
de&longs;criptas. quoniam autem verticem habet acutum, ad cunei quoq;
rationem commodè referri poterit.
DE CVNEO.
Aristoteles in quæ&longs;tioni
bus Mechanicis quæ&longs;tione deci
ma&longs;eptima a&longs;&longs;erit, cuneum &longs;cin
dendo ponderi duorum vicem
pror&longs;us gerere vectium &longs;ibi inui
cem contrariorum hoc
Sit cuneus ABC, cu
ius vertex B, & &longs;it AB
æqualis BC; quod au
tem &longs;cindendum e&longs;t,
&longs;it DEFG; &longs;itq; pars
cunei HB k intra DE
FG, & HB æqualis
&longs;it ip&longs;i Bk. percutiatur
(vt fieri &longs;olet) cuneus
in AC, dum cuneus in
AC percutitur, AB fit
vectis, cuius fulcimen
tum e&longs;t H, & pondus in
B. eodemq; modo CB
fit vectis, cuius fulci
mentum e&longs;t K, & pondus &longs;imiliter in B. &longs;ed dum percutitur cu
neus, maiori adhuc ip&longs;ius portione ip&longs;um DEFG ingreditur,
quàm prius e&longs;&longs;et: &longs;it autem portio hæc MBL; &longs;itq; M B ip&longs;i BL
æqualis. & cùm MB BI &longs;int ip&longs;is HB BK maiores; erit ML maior
dum igitur ML
erit in &longs;itu Hk; opor
& D moueatur ver&longs;us
O, G autem ver&longs;us N:
& quò maior pars cu
nei intra DEFG ingre
dietur, eò maior fiet
&longs;ci&longs;sio; & DG ma
gis adhuc impellentur
ver&longs;us ON. pars igi
tur KG eius, quod &longs;cin
ditur, mouebitur à ve
cte AB, cuius fulcimen
tum e&longs;t H, & pondus
in B; ita vt punctum B ip&longs;ius vectis AB impellat partem KG. & pars HD mouebitur à vecte CB, cuius fulcimentum e&longs;t k; ita
vt B vecte CB partem HD impellat.
Cùm autem tria &longs;int vectium genera, vt &longs;upra
o&longs;ten&longs;um e&longs;t; idcirco conuenientius erit forta&longs;&longs;è
cuneum hoc modo con&longs;iderare.
Ii&longs;dem po&longs;itis, intelligatur vectis AB, cuius fulcimentum B, &
pondus in H, vt in &longs;ecunda huius de vecte diximus. &longs;imiliter ve
ctis CB, cuius fulcimentum B, & pondus in K; ita vt pars HD
moueatur à vecte AB, cuius fulcimentum e&longs;t B, & pondus in H;
ita vt punctum H ip&longs;ius vectis AB impellat partem HD. &longs;imi
li quoq; modo pars KG moueatur à vecte CB, cuius fulcimentum
e&longs;t B, & pondus in k, it aut k ip&longs;ius uectis CB partem k G mo
ueat. quod quidem for&longs;itan rationi magis con&longs;entaneum erit.
Sit enim cuneus ABC;
&longs;intq; duo pondera &longs;epa
rat a DEFG, & HIkL,
intra quæ &longs;it pars cunei
DBH, cuius uertex B
medium inter utrumq; &longs;i
tum obtineat. percutia
tur autem cuneus, ita ut
magis adhuc intra pon
dera propellatur, &longs;icuti
prius dictum e&longs;t; ponde
ra enim &longs;unt, ac &longs;i unum tantùm continuum e&longs;&longs;et GFkL, quod
&longs;cindendum e&longs;&longs;et: eodem enim modo pars DG, dum cuneus
ulterius impellitur, mouebitur uer&longs;us M; & pars HL uer&longs;us N.
Moueatur itaq; pars DG uer&longs;us M, & pars HL uer&longs;us N, B uerò
dum ulterius progreditur, &longs;emper medium inter utrunq; pondus
remaneat. dum autem DG à cuneo mouetur uer&longs;us M; patet B
non mouere partem DG uer&longs;us M uecte CB, cuius fulcimentum
H;
cto uectis D uecte AB, cuius fulcimentum B; punctum enim D tan
git pondus, & in&longs;trumenta mouent per contactum. Similiter
HL mouebitur ab H uecte CB, cuius fulcimentum B; & uterq;
uectis utriq; re&longs;i&longs;tit in B, ita ut B potius fulcimenti uice fungatur,
quàm mouendi ponderis. quod ip&longs;um hoc quoq; modo manife
&longs;tum erit.
Sit, quod &longs;cindendum e&longs;t A
BCD
gulum; &longs;intq; duo vectes æqua
les EF GF, & partes vectium
HF KF &longs;int intra ABCD; &longs;itq;
HF æqualis Fk, & HA æqua
lis KB. Oporteat verò vecti
bus EF GF &longs;cindere ABCD
ab&longs;q; percu&longs;sione, videlicet &longs;int
potentiæ mouentes in EG æqua
les. vt autem &longs;cindatur ABCD,
oportet partem HA moueri uer
&longs;us M. & kB ver&longs;us N; &longs;ed dum vectes mouentur, putá alter in
M, alter verò in N; nece&longs;&longs;e e&longs;t, vt punctum F immobile rema
neat; in illo enim fit vectium occur&longs;us. quare F erit fulcimen
tum vtriu&longs;q; vectis, & FG mouebit partem kB, cuius fulcimen
tum erit F, & potentia mouens in G; & pondus in k. &longs;imi
liter pars HA mouebitur à vecte EF, cuius fulcimentum F, po
tentia in E, & pondus in H.
Si autem k H e&longs;&longs;ent fulcimenta immobilia, & pondera in F;
dum vectis FG conatur mouere pondus in F, tunc ei re&longs;i&longs;tit ve
ctis EF, qui etiam conatur mouere pondus in F ad partem op
po&longs;itam; &longs;ed quoniam potentiæ &longs;unt æquales, & cætera æqualia;
ergo in F non fiet motus: æquale enim non mouet æquale. patet
igitur in F maximam fieri vectium &longs;ibi inuicem occurrentium re&longs;i
&longs;tentiam, ita ut F &longs;it quoddam immobile. Quare con&longs;iderando
cuneum,
tius utitur hoc &longs;ecundo modo, quàm primo.
Quoniam autem totus cuneus &longs;cindendo mo
uetur, po&longs;&longs;umus idcirco eundem alio quoq; mo
do con&longs;iderare; videlicet dum ingreditur id,
pra planum horizonti inclinatum mouere.
