CASATI
MECHANICA.
CASATI
PLACENTINI
SOCIET. JESU
MECHANICORUM
LIBRI OCTO,
IN QUIBUS UNO EODEMQUE
demon&longs;trantur,
proponitur.
Apud ANISSONIOS, JOAN, POSUEL
& CLAUDIUM RIGAUD.
D C. LXXXIV.
CUM PRIVILEGIO REGIS.
GALLIARUM
ET NAVARRÆ REGI
LUDOVICO
MAGNO.
REX INVICTISSIME,
Mechanicis lucubrationem,
ignotus homo, vix forta&longs;&longs;e cre
dibili confidentiâ, &longs;i&longs;to: Sed
quâ Regiâ comitate omnium
animos concilias, eâdem &longs;u&longs;tentor, ne repul&longs;am
timeam. In Te Orbis univer&longs;i conjecti &longs;unt oculi,
quos Tuæ Gloriæ &longs;plendor allicit: à communi feli-Amplißima
Tua in Societatem no&longs;tram merita, quorum nullam
partem, ne cogitandâ quidem gratiâ, con&longs;equi
po&longs;&longs;umus, hoc &longs;altem officij ab univer&longs;o Ordine re
petunt, ut &longs;inguli, quem cordi penitißimè impre&longs;&longs;um
ge&longs;tamus non ingrati LVDOVICVM, in
libris palàm in&longs;criptum velimus. Me verò Natu
ræ atque Artis mutuam &longs;ocietatem coëuntium in
Machinis, ferè dixerim, miracula contemplari
a&longs;&longs;uetum rapuere ad mirabundum, quæ ip&longs;e patra&longs;ti,
& bello, & pace, egregia atque præclara facinora
non modò mirabilia, &longs;ed prodigiis &longs;imilia. Neque
illa quidem aut ex rerum magnitudine ac difficul
tate, aut ex multiplicato numero, aut ex di&longs;simi
lium varietate, aut ex &longs;erie non interruptâ, me
tienda duxi, quamquam & in his admirabilitatis
plurimum in&longs;it: Verùm longè omnem admirationem
multúmque &longs;uperare mihi videtur, quòd paucis
lu&longs;tris vel &longs;æcula complexus, unus pluribus Regibus
par, tot, tantáque perficere valui&longs;ti. Ingentis pon
deris gravitatem vincit adhibita Machina, &longs;ed
diuturno impul&longs;u agitanda, ut proficiat aliquid: At
plurima immen&longs;is munita difficultatibus exiguo tem
poris &longs;patio expugnare, atque ad optatum exitum
perducere, ita Tuum e&longs;t, REX INVICTISSIME,
ut quemadmodum rerum ge&longs;tarum gloriâ, ac nomi
nis celebritate, nemini &longs;uperiorum Regum &longs;ecundus
&longs;i&longs;tat ve&longs;tigiis, ventura &longs;æcula &longs;perare vix audeant.
Patere igitur pro &longs;ummâ, quâ præditus es, huma
nitate, qualemcumque hanc rerum Mechanicarum
tractationem Regio in&longs;igniri Nomine, ut, quos
meas ha&longs;ce commentationes legere non piguerit,
vel hinc di&longs;cant, aliud e&longs;&longs;e non imitabile genus
Facultatis, quâ ingentia citò perficiantur, &longs;i
LVDOVICI MAGNI mens acce&longs;&longs;erit.
Incolumem Te diu &longs;ervet DEVS Catholicæ Fi
dei incremento, Regníque Tui felicitati; audiát
que bonorum omnium Largitor vota, quæ pro Ma
je&longs;tate Tuâ &longs;upplex nuncupat
Parmæ Kal, Maij 1683.
Humillimus atque Ob&longs;equenti&longs;&longs;imus
Servus
PAULUS CASATUS è SOC. JESU.
in Provincia Veneta.
EGo Octavius Rubeus Societatis Je&longs;u in Provincia Veneta
Præpo&longs;itus Provincialis, pote&longs;tate ad id mihi factâ ab
Adm. R. P. N.
Præpo&longs;ito Generali Jo.
Paulo Oliva, faculta
tem facio, ut Opus in&longs;criptum,
Authore P. Paulo Ca&longs;ato Societatis No&longs;træ Sacerdote,
Societatis Doctorum hominum judicio approbatum, typis
mandetur, &longs;i ita iis, ad quos pertinet, videbitur. Cujus rei
gratiâ has litteras meâ manu &longs;ub&longs;criptas, & &longs;igillo officij mei
munitas dedi. Parmæ 23. Februarij 1681.
OCTAVIUS RUBEUS.
LUDOVICUS MAGNUS Galliarum & Navarræ Rex Chri&longs;tiani&longs;&longs;imus,
Diplomate &longs;uo &longs;anxit, nequis per univer&longs;os Regnorum &longs;uorum fines
intra decem proximos annos à die publicationis exemplarium computandos,
imprimat &longs;eu typis excudendum curet & venale habeat Opus quod in&longs;cribi
tur, Ie&longs;u
Ani&longs;&longs;onios Bibliopolas Lugdunen&longs;es, aut illos quibus ip&longs;imet conce&longs;&longs;erint.
Prohibuit in&longs;uper eadem auctoritate Regia omnibus &longs;uis &longs;ubditis, idem
Opus extra Regni &longs;ui limites imprimendum curare, & impre&longs;&longs;um divende
re, vel quempiam ubicumque fuerit ad id agendum impellere; ac in&longs;tigare
&longs;ine con&longs;en&longs;u dictorum ANISSONIORUM; Qui &longs;ecus faxit, confi&longs;ca
tione librorum, aliaque gravi pœnâ multabitur, uti latius patet in diplo
mate regio. Dabatur Ver&longs;alis die vige&longs;ima prima Januarij anno Dom. 1684.
JUNQUIERES.
MECHA
SERO in lucem prodit hæc Me
chanicorum tractatio, & vix fide
me abduco, quam dedi, cùm Di&longs;
&longs;ertationes de
qua&longs;i Prodromum emi&longs;i ante plures
annos: &longs;cilicet à &longs;tudiis tunc ab&longs;tra
ctus, utpote alieni juris, & ad mu
nera his non affinia tran&longs;latus, mul
tam &longs;alutem & Mathematicis di&longs;ciplinis & Phy&longs;icis dicere
coactus &longs;um; adeò ut demum tot elap&longs;is annis urgente jam
&longs;enio cogitationem omnem abjecerim de huju&longs;modi com
mentationibus, diffidens me po&longs;&longs;e ad hanc &longs;criptionem
&longs;atis temporis invenire, quin eam proxima mors interci
peret, & &longs;u&longs;ceptum alieni&longs;&longs;imo tempore laborem irritum
faceret. Adde quòd (pro meâ negligentiâ, quæ calamo
parcit) temporis diuturnitate deletæ ex animo pleræque
imagines vix tenue ve&longs;tigium reliquerant, cui novis indu
ctis coloribus eas redintegrari po&longs;&longs;e confiderem. Amico
rum tamen officio&longs;is &longs;timulis me urgeri pa&longs;&longs;us &longs;um, ut &longs;ub
ci&longs;ivis, quæ incurrebant, temporibus tentarem, an de&longs;ti
natam animo tractationem, cujus brevem Synop&longs;im au
ditoribus meis in Romano Collegio, anno labentis &longs;æculi
decimi &longs;eptimi quinquage&longs;imo quarto, tradideram, re
dordiri, & aliquâ ratione perficere liceret. Licuit autem,
præter &longs;pem, toties dimi&longs;&longs;um calamum re&longs;umere, ut tan-
&longs;i&longs;&longs;e invenerim, quod Mathematicarum di&longs;ciplinarum can
didatis profuturum amici cen&longs;uerunt, &longs;i publici juris fieret.
Quapropter alienæ utilitati &longs;erviendum potiùs fuit, quàm
meæ voluntati.
Verùm nete moveat, Amice Lector, quòd Mechanici
in&longs;cribantur libri, cùm tamen aliqua ad Centrobaryca, ali
qua ad Statica pertineant. Cùm enim hæc ad pleniorem
eorum intelligentiam, quæ de Machinis di&longs;putanda erant,
referantur, nomen à &longs;copo de&longs;umendum fuit: Nec decrat
ex Ari&longs;totele (&longs;i tamen ip&longs;i tribuenda &longs;it illa tractatio) &longs;uf
fragium, qui Mechanicas Quæ&longs;tiones in&longs;crip&longs;it libellum,
in quo non de &longs;olis Mechanicis facultatibus agitur.
Methodum ne culpes, quòd non in Theoremata &
Propo&longs;itiones rem totam dige&longs;&longs;erim, &longs;ed in Capita di&longs;tri
buerim, & quidem aliquando longiu&longs;cula: Brevitati nimi
rum &longs;tudens non amavi codicem titulis implere, ne fortè,
ad o&longs;tendendam con&longs;equentium cum præcedentibus con
nexionem, cogerer idem &longs;æpiùs inculcare. Facilius au
tem duxi ea, quæ conjuncta &longs;unt, uno eodemque ca
pite complecti, ut ex ipsâ verborum con&longs;ecutione re
rum cognatio innote&longs;cat. Præterquam quod, &longs;i formâ
illâ Mathematicis familiari u&longs;us fui&longs;&longs;em, animum forta&longs;&longs;e
induxi&longs;&longs;es, me mihi ineptè blandiri, & qua&longs;i Geometri
cas ratiocinationes obtrudere ea, quæ &longs;atis probabili con
jecturâ &longs;tabilire conatus &longs;um. Quamvis enim non pauca
attulerim, quæ Geometricas demon&longs;trationes recipiunt,
nec mihi videar p&longs;eudographis &longs;yllogi&longs;mis deceptus; quia
tamen & apud Phy&longs;icos & apud Mathematicos agenda
erat cau&longs;a, multa fuere ad Philo&longs;ophicas rationes revocan
da; & quidem, quoad ejus fieri potuit, à receptis in &longs;cho-
temerè &longs;ine argumentis proferrem, aut ne longiùs ab in
&longs;tituto recederem, &longs;i quid novi, quæ&longs;itâ veri &longs;imilitudine,
molirer. Hoc videlicet mihi poti&longs;&longs;imum curæ fuit, ut Phy
&longs;icam admirandorum per Machinas motuum cau&longs;am in
ve&longs;tigarem: in Phy&longs;icis autem modum &longs;ciendi Geome
tricum inquirens, ne ab Ari&longs;totele redarguerer, timerem.
Quare alia Geometricè, alia Phy&longs;icè tractata æquo animo
patere.
Stylum autem quid excu&longs;em?
Non e&longs;t, fateor, con
&longs;tans & perpetuus, &longs;uíque &longs;imilis: tum quia non eadem
&longs;emper &longs;ubjecta materia e&longs;t, tum quia, prout tempus fe
rebat, animum inæqualiter affectum ad &longs;cribendum at
tuli; nec poterat æquabiliter fluere toties interci&longs;a oratio.
Unum e&longs;t inter cætera, quod forta&longs;&longs;e de&longs;ideres, nimi
rum illorum, qui de hoc eodem argumento &longs;crip&longs;erunt,
&longs;ententias explicari, & quæ à me dicuntur, eorum autho
ritate muniri. Plurimum &longs;anè mihi lucis afful&longs;i&longs;&longs;et ex do
ctorum virorum Commentariis, neque contemnenda or
namenti acce&longs;&longs;io hujus meæ lucubrationis tenuitati fieret ex
diver&longs;is Authorum opinionibus: Verùm ut nunc res&longs;e ha
bet, opportunâ librorum &longs;upellectile de&longs;titutus authorum
mentionem facere plenam non potui, jejunam non debui,
ne quis per Mihi autem
non ea e&longs;t memoriæ firmitas, quæ, quid aliquando lege
rim, aut ubi legerim, &longs;atis explicatâ recordatione &longs;uggerat.
Quòd &longs;i placui&longs;&longs;et, corrogatis aliunde libris, magnificam
hanc eruditionis pompam meæ qualicumque commenta
tioni adhibere, non &longs;atis otii ad legendum &longs;uppetebat, &
nimium temporis po&longs;tula&longs;&longs;et &longs;criptio, &longs;i exponendæ pri
mùm, dein confirmandæ aut refellendæ fui&longs;&longs;ent aliorum
pro meâ con&longs;uetudine breviter &longs;implicitérque &longs;cribere,
vix aliquando tactâ alicujus Authoris opinione, quam in
adver&longs;ariis jampridem notatam inveni.
Nec te pluribus volo, Amice Lector.
Multa habebis,
quæ pro tuâ humanitate mihi condones, plura quæ ama
nuen&longs;i, plurima forta&longs;&longs;e quæ Typographo, ubi præ&longs;ertim
de Numeris, & de Majori aut Minori Ratione &longs;ermo e&longs;t;
facilis enim contingit o&longs;citanti hallucinatio, ut ab Auto
grapho aberret exemplar, & Numerus numero, verbum
verbo commutetur: Non ægrè tamen ex adjunctis peti
poterit correctio. In iis verò, in quibus à me per impru
dentiam peccatum fuerit, à tuâ Sapientiâ facilè patiar me
dedoceri. Vale.
ELENCHUS
De Centro Gravitatis.