Sit planum horizonti æquidi&longs;tans tran&longs;iens per AB; &longs;it cuneus
CDB, & CD æqualis ip&longs;i DB; & latus cunei DB &longs;it &longs;emper in
&longs;ubiecto plano. &longs;it deinde pondus AEFG immobile in A; &longs;itq;
pars cunei EDH &longs;ub AEFG. Quoniam enim dum percutitur cu
neus in CB, maior pars cunei ingreditur &longs;ub AEFG, quàm &longs;it
EDH; &longs;it hæc pars IDH. & quoniam latus cunei DB &longs;emper
e&longs;t in &longs;ubiecto plano per AB ducto horizonti parallelo, tunc quan
do pars cunei kDI erit &longs;ub AEFG; erit punctum k in H, & I
&longs;ub E. &longs;ed Ik maior e&longs;t HE; punctum igitur E &longs;ur&longs;um motum
erit. & dum cuneus &longs;ub AEFG ingreditur, punctum E &longs;ur&longs;um
&longs;uper latus cunei EI mouebitur, eodemq; modo &longs;i cuneus vlterius
progredietur, &longs;emper punctum E &longs;uper latus cunei DC mouebitur:
punctum igitur E ponderis &longs;uper planum DC mouebitur horizonti
inclinatum, cuius inclinatio e&longs;t angulus BDC. quod demon
&longs;trare oportebat.
In hoc exemplo, con&longs;iderando cuneum in&longs;tar vectis mouen
tem, manife&longs;tum e&longs;t, cuneum BCD pondus AEFG vecte CD
mouere; ita vt D &longs;it fulcimentum, & pondus in E. non autem ve
cte BD, cuius fulcimentum H, & pondus in D.
Vt autem res clarior reddatur, alio vtamur
exemplo.
Sit planum hori
zonti æquidi&longs;tans
tran&longs;iens per AB; &longs;it
cuneus CAB, cuius
latus AB &longs;it &longs;emper
in &longs;ubiecto plano; &longs;it
qué pondus AEFG,
quod nullum alium
habeat motum, ni&longs;i
&longs;ur&longs;um, & deor&longs;um ad rectos angulos horizonti; ita vt ducta IGk
&longs;ubiecto plano, ip&longs;iqué AB perpendicularis, punctum G &longs;it &longs;em
per in linea IGk. & quoniam dum cuneus percutitur in CB, to
tus &longs;uper AB vlterius progreditur; pondus AEFG eleuabitur ex Moueatur cuneus ita, vt E tandem per
ueniat in C, & po&longs;itio cunei ABC &longs;it MNO, & po&longs;itio pon
deris AEFG &longs;it PMQI, & G &longs;it in I. Quoniam itaq; dum cu
neus &longs;uper lineam BO mouetur, pondus AEFG &longs;ur&longs;um moue
tur à linea AC. & dum cuneus ABC vlterius progreditur, &longs;em
per pondus AEFG magis à latere cunei AC eleuatur: pondus igi
tur AEFG &longs;uper planum cunei AC mouebitur; quod quidem
nihil aliud e&longs;t, ni&longs;i planum horizonti inclinatum, cuius inclinatio
e&longs;t angulus BAC.
Hic motus facilè ad libram, vectemq; reducitur.
quod enim
&longs;uper planum horizonti inclinatum mouetur ex nona Pappi octa
ui libri Mathematicarum collectionum reducitur ad libram. ea
dem enim e&longs;t ratio, &longs;iue manente cuneo, vt pondus &longs;uper cunei
latus moueatur; &longs;iue eodem etiam moto, pondus adhuc &longs;uper ip
&longs;ius latus moueatur; tamquam &longs;uper planum horizonti incli
natum.
Ea verò, quæ &longs;cinduntur, quomodo tam
quam &longs;uper plana horizonti inclinata mouean
tur, o&longs;tendamus.
Sit cuneus ABC,
& AB ip&longs;i BC æqua
lis. Diuidatur AC
bifariam in D, conne
ctaturq; BD. &longs;it dein
de linea EF, per quam
tran&longs;eat planum hori
zonti æquidi&longs;tans; &longs;itq;
BD in eadem linea EF;
& dum cuneus percuti
tur, dumq; mouetur ver
&longs;us E, &longs;emper BD &longs;it in linea EF. quod verò &longs;cindendum e&longs;t
&longs;it GHLM, intra quod &longs;it pars cunei kBI. manife&longs;tum e&longs;t,
mouetur, partem kG
ver&longs;us N moueri; & par
tem HI uer&longs;us O. per
cutiatur cuneus, ita vt
AC &longs;it in linea NO;
tunc k erit in A, & I in
C: & k ex &longs;uperius di
ctis motum erit &longs;uper
kA, & I &longs;uper IC. quare dum cuneus mo
uetur, pars KG &longs;uper BA latus cunei mouebitur, & pars IH &longs;uper
latus BC. pars igitur kG &longs;uper planum mouetur horizonti incli
natum, cuius inclinatio e&longs;t angulus FBA. &longs;imiliter IH moue
tur &longs;uper planum BC in angulo FBC. Partes ergo eius, quod
&longs;cinditur &longs;uper plana horizonti inclinata mouebuntur. & quam
quam planum BC &longs;it &longs;ub horizonte; pars tamen IH &longs;uper IC mo
uetur, tamquam &longs;i BC e&longs;&longs;et &longs;upra partes
enim eius quod
uentur; eadem ergo erit ratio motus partis IH, ac partis KG. &longs;i
militer eadem e&longs;t ratio, &longs;iue EF &longs;it horizonti æquidi&longs;tans, &longs;iue
horizonti perpendicularis, vel alio modo. nece&longs;&longs;e e&longs;t enim poten
tiam cuneum mouentem eandem e&longs;&longs;e, cùm cætera eadem rema
neant. eadem igitur erit ratio.
Po&longs;t hæc con&longs;iderandum e&longs;t, quæ nam &longs;int ea, quæ efficiunt,
vt aliquod facilius moueatur, &longs;iue &longs;cindatur. quæ quidem duo
&longs;unt.
Primum, quod efficit, vt aliquod facilè &longs;cin
datur, quod etiam ad e&longs;&longs;entiam cunei magis per
tinet, e&longs;t angulus ad verticem cunei; quò enim
minor e&longs;t angulus, eò facilius mouet, ac &longs;cindit.
Sint duo cunei ABC DEF, & angulus
ABC ad verticem minor &longs;it angulo DEF.
dico aliquod facilius moueri, &longs;iue &longs;cindi à cu
neo ABC, quàm à DEF. diuidantur AC
DF bifariam in G H punctis; connectan
turq; BG, & EH. Quoniam enim partes
eius, quod &longs;cinditur à cuneo ABC, &longs;u
per planum horizonti inclinatum mouen
tur, cuius inclinatio e&longs;t GBA: quæ ve
rò à cuneo DEF, &longs;uper planum horizonti
inclinatum mouentur, cuius inclinatio e&longs;t
HED; & angulus GBA minor e&longs;t angulo HED; cùm
CBA minor &longs;it DEF: & ex nona Pappi octaui libri mathe
maticarum collectionum, quod mouetur &longs;uper planum AB faci
lius mouebitur, & à minore potentia, quàm &longs;uper ED; Quod
ergo &longs;cinditur à cuneo ABC facilius, & à minore potentia &longs;cin
detur, quàm à cuneo DEF. &longs;imiliter o&longs;tendetur, quò magis an
gulus ad verticem cunei erit acutus, eò facilius aliquod moueri,
ac &longs;cindi. quod demon&longs;trare oportebat.