MACHINARUM vires, quibus innatæ corporum in
motum aut quietem
propen&longs;ioni ob&longs;i&longs;timus, explo
raturus, præterire non po&longs;&longs;um gravitatem ip&longs;am:
ne &longs;cilicet ignoretur, quid arte vincendum &longs;it. Ideò
primum hunc Librum Centro gravitatis tribuen
dum cen&longs;ui, cùm plura ex illo pendeant examinanda in po&longs;te
rioribus. Neque tamen hîc &longs;ubtili&longs;&longs;imam illam &longs;tatices partem
per&longs;equar, quæ in corporibus &longs;ingulis gravitatis centrum in
ve&longs;tigat: id enim, & abundè ab aliis præ&longs;titum, & mihi in hac
tractatione minimè nece&longs;&longs;arium; quippe cui &longs;atisfuerit cen
trum illud phy&longs;icè per&longs;pectum habere, quatenus præcaven
dum e&longs;t, ne alienâ ponderis ad machinam applicatione longè
alia fiat momentorum ratio, quàm oporteat. Ut autem Centri
gravitatis notitia clarior habeatur, non inutile ducam quæ&longs;tio
nes aliquot ad illud enucleatiùs explicandum pertinentes ad
dere, ut ip&longs;is etiam tyronibus fiat &longs;atis: quamquam enim illis
machinalis &longs;cientia carere po&longs;&longs;e alicui forta&longs;&longs;e videatur, rem
tamen penitiùs intro&longs;piciens eas extrà mechanicæ con&longs;idera
tionis fines po&longs;itas non e&longs;&longs;e cogno&longs;cet.
QUoniam hæc rerum univer&longs;itas corpora diver&longs;æ inter &longs;e
rationis complectitur, eorum ordo aliquis nece&longs;&longs;ariu, fuit
fuit, ut &longs;ingulis à natura ea tribueretur facultas, quâ & &longs;e &longs;uo
in loco, hoc e&longs;t, juxta in&longs;itam propen&longs;ionem &longs;ibi debito, con
&longs;ervare po&longs;&longs;int, & ad illum &longs;e ip&longs;a promovere, &longs;i fortè indè
dimota fuerint. Quia verò æqualia non ni&longs;i æqualiter, &longs;imili
que ratione di&longs;ponenda erant, nullum autem corpus præter
&longs;phæram habet perfectam in partium di&longs;po&longs;itione æqualitatem,
debuerunt corpora omnia orbem unum con&longs;tituere. At in
&longs;phæra punctum unum e&longs;t, à quo æqualibus radiis extremæ
&longs;uperficiei partes removentur: igitur ex ordine ad punctum
hoc, quod Centrum dicitur, comparanda &longs;unt corpora; qua
tenus cùm naturâ impellente moventur, ut in loco &longs;ibi debito,
à quo per vim &longs;ejuncta fuere, demum con&longs;i&longs;tant, vel ad cen
trum hoc accedunt, vel ab eo recedunt.
Et quidem &longs;i ad centrum accedant, gravitare dicuntur, &longs;i
verò recedant, levitare: & quæ propiora centro con&longs;i&longs;tunt,
graviora, quæ autem remotiora, leviora quoque cen&longs;entur
&longs;ecundùm &longs;peciem gravitatis, & levitatis: quicquid &longs;it quod
æqualia e&longs;&longs;e po&longs;&longs;int &longs;ecundùm gravitatem ab&longs;olutam, aut etiam
&longs;æpè contingat minus habere gravitatis ab&longs;olutæ id, quod e&longs;t
gravius &longs;ecundùm &longs;peciem. Sic libra plumbi æqualis e&longs;t libræ
aquæ, immò minor centum libris aquæ; quia tamen plum
bum infra aquam de&longs;cendens fit centro vicinius, etiam gra
vius e&longs;t &longs;ecundùm &longs;peciem. Quod &longs;i comparare velis duo cor
pora &longs;olida, quæ &longs;ibi &longs;ua duritie ita ob&longs;i&longs;tunt, ut neutrum intra
alterum moveri po&longs;&longs;it tanquam in medio; illud e&longs;&longs;e &longs;ecundùm
&longs;peciem gravius affirmabis, quod datâ paritate molis cum alio
corpore, cum quo comparatur, &longs;taterâ expen&longs;um in eodem
medio, in quo utrumque gravitat puta in aëre, plus habere
ponderis deprehendes. Sic aurum e&longs;t ferro gravius in &longs;pecie,
quia ex æqualibus molibus auri & ferri, aurea e&longs;t pondero&longs;ior.
Generatim autem loquendo ea &longs;unt in &longs;pecie graviora, quæ
&longs;unt den&longs;iora, ea verò in &longs;pecie leviora, quæ rariora: nam &
inflata ve&longs;ica ob aërem con&longs;tipatum gravior e&longs;t, quàm flaccida;
& Æolipilam candentem, aëre intus vi caloris raro, leviorem
primùm, po&longs;teà, ubi refrixerit, graviorem e&longs;&longs;e experimento
didicimus, aëre a&longs;&longs;umptam raritatem abjiciente. Cùm enim
radij à &longs;phæræ centro ad &longs;uperficiem ducti longiùs à &longs;e invi-
&longs;ub&longs;tantiæ &longs;ub minori mole, in angu&longs;tiore &longs;patio collocarentur;
ea verò, quæ &longs;ub majoribus dimen&longs;ionibus continentur, am
pliora &longs;patia occuparent, ubi radij magis di&longs;tant: ut videlicet
hac ratione æqua &longs;ub&longs;tantiæ di&longs;tributio fieret in totâ &longs;phærâ.
Hinc vides, cur idem corpus, eo ip&longs;o quod rarum fit, a&longs;cendat,
ut aqua in vaporem re&longs;oluta (ni&longs;i aliunde ad de&longs;cendendum
determinetur, ut aurum fulminans) quia materies eadem &longs;ub
majoribus dimen&longs;ionibus petit longiùs abe&longs;&longs;e à centro, ibiquè
tanti&longs;per conquie&longs;cit, dum con&longs;tipata, atque minorem in mo
lem redacta, iterum de&longs;cendat.
Quare centrum hoc, quod motus, vel quies corporum re&longs;pi
cit, dicitur
e&longs;&longs;e cum centro univer&longs;i: vel &longs;altem (ne parùm utili nos di&longs;pu
tatione torqueamus) centrum eorum, quæ in hac &longs;phærâ ele
mentari gravia, aut levia dicuntur, idem e&longs;t cum centro ter
raquei hujus globi, ut quotidiana docet experientia: quicquid
&longs;it, an pars lunaris globi, &longs;i à lunâ &longs;ejungeretur, reditura e&longs;&longs;et
ad lunam, ut ad centrum &longs;ui motus. Tam itaquè, quæ huju&longs;mo
di centro proxima &longs;unt, deor&longs;um po&longs;ita dicuntur, &longs;ur&longs;um verò,
quæ ab eo longiùs collocata &longs;unt. Hinc telluris &longs;uperficiei in
&longs;i&longs;tentes caput &longs;ur&longs;um, pedes deor&longs;um habere dicimur. Ille
verò, quamvis rectus, & pedes, & caput &longs;ur&longs;um haberet, cu
jus umbilicus huic centro univer&longs;i congrueret. Per quod pa
riter centrum &longs;i &longs;cala ducta intelligatur, duo po&longs;&longs;ent &longs;ibi non
occurrere invicem, licet alter a&longs;cenderet, alter de&longs;cenderet;
hic &longs;iquidem accederet ad centrum, ille inde recederet: per
eam verò po&longs;&longs;et uterque a&longs;cendere, & tamen licet, æquali mo
tu moverentur, &longs;emper invicem di&longs;tarent magis, quò à centro
ad oppo&longs;itas partes recederent.
INter ea, quæ planè homogenea &longs;unt, ordo e&longs;&longs;e non pote&longs;t à
naturâ in&longs;titutus: hinc &longs;i nulla e&longs;&longs;et corporum di&longs;&longs;imilitudo,
conflaretur, nulla quoque e&longs;&longs;et aut gravitas, aut levitas. Quid
enim hæc potiùs pars, nulla naturæ conditione à cæteris di&longs;cre
ta, petat abe&longs;&longs;e à centro, illa verò exigat in eo conquie&longs;cere?
verùm quia multiplici corporum genere coagmentata rerum
univer&longs;itas inconcinna e&longs;&longs;e non potuit, &longs;uum cuique locum na
tura tribuit, in quo &longs;e &longs;i&longs;teret, ut infra hæc quidem de&longs;cende
ret, &longs;uprà illa verò a&longs;cenderet, &longs;i quando &longs;ibi invicem con
tigua fierent ordine præpo&longs;tero, nec ullus e&longs;&longs;et motui obex.
Cùm itaque corpora &longs;ingula in&longs;itam habeant propen&longs;ionem
(ab Ari&longs;totele dicitur
ver&longs;o; con&longs;tat præter de&longs;cendentium gravitatem dari etiam po
&longs;itivam levitatem, quâ corpus aliquod &longs;e ip&longs;um promovet ad
&longs;uperiores partes univer&longs;i à centro magis di&longs;tantes, neque &longs;o
lùm admittendam levitatem negativam, quâ corpora minùs
gravia cen&longs;entur levia, &longs;i eorum cum gravioribus fiat compa
ratio. Nam &longs;i ea, quæ levia dicuntur, eatenus dicas a&longs;cendere,
quatenus à gravioribus in inferiorem locum de&longs;cendentibus
propelluntur; mihi æquè liberum erit tollere omnem po&longs;iti
vam gravitatem, &longs;olâ levitate admi&longs;sâ; & omnia pariter &longs;ol
vam dicendo ea gravia cen&longs;eri, quæ minùs levia &longs;unt, atque
ideò tantùm de&longs;cendere, quòd extrin&longs;ecùs à levioribus a&longs;cen
dentibus loco pul&longs;a detrudantur, non quòd ab internâ faculta
te deor&longs;um impellantur. Quod &longs;i vel gravitas de medio tollen
da &longs;it, vel levitas, &longs;atius e&longs;t levitatem relinquere; naturâ vi
delicet ad altiora &longs;emper, & perfectiora a&longs;pirante, nec adeò
contendente de infimo loco. Quare cùm per gravitatem &longs;olam
æquè ac per &longs;olam levitatem motus i&longs;ti explicentur, cætero qui
autem ingenita &longs;it unicuique corpori &longs;ui loci exigentia; utram
que admittere rationi maximè con&longs;entaneum fuerit.
Vitreum globum vacuum, qui in tubulum recurvum de&longs;i
nat, quoad fieri pote&longs;t, calefactum, ut inclu&longs;us aer rare&longs;cat,
Hermeticè claude: tum adjiciatur congruens plumbi gravitas,
quâ infra aquam deprimatur. Sit autem globus, unà cum ad
jecto plumbo, connexus cum exqui&longs;itæ libræ brachio, aut lan
ce, ejú&longs;que gravitas intrà aquam exploretur: ubi gravitas in
notuerit, adhuc &longs;ub aquâ retineatur globus, &longs;ed longiore for
cipe extremum tubuli caput occlu&longs;um frangatur: & animad-
jus incrementum indicabitur ab addito in oppo&longs;itâ lance pon
dere ad con&longs;tituendum æquilibrium. Cùm itaque idem maneat
vitrum, idémque plumbum, & nulla facta &longs;it alicujus gravita
tis acce&longs;&longs;io, illud unum &longs;upere&longs;t, quòd aör rarus intrà globum
conclu&longs;us levior, quàm idem aör, aperto tubulo, &longs;ibi re&longs;titu
tus, plus elidit gravitatis plumbi & vitri; atque moles compo
&longs;ita ex plumbo, vitro, & aëre raro, &longs;ecundùm &longs;peciem levior
e&longs;t, quàm moles ex plumbo, vitro, & aëre non raro. Aër igi
tur intra aquam ita levis e&longs;t, ut aliquid gravitatis imminuat:
Nam &longs;i globum eundem ex aquâ extractum, omni aëre exclu
&longs;o, aquâ repleveris, & iterum eodem plumbo adjecto eju&longs;dem
gravitatem intrà aquam examinaveris, illam adhuc majorem
deprehendes; quia &longs;cilicet nulla levitas aöris ade&longs;t, quæ ali
quam deterat gravitatem, &longs;ed illa &longs;olùm perire videtur, quam
infert di&longs;crimen gravitatum &longs;ecundùm &longs;peciem, ut ex Hy
dro&longs;taticis con&longs;tat. Neque &longs;u&longs;piceris hæc gravitatum incre
menta oriri ex aquâ &longs;ubeunte per apertum tubulum, cùm aër
a&longs;&longs;umptam ex calore raritatem abjicit, &longs;e in naturalem &longs;uam
molem re&longs;tituens, &longs;ivè, aëre pror&longs;us exclu&longs;o, ex aquæ globum
implentis gravitate. Si enim vitrum aliud aut nullius, aut mo
dici&longs;&longs;imæ aquæ capax, &longs;ed eju&longs;dem in aëre ponderis cum a&longs;
&longs;umpto globo, &longs;imiliter in aquâ expendas, eandem invenies
gravitatem, &longs;ive multâ, &longs;ive modicâ aquâ repletum fuerit.
Non igitur aqua intrà aquam gravitatem auget.