Po&longs;&longs;umus etiam hoc alia ratione o&longs;tendere
con&longs;iderando cuneum, vt vectibus &longs;ibi inuicem
aduer&longs;is mouet, &longs;icuti &longs;ecundo modo dictum e&longs;t.
hoc autem prius o&longs;tendere oportet.
Sit vectis AB, cuius fulcimentum
&longs;it B immobile; quod autem mouen
dum e&longs;t, &longs;it CDEF rectangulum ita
accommodatum, vt deor&longs;um ex par
te FE moueri non po&longs;sit; & punctum
E &longs;it immobile, & tamquam centrum;
ita vt punctum D moueatur per cir
cumferentiam circuli DH, cuius cen
trum &longs;it E. & C per circumferentiam
CL, ita vt iuncta CE &longs;it eius &longs;emi
diameter. tangat in&longs;uper CDEF ve
ctem AB in C, atq; vectis AB moueat pondus CDEF, & po
tentia mouens &longs;it in A, fulcimentum B, & pondus in C. &longs;it
deinde alius vectis MCN, qui etiam moueat CDEF, cuius ful
cimentum immobile &longs;it N; potentia mouens in M, & pondus
&longs;imiliter in C; &longs;itq; CN æqualis ip&longs;i CB, & CM ip&longs;i CA; al
ternatimq; moueatur pondus CDEF vectibus AB MN. dico
CDEF facilius ab eadem potentia moueri vecte AB, quàm ve
cte MN.
Fiat centrum B, & interuallo BC circumferentia de&longs;cribatur
CO. &longs;imiliter centro N, interuallo quidem NC, circumferen
tia de&longs;cribatur CP. Quoniam enim dum vectis AB mouet CD
EF, punctum vetis C mouetur &longs;uper circumferentiam CO; cùm
&longs;it B fulcimentum, & centrum immobile. &longs;imiliter dum vectis
MN mouet CDEF, punctum C mouetur per circumferentiam
CP; dum igitur vectis AB mouet CDEF, conatur mouere pun
ctum C ponderis &longs;uper circumferentiam CO; quod quidem effi
cere non pote&longs;t: quia C mouetur &longs;uper circumferentiam CL. qua
re in motu vectis AB &longs;ecundùm partem ip&longs;i re&longs;pondentem, ac mo
tu ponderis &longs;ecundum C facto, contingit repugnantia quædam;
in diuer&longs;as enim partes mouentur. &longs;imiliter dum vectis MN mo
uet CDEF, conatur mouere C &longs;uper circumferentiam CP; at
que ideo in hoc etiam vtroq; motu &longs;imilis oritur repugnantia.
quoniam autem circumferentia CO propior e&longs;t circumferentiæ
CL, quam &longs;it CP; hoc e&longs;t propior e&longs;t motui, quem facit pun
ctum C ponderis; ideo minor erit repugnantia inter motum vectis
motum eiu&longs;dem C. quod etiam patet, &longs;i intelligatur CF hori
zonti perpendicularis, tunc enim circumferentia CP magis ten
dit deor&longs;um, quàm CO; & CL tendit &longs;ur&longs;um. & ideo minor fit re
pugnantia inter vectem AB, & motum C, quàm inter
motum C. &longs;ed vbi minor repugnantia ibi maior facilitas.
ergo faci
lius mouebitur CD EF vecte AB, quàm vecte MN. quod demon
&longs;trare oportebat.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, quò minor e&longs;t an
gulus à linea CF, vel CE, vel CD contentus;
hoc e&longs;t, quò minor e&longs;t angulus BCF, vel BCE,
vel etiam BCD, eò facilius pondus moueri.
quod quidem eodem modo o&longs;tendetur.
Quod autem propo&longs;itum e&longs;t, &longs;ic demon
&longs;trabimus.
Sint cunei ABC DE
F, & angulus ABC mi
nor &longs;it angulo DEF, &
AB BC DE EF &longs;int in
ter &longs;e &longs;e æquales. Sint de
inde quatuor pondera æ
qualia GH IL NO QR
rectangula; &longs;intq; LM
kH in eadem recta linea:
&longs;imiliter RS PO in recta linea; erunt GK IM parallelæ, & NP
QS parallelæ. &longs;it IBG pars cunei intra pondera GH IL; & cu
nei pars QEN intra pondera NO QR; &longs;intqué IB BG QE
EN inter &longs;e &longs;e æquales. dico pondera GH IL facilius ab eadem
ABC, quàm pondera
NO QR cuneo DEF.
Diuidantur AC DF
bifariam in TV, iungan
turq; TBVE, erunt an
guli ad T, & V recti. con
nectatur IG, quæ &longs;ecet
BT in X. Quoniam e
nim IB e&longs;t æqualis BG, & BA æqualis BC; erit IA ip&longs;i GC quare vt BI ad IA, ita e&longs;t BG ad GC.
parallela igitur
ac propterea anguli ad X &longs;unt recti: &longs;ed & an
TB æquidi&longs;tans e&longs;t ip&longs;is Gk IM. angulus igitur TBC æqua
lis e&longs;t angulo BGK, & TBA ip&longs;i BIM æqualis. &longs;imiliter demon
&longs;trabimus angulum VEF æqualem e&longs;&longs;e ENP, & VED æqualem
EQS. cùm autem angulus ABC minor &longs;it angulo DEF; erit
& angulus TBC minor VEN. quare & BGk minor ENP.
&longs;imili modo BIM minor EQS.
quoniam autem cuneus ABC
duobus mouet vectibus AB BC, quorum fulcimenta &longs;unt in B;
& pondera in GI: &longs;imiliter cuneus DEF duobus vectibus mouet
DE EF, quorum fulcimenta &longs;unt in E; & pondera in N Q: per
præcedentem pondera GH IL facilius vectibus AB BC mo
uebuntur, quàm pondera NO QR vectibus DE EF. ponde
ra ergo GH IL facilius cuneo ABC mouebuntur, quàm ponde
ra NO QR cuneo DEF. & quia eadem e&longs;t ratio in mouendo,
atq; in &longs;cindendo; facilius idcirco aliquod cuneo ABC &longs;cindetur
quàm cuneo DEF. &longs;imiliterq; o&longs;tendetur, quò minor e&longs;t angu
lus ad verticem cunei, eò facilius aliquod moueri, vel &longs;cindi. quod
demon&longs;trare oportebat.
Præterea quæ mouentur à cuneo DEF, per maiora mouentur
&longs;patia; quàm ea, quæ à cuneo ABC. nam vt DF &longs;it intra QN,
& AC &longs;it intra IG; nece&longs;&longs;e e&longs;t, vt QN per &longs;patia moueantur
maiora; &longs;cilicet vnum dextror&longs;um, alter &longs;ini&longs;tror&longs;um, quàm IG;
cùm DF maior &longs;it AC; dummodo totus cuneus intra pondera inà potentia verò facilius eodem tempore mouetur ali
quod per minus &longs;patium, quàm per maius; dummodo cætera, qui
bus fit motus, &longs;int æqualia: &longs;i ergo eodem tempore AC DF in
IG QN
les; facilius à potentia mouebuntur GI cuneo ABC, quàm QN
cuneo DEF. quare facilius pondera GH IL à potentia mouebun
tur cuneo ABC, quàm pondera NO QR cuneo DEF. &longs;imiliter
qué o&longs;tendetur, quò angulus ad verticem cunei minor e&longs;&longs;et, eò fa
cilius pondera moueri, vel &longs;cindi.
Secundum, quod efficit, vt aliquod facilius
&longs;cindatur, e&longs;t percu&longs;sio; qua cuneus mouetur, &
mouet; hoc e&longs;t percutitur, ac &longs;cindit.
Sit cuneus A, quod &longs;cinditur B, quod
percutit C; quod quidem, vel ex &longs;e ip&longs;o,
vel à regente, atq; ip&longs;um mouente poten
tia percutit, atq; mouet. &longs;i quidem ex
&longs;e ip&longs;o, Primùm quò grauius erit, eò
maior fiet percu&longs;sio. quinetiam, quò
longior fuerit di&longs;tantia inter AC, maior
itidem fiet percu&longs;sio. graue enim vnum
quodq; dum mouetur; grauitatis ma
gis a&longs;&longs;umit motum, quàm quie&longs;cens: &
adhuc magis quo longius mouetur.
Si verò C ab aliqua moueatur po
tentia, vt &longs;i per manubrium DE mo
ueatur; primùm quò grauius erit C,
deinde quò longius erit DE, eò ma
ior fiet percu&longs;sio. &longs;i enim ponatur po
tentia mouens in E, erit C magis di
&longs;tans à centro & ideo citius mouebi
tur. vt in quæ&longs;tionibus Mechanicis
latè mon&longs;trat Ari&longs;toteles; nec non
ex iis, quæ in tractatu de libra di
cta fuere, patere pote&longs;t, quò magis
pondus C à centro di&longs;tat, eò grauius reddi. quod ip&longs;um etiam va
lidiori pellet impul&longs;u virtute in E potentiore exi&longs;tente.
Hoc verò &longs;ecundùm e&longs;t, quod efficit, vt hoc in&longs;trumento ma
gna moueantur, &longs;cindanturq; pondera. percu&longs;sio enim vis e&longs;t ua
lidi&longs;sima, vt ex decimanona
patet. &longs;i enim &longs;upra cuneum maximum imponatur onus; tunc cu
neus nihil ferè efficiet, præ&longs;ertim ictus comparatione. quod &longs;i ad
huc ip&longs;i cuneo vectem, vel cochleam, vel quoduis aliud huiu&longs;mo
di aptetur in&longs;trumentum ad cuneum ponderi intimius propellen
dum, nullius ferè momenti præ ictu continget effectus. cuius qui
corpus A
partem detrahere qui&longs;piam voluerit, pu
tá partem anguli B; tunc malleo ferreo
ab&longs;q; alio in&longs;trumento percutiendo in B,
facilè aliquam anguli B partem franget.
quod quidem nullo alio in&longs;trumento
percu&longs;sionis munere carente, ni&longs;i maxi
ma cùm difficultate efficere poterit; &longs;iue
fuerit vectis, &longs;iue cochlea, &longs;iue quoduis aliud huiu&longs;modi. quare
percu&longs;sio in cau&longs;a e&longs;t, quo magna &longs;cindantur pondera. cùm autem
&longs;ola percu&longs;sio tantam vim habeat, &longs;i ei aliquod adiiciamus in&longs;tru
mentum ad mouendum, &longs;cindendumq; accomodatum, admiran
da profectò videbimus. In&longs;trumentum huiu&longs;
modi cuneus e&longs;t, in quo duo (quantum ad ip
&longs;ius formam attinet) con&longs;ideranda occurrunt.
Alterum e&longs;t, cuneum ad &longs;u&longs;cipiendam, &longs;u&longs;tinen
damq; percu&longs;sionem apti&longs;simum e&longs;&longs;e; alterum
e&longs;t quòd propter eius in altera parte &longs;ubtilita
tem facilè intra corpora ingreditur, vt manife
&longs;tè patet. Cuneus ergo cum percu&longs;sione ip&longs;ius
efficit, vt in mouendis, &longs;cindendi&longs;q; ponderi
bus ferè miracula cernamus.
Ad huiu&longs;modi facultatis in&longs;trumentum, ea
quoquè omnia commodè referri po&longs;&longs;unt, quæ
percu&longs;sione, &longs;iue impul&longs;u incidunt, diuidunt,
perforant, huiu&longs;modiq; alia obeunt munera. vt
en&longs;es, gladii, mucrones, &longs;ecures, & &longs;imilia. &longs;erra
quoq; ad hoc reducetur; dentes enim percu
tiunt, cuneiq; in&longs;tar exi&longs;tunt.
DE COCHLEA.
Pappvs in eodem octauo libro
multa pertractans de cochlea, do
cet quomodo conficienda &longs;it; &
quomodo magna huiu&longs;modi in
&longs;trumento moueantur pondera;
nec non alia theoremata ad eius
cognitionem valdè vtilia. Quoniam autem in
ter cætera pollicetur, &longs;e o&longs;tendere velle, co
chleam nihil aliud e&longs;&longs;e præter a&longs;&longs;umptum cu
neum percu&longs;sionis expertem vecte motionem
facientem; hoc autem in ip&longs;o de&longs;ideratur; pro
pterea idip&longs;um o&longs;tendere conabimur, nec non
eiu&longs;dem cochleæ ad vectem, libramq; reductio
nem; vt ip&longs;ius tandem completa habeatur co
gnitio.
Sit cuneus ABC, qui circa cylindrum DE circumuoluatur: &longs;itq;
IGH cuneus circa cylindrum reuolutus, cuius vertex &longs;it I. &longs;it de
inde cylindrus cum circumpo&longs;ito cuneo ita accomodatus, vt ab&longs;q;
vllo
&longs;itq; LMNO, quod &longs;cindendum e&longs;t; quod etiam ex parte MN
&longs;it immobile: vt in iis, quæ &longs;cinduntur, fieri &longs;olet: & &longs;it vertex
I intra RS. circumuertatur kF, & perueniat ad kP; dum autem kF
circumuertitur, circumuertitur etiam totus cylindrus DE, & cu
neus IGH: quare dum KF erit in kP, vertex I non erit amplius
intra RS, &longs;ed cunei pars alia, vt TV: &longs;ed TV maior e&longs;t, quàm
RS; &longs;emper enim pars cunei, quæ magis à vertice di&longs;tat, maior
e&longs;t ea, quæ ip&longs;i e&longs;t propinquior: vt igitur TV &longs;it intra RS, opor
tet, vt R cedat, moueaturq; ver&longs;us X, & S ver&longs;us Z, vt faciunt
ea, quæ &longs;cinduntur. totum ergo LMNO &longs;cindetur.