Sed illud, ut reliqua &longs;ileam, non leviter &longs;uadere pote&longs;t cor
pora &longs;uis nutibus non deor&longs;um tantùm, &longs;ed etiam &longs;ur&longs;um co
nari, quod mihi haud ita pridem aliud inve&longs;tiganti contigit
ob&longs;ervare. Cum enim animadverti&longs;&longs;em aliquando, quàm di&longs;
par e&longs;&longs;et gravitas aquæ dimidiam &longs;itulam implentis, &longs;i illa in &longs;u
perficie horizontali libraret &longs;e&longs;e, ac quandò &longs;uppo&longs;ita ligneo
globo firmiter cum &longs;uperiore tigillo cohærenti altiùs ad latera
a&longs;&longs;urgebat locum globo concedens, quem tamen non &longs;u&longs;tine
bat; &longs;ubiit animum cupido tentandi, an bubula ve&longs;ica inflata
tran&longs;ver&longs;is virgulis infra va&longs;is labra depre&longs;&longs;a ita, ut eam aqua
circumplecteretur, vim haberet pariter augendi momenta gra
vitatis; aquam &longs;iquidem cogebat a&longs;&longs;urgere ad altitudinem ma
jorem perpendicularem, ac quandò, ve&longs;icâ liberè innatante,
Inveni tamen nullum planè ob&longs;ervari po&longs;&longs;e in
gravitate di&longs;crimen, quamvis tam ampla e&longs;&longs;et ve&longs;ica, ut facilè
dimidiam va&longs;is capacitatem impleret: in utroque enim ca&longs;u pon
dus fuit lib. 44 1/2. Id mihi, fateor, accidit præter opinionem:
Nam &longs;i ex pariete extet tigillus, cui adnectatur
cylindrus P, aut ve&longs;ica ritè firmata, ferè im
plens capacitatem va&longs;is AB, va&longs;que illi &longs;up
ponatur ita, ut aqua deinde infu&longs;a po&longs;&longs;it libe
rè cylindro circumfundi; percipies onus lon
gè majus, quàm pro gravitate aquæ infu&longs;æ,
&longs;i permitteretur &longs;ub&longs;idere: & &longs;i vas ex &longs;taterâ
pendeat, adducto reductóve &longs;acomate appa
rebunt momenta gravitatis longè majora, quàm &longs;i tota illa
aqua fundum peteret, & cylindri pars, quæ priùs immerge
batur, ab&longs;ci&longs;&longs;a, aut ve&longs;ica innataret. Intelligebam id ex majori
altitudine perpendiculari aquæ &longs;upra eandem ba&longs;im oriri; nam
depre&longs;&longs;o va&longs;e ita, ut paulatim cylindus emergat, & aqua &longs;ub
&longs;idat, &longs;emper minuitur pondus: idem futurum &longs;perabam, &longs;i
ve&longs;ica intra aquam non ab extrin&longs;eco obice detineretur, &longs;ed à
virgulis cum va&longs;e ip&longs;o connexis; quandoquidem aqua ad ean
dem pariter altitudinem a&longs;&longs;urgebat &longs;uper ba&longs;im eandem: at
&longs;pem fefellit eventus. Nec alia mihi &longs;e obtulit probabilior ra
tio, quàm ut exi&longs;timarem aquam altiorem vehementius qui
dem deor&longs;um niti, ve&longs;icam tamen leviorem altiùs depre&longs;&longs;am,
conantem &longs;ur&longs;um, æqualiter contendere, ut emergeret; cùm
verò ni&longs;us i&longs;te &longs;ur&longs;um oppo&longs;itas virgulas, atque adeò vas cum
illis connexum urgeret, elidi adver&longs;um impetum deor&longs;um,
qui à majore altitudine perpendiculari aquæ oriebatur, & &longs;o
lum remanere conatum ex ip&longs;orum corporum &longs;ub&longs;tantiâ pro
manantem, quæ &longs;icut eadem &longs;emper erat, &longs;ivè innataret ve&longs;i
ca, &longs;ivè per vim immergeretur, ita eadem obtinebat gravita
tis momenta. Quo experimento (quamquam non me lateat,
quid pro &longs;e afferre hîc po&longs;&longs;ent aliter &longs;entientes) vi&longs;us mihi
&longs;um deprehendere non ob&longs;curum po&longs;itivæ levitatis ve&longs;ti
gium.
Ut autem levitatem corporibus adimendam a&longs;&longs;ererent in
genio&longs;i Academici, hoc poti&longs;&longs;imum ducti &longs;unt experimento.
plano horizontali D, E,
perpendicularem &longs;tatue
runt; & ut cylindri ba
&longs;is &longs;ubjecto plano exactè
congrueret, laminas duas
accurati&longs;&longs;imè lævigatas,
tùm cylindri ba&longs;i, tùm
&longs;ubjecto plano firmiter
adnexas voluerunt. Tùm
ne aër facilè inter utrum
que &longs;ubiret, erecto &longs;upra
aut cerâ aggerulo,
tumCylindrum extremo libræ jugo G, alligâ
runt, addito in oppo&longs;itâ libræ extremitate H pondere L cylin
dri pondus adæquante; quod utique cylindrum elevare non po
te&longs;t. Additum igitur e&longs;t & aliud pondus M u&longs;que eò, dum cy
lindrus à &longs;ubjecto plano avelleretur, & fuit librarum circiter
trium: quam men&longs;uram arguunt e&longs;&longs;e re&longs;i&longs;tentiæ cylindri con
tiguo plano adhærentis metu vacui. His peractis concavum
vas cylindricum NOP, æqualis aut majoris altitudinis parâ
runt, laminâ pariter perpolitâ va&longs;is fundo adnexâ, cui impo
&longs;itus fuit cylindrus, adeoque adhæ&longs;it, ut, pleno-mercurij
va&longs;e, omninò non avelleretur, ut innataret; &longs;ed tunc demum
argento vivo innatavit, cùm per vim à va&longs;is fundo avul&longs;us e&longs;t
cylindrus: cui, ut iterum fundum peteret, & argento vivo
immergeretur, imponendum fuit pondus Q librarum circiter
quinque. Vis ergò levitatis ligni in mercurio (&longs;i qua levitas
e&longs;&longs;et) æ&longs;timanda e&longs;&longs;et ut quinque, cùm vis adhæ&longs;ionis metu
vacui &longs;olùm inventa &longs;it ut tria: debui&longs;&longs;et igitur levitas ita præ
valere, ut adhæ&longs;ionem vinceret, & cylindrus &longs;ponte elevaretur.
Non e&longs;t itaque levitas, quæ ligneum cylindrum innatare cogit,
&longs;ed mercurij gravitas major ip&longs;a e&longs;t, quæ lignum elevat, cum
primùm locus patet, in quem de&longs;cendat.
Sed antequam experimentum hoc ad examen revocemus,
ut innote&longs;cat, quid hinc confici po&longs;&longs;it ad levitatem excluden
dam, haud ægrè permi&longs;erim, cùm in abeuntis &longs;uâ &longs;ponte cor-
illud urgeri po&longs;&longs;e, ut velociùs moveatur: duo &longs;cilicet corpora
diver&longs;æ &longs;ecundùm &longs;peciem gravitatis &longs;i fuerint perturbatè di&longs;
po&longs;ita intrà medium, in quo utrumque gravitat, nil mirum, &longs;i
à graviore majori ni&longs;u conante extrudatur minùs grave: id
quod etiam de duobus levibus dicendum perturbatè di&longs;po&longs;itis
in medio, ubi utrumque levitat: duobus enim &longs;imul currenti
bus, ab eo qui ponè &longs;ub&longs;equitur, &longs;i majoribus viribus polleat,
priorem urgeri atque impelli palam e&longs;t, quamquam motus uni
ver&longs;us impul&longs;ioni tribuendus non &longs;it. Ita quoque a&longs;cendentem
in mercurio ligneum cylindrum à de&longs;cendente mercurio &longs;ur
&longs;um urgeri aliquatenus po&longs;&longs;e non diffitebor, &longs;icut & mercu
rium ip&longs;um repugnare, ne &longs;ur&longs;um propellatur, atque ab eodem
lignum innatans prohiberi, ne de&longs;cendat: hinc tamen non
&longs;equitur ligni a&longs;cendentis motum, aut innatantis quietem,
prægravis mercurij viribus omnino ad&longs;cribi jure debere, nam,
& &longs;ua vis a&longs;cendendi, atque con&longs;i&longs;tendi, ligno ip&longs;i tribuen
da e&longs;t.
Quid quòd ip&longs;æ innatantis cylindri portiones, altera quidem
mercurio immer&longs;a, altera verò extans, levitatem ip&longs;i ligno in
&longs;itam declarant? Quid enim partis immer&longs;æ ad extantem (&longs;i
moles &longs;pectetur) ea ratio e&longs;t, quæ &longs;pecificæ gravitatis ligni ad
differentiam gravitatum ligni, atque mercurij? ni&longs;i quia por
tionis mercurio immer&longs;æ levitas, atque extantis in aëre gravi
tas, æquilibritatem con&longs;tituent; quemadmodum in
chinis mota differt.Hanc porrò æqua
litatem Algebricè &longs;ic o&longs;tendo. Ratio gravitatis ligni ad gravi
tatem mercurij &longs;it ut S. ad R; differentia e&longs;t R—S. Ponatur
cylindri pars immer&longs;a. A. Quia igitur ut &longs;pecifica gravitas
corporis innatantis ad differentiam gravitatum, hoc e&longs;t ut
S ad R — S, ita pars cylindri immer&longs;a A, ad extantem
(R in A—S in A/S); Si pars extans in aëre in &longs;uam gravitatem S du
catur, pars verò immer&longs;a A in differentiam gravitatum R—S,
hoc e&longs;t in — R + S, quia e&longs;t deficiens, efficitur hinc quantitas
R in A — S in A, hinc verò — R in A + S in A, quæ &longs;e invi
cem elidunt. Æqualia igitur &longs;unt levitatis, & gravitatis mo
menta. Sit enim exempli causâ gravitas ligni ad gravitatem
E&longs;t igitur cylindri
pars immer&longs;a eju&longs;dem (5/13), extans verò (8/13): at portio immer&longs;a de
ficit à gravitate mercurij &longs;ecundùm &longs;peciem ut 8; igitur (5/13) in - 8
dant (40/13): item partis extantis gravitas in aëre e&longs;t S; igitur (8/13)
in 5 dant (40/13): confligunt itaque inter &longs;e pari conatu levitas (-40/13) &
gravitas (40/13), adeóque fit con&longs;i&longs;tentia & innatat lignum.
Sed jam ad propo&longs;iti experimenti examen de&longs;cendamus.
Aio
cylindri re&longs;i&longs;tentiam ex adhæ&longs;ione metu vacui non &longs;atis explo
ratam fui&longs;&longs;e per libram; hæc enim dum ex pondere M deor&longs;um
inclinatur, extremitas G &longs;ur&longs;um elevata arcum de&longs;cribit, ac
proinde cylindri a&longs;cendentis motus non e&longs;t per lineam horizon
tali plano perpendiculariter in&longs;i&longs;tentem, &longs;ed per inclinatam:
Quare cùm A. versùs I libræ centrum trahatur, cylindri ba&longs;is
non incipit elevari parallela horizonti, &longs;ed cum inclinatione, ita
ut C priùs elevetur, quàm B: ea autem, quæ &longs;ibi invicem adhæ
re&longs;cunt, multò faciliùs divelli manife&longs;tum e&longs;t, &longs;i id cum inclina
tione fiat, quàm &longs;i &longs;ervandus &longs;it paralleli&longs;mus. Adde in hac in
clinatione faciliùs adhuc divelli cylindrum à &longs;uppo&longs;ito plano,
quò longior cylindrus fuerit; habet &longs;cilicet rationem vectis,
cujus potentia e&longs;t in A, hypomochlion in B, re&longs;i&longs;tentia vin
cenda in C. Quare pondus M non aptè metitur re&longs;i&longs;tentiam,
quæ oritur ex corporum adhære&longs;centiâ, metu vacui, &longs;ed hæc
multò major e&longs;t, &longs;i ad perpendiculum motus fieri debeat;
quemadmodum & fieri oporteret, &longs;i in va&longs;e NOP mercurij
pleno cylindrus fundo adhærens rectâ a&longs;cenderet. Quamvis
igitur pondus Q librarum quinque admitteretur men&longs;ura levi
tatis, non continuò argui pote&longs;t hujus exce&longs;&longs;us &longs;upra re&longs;i&longs;ten
tiam adhæ&longs;ionis. Quin immo affirmare au&longs;im, &longs;i libræ loco
adhibita fui&longs;&longs;et amplior trochlea, & ex funiculo ejus orbitam
diculum pependi&longs;&longs;ent, non &longs;atis futurum fui&longs;&longs;e
trium, &longs;ed multò majus adhibendum fui&longs;&longs;e, ut cylindri re&longs;i&longs;ten
tiam &longs;uperaret; fui&longs;&longs;et enim avellenda ba&longs;is &longs;ervato paralleli&longs;mo.