&longs;imiliter
què demon&longs;trabimus, dum manubrium kP erit in kQ, tunc GH
e&longs;&longs;e intra RS: & vt GH &longs;it intra RS, nece&longs;&longs;e e&longs;t, vt R &longs;it in X,
& S in Z; ita vt
&longs;cindetur. &longs;ic igitur patet, dum kF circumuertitur, &longs;emper R moue
ri ver&longs;us X, atq; S ver&longs;us Z: & R &longs;emper &longs;uper ITG moueri, S au
tem &longs;uper IVH, hoc e&longs;t &longs;uper latera cunei circa cylindrum circum
uoluti.
PROPOSITIO I.
Cuneus hoc modo circa cylindrum accommo
datus, nihil e&longs;t aliud; ni&longs;i cochlea duas habens he
lices in vnico puncto inuicem coniunctas.
Sit cuneus ABC; & AB
ip&longs;i BC æqualis. diuidatur
AC bifariam in D, iunga
turq; BD; erit BD ip&longs;i AC
perpendicularis; & AD
ip&longs;i DC æqualis, triangu
lumq; ABD triangulo C
BD æquale. fiant deinde
triangula rectangula EFG
HIk non &longs;olum inter &longs;e,
verùm etiam vtriq; ADB
& CDB æqualia. &longs;itq; cy
lindrus LMNO, cuius perimeter &longs;it æqualis vtriq; FG kI. &
LMNO &longs;it parallelogrammum per axem. fiatq; MP æqualis
FE; & PN æqualis HI. ponaturq; HI in NP, circumuolua
turq; triangulum HIk circa cylindrum; & &longs;ecundùm kH helix
de&longs;cribatur NQP, vt Pappus quoq; docet in octauo libro propo
&longs;itione vige&longs;ima quarta. &longs;imiliter ponatur EF in MP, circum
uoluaturq; triangulum EFG circa cylindrum; de&longs;cribaturq; per
EG helix PRM. cùm itaq; PMPN &longs;int æquales EFHI, erit
MN æqualis ip&longs;i AC, & cùm helices PRM PQN &longs;int æquales
lineis EGHk; helices igitur ip&longs;is ABBC æquales erunt. cu
neus ergo ABC totus circumuolutus erit circa cylindrum LMNO.
vt docet Pappus &longs;ecundùm
latitudinem cunei; & hoc
modo cuneus vná cum cy
lindro nihil aliud erit,
quàm cochlea duas habens
helices PRMPQN cir
ca cylindrum LN in vnico
puncto P inuicem coniun
ctas. quod demon&longs;trare o
portebat.
COROLLARIVM.
Hinc manife&longs;tum e&longs;&longs;e pote&longs;t, quomodo heli
ces in ip&longs;a cochlea de&longs;cribi po&longs;sint.
Quomodo autem pondera &longs;uper helices co
chleæ moueantur, o&longs;tendamus.
Sit (veluti prius) cuneus IGH circa cylindrum DE reuolutus,
cuius vertex &longs;it I. apteturq; cylindrus ita, vt liberè vna cum &longs;uo
axe circumuertatur. &longs;intq; duo pondera MN cuiu&longs;cunq; figuræ
voluerimus, ita tamen aptata, vt moueri non po&longs;sint, ni&longs;i &longs;uper &longs;intq; MN
iuxta cunei verticem I. Circumuertatur KF, & perueniat ad kP:
dum autem kF erit in kP, tunc TV erit intra pondera MN; &longs;i
cut &longs;upra diximus. M igitur ver&longs;us L mouebitur, & N ver&longs;us O.
&longs;imiliter o&longs;tendetur, dum kP erit in KQ, tunc GH e&longs;&longs;e intra pon
dera MN; & M erit in X, & N in Z; ita vt XZ &longs;it æqualis GH. quare dum kF circumuertitur, &longs;emper pondus N mouetur ver&longs;us
O, & &longs;uper helicem IRS; M verò &longs;uper aliam helicem.
Similiter &longs;i cochlea plures habeat hæ
lices, vt in &longs;ecunda figura, pondus A,
dum cochlea circumuertitur, &longs;emper &longs;u
per helices BCDEFG mouebitur;
dummodo pondus A aptetur ita vt mo
ueri non po&longs;sit, ni&longs;i &longs;uper rectam HI ip&longs;i
cylindro æquidi&longs;tantem. eodem enim
modo, quo &longs;uper primam mouetur heli
cem, mouetur etiam &longs;upra &longs;ecundam,
& tertiam, & cætera. quotcunq; enim
fuerint helices, nihil aliud &longs;unt, quàm
latus cunei circa idem cylindrum iterum
atq; iterum circumuolutum. & &longs;iue co
chlea fuerit horizonti perpendicularis,
&longs;iue horizonti æquidi&longs;tans, vel alio mo
do collocata, nihil refert: &longs;emper enim
eadem erit ratio.
Si verò (vt in tertia figura) &longs;upra cochleam imponatur aliquod,
vt B, quod quidem tylum vocant, ita accommodatum, vt inferio
ri parte helices habeat concauas ip&longs;i cochleæ appo&longs;itè admodum
congruentes; per&longs;picuum &longs;atis e&longs;&longs;e poterit, ip&longs;um B, dum
circumuertitur, &longs;uper helices cochleæ eo pror&longs;us modo moueri;
quo pondus iuxta primam
tetur, vt docet Pappus in octauo libro; ita &longs;cilicet vt tantùm an
tè, retrouè axi cylindri æquidi&longs;tans moueatur.
Et &longs;i loco tyli, quod helices habet concauas in parte inferiori, con
&longs;tituatur, vt in quarta figura, cylindrus concauus vt D, & in eius
concaua &longs;uperficie de&longs;cribantur helices, incidanturq; ita, vt aptè
in &longs;uperficie concauia cylindri, &longs;icuti fit in conuexa) &longs;i deinde co
chlea in &longs;uis polis firmetur, &longs;cilicet in &longs;uo axe, circumuertaturq;;
patet D ad motum circumuer&longs;ionis cochleæ quemmadmodum ty
lum moueri. nec non &longs;i D in EF firmetur, ita vt immobilis ma
neat, dum circumuertitur cochlea; &longs;uper helices cylindri D, ad
motum &longs;uæ circumuer&longs;ionis dextror&longs;um, vel &longs;ini&longs;tror&longs;um factæ;
tùm in anteriorem, tùm in po&longs;teriorem partem mouebitur. cylin
drus autem D hoc modo
fæmina nuncupatur.
Si autem cochleæ (vt in quinta figura) tympanum C dentibus
obliquis dentatum apponatur, vt docet Pappus in eodem octauo li
bro; vel etiam rectis; ita tamen con&longs;tructis, vt facilè cum cochlea
conueniant: &longs;imiliter manife&longs;tum e&longs;t ad motum cochleæ circumuer
ti etiam tympanum C. eodemq; modo tympani dentes &longs;uper he
lices cochleæ moueri. & hæc dicitur cochlea infinita, quia & co
chlea, & tympanum dum circumuertuntur, &longs;emper eodem modo
&longs;e &longs;e habent.
Hæc diximus, vt manife&longs;tum &longs;it cochleam in mouendo pondere
cunei munere ab&longs;q; percu&longs;sione fungi. Illud enim remouet à loco,
vbi erat; quemadmodum cuneus remouet ea, quæ mouet, ac &longs;cindit.
omnia enim hæc à cochlea mouentur, &longs;icuti pondus A in &longs;ecun
da figura, & M in prima.
Quoniam autem duplici ratione mouentem cuneum con&longs;iderari
po&longs;&longs;e o&longs;tendimus, videlicet vt mouet vectibus, vel vt e&longs;t planum
horizonti inclinatum, dupliciter quoq; cochleam con&longs;iderabimus;
& primùm vt vectibus mouet, vt in prima figura circumuertatur
kF, & perueniat in KP; tunc, &longs;icut dictum e&longs;t, TV erit intra pon
dera MN. & &longs;icut con&longs;ideramus vectes in cuneo, eodem quoq;
modo eos con&longs;iderare po&longs;&longs;umus in cochlea hoc pacto. erit &longs;cilicet
IVH vectis, cuius fulcimentum I, & pondus in V. &longs;imiliter ITG ve
ctis, cuius fulcimentum I, & pondus in T. potentiæ verò mo
uentes GH e&longs;&longs;e deberent; &longs;ed &longs;icuti in cuneo potentia mouens
e&longs;t percu&longs;sio, quæ mouet cuneum; idcirco erit, ubi potentia mo
uet cochleam; &longs;cilicet in P manubrio kP. cochlea enim &longs;ine per
cu&longs;sione mouetur. Hæc autem con&longs;ideratio propter vectes infle
xos impropria for&longs;itan e&longs;&longs;e videbitur; Quocirca &longs;i id, quod moue
tur à cochlea, &longs;upra planum horizonti inclinatum moueri intelli
gatur; erit quidem huiu&longs;modi con&longs;ideratio (cùm ip&longs;i quoq; cuneo
conueniat) figuræ ip&longs;ius cochleæ magis conformis.
PROPOSITIO II.
Si fuerit cochlea AB helices habens æquales
CDEFG. Dico has nihil aliud e&longs;&longs;e præter pla
num horizonti inclinatum circa cylindrum re
uolutum.
Sit cochlea AB horizonti perpendicularis duas habens helices
CDEFG. exponatur HI æqualis GC, quæ bifariam diui
datur in k; erunt Hk kI non &longs;olum inter &longs;e &longs;e, verùm etiam
ip&longs;is GE EC æquales, & ip&longs;i HI ad rectos angulos ducatur LI;
& per LI intelligatur planum horizonti æquidi&longs;tans; &longs;itq; LI du
pla perimetro cylindri AB, quæ bifariam diuidatur in M; erunt
IM ML cylindri perimetro æquales. connectatur HL, & à pun
cto M ducatur MN ip&longs;i HI æquidi&longs;tans, coniungaturq; KN. quo
niam enim &longs;imilia &longs;unt inter &longs;e &longs;e triangula HILNML, cùm
NM &longs;it æquidi&longs;tans HI; erit LI ad IH, vt LM ad MN: &
permutando vt IL ad LM; ita HI ad NM. &longs;ed IL dupla e&longs;t ip&longs;ius
LM; ergo & HI dupla erit MN. &longs;ed e&longs;t etiam dupla ip&longs;ius kI,
quare kI NM inter &longs;e æquales erunt. & quoniam anguli ad MI
&longs;unt recti; erit kM parallelogrammum rectangulum, & kN æqua
lis erit IM. quare KN perimetro cylindri AB æqualis erit. pona
tur itaq; HI in GC, erit Hk in GE. circumuoluatur deinde trian
gulum HkN circa cylindrum AB, de&longs;cribet HN helicen GFE;
cùm NK perimetro cylindri &longs;it æqualis; & punctum N erit in E;
& MN in CE. & quia ML æqualis e&longs;t perimetro cylindri; cir
cumuoluatur rur&longs;us triangulum NML circa cylindrum AB, NL
de&longs;cribet helicen EDC. quare tota LH duas de&longs;cribet helices
CDEFG. patet igitur has helices cochleæ nihil aliud e&longs;&longs;e, ni
&longs;i planum horizonti inclinatum; cuius inclinatio e&longs;t angulus HLI
circa cylindrum circumuolutum, &longs;upra quod pondus mouetur.
quod demon&longs;trare oportebat.
Quomodo autem hoc ad libram reducatur
nona octaui libri eiu&longs;dem Pappi.
Po&longs;tquam vidimus quomodo pondera huiu&longs;modi moueantur
in&longs;trumento; nunc con&longs;iderandum e&longs;t, quæ nam &longs;int ea, quæ effi
ciunt, vt pondera facilè moueantur: hæc autem duo &longs;unt.
Primùm quidem, quod efficit, vt facilè pon
dus moueatur, quod etiam ad e&longs;&longs;entiam cochleæ
magis pertinere videtur; e&longs;t helix circa co
chleam. vt &longs;i circa datam cochleam AB duæ
&longs;int helices inæquales CDA EFG, &longs;itq; AC mi
nor EG. Dico idem pondus facilius &longs;uper heli
cen CDA moueri, quàm &longs;uper EFG.