Quantum autem virium, ferè &longs;upra fidem, habeat vacui
horror ad corpora retinenda, &longs;atis apertè declarant gravia, quæ
&longs;u&longs;penduntur. Ego &longs;anè vidi marmoreum mortarium commu
nis magnitudinis &longs;atis vulgari artificio &longs;u&longs;pendi vitreo cyatho:
ad formandos panes recens macerata, & aquâ ita &longs;ubacta, ut
illi tenaciter cohæreret: tum vitreo calici injecta &longs;tuppa admo
to igne exar&longs;it, applicitu&longs;que calix ma&longs;&longs;æ eam attraxit, &longs;icut &
medicorum cucurbitulæ carnem attrahunt: quare accepto ca
licis vitrei pede facile fuit mortarium elevare, & &longs;u&longs;pendere.
Quod &longs;i marmoreum mortarium ex metu vacui in aëre pendu
lum hæ&longs;it, quid mirum &longs;i & ligneus cylindrus &longs;ubjecto plano
adhære&longs;cens in mercurio &longs;tetit?
Nondum itaque ex hoc experimento, aut ex &longs;imilibus, ubi
metu vacui &longs;uos motus moliri corpora non po&longs;&longs;unt, &longs;atis habe
mus argumenti, quo levitatem, &longs;olâ gravitate retentâ, expun
gamus. Huju&longs;modi e&longs;t illud, ubi in lignei va&longs;is fundo exca
vatur &longs;caphium, cui exqui&longs;itè congruat eburneus globus, qui
&longs;uperinfu&longs;o hydrargyro non a&longs;cendit. Neque enim ideò non
a&longs;cendit, quia rima nulla patet argento vivo, per quam &longs;ubiens
extrudat eburneum globum, &longs;ed quia ita &longs;ibi exqui&longs;itè con
gruunt ebur, & lignum, ut vis ip&longs;a a&longs;cendendi vincere non va
leat vim adhære&longs;centiæ. Nam & eadem vis in aere &longs;u&longs;pendit
corpora gravia, ne de&longs;cendant. Quamvis autem non totum
hemi&longs;phærium globi eburnei, &longs;ed &longs;olùm ejus maximus circu
lus congrueret excavato ligno, & cavitas ip&longs;a aëre repleretur,
non propterea tollitur vis adhære&longs;centiæ illius annularis; quia
&longs;cilicet vis a&longs;cendendi in hydrargyro tanta non e&longs;t, ut valeat
inclu&longs;um ibi aërem di&longs;trahere, &longs;icut opus e&longs;&longs;et ad incipiendum
motum citra periculum vacui, & præterea &longs;uperanda e&longs;t re
&longs;i&longs;tentia hydrargyri dividendi; corpora enim in motu divi
dunt medium, pro cujus cra&longs;&longs;itudine re&longs;i&longs;tentiam experiuntur.
Adde hemi&longs;phærium inferius in aëre tanquam in loco po&longs;itum
gravitare non minùs, quàm hemi&longs;phærium &longs;uperius levitet in
hydrargyro; proinde nil mirum, &longs;i globus non a&longs;cendat. Quod
&longs;i aëre exclu&longs;o locum illum impleveris hydrargyro, & ebur
neum globum ita foramini aptaveris, ut illi exqui&longs;itè congruat;
&longs;i in &longs;uperinfu&longs;o hydrargyro globus non a&longs;cendat, indicio e&longs;t
ita globum e&longs;&longs;e foramini infixum, ut neque valeat elevari à &longs;ub
jecto hydrargyro in &longs;caphij formam per vim excavato: neque
enim facilè mihi per&longs;uadebis &longs;pecificarum gravitatum diffe
rentiam exigere, ut hemi&longs;phærium integrum præcisè extet:
multò minùs id in hydrargyro præ&longs;tare pote&longs;t, ut vacuum
evitetur.
At, inquis, fi&longs;tulam quadricubitalem &longs;piritu vini plenam
cum globulo innatante &longs;i clau&longs;eris, & inverteris deor&longs;um,
a&longs;cendet globulus &longs;patio 200 vibrationum perpendiculi; in eâ
dem verò fi&longs;tulâ communis, & &longs;implicis aquæ plenâ a&longs;cendet
&longs;ubduplo tempore 100 vibrationum. Cur hoc?
ni&longs;i quia aqua
ut pote gravior validiùs extrudit globulum, quàm &longs;piritus vini.
Nihilominus: &longs;i gravia in levibus magis gravitant, & velociùs
de&longs;cendunt, quò major e&longs;t &longs;pecificarum gravitatum differen
tia; vici&longs;&longs;im levia in gravibus magis levitant, & velociùs
a&longs;cendunt, quò major e&longs;t &longs;ecundùm &longs;peciem levitatis differen
tia: Atqui &longs;piritus vini magis accedit ad &longs;pecificam levitatem
innatantis globuli, aqua autem magis differt; in aquâ igitur
globulus magis levitat, & velociùs a&longs;cendit, &longs;icut lapis in aëre
velociùs de&longs;cendit quàm in aqua, aut in melle.
Addis iterum.
Vitreo va&longs;culo, cui longior fi&longs;tula adhæreat,
fomitem cum filo &longs;ulphurato ope fili ferrei ingere, ut vitrum
tangat: totum imple hydrargyro, & conver&longs;o deor&longs;um o&longs;culo
de&longs;cendit hydrargyrus; atque &longs;ub&longs;i&longs;tit in altitudine cubiti, &
quadrantis: admotâ lucernâ vitrarij vitrum calefiat, ut fomes
cum filo &longs;ulphurato accendatur: fumus de&longs;cendit, nec ni&longs;i
aperto &longs;uperiore va&longs;is o&longs;culo a&longs;cendit, aëre videlicet &longs;ubeunte,
à quo extrudatur &longs;ur&longs;um. Nego fumum ab aëre &longs;ur&longs;um extru
di, &longs;ed qui gravior &longs;piritu raro mercurij in illo de&longs;cendebat,
ubi aërem tangit, ut pote levior in illo a&longs;cendit.
Non au&longs;im tamen in lapide, qui gravitatem in aquâ & aëre,
levitatem in mercurio, aut plumbo liquente obtinet, duplicem
&longs;tatuere virtutem, quarum altera &longs;ur&longs;um, altera deor&longs;um con
nitatur: Cum enim impetus motum efficiens (ut infrà con&longs;ta
bit) eju&longs;dem naturæ &longs;it, in quamcunque demum orbis plagam
dirigatur motus; &longs;atis video ab uno eodemque principio, pro
variâ contigui corporis conditione, a&longs;cen&longs;um, de&longs;censúmve
prodire po&longs;&longs;e. Quandoquidem motus, qui in eadem lineâ per
ficitur, &longs;imiles planè includit ubicationes &longs;ucce&longs;&longs;ivè acqui&longs;i
tas, &longs;ivè a&longs;cen&longs;us &longs;it, &longs;ivè de&longs;cen&longs;us, ordine tantùm in earum
adeptione, commutato. Quare cum a&longs;cen&longs;us à de&longs;cen&longs;u hoc
po&longs;t alias propè finem motûs, &longs;i fui&longs;&longs;et centro propior quàm
mercurius, eam acquirat &longs;ub initium motûs ante alias, &longs;i in
mercurij locum aër aut aqua &longs;urrogetur centro vicinior quàm
lapis: ad ordinem hunc permutandum non videtur nece&longs;&longs;aria
virtutis motricis di&longs;&longs;imilitudo; nihil quippe producitur di&longs;&longs;imi
le. Sed &longs;i quis &longs;ufficere dicat conditionum varietatem, nihil
ab&longs;onum fortè loquatur: debuit enim una virtus activa in &longs;ui
effectus productione non uni tantùm conditioni alligari, &longs;ed pro
earum varietate modum quoque operandi mutare po&longs;&longs;e, modò
præ&longs;titutos fines, quoad &longs;ub&longs;tantiam, non tran&longs;iliret.
Neque arbitror hoc tantùm &longs;en&longs;u negatam ab aliquibus levi
tatem po&longs;itivam; potui&longs;&longs;ent enim æquè negare gravitatem, ad
mi&longs;&longs;a &longs;olùm potentia motrice. Sed &longs;i vis i&longs;ta &longs;e movendi deor
&longs;um gravitas po&longs;itiva dicenda e&longs;t, cùm eadem &longs;it virtus &longs;e mo
vendi &longs;ursùm, cur levitas po&longs;itiva non fuerit? Qui enim levita
tem à gravitate &longs;ejunctam negat, non illicò levitatem expun
git: quemadmodum Angelos intelligentiâ aut voluntate dimi
nutos non a&longs;&longs;erunt ij, qui vitalium facultatum di&longs;tinctionem
non agno&longs;cunt. Nullum igitur corpus &longs;impliciter, & ab&longs;olutè
grave dicendum e&longs;t, ni&longs;i quod cæteris omnibus ita petat &longs;ube&longs;&longs;e,
ut nequeat raritatem a&longs;&longs;umere, vi cujus evadat levius corpore
&longs;imili quidem &longs;ecundùm naturam, di&longs;&longs;imilis tamen raritatis:
nullum &longs;impliciter, & ab&longs;olutè leve, ni&longs;i quod ita exigat extre
mam orbis laciniam occupare, ut nunquam con&longs;tipari po&longs;&longs;it, ac
fieri gravius proximo corpore rariore. Reliqua omnia non ni&longs;i
comparatè gravia, aut levia dici po&longs;&longs;unt: &longs;ic plumbum grave e&longs;t
in aëre, grave in aqua, at pariter leve in mercurio, leve &longs;i cum
auro conferatur.
Hinc corpus in loco &longs;ibi debito con&longs;titutum, sèque ibi con
&longs;ervans (extra tamen &longs;phæræ centrum, nec in extimâ orbis ele
mentaris &longs;uperficie) ob idip&longs;um, quia ob&longs;i&longs;tit non tantùm, ne
infra &longs;ubjectum corpus deprimatur, verùm etiam, ne in locum
&longs;uperioris attollatur, & levitare &longs;imul dicendum e&longs;t, & gravi
tare. At &longs;i in alienum locum transferatur, quia in medio levio
re ita repugnat a&longs;cen&longs;ui, ut petat de&longs;cendere, &longs;olùm gravitat;
quia verò in graviore ita depre&longs;&longs;ioni reluctatur, ut exigat ad
&longs;uperiora evadere, &longs;olùm levitat. Quod &longs;i corpora huju&longs;modi
do illa in locum non &longs;uum tran&longs;lata aut de&longs;cendere expetunt,
aut a&longs;cendere, vel re etiam ipsâ de&longs;cendunt, aut a&longs;cendunt,
non admodum repugnabo; modò conatum illum, quo &longs;e &longs;uo
tutantur in loco, gravitationem, & levitationem &longs;altem in actu
primo, aut pariter a&longs;&longs;eras, aut pariter neges.
Porrò motus omnis gravium, & levium &longs;icut in vacuo exer
ceri non pote&longs;t (ut in
in medio fit, vel tardiùs, vel citiùs, tùm pro majori vel minori
ip&longs;ius medij re&longs;i&longs;tentia ad &longs;ci&longs;&longs;ionem partium magis, vel minùs
connexarum, tùm comparatâ gravitate &longs;eu levitate mobilis
cum levitate &longs;eu gravitate medij. Hinc e&longs;t gravibus minus
re&longs;i&longs;tere leviora, magis verò, quæ minùs levia, cæteris pari
bus: &longs;ic aër minùs re&longs;i&longs;tit lapidi cadenti, quàm &longs;i idem lapis in
ciperet moveri in aquâ, quæ minùs levis e&longs;t, quàm aër.
Ex oppo&longs;ito autem levibus graviora minùs re&longs;i&longs;tunt, quæ au
tem minùs gravia, magis re&longs;i&longs;tunt: &longs;ic exhalatio ex fundo
aquæ, in vitreâ phialâ ad ignem expo&longs;itâ, per aquam a&longs;cendit
velociùs, quàm deinde extra aquam po&longs;ita a&longs;cendat in aëre,
ubi fumeam naturam induerit. Unde patet non adeò &longs;olidum
ab aliquibus ex hoc experimento &longs;umi argumentum negandi
po&longs;itivam levitatem. Quæ enim de gravibus ex comparatione
cum levibus dicuntur, ea de levibus, proportione &longs;ervatâ, di
cenda &longs;unt, &longs;i cum gravibus conferantur. Cur autem gravibus
leviora, levibus graviora minùs re&longs;i&longs;tant, ratio e&longs;t, quia mo
bile movetur in medio propter di&longs;&longs;imilitudinem; nam &longs;i corpus
contiguum e&longs;&longs;et, &longs;imile non moveretur; quando igitur major
e&longs;t di&longs;&longs;imilitudo, debet velociùs moveri, &longs;egniùs autem, & len
tiùs, quò propiùs abe&longs;t à &longs;imilitudine, donec in &longs;imili demum
quie&longs;cat.
E&longs;t itaque in corporibus gravitas, & levitas, vi cujus motus ali
quos juxta naturæ propen&longs;ionem perficiunt, ut certo denique in
loco con&longs;i&longs;tant, eju&longs;demque vi re&longs;i&longs;tunt, ne oppo&longs;itis motibus
cieantur, & à &longs;uæ quietis loco avellantur. Quamvis autem
maneat gravitas aut levitas, non
(Græcis
ne e&longs;t; hæc enim, ut infra patebit, ut plurimum ex po&longs;itione, &
&longs;itu mutatur, vel comparatè ad
QUamvis non minùs levitate, quàm gravitate prædita &longs;int
corpora, quia tamen frequentiùs gravitatem vincere co
namur, quàm levitatem; ideò illa poti&longs;&longs;imùm cadit &longs;ub con
templationem &longs;cie&ngrave;tiæ Machinalis: vix enim aliquando con
tingere poterit, ut opus &longs;it infra aquam corpus aliquod leve
per vim deprimere. Hinc factum e&longs;t, ut de &longs;olo gravitatis cen
tro &longs;ermo communiter &longs;it, levitatis autem centrum &longs;ilentio
obvolvatur: quia nimirùm quæ de gravitate de&longs;cendente ex
plicantur, ea de levitate a&longs;cendente, pro rata portione, dicta
facile intelliguntur.