Compleatur cuneus
ADCHI, hoc e&longs;t de
&longs;cribatur helix CHI
æqualis CDA, & ver
tex cunei &longs;it C. &longs;imili
ter compleatur cuneus
GFEKL, cuius ver
tex E. exponatur de
inde recta linea MN,
quæ &longs;it ip&longs;i AC æqua
lis, cui ad rectos angu
los ducatur NP, quæ &longs;it
æqualis perimetro cy
lindri AB: & conne
ctatur
per ea, quæ dicta &longs;unt,
ip&longs;i CDA æqualis.
producatur deinde M
N in O, fiatq; ON æ
qualis MN, coniunga
turq; OP; erit OPM cuneus cuneo ADCHI æqualis. &longs;imili
neus STQ æqualis cu
neo GFEkL; erit TR
ip&longs;i PN, & perime
tro cylindri æqualis; &
QR æqualis GE. cùm autem GE ma
ior &longs;it AC; erit & RQ
maior MN. &longs;ecetur
RQ in V; fiatq; RV
ip&longs;i MN æqualis, &
coniungatur TV; erit
triangulum TVR tri
angulo MPN æquale:
duæ enim TR RV
duabus PN NM &longs;unt
æquales, & anguli,
quos continent, &longs;unt
æquales, nempe recti;
angulo NPM æqualis erit. quare angulus MPN minor e&longs;t angu
lo QTR; & horum dupli, angulus &longs;cilicet MPO minor angulo
QTS. quoniam autem cuneus, qui angulum ad verticem mino
rem habet, facilius mouet, ac &longs;cindit, quàm qui habet maiorem;
cuneus ergo MPO facilius mouebit, quàm QTS. facilius igitur
pondus à cuneo ADCHI mouebitur, quàm à cuneo GFEkL.
pondus ergo &longs;uper helicen CDA facilius mouebitur, quàm &longs;uper
EFG. eodemq; modo o&longs;tendetur, quò minor erit AC, eò faci
lius pondus moueri. quod demon&longs;trare oportebat.
ALITER.
Sit data cochlea AB duas habens helices æquales CDEFG; &longs;it
deinde alius cylindrus
&longs;i CG æqualis; diuidaturq; OP in tres partes æquales OR RT
TP, & tres de&longs;cribantur helices OQRSTVP; erit vnaquæq; OR RT
TP minor CE, & EG: tertia enim pars minor e&longs;t dimidia. dico
idem pondus facilius &longs;uper helices OQRSTVP moueri, quàm &longs;u
per CDEFG. exponatur HIL triangulum orthogonium, ita vt
HI &longs;it ip&longs;i CG æqualis, & IL duplo perimetri cylindri AB æqua
lis, & per
æqualis CDEFG; & Hexponatur
&longs;imiliter
qualis, quæ etiam æqualis erit CG, & HI; &longs;itq; ZY cylindri pe
rimetro tripla, erit XY æqualis OQRSTVP. diuidatur ZY in
tres partes æquales in
lindri
per con&longs;equens ip&longs;is IM, & ML. connectatur X<35>.
& quoniam
duæ HI IL duabus XZ Z<35> &longs;unt æquales, & angulus HIL re
ctus æqualis e&longs;t angulo XZ<35> recto; erit triangulum HIL trian
gulo XZ<35> æquale; & angulus HLI angulo X<35>Z æqualis; & &longs;ed quoniam angulus X<35>Z maior e&longs;t angu
lo ac propterea
HL magis horizonti inclinat, quàm XY. quare
potentia &longs;uper
lè elicitur ex cùm
aliud &longs;int, quàm
ca cylindrum
aliud, quàm planum HL horizonti inclinatum in angulo HLI cir
ca cylindrum AB circumuolutum; facilius ergo pondus &longs;uper he
Si autem OP diuidatur in quatuor partes æquales, de&longs;cribantur
què circa
per has quatuor, quàm &longs;uper tres OQRSTVP. & quò plures
erunt helices, eò facilius pondus mouebitur. quod demon&longs;trare
oportebat.
Tempus verò huius motus facilè patet, helices enim CDEFG
&longs;unt æquales HL; helices verò OQRSTVP &longs;unt æquales
XY: &longs;ed XY maior e&longs;t HL; ideo fiat Y
tur duo pondera &longs;uper lineas LHY
tates motuum &longs;int æquales, citius pertran&longs;ibit quod mouetur &longs;uper
LH, quàm quod &longs;uper Yin eodem enim tempore erunt
in Hquare tempus eius, quod mouetur &longs;uper helices OQRS
TVP, maius erit eo, quod e&longs;t men&longs;ura eius, quod mouetur &longs;uper C
DEFG. & quò plures erunt helices, eò maius erit tempus.
cùm au
tem datæ &longs;int lineæ HI&longs;imiliter &
erit. quare & harum proportio data erit.
temporum igitur propor
tio eorum, quæ &longs;uper helices mouentur data erit.
Alterum, quod efficit, vt pondera facilè mo
ueantur, &longs;unt &longs;cytalæ, aut manubria, quibus co
chlea circumuertitur.
Sit cochlea habens helices ABCD, quæ etiam &longs;cytalas ha
beat EFGH foraminibus cochleæ impo&longs;itas. &longs;it infra helices
cylindrus MN, in quo non &longs;int inci&longs;æ helices; & circa cylindrum
funis circumuoluatur trahens pondus O, quod ad motum &longs;cytala
rum EFGH moueatur, ac &longs;i ergatæ in&longs;trumento traheretur. du
catur (per ea quæ prius dicta &longs;unt de axe in peritrochio) Lk &longs;cy
talæ æqualis, axiq; cylindri perpendicularis, eumq; &longs;ecans in I:
patet quò longior &longs;it LI, & quò breuior &longs;it Ik, pondus O facilius
moueri. e&longs;t autem animaduertendum, quòd dum cochlea mouet
pondus, &longs;i mente concipiatur, quòd loco trahendi pondus O fune,
pondus &longs;uper helices ABCD moueat; pondus quoq; in k, quod
&longs;it R, &longs;uper helices etiam facilius mouebit. e&longs;t enim LK vectis, cuius
facilius enim mouetur pon
dus vecte Lk, quàm &longs;ine vecte; quia LI &longs;emper maior e&longs;t Ik.
in L vecte Lk &longs;uper helicen Ck: vel quod idem e&longs;t, &longs;icut etiam
&longs;upra diximus, &longs;i pondus R aptetur ita, vt moueri non po&longs;sit, ni
&longs;i &longs;uper rectam PQ axi cylindri æquidi&longs;tantem; circumuertaturq;
cochlea, potentia exi&longs;tente in L; mouebitur pondus R &longs;uper he
licen CD eodem modo, ac &longs;i à vecte Lk moueretur. idem enim
e&longs;t, &longs;iue pondus manente cochlea &longs;uper helicen moueatur; &longs;iue he
lix circumuertatur, ita vt pondus &longs;uper ip&longs;am moueatur. cùm
ab eadem potentia in L moueatur. &longs;imiliter o&longs;tendetur, quò lon
gior &longs;it LI, adhuc pondus facilius &longs;emper moueri. à minori enim
potentia moueretur. quod erat propo&longs;itum.
Tempus quoq; huius motus manife&longs;tum e&longs;t, quò enim longior
e&longs;t LI, eò tempus maius erit: dummodo potentiæ motuum &longs;int
in velocitate æquales; &longs;icuti dictum e&longs;t de axe in peritrochio.
COROLLARIVM.
Ex his manife&longs;tum e&longs;t.
quò plures &longs;unt heli
ces; & quò longiores &longs;unt &longs;cytalæ, &longs;iue manu
bria, pondus ip&longs;um facilius quidem, tardius au
tem moueri.