Ad centrum terræ (quod & centrum gravium ac levium
dicimus) properant corpora quæcumque gravia in medio le
viore con&longs;tituta &longs;ibi redduntur, ut motus &longs;uos perficiant. Quo
niam verò natura finem propo&longs;itum per media, quæ pote&longs;t, bre
vi&longs;&longs;ima pro&longs;equitur, ambages, & diverticula fugiens; mo
ventur per lineam rectam, ut pote brevi&longs;&longs;imam, ni&longs;i externo
aliquo impedimento cogantur à rectitudine deflectere: Hæc
autem recta linea intelligi debet ex terræ centro ducta ad cor
pus ip&longs;um, quod movetur; ac proinde tùm in &longs;phæricam &longs;u
perficiem, tùm in planum Horizon
tis ad perpendiculum cadit. Sed quia
corpus, quod deor&longs;um contendit,
plures habet partes, quibus con&longs;tat,
&longs;ingulas &longs;uâ gravitate præditas, lineæ
verò à &longs;ingulis hi&longs;ce partibus exeun
tes in terræ centro concurrunt; fieri
non pote&longs;t, ut &longs;ervatâ corporis figu
râ, atque continuo partium nexu non
di&longs;&longs;oluto, per rectam &longs;uam lineam ad
centrum ductam unaquæque pars
de&longs;cendat. Si enim parallelepipe
dum AB in aëre dimittatur, ut &longs;pon-
rat, quin oppo&longs;itum extremum B à recta BC longi&longs;&longs;ime rece
dat, & contra: utramque verò extremitatem &longs;imul A & B
rectâ in centrum C tendere non po&longs;&longs;e e&longs;t manife&longs;tum: Quare
cum &longs;ibi invicem ob&longs;i&longs;tant æqualiter, ob gravitatis æqualita
tem, eas ex perpendicularibus AC, BC æqualiter &longs;ecedere
oportet ad latera, atque parallelas BE, AF de&longs;cendendo de&longs;
cribere. Eadem e&longs;t ratio de cæteris partibus æquali intervallo
&longs;ejunctis à medio D; omnes enim à &longs;uis perpendiculis rece
dunt, præter punctum medium D, cujus perpendicularis
DC parallela e&longs;t lineis à reliquis partibus in motu de&longs;criptis.
Ex omnibus itaque particulis datum grave componentibus, eæ
&longs;olùm, quæ puncto D imminent, per rectam DC in centrum
moventur; quæ tàm plano horizontis in C, quàm &longs;uperficiei
&longs;phæricæ in H perpendicularis e&longs;t; cæteræ verò parallelæ BE,
AF perpendiculares quidem in horizontem cadunt, &longs;ed &longs;phæ
ricam &longs;uperficiem obliquè &longs;ecant.
Jam verò &longs;i eju&longs;dem parallelepipedi aliud planum AO hori
zonti parallelum moveri versùs C intelligas, erit in eo &longs;imiliter
aliud punctum unicum, quod rectam DC percurrat; & intra
corporis &longs;oliditatem unica linea puncto illi imminens viâ eâdem
in centrum perget non declinans à perpendiculo: cæteræ partes,
tam quæ ad
ponè, &longs;ibi mutuò adver&longs;antes à recto in
æqualiter. Cum itaque, in priori po&longs;itione, linea puncto D
imminens, e&longs;&longs;et in communi &longs;ectione planorum, quorum alte
rum partes dextras à &longs;ini&longs;tris, alterum anteriores à po&longs;terioribus
æqualiter &longs;ecernebat; in &longs;ecundâ autem po&longs;itione linea à per
pendiculo non recedens &longs;it quoquè in duorum planorum com
muni &longs;ectione, quibus pariter corporis gravitas in æquas tribui
tur partes; unum verò ex planis &longs;ecantibus &longs;it utrique po&longs;itioni
commune; unicum e&longs;t punctum tribus planis commune, in quo
binorum planorum &longs;ectiones &longs;e invicem &longs;ecant, & &longs;it ex. gr.
punctum I; quod unicum rectâ pergit in centrum C, quemcum
que tandem &longs;itum in motu obtineat corpus datum AB, ip&longs;um
enim e&longs;t duabus illis lineis commune, quæ in &longs;ingulis po&longs;itioni
bus ad &longs;ui perpendiculi latera non recedunt: cætera illarum li
nearum puncta, mutatâ po&longs;itione corporis, lineam quoque mo
tûs mutant.
Illud itaquè punctum in quocumque corpore gravi, quod
&longs;emper in motu de&longs;cribit lineam rectà in terræ centrum
ductam, dicitur
gravitatis conjungit cum terræ centro,
&longs;ecundùm quam videlicet dirigitur motus, & dimentienda e&longs;t
corporis à centro terræ di&longs;tantia, &longs;i quatenus grave con&longs;idere
tur. Porrò punctum I centrum gravitatis dicitur, quia centri
nomen tribuitur puncto, quod e&longs;t medium: & quemadmodum
magnitudinis alicujus centrum vocatur punctum illud, quod
æquales magnitudines circun&longs;tant, &longs;i partes, quæ ex adver&longs;o
&longs;unt, accipiantur; ita in gravibus centrum gravitatis dicitur,
quod æquales gravitates, vel æqualia gravitatum momenta cir
cun&longs;tant. Quod &longs;i punctum I non haberet hinc, & hinc æqua
les gravitatum vires, ab alterutrâ parte præ&longs;tante viribus pro
pelleretur in latus extra lineam directionis, à quâ nunquam re
cedit, &longs;i liberè moveatur. Cave tamen, ne partium æqualita
tem dimetiaris linearum longitudine à céntro gravitatis exeun
tium, ita ut &longs;ingulas lineas æqualiter dividendas putes; &longs;ed to
tum corpus debet intelligi divi&longs;um bifariam à plano per cen
trum gravitatis ip&longs;ius corporis, & per centrum gravium ac le
vium tran&longs;eunte, ita ut &longs;i planum à dextrâ in &longs;ini&longs;tram ductum
&longs;ecernat partes anteriores à po&longs;terioribus, æqualia &longs;int gravita
tum momenta antè, & ponè; &longs;i aliud planum per eandem di
rectionis lineam ductum partes dextras à &longs;ini&longs;tris di&longs;tinguat pa
ria &longs;imiliter hinc & hinc gravitatum momenta relinquat.
Gravitatum, inquam, momenta, non gravitates; ne locus
pateat æquivocationi; neque enim quoties æqualia &longs;unt mo
menta, toties æquales &longs;unt gravitates hinc & hinc centrum gra
vitatis complectentes, ut patebit ex iis, quæ de æquilibrio dice
mus. Unde fit in iis tantùm corporibus, quæ partibus unius eju&longs;
demque naturæ, ac ductu perpetuo &longs;imiliter con&longs;titutis,
idem e&longs;&longs;e centrum gravitatis atque magni
tudinis; reliqua certis regulis non circum
&longs;cripta, aut ex variis naturis compo&longs;ita, in
alio puncto, molis centrum habere, in alio,
gravitatis. Si enim duo &longs;olida VT, cujus
centrum gravitatis, & magnitudinis R,
& MN, cujus centrum S, æqualia &longs;ecun-
totius molis compo&longs;itæ in I, ubi planum tran&longs;iens per VN &longs;e
cat lineam RS jungentem centra &longs;ingularum gravitatum æqua
lium, &longs;ed erit in L, ubi recta RS bifariam dividitur: planum
autem per centrum terræ, & punctum L ductum non ita &longs;ecat
hanc molem, ut &longs;int æquales hinc, & hinc gravitates, quamvis
æqualia &longs;int gravitatum inæqualium momenta, quæ ex figuræ
po&longs;itione poti&longs;&longs;imùm pendent. Quod &longs;i corporis VT gravitas
ad corporis MN gravitatem, eam haberet rationem, quam SI
ad IR, e&longs;&longs;et I gravitatis centrum molis compo&longs;itæ, quæ à plano
per terræ centrum, & punctum I ducto non in gravitates æqua
les, &longs;ed in momenta æqualia divideretur; ut in loco inferiùs ex
plicabitur.
Ob&longs;erva autem non &longs;emper centrum gravitatis e&longs;&longs;e in ip&longs;o
corpore gravi, ut patet in corporibus annularibus, aut angulos
cavos habentibus, in quibus nullum e&longs;t punctum per quod tran
&longs;euntia plana quæcunque dividant in æquas partes momenta
gravitatum: ita tamen e&longs;t extra corporis cavi &longs;oliditatem, ut &longs;it
intra ip&longs;am cavitatem punctum, ex quo &longs;i intelligatur annulus,
vel fru&longs;tum annulare &longs;u&longs;pendi, manet po&longs;itionem habens hori
zonti parallelam, cum habeat æqualia hinc, & hinc gravita
tum momenta. Quod &longs;i corpus in cavos angulos &longs;inuatum ha
beat particulam aliquam procurrentem, pote&longs;t contingere, ut
in illius particulæ extremo &longs;it totius molis centrum gravitatis:
&longs;ic brevioris alicujus bacilli extremitati alteri &longs;i duos cultros in
fixeris, ut &longs;inguli cum bacillo hinc, & hinc angulum acutum
ad ea&longs;dem partes con&longs;tituant, ita inclinari po&longs;&longs;unt, ut extremo
ungue &longs;uppo&longs;ito reliquæ bacilli extremitati tota illa moles &longs;u&longs;ti
neatur citrà periculum cadendi, cùm gravitatis centrum in illa
extremitate, intrà cavitatem, quam inclinati cultri faciunt,
æqualia habeat ex omni parte gravitatum momenta, &longs;i planum
&longs;ecans per illud tran&longs;eat.
COrpora non intelliguntur gravitare ni&longs;i in alieno loco;
quando &longs;cilicet corpus contiguum inter illa & centrum
terræ interjectum, quod medii rationem habere pote&longs;t, levius
e&longs;t; petit enim infra illud e&longs;&longs;e: ni&longs;um autem hunc deorsùm Sed quoniam ni&longs;us i&longs;te videtur idcircò
à naturâ in&longs;titutus, ut perturbatus corporum ordo re&longs;tituatur;
&longs;i ex fine ratio petenda &longs;it, &longs;atis apparet corpora gravia ćentro
terræ vicina minùs gravitare. Quemadmodum enim quotie&longs;
cunque aliquis à propo&longs;ito fine magis di&longs;tat, eò magis anxius
e&longs;t, atque &longs;olicitus de mediis ad illum a&longs;&longs;equendum nece&longs;&longs;ariis,
& animo æquiore toleratur modica, quàm multa violentia; ita
natura minorem ordinis debiti perturbationem &longs;entiens, &longs;i gra
ve parùm ab&longs;it, quàm &longs;i longè abe&longs;&longs;et, à loco, ubi juxta inge
nitam propen&longs;ionem exigit con&longs;i&longs;tere, minùs &longs;olicita e&longs;&longs;e debet
de illo re&longs;tituendo, nec adeò vehementi conatu, hoc e&longs;t gravi
tatione, illud urgere debet in locum &longs;uum.
Ad hæc omnibus aperti&longs;&longs;imè liquet eò majore naturæ impe
tu corpora deorsùm niti, quò levius e&longs;t corpus, in quo tan
quam in medio perficiendus e&longs;t motus, &longs;i dimittantur. Sic à
&longs;axo in aëre pendente manum deorsùm validiùs trahi &longs;enti
mus, quàm ab eodem aquæ immer&longs;o trahatur, & multò lan
guidiùs conatur deor&longs;um lapis in melle de&longs;cendens, quàm in
aqua; quia videlicet aqua levior e&longs;t melle, & aër levior aquâ.
Hinc e&longs;t quod, &longs;i medij partes fuerint diversâ gravitate prædi
tæ, pars centro terræ propior etiam erit gravior; atque ideò
corpus in parte medij graviore minùs gravitabit propè centrum
terræ, quàm procul. E&longs;&longs;e autem eju&longs;dem medij non commoti
partes graviores in imo, omnium ferè hominum &longs;en&longs;us e&longs;t:
quotus enim qui&longs;que e&longs;t, qui ne&longs;ciat mellis optimam partem
e&longs;&longs;e, quæ in va&longs;is fundo, vini quæ in medio, olei quæ in &longs;um
mo? id autem verum non e&longs;&longs;et, ni&longs;i liquoris eju&longs;dem partes
e&longs;&longs;ent diversâ gravitate delatæ in loca à terræ centro di&longs;pari-
quò propiùs aëris levitatem &longs;pirituum &longs;ubtilitate æmulatur,
ideò quod in &longs;ummo va&longs;e innatat, optimum e&longs;t: At vini &longs;ua
vitas in exqui&longs;itâ &longs;ui tartari &longs;ufficienti humore diluti cum &longs;pi
ritibus permi&longs;tione con&longs;i&longs;tens medium locum in va&longs;e exigit,
&longs;icut media e&longs;t illius gravitas inter vagantium &longs;pirituum levita
tem, & fæculenti tartari gravitatem: Mellis demùm dulcedo
ex &longs;ui &longs;alis, &longs;eu &longs;acchari, copiâ proveniens iis partibus poti&longs;&longs;i
mum ine&longs;t, quæ multo &longs;ale refertæ graviores quoquè &longs;unt, &
in fundo &longs;ub&longs;idunt. Nec e&longs;t iis abroganda fides, qui in alti&longs;&longs;i
mo mari adeò gravem aquam à &longs;e deprehen&longs;am alicubi te&longs;tan
tur, ut &longs;upta reliquum maris fundum ambulantes ad alti&longs;&longs;i
mam fo&longs;&longs;am venerint, in quam penetrare &longs;æpiùs irrito conatu
tentârint: his enim non ægrè fidem habeo, qui aërem in imis
vallibus cra&longs;&longs;iorem atquè graviorem, in &longs;ummis verò montibus
puriorem atque leviorem ab omnibus admitti video. Cum ita
que (&longs;i ex notis ad minùs nota progredi philo&longs;ophando liceat)
propè centrum gravium ac levium medij partes graviores &longs;int,
quam procul ab illo; minor e&longs;t gravitatio corporum, &longs;i centro
propiora fiant, ac quando longè ab illo remota detinebantur.
Hinc autem re&longs;ponderi pote&longs;t quærentibus, cur in fodinis lon
gè faciliùs crudi metalli ma&longs;&longs;a moveatur, quàm in &longs;uperficie
terræ: aër &longs;cilicet profundis illis cuniculis inclu&longs;us gravior mul
tò ac cra&longs;&longs;ior e&longs;t aëre i&longs;to, quem in&longs;piramus, atque adeò ibi
metallum minùs gravitat.
Quòd &longs;i libeat minorem hanc gravitationem experimento
deprehendere, &longs;ume vitream fi&longs;tulam &longs;upernè clau&longs;am longio
rem pedibus tribus Romanis, eam imple argento vivo, digito
que o&longs;culum accuratè claudens inverte, ac argento vivo &longs;ub
jecti va&longs;is immerge; tùm amoto digito de&longs;cendet mercurius in
fi&longs;tulâ, iterúmque a&longs;cendet, & in certâ demum altitudine per
pendiculari quie&longs;cet. Ob&longs;ervatâ igitur altitudine perpendicu
lari, quam mercurius obtinet, &longs;i in imâ valle experimentum
in&longs;tituatur, eâque comparatâ cum altitudine perpendiculari,
in qua con&longs;i&longs;tit, cùm in &longs;ummo montis alti&longs;&longs;imi vertice expe
rimentum idem &longs;umitur, animadvertes altitudinem mercurij
per vim in fi&longs;tulâ &longs;u&longs;pen&longs;i minorem e&longs;&longs;e in &longs;ummo monte, quàm
quàm in va&longs;e, e&longs;t in aëre fi&longs;tulam ambiente tanquam in loco;
in aëre autem leviori cùm magis gravitet, in minori etiam al
titudine perpendiculari con&longs;i&longs;tit. Experimentum hoc in valle,
& in monte &longs;umere mihi otium non fuit, quamvis in eo &longs;æ
piùs me exercuerim: &longs;ed de illius veritate ambigere non &longs;inunt
te&longs;tes in Galliâ luculenti&longs;&longs;imi, qui di&longs;crimen hoc in mercuri;
altitudine ob&longs;ervârunt in altioribus montibus.
Verùm, ex alio præteteà capite imminui debet gravitatio
corporum in minori à centro remotione, habitâ &longs;olùm ratione
&longs;itûs. Cùm enim totius corporis gravitatio conflata &longs;it ex &longs;in
gularum partium impetu, quo deor&longs;um nituntur, manife&longs;tum
e&longs;t &longs;ingulis partibus languidiùs deor&longs;um conantibus, totius cor
poris gravitationem e&longs;&longs;e pariter languidiorem. Quoniam verò
quicquid in motu cogitur à recto &longs;ecundùm naturam tramite
deflectere, lentiùs atque remi&longs;&longs;iùs pergit ad præ&longs;titutum mo
tûs terminum; particulæ autem corporis &longs;olidi gravis, propio
res centro factæ, magis à &longs;uo perpendiculo, &longs;ibi invicem ad
ver&longs;antes, declinant; &longs;atis con&longs;tat &longs;ingulas fractis quodammo
do viribus languentes plurimum de conatu remittere. Si enim
&longs;olidum AB fiat centro vicinius ita,
ut A &longs;it in K, & B in L, lineæ di
rectionis partium extremarum &longs;unt
KC, LC: at coguntur per lineas
KF, LE parallelas de&longs;cendere,
fiuntque anguli CKF, CLE ex
terni majores internis CAK, CBL
per 16. l. 1. magis igitur in K & L
recedunt à perpendiculo, quàm re
cederent in A & B. Quia itaque
pars in K exi&longs;tens magis impeditur
ab oppo&longs;itâ extremitate, quæ in L,
ne per KC de&longs;cendat (ni&longs;i enim
pars, quæ in L, urgeret oppo&longs;itam
tentans per LC de&longs;cendere, non cogeretur pars in K exi&longs;tens
adeò recedere à &longs;uâ directionis lineâ) minori etiam impetu
deor&longs;um fertur. E&longs;t autem eadem de reliquis partibus ratio,
gravitatis; &longs;ingulæ enim ad centrum terræ accedentes magis à
&longs;uo perpendiculo recedunt, minú&longs;que deor&longs;um gravitant. Quî
igitur fieri po&longs;&longs;it, ut debilitato &longs;ingularum particularum cona
tu, atque impetu deor&longs;um, non minuatur pariter totius cor
poris gravitatio, &longs;i fiat centro vicinius?
Illud tamen non diffiteor, quod &longs;i medij levitates, aut angu
lorum CLE, CBL inclinationes eo tantùm di&longs;crimine &longs;ecer
nantur, quod omnem &longs;en&longs;um fugiat, vel &longs;altem ex medij gra
vitate, & anguli magnitudine conjunctim &longs;umptis oriri non
po&longs;&longs;it varietas, quæ &longs;ub &longs;en&longs;um cadat; neque percipietur gra
vitationis differentia in majori vicinitate. Sed hoc non facit,
quin inter gravitationes di&longs;crimen intercedat; neque enim
continuò, &longs;i quid &longs;en&longs;um latet, id omninò non e&longs;&longs;e dicendum
e&longs;t: contingere &longs;i quidem pote&longs;t motum aliquem ita &longs;en&longs;im, &
&longs;ine &longs;en&longs;u fieri, ut non ni&longs;i elap&longs;o temporis &longs;patio demùm inno
te&longs;cat. Sic &longs;i vinum, cujus gravitas vix minor &longs;it gravitate aquæ
arte &longs;atis notâ affuderis aquæ ita, ut innatet, & &longs;upremam va
&longs;is partem occupet, aliudque vas &longs;imili vino plenum, &longs;ed paulò
altius, habeas, tum ex libra centrum motûs habente in cen
tro gravitatis jugi pendeant æqualia pondera intrà vinum
utriu&longs;que va&longs;is; fiet utique ponderum æquilibrium, & con
&longs;i&longs;tent eo in &longs;itu, quem illis dederis: at &longs;i alterum libræ extre
mum ita deprimas, ut pondus, quod ex eo pendet, ex vino ad
aquam vix graviorem tran&longs;eat, reliquo pondere intra vinum
manente; initio quidem non apparebit motus libræ &longs;e re&longs;ti
tuentis, quia pondus in vino non excedit gravitationem pon
deris æqualis in aquâ ni&longs;i eo exce&longs;&longs;u, quo gravitas aquæ &longs;upe
rat gravitatem vini; hic autem exce&longs;&longs;us cum minimus &longs;it, mo
tum quoque efficiet, quem ægrè à quiete di&longs;cernas, ni&longs;i ubi
po&longs;t aliquod tempus deprehenderis pondus altius de&longs;cendi&longs;&longs;e,
depre&longs;&longs;ius autem a&longs;cendi&longs;&longs;e. Haud &longs;ecus philo&longs;ophandum e&longs;t
de majore, aut minore corporum gravitatione, &longs;i di&longs;paribus in
tervallis à terræ centro removeantur, diutiùs enim propè cen
trum incumbere poterunt &longs;u&longs;tinenti, quàm procul: id quod
&longs;atis erit ad minorem gravitationem patefaciendam, quæ non
&longs;tatim innote&longs;cat.
Hæc autem non leviter confirmari videntur ex iis, quæ quo
tidiè ferè videmus; nam &longs;i circinus, quo circulos de&longs;cribere
&longs;olemus, cadat, &longs;emper nodus prævertit cu&longs;pides, & prior ter
ram ferit; ni&longs;i fortè nodus ad perpendiculum immineat cru
ribus: & omnia ferè corpora, quæ centrum gravitatis ex una
parte habent, &longs;i ex modicâ altitudine dimittantur, videntur
quidem cadere parallela; &longs;ed ex majori altitudine &longs;i de&longs;cen
dant, pars gravior prior terram attingit. Sit enim corpus ES,
cujus gravitatis centrum H, linea
directionis HA; &longs;i horizonti paral
lelum de&longs;cenderet, per rectas EI,
SR parallelas lineæ directionis mo
veretur; id quod in modicâ tantùm
altitudine contingere videtur, quia
nondum facta e&longs;t ea gravitationis
imminutio in extremitate S, quæ
percipi po&longs;&longs;it. Si enim E per EI
de&longs;cenderet, S verò per SR, an
gulus IEA æqualis alterno EAH
per 29. lib. 1. minor e&longs;&longs;et angulo
RSA, qui æqualis e&longs;t alterno
HAS; nam ex hypothe&longs;i minùs
di&longs;tat E, quàm S, à centro gravi
tatis H, & e&longs;t angulus EAH minor angulo HAS; pars igi
tur S magis deflecteret à &longs;uo perpendiculo SA, quàm E de
flecteret ab EA; cùm itaque S magis in latus propelleretur,
plus etiam de conatu deor&longs;um remitteret, quàm E; atque adeò
non po&longs;&longs;et æqualiter de&longs;cendere ac moveri, contra hypothe&longs;im
paralleli&longs;mi. Dicendum e&longs;t igitur non per parallelas EI, SR
fieri motum, &longs;ed intra illas paulatim partem E graviorem præ
currere: quia &longs;cilicet partes omnes extra lineam directionis
AH con&longs;titutæ dum removentur à &longs;uo perpendiculo, aliquid
amittunt de impetu, quo deor&longs;um nituntur, propiores quidem
minus, remotiores autem plus; pars &longs;i quidem G in principio
motûs de&longs;cendens parallela lineæ directionis per GM facit an
gulum AGM internum per 16.lib.1. minorem externo GMS,
qui per 29. 1. e&longs;t æqualis alterno MSR. Quia ergo AGM
amittit, quàm pars S; & quamvis initio di&longs;crimen hoc non
percipiatur, demum fit, ut additis pluribus differentiis mani
fe&longs;tè appareat partem S minùs gravitare, quia tardiùs deor
&longs;um movetur; & tandem ip&longs;a &longs;equitur partem E præcur
rentem, po&longs;tquam minori illâ gravitatione permi&longs;it parti E,
ut propiùs accederet ad lineam directionis, fieretquè quæ
dam virtualis conver&longs;io circa centrum gravitatis H, in qua
extremitas E occuparet infimum locum, S autem &longs;upre
mum. Quare cùm nos doceat experientia partem HS
æquiponderantem parti HE, &longs;i &longs;u&longs;pendantur ex H, in mo
tu tamen minùs gravitare, quàm oppo&longs;itam, ideóque fieri
illam conver&longs;ionem, ut pars E fiat inferior; neque aptior
a&longs;&longs;ignari po&longs;&longs;it ratio, quàm quæ petitur ex rece&longs;&longs;u partium
majori à &longs;uo perpendiculo: &longs;atis liquet, quantum momenti
habeat hæc declinatio à perpendiculo ad minuendam gra
vitationem. Ex majori igitur declinatione à lineâ perpen
diculari, quæ con&longs;equitur corpus con&longs;titutum non adeò
procul à centro terræ ut priùs, non ineptè arguitur minor
corporis gravitatio in eo &longs;itu, &longs;i cætera &longs;int paria: neque
enim comparo corpus, quod per motum de&longs;cendit, per&longs;e
verans in &longs;uo motu, cum corpore in loco altiori tran&longs;eun
te à quiete ad motum; nam tunc ex impetu per motum
concepto major e&longs;t gravitatio in loco inferiore, quàm in &longs;u
periore: &longs;ed tantùm corpora invicem comparo, vel pariter
quie&longs;centia, vel æquali tempore mota, illudque, quod ter
ræ vicinius e&longs;t, a&longs;&longs;ero, vel minori ni&longs;u conari à quiete in
loco alieno tran&longs;ire ad motum, vel æquali tempore, quo præ
ce&longs;&longs;it motus, minus impetus acqui&longs;ii&longs;&longs;e ac minoribus viribus
motum continuare.