Virtus deniq; mouentis, atq; in &longs;cytalis con
&longs;titutæ potentiæ, hinc manife&longs;ta fiet.
Sit datum A centum; &longs;it planum horizonti inclinatum CD in
angulo DCE. inueniatur ex eadem nona Pappi quanta vi pondus
A &longs;uper CD mouetur; quæ &longs;it decem. exponatur cochlea LM
helices habens GHIK &c. in angulo ECD; per ea, quæ dicta
&longs;unt, potentia decem pondus A &longs;uper helices GHIk mouebit. &longs;i
autem hac cochlea volumus pondus A mouere, & potentia mo
uens &longs;it vt duo. ducatur NP axi cochleæ perpendicularis, axem
&longs;ecans in O; fiatq; PO ad ON, vt vnum ad quinq; hoc e&longs;t duo ad Quoniam enim potentia mouens pondus A in P, ide&longs;t
&longs;uper helices e&longs;t vt decem, cui potentiæ re&longs;i&longs;tit, & æqualis e&longs;t po
tentia in N vt duo; e&longs;t enim NP vectis, cuius fulcimentum e&longs;t
O. potentia ergo vt duo in N pondus A &longs;uper helices cochleæ
mouebit. efficiantur igitur &longs;cytalæ, &longs;iue manubria, quæ v&longs;q; ad N
tum cochlea
Si igitur &longs;it cochlea QR helices habens in angulo DCE, & cir
ca ip&longs;am &longs;it eius mater S, quæ &longs;i pependerit centum, adiiciatur ST
manubrium quoddam, &longs;iue &longs;cytala; ita vt T in eadem proportio
ne di&longs;tet ab axe cylindri, vt NOP; patet potentiam vt duo in T
mouere S &longs;uper helices cochleæ. nihil enim aliud e&longs;t S, ni&longs;i pon
dus &longs;uper helices cochleæ motum. &longs;imiliter &longs;i S &longs;it immobilis, cir
cumuertaturq; cochlea manubrio, &longs;iue &longs;cytala QX in eadem pro
portione con&longs;ecta; fueritq; cochlea centum pondo (quòd qui
dem, vel ex &longs;e ip&longs;a, vel cum pondere V cochleæ appen&longs;o, vel cum
pondere Y cochleæ &longs;uper impo&longs;ito centum pependerit) manife
&longs;tum e&longs;t potentiam vt duo in X mouere cochleam QR &longs;uper he
lices intra matricem cochleæ inci&longs;as. atq; ita in aliis, quæ cochleæ
in&longs;trumento mouentur; proportionem potentiæ ad pondus inue
niemus.
COROLLARIVM.
Ex hoc manife&longs;tum e&longs;t, quomodo datum pon
dus à data potentia cochlea moueatur.
Illud quoq; præterea hoc loco ob&longs;eruandum occurrit; quò plu
res erunt matricis cochleæ helices, eò minus in pondere mouen
do cochleam pati. &longs;i enim matrix vnicam duntaxat helicen po&longs;&longs;e
derit, tunc pondus vt centrum à &longs;ola cochleæ &longs;u&longs;tinebitur helice;
&longs;i verò plures, in plures quoque, ac totidem cochleæ heli
ces ponderis grauitas di&longs;tribuetur; vt &longs;i quatuor contineat helices,
tunc quatuor vici&longs;sim cochleæ helices vniuer&longs;o ponderi &longs;u&longs;tinendo
incumbent; &longs;iquidem vnaquæquè quartam totius ponderis portio
nem &longs;u&longs;tentabit. quòd &longs;i adhuc plures contineat helices, ponderis
quoq; totius in plures, atque ideo minores portiones fiet di&longs;tri
butio.
O&longs;ten&longs;um e&longs;t igitur pondus à cochlea moueri
tamquam à cuneo percu&longs;sionis experte: loco e
nim percu&longs;sionis mouet vecte, hoc e&longs;t &longs;cytala, &longs;i
ue manubrio.
His demon&longs;tratis liquet, quomodo
dus à data potentia moueri po&longs;sit. quòd &longs;i vecte
hoc a&longs;&longs;equi volumus; po&longs;&longs;umus & dato vecte da
tum pondus data potentia mouere. quod quidem
in nullis ex aliis fieri po&longs;&longs;e ab&longs;olutè contingit: &longs;iue
&longs;it cochlea, &longs;iue axis in peritrochio, &longs;iue trochlea.
non enim datis trochleis, neq; dato axe in peri
trochio, neq; data cochlea, datum pondus à data
potentia moueri pote&longs;t, cùm potentia in his &longs;em
per &longs;it determinata: &longs;i igitur
mouere debeat, hac minor &longs;it data, nunquam pon
dus mouebit. po&longs;&longs;umus tamen dato axe, & tympa
no ab&longs;q; &longs;cytalis datum pondus data
uere; cùm &longs;cytalas con&longs;truere po&longs;simus, ita vt &longs;e
midiameter tympani dati vná cum longitudine
&longs;cytalæ ad axis &longs;emidiametrum
portionem. quod idem cochleæ contingere po
te&longs;t, &longs;cilicet datum pondus data cochlea &longs;ine ma
nubrio, vel &longs;cytala, data potentia mouere. co
gnita enim potentia, quæ pondus &longs;uper helices
moueat, po&longs;&longs;umus manubrium, &longs;iue &longs;cytalam ita
vim habeat, quam potentia pondus &longs;uper helices
mouens cùm autem hoc datis trochleis nullo mo
do fieri po&longs;sit. datum tamen pondus data poten
tia trochleis infinitis modis mouere po&longs;&longs;umus.
datum verò pondus data potentia cunei in&longs;tru
mento mouere, hoc minimè fieri po&longs;&longs;e clarum e&longs;
&longs;e videtur; non enim data potentia datum pon
dus &longs;uper planum horizonti inclinatum mouere
pote&longs;t, neq; datum pondus à data potentia moue
bitur vectibus &longs;ibi
dum in cuneo in&longs;unt; cùm in vectibus cunei pro
pria, veraq; vectis proportio &longs;eruari non po&longs;sit.
vectium enim fulcimenta non &longs;unt immobilia,
cùm totus cuneus moueatur.
Poterit deinde quis &longs;truere machinas, atq; eas
ex pluribus componere; vt ex trochleis, & &longs;uc
culis, vel ergatis, pluribu&longs;uè dentatis tympanis,
uel quocunq; alio modo; & ex ijs, quæ diximus; fa
cilè inter pondus, & potentiam proportionem
inuenire.
FINIS.
Locorum aliquot, quæ inter imprimendum deprauata
&longs;unt, emendatior lectio.
¶ 15,
9, Lemma in
¶ 123, Monteregio
¶ 127, ex Cor.
REGISTRVM.
<12><12><12> ABCDEFGHIKLMNOPQRSTVX
YZ, Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk.
Omnes duerni.
PISAVRI
Apud Hieronymum Concordiam.
M. D. LXXVII.