Ex his quæ de gravibus hactenus di&longs;putata &longs;unt, aliquis
forta&longs;sè inferat levia à centro remotiora minùs levitare, &longs;i
cut gravia centro propiora minùs gravitant. Verùm res e&longs;t
pen&longs;iculatiùs examinanda, nec &longs;impliciter ex oppo&longs;itis gra
vium, ac levium naturis definienda, qua&longs;i ob id ip&longs;um,
quia &longs;ibi gravitas atque levitas adver&longs;antur, contraria ha
berent omnia con&longs;equentia. Et quidem quod &longs;pectat ad
&longs;ed potiùs augetur in majoribus à terræ centro intervallis;
ubi minùs à &longs;uo perpendiculo declinant partes centrum le
vitatis circun&longs;tantes, & idcirco minùs de conatu remit
tunt, quò nituntur ad &longs;upe
riora evadere. Sit namque
Globus HG, cujus centrum
levitatis M, & linea di&longs;cretio
nis OMN; cui parallelæ
&longs;unt HD & GF, quas de&longs;
cribunt a&longs;cendendo extremi
tates H & G, & motum eum
dem continuabunt, &longs;i globus
in N tran&longs;latus intelligatur.
Quando igitur globus e&longs;t in M,
extremitas H recedit à per
pendiculo OI, & cum eo
facit angulum IHT; quan
do autem e&longs;t in N, extremi
tas T a&longs;cendens per TD fa
cit cum perpendiculo OR an
gulum RTD, qui per 15.lib.1.
æqualis e&longs;t angulo HTO ad
verticem, hic autem, inter
nus cum &longs;it, per 16. 1. minor
e&longs;t externo IHT. E&longs;t ergo
RTD minor angulo IHT,
atque ideò plus habet mo
menti &longs;ur&longs;um, ubi minus à
recto &longs;ecundum naturam tra
mite deflectit.
Di&longs;crimen hoc momentorum ab angulorum inæqualitate
proveniens optimè intelligit natura, quæ ita motum perfi
cit, ut, &longs;i duo inæqualiter levia coagmentata fuerint, le
vius præcurrat. Sic &longs;i A cortex &longs;uberis coagmentetur ligno
fagino B, & intra aquam mediocriter profundam horizon
taliter collocetur &longs;olidum DC, ita per lineam directio-
levitatis, ut demum A in
loco &longs;uperiore, B autem in
inferiore con&longs;tituatur, ex
tremo D per rectam DO
a&longs;cendente: Quo in motu
natura magnum invenit
compendium. Quia enim
partes centro levitatis vi
ciniores magis levitant,
quòd linea parallela lineæ
directionis faciat minorem
angulum cum earum per
pendiculo (&longs;ic &longs;i linea di
rectionis &longs;it FL, eique pa
rallelæ NG, RX, angu
lus NGX internus per
29. 1. e&longs;t æqualis externo
RXY, at PGX externus per 16. 1. major e&longs;t interno GXF,
hoc e&longs;t VXY ad verticem, ergo PGX major e&longs;t angulo
VXY, & &longs;i uterque auferatur ex æqualibus NGX, RXY,
remanet NGP minor angulo RXV, ideoque G magis levi
tat, quam X) ex majore impedimento, quod initio motûs ha
betur ob anguli HDI magnitudinem, dum pars D minùs le
vitat, centrum levitatis per SO a&longs;cendens inclinat corpus DC,
& extremitas D in recta DO con&longs;tituitur, in qua longê ci
tiùs minuuntur impedimenta, quàm &longs;i per parallelam DI
a&longs;cenderet: vix enim a&longs;cendit in E, cum impedimenta &longs;unt
æquè diminuta, ac &longs;i a&longs;cendi&longs;&longs;et in I; quandoquidem angu
lus KEI per 29. 1. e&longs;t æqualis alterno EID, atque adeò
etiam angulo, quem in I faceret parallela DI cum perpendi
culo; e&longs;t igitur angulus KEI minor quocunque alio angulo,
qui fieret in punctis intermediis lineæ DI; &longs;ed quoniam cen
trum levitatis a&longs;cendendo acqui&longs;ivit majorem impetum, quàm
extremitas in E exi&longs;tens, per vim illam rapit extra paralle
lam EK, trahitque per lineam EO, & perpendiculum facit
angulum &longs;emper minorem cum lineâ directionis; unde fit
partem inferiorem &longs;emper faciliùs trahi, quo minùs in diver&longs;a
norem angulum facit linea motûs DO; donec demùm to
tum &longs;olidum obtineat &longs;itum perpendicularem; quod initio erat
in æquilibrio.
Cæterum, quamvis habitâ ratione &longs;itûs, levia altiora magis
levitent, &longs;ivè parallela horizonti jaceant extrema, &longs;ivè incli
nata, ratione tamen medij, quod in &longs;uperioribus e&longs;t levius,
quàm in inferioribus, minùs levitant: experientia enim o&longs;ten
dit ea lentiùs a&longs;cendere, quæ propiùs accedunt ad medij na
turam &longs;ecundùm levitatem: nam ex tribus globulis &longs;phæricis,
quorum diameter unc. 2 1/5 pedis Romani, cereus erat ponderis
drachmarum 24, faginus drachm. 22, vitraëreus drachm. 7.
in aëre expen&longs;i, &longs;ed eorum motus in aquâ ad altitudinem pe
dum 14, valdè inæqualis fuit, numeratis vibrationibus eju&longs;
dem perpendiculi; cereus &longs;iquidem a&longs;cendit lenti&longs;&longs;imè vibra
tionibus 88, faginus vibrationibus 37, vitraëreus vibrationi
bus 33: unde patet cereum, qui minimùm ab aquâ differt in
pondere (aquæ etenim molis æqualis e&longs;t drachm. 25 3/5) minùs
in eâ levitare. Sicut igitur diver&longs;a levia in eodem medio inæ
qualiter levitant, &longs;ic idem leve in medio di&longs;&longs;imili inæqualiter
levitabit pro majore aut minore levitatum di&longs;&longs;imilitudine.
Conveniunt itaque gravia, & levia, quod hæc procul à cen
tro offendentia medium levius minùs levitant, illa propè cen
trum habentia medium gravius minùs gravitant. Differunt au
tem ratione po&longs;itionis, quia, in loco remotiore à centro, per
pendicula omnia concurrunt ad angulos magis acutos, minú&longs;
que differunt à lineâ rectâ, ideo qua&longs;i collatis viribus magis
gravitant, & magis levitant; at prope centrum cum perpendi
cula magis in diver&longs;a abeant, & levia minùs levitant, & gravia
minùs gravitant. Porrò hanc &longs;imilitudinem gravitationis gra
vium, & levitationis levium in eodem loco, à me vocari di&longs;cri
men, & differentiam, quia habita ratione oppo&longs;itorum videba
tur leve remotius debere minùs levitare, &longs;icut grave propius
minùs gravitat, ne te moveat; litem de verbo non faciam.
inveniatur.
OPus mechanicum plerunque non indiget puncto illo,
quod intra corporum &longs;oliditatem latet, ac centrum gra
vitatis definivimus; &longs;ed &longs;atis e&longs;t &longs;i in extimâ corporis &longs;uperfi
cie innote&longs;cat punctum, aut linea imminens ip&longs;i gravitatis
centro, pro ratione &longs;itûs, in quo corpus grave con&longs;i&longs;tere cu
pimus. Ideo geometricum laborem inveniendi punctum illud
intimum Centrobarycæ relinquens, mechanica tantùm inqui
&longs;itione, & qua&longs;i tentans, perve&longs;tigo punctum illud, aut li
neam in corporis &longs;uperficie, cui re&longs;pondet planum per lineam
directionis ductum, & &longs;ecans corpus in certo &longs;itu con&longs;titu
tum. Et quidem &longs;i corpus &longs;phæricum fuerit ex partibus eju&longs;
dem naturæ conflatum, aut &longs;altem ex partibus heterogeneis
quidem, &longs;ed circa &longs;phæræ centrum &longs;imiliter di&longs;po&longs;itis ita, ut
intima &longs;phærula folliculis quibu&longs;dam obvolvatur; quia idem
e&longs;t molis atque gravitatis centrum, punctum quodcumque in
&longs;phærica &longs;uperficie a&longs;&longs;umatur, aptum erit; &longs;ingula enim &longs;i
milem habent po&longs;itionem. Sin autem aut &longs;phæræ &longs;egmentum,
aut &longs;phæra ex partibus heterogeneis inæqualiter di&longs;po&longs;itis fue
rit; imponatur plano horizontali accuratè levi, & maximè æqua
bili; & quod punctum tangetur à &longs;uppo&longs;ito plano, ubi motus
omnis ce&longs;&longs;averit, illud e&longs;t, quod poti&longs;&longs;imùm quæritur, ac
punctum &longs;uperius, quod huic è regione e&longs;t, erit pariter aptum
ad propo&longs;itum finem.
Quod &longs;i cylindricum fuerit oblatum corpus, aut pri&longs;ma quod
cunque continuo, & &longs;imili ductu productum; &longs;ecetur bifariam
longitudo, & punctum habebitur cylindri centro gravitatis
re&longs;pondens: pri&longs;matis autem &longs;ingula plana parallelogramma &longs;i
dividantur in æquas tum longitudinis, tum latitudinis partes,
planum per inventa puncta ductum tran&longs;ibit per centrum
gravitatis pri&longs;matis, dividet enim in partes æquales, & &longs;imi
liter po&longs;itas, unde oritur momentorum gravitatis æqualitas.
Ut &longs;i parallelepipedi BC plana ita
dividantur, ut habeant puncta me
dia I, & O, & per ea agatur pla
num, con&longs;tat æqualia e&longs;&longs;e momenta
gravitatis partium IB, & IC, cùm
nullo ex capite po&longs;&longs;it oriri momento
rum inæqualitas. At &longs;i non facies parallelogrammæ pri&longs;matis
dividendæ &longs;int, &longs;ed potius ba&longs;is, quæ &longs;æpè varia e&longs;t, & irre
gularis, tunc inveniendum e&longs;t in ea punctum, in quo &longs;ibi oc
currunt &longs;ectiones planorum &longs;ecantium datum corpus in mo
menta æqualia, illudque re&longs;pondet centro gravitatis intra &longs;o
liditatem exi&longs;tenti.
Sit autem primò ba&longs;is pri&longs;matis
trigona AHI; dividatur unum
ex lateribus ex. gr. HI bifariam
in G, planum enim tran&longs;iens per
A & G, atque bifariam &longs;ecans pa
rallelogrammum HV tran&longs;ibit per
centrum gravitatis pri&longs;matis trigo
ni. Nam &longs;i datum pri&longs;ma &longs;ecetur
pluribus planis parallelis plano
HV facientibus &longs;ectiones ML,
BO, NS, CE, & ex harum
&longs;ectionum extremis exeant alia
plana &longs;ecantia parallela plano AG;
ab&longs;cinduntur ex pri&longs;mate dato pa
rallelepipeda LF, OK &c. quæ à plano AG dividuntur in
partes GL, GM æquales ac &longs;imiliter po&longs;itas; item DO, DB, &c.
Igitur &longs;ingula in eodem plano AG habent gravitatis centrum,
ac proinde tota moles ex iis parallelepipedis compo&longs;ita in eo
dem plano habet centrum gravitatis. Quoniam verò, &longs;i adhuc
plana &longs;ecantia frequentiora &longs;int, plura fiunt parallelepipeda,
quorum omnium moles compo&longs;ita adhuc minus differt à mole
totius pri&longs;matis dati, ita ut toties multiplicari po&longs;&longs;it bi&longs;ectio,
ut demum relinquatur differentia minor quacunque minimâ
mole excogitabili; hinc fit molem compo&longs;itam ex parallelepi
pedis illis infinitis (&longs;ic loqui liceat, quia non e&longs;t certus eorum
numerus explicabilis) habere centrum gravitatis in plano AG;
pedis habere in eodem plano AG centrum &longs;uæ gravitatis,
quandoquidem non differt ab illis ni&longs;i differentiâ minore qua
cumque minimâ excogitabili. Sunt igitur partium AGH,
AGI momenta æqualia; quia &longs;i inæqualia e&longs;&longs;ent haberent
differentiam, qua po&longs;&longs;et dari minor (neque enim e&longs;&longs;et indivi
dua) hæc autem differentia &longs;i e&longs;&longs;et, alia non e&longs;&longs;et, quàm quæ
intercedit inter pri&longs;ma datum, & omnia parallelepipeda, cu
jus differentiæ inæquales partes e&longs;&longs;ent in AGH, & AGI:
igitur differentia partium AGH, AGI e&longs;&longs;et minor diffe
rentiâ pri&longs;matis, & omnium parallelepipedorum; nam e&longs;&longs;e non
pote&longs;t major, vel illi æqualis: &longs;ed jam ex hypothe&longs;i differentia
inter molem compo&longs;itam ex omnibus parallelepipedis, & pri&longs;
ma, e&longs;t minor quacumque minimâ datâ, ergo &longs;i e&longs;&longs;ent inæ
qualia momenta partium AGH, AGI haberent differen
tiam minorem, & non minorem eâdem differentiâ inter pri&longs;
ma & omnia parallelepipeda. Non &longs;unt igitur inæqualia.
Res
autem forta&longs;sè &longs;ic breviùs explicabitur; &longs;i partes AGH, AGI
non &longs;unt æquales, &longs;it AGH minor quàm AGI, differentiâ Y.
Tot autem fiant bi&longs;ectiones, ut parallelepipeda relinquant
differentiam minorem quàm Y. Quia ergo parallelepipeda
in AGI habent differentiam minorem quàm Y, à parte pri&longs;
matis AGI, illa &longs;unt majora quàm pars pri&longs;matis AGH,
quæ deficit à parte AGI differentiâ Y. Atqui parallelepepida
in AGH &longs;unt æqualia parallelepipedis in AGI, ergo etiam
parallelepipeda in AGH majora &longs;unt, quàm tota pars AGH,
quod e&longs;t manife&longs;tè fal&longs;um. Non e&longs;t igitur altera pars major,
altera minor. Porrò ex continua bi&longs;ectione laterum AC,
& CN &c. relinqui &longs;emper minorem differentiam, hoc e&longs;t &longs;e
mi&longs;&longs;em præcedentis differentiæ, con&longs;tat, quia &longs;i AC &longs;ecetur
in P, & ducantur plana parallela planis AG, & HV, dividi
tur CT bifariam in Q, & e&longs;t TP parallelepipedum ablatum
duplum pri&longs;matis trigoni CPQ, cui æquale e&longs;t pri&longs;ma APX;
adeóque duobus hi&longs;ce pri&longs;matis æquale e&longs;t ablatum parallele
pipedum TP, quod e&longs;t &longs;emi&longs;&longs;is differentiæ ATC, quæ priùs
relinquebatur: & eadem e&longs;t de cæteris ratio. Quare &longs;i ex datâ
quantitate auferatur &longs;emi&longs;&longs;is, & iterum &longs;emi&longs;&longs;is re&longs;idui, & &longs;ic
in infinitum, nece&longs;&longs;e e&longs;t aliquando eò devenire, ut re&longs;idua
ex prop. 1. lib. 10. Eucl. Ideo fieri non pote&longs;t, ut pri&longs;mate di
vi&longs;o à plano AG, altera pars excedat momenta alterius quan
titate Y, quia tot po&longs;&longs;unt ab&longs;cindi purallelepipeda, ut relin
quatur differentia illorum à pri&longs;mate minor, quàm &longs;it Y: pla
num autem AG æqualiter dividit momenta parallelepipedo
rum, igitur cum tota re&longs;idua differentia minor &longs;it quam Y,
e&longs;&longs;e omnino non pote&longs;t, ut altera pars habeat exce&longs;&longs;um quan
titati Y re&longs;pondentem &longs;i enim quantitates illæ differrent, po&longs;
&longs;et dari quantitas minor illarum differentiâ; &longs;ed non pote&longs;t hu
ju&longs;modi minor quantitas dari, nam quælibet data e&longs;t major,
igitur non differunt, &longs;ed &longs;unt æquales.
His ita con&longs;titutis facilè definitur punctum centro gravitatis
imminens in ba&longs;i pri&longs;matis: quia enim o&longs;ten&longs;um e&longs;t planum
ab angulo per medium latus oppo&longs;itum ductum tran&longs;ire per
centrum gravitatis, & dividere in momenta æqualia totum
pri&longs;ma, centrum gravitatis erit non &longs;olùm in plano AG, &longs;ed
etiam in plano IN propter eandem rationem. Punctum igi
tur D, in quo occurrunt &longs;ibi communes
&longs;ectiones planorum &longs;ecantium, & ba&longs;is, e&longs;t
punctum, quod quæritur, imminens centro
gravitatis. Punctum D autem &longs;ecare rectam
NI ita, ut ND ad DI &longs;it ut 1 ad 2, &longs;ic
o&longs;tenditur. Ducatur recta NG, quæ per 2. lib. 6. e&longs;t paral
lela ip&longs;i AI; ergo ut HG ad HI, ita NG ad AI per 4. lib. 6.
ergo NG ad AI e&longs;t ut 1 ad 2: ergo triangula NGA, AGI
&longs;unt ut 1 ad 2, per 1. lib. 6. Cum autem ut ND ad DI,
ita NDA ad DIA, & NDG ad DIG per 1. 6. erit
etiam, ex 12. lib. 5. ut ND ad DI, ita NGA ad AGI,
hoc e&longs;t 1 ad 2. Eadem ratione o&longs;tenditur GD ad DA e&longs;&longs;e,
ut 1 ad 2. Vel etiam breviùs: Quia enim NG, AI &longs;unt pa
rallelæ, triangula NDG, ADI &longs;unt &longs;imilia propter angulo
rum æqualitatem; ergo ut NG ad AI, hoc e&longs;t ut 1 ad 2,
ita GD ad DA, & ND ad DI. Quare &longs;atis erit latus unum
trianguli bifariam &longs;ecare, & ab oppo&longs;ito angulo rectam duco
re; cujus tertia pars ver&longs;us ba&longs;im divi&longs;am dabit centrum gravi
tatis trianguli.
Jam verò &longs;i ba&longs;is pri&longs;matis quadrangula fuerit parallelogram-
tran&longs;eunt omnia plana dividentia æqualiter corporis dati mo
menta, cum &longs;int partes utrinque æquales, & &longs;imiliter po&longs;itæ.
Et ob eandem rationem &longs;i ba&longs;is pri&longs;matis fuerit aliqua ex figu
ris ordinatis, &longs;eu æquilateris; centrum figuræ e&longs;t punctum im
minens centro gravitatis; planum &longs;i
per
literque po&longs;itas; atque adeò momenta hinc, & hinc &longs;unt æqualia.
At &longs;i ba&longs;is trapezia fuerit, duc utramque
diametrum EC, & BD: tum in ba&longs;i trigo
nâ BCD pri&longs;matis partialis inveniatur
punctum centro gravitatis re&longs;pondens (pun
ctum hoc deinceps, brevitatis gratiâ, dice
tur centrum gravitatis, quamvis per abu&longs;ionem) & &longs;it H; & in
oppo&longs;ita ba&longs;i trigona reliqui pri&longs;matis BDE pariter invenia
tur punctum F; & per utrumque punctum tran&longs;eat planum
FH; nam in hoc eodem plano e&longs;t centrum gravitatis totius
pri&longs;matis trapezij, quod dividitur in momenta æqualia: hoc &longs;i
quidem planum tran&longs;iens per H gravitatis momenta æqualia
habet hinc, & hinc in pri&longs;mate trigono BDC; &longs;imiliter cum
tran&longs;eat per F, habet hinc, & hinc momenta æqualia gravitatis
pri&longs;matis trigoni BED: &longs;i igitur æqualia æqualibus jungantur,
planum idem æqualiter partitur momenta gravitatis pri&longs;matis
trapezij EDCB, & in eo e&longs;t centrum gravitatis illius. Eadem
ratione in ba&longs;i trigona EBC inveniatur punctum G, & in ba&longs;i
EDC punctum S, per quæ &longs;i agatur planum GS, in eo pariter
erit
E&longs;t igitur centrum gravitatis in communi
&longs;ectione planorum FH, & GS; ac proinde
punctum I illud e&longs;t, quod quæritur. Aliter
etiam, & facillimè in ba&longs;i trapezia ABCD
invenitur centrum gravitatis: ductis enim
diametris AC, BD, altera diameter ex. gr.
AC bifariam &longs;ecetur in E, ducanturque rectæ
DE, BE; trianguli ADC centrum gravi
tatis e&longs;t in recta DE, & quidem in F, ita ut EF
&longs;it tertia pars totius ED, ut con&longs;tat ex paulò
ante demon&longs;tratis. Ducatur igitur FG pa-
tis trianguli ABC, quia per 2. lib. 6. ut EF ad FD, ita EG
ad GB; Quia ergo diameter AC &longs;ecatur in H, &longs;umatur FO
æqualis ip&longs;i GH, & e&longs;t O centrum gravitatis trapezij, e&longs;t enim
triangulum ABC ad triangulum ADC, ut FO ad OG, hoc
e&longs;t ut HG ad HF. E&longs;t autem HG ad HF ut BI ad ID pro
pter paralleli&longs;mum linearum GF, BD. Porrò con&longs;tat triangu
lum ABC ad triangulum ADC e&longs;&longs;e ut BI ad ID, nam trian
gula ABI, ADI &longs;unt ut ba&longs;es BI, DI, item BCI, DCI &longs;unt
ut eædem ba&longs;es BI, DI per 1. lib. 6; igitur, & totum triangu
lum ABC ad totum ADC e&longs;t ut BI ad DI: igitur, & trian
gulum ABC ad triangulum ADC e&longs;t ut FO ad OG.
Hinc facilis patet via ad inve&longs;ti
gandum idem punctum in ba&longs;i pri&longs;
matis pentagoni BDEAC. Pri
mùm enim ducto plano per BE, in
veniatur in ba&longs;i trigonâ BDE
punctum R, & in ba&longs;i BEAC qua
drangulâ punctum P; & ducto plano
per RP, in eo erit centrum gravi
tatis pri&longs;matis pentagoni, cum in eo
dem &longs;int centra gravitatis partium.
Deinde ducto per D & A plano, inveniatur in ba&longs;i trigona
DEA punctum L centrum gravitatis, & in ba&longs;i quadrangu
lâ ACBD punctum M centrum gravitatis: in plano pariter
ducto per ML e&longs;t centrum gravitatis totius pri&longs;matis pentago
ni, quod proinde e&longs;t in communi planorum per PR, & LM
ductorum &longs;ectione; atque adeò punctum, quod quæritur, e&longs;t O.
Eadem e&longs;t methodus in pri&longs;mate hexagono; ducto enim plano
dividente in duo pri&longs;mata, quorum alterum e&longs;t trigonum, al
terum pentagonum, inveniatur utriu&longs;que centrum gravitatis,
& per inventa puncta agatur planum. Deinde iterum alio pla
no &longs;ecetur in duo pri&longs;mata, quorum alterum pariter &longs;it trigo
num, alterum pentagonum, & per inventa &longs;ingularia gravi
tatum centra agatur planum: duo &longs;iquidem plana ducta per
centra gravitatis partium, tran&longs;eunt pariter per centrum gra
vitatis totius, quod e&longs;t in communi eorum &longs;ectione. Eademque
de reliquis pri&longs;matis e&longs;t ratio.
Sed hæc indica&longs;&longs;e &longs;ufficiat, quæ operi Mechanico &longs;atis e&longs;&longs;e
po&longs;&longs;unt in omnibus ferè pri&longs;matis: Si enim ba&longs;is non fuerit
planè rectilinea, in&longs;cripto polygono rectilineo, quod mini
mùm differat à plano ba&longs;is, quæres ejus centrum gravitatis,
methodo jam tradita; illoque u&longs;urpato tanquam vero dati pri&longs;
matis centro quæ&longs;ito, minimùm aberrabis; aliquando tamen
aberrabis, aliquando continget, ut inventum cum quæ&longs;ito
conveniat. Quod &longs;i accuratiori inve&longs;tigatione opus fuerit:
quemadmodum in cæteris corporibus, quæ continuum ductum
non habent, &longs;ed inæquali cra&longs;&longs;itudine cre&longs;cunt, aut decre&longs;
cunt, ut in obeli&longs;cis, aut pyramidibus truncatis, reliqui&longs;què
planè inordinatis molibus; tunc ad geometricam Centrobary
ces methodum confugiendum e&longs;t; quam hic ego non per&longs;e
quor. Praxes igitur aliquæ proponendæ &longs;unt, quibus centrum
gravitatis phy&longs;icè per&longs;pectum habere po&longs;&longs;imus in corporibus,
quorum frequentior, vulgari&longs;que u&longs;us e&longs;&longs;e pote&longs;t.
Prima praxis &longs;it ad inveniendum gra
vitatis centrum in cingulis, quæ laminis
quoque communis e&longs;&longs;e pote&longs;t. Sit datum
cingulum AH, quod primùm &longs;u&longs;penda
tur ex H, & inde pendens perpendicu
lum &longs;ecet oppo&longs;itum latus IA in C; note
tur igitur punctum C. Deinde iterum
&longs;u&longs;pendatur ex R, & perpendiculum ca
dat in punctum F, quod notetur. His cognitis ducatur filum
ex R in F, ibique intentum alligetur; aliud filum &longs;imiliter
ex H in C ducatur, & &longs;ecans in S filum RF, dabit punctum S
quæ&longs;itum centrum gravitatis: ex quo &longs;i &longs;u&longs;penderetur datum
cingulum, maneret horizonti parallelum. Quod &longs;i e&longs;&longs;et corpus
talis figuræ, ut &longs;patium non clauderet, &longs;ed haberet angulum
cavum, aut e&longs;&longs;et fru&longs;tum annulare, eadem e&longs;t methodus factâ
&longs;u&longs;pen&longs;ione illius ex duobus punctis, ex quibus perpendiculum
cadere po&longs;&longs;it intrà corporis &longs;uperficiem; in qua &longs;i notentur
puncta, per quæ tran&longs;it, & ducantur fila, ut priùs, eorum com
munis &longs;ectio dabit quæ&longs;itum centrum gravitatis. Hinc &longs;i vel la
mina e&longs;&longs;et perforanda, ut axi infigeretur, vel cingulum e&longs;&longs;et
axi imponendum, in utrâque &longs;uperficie oppo&longs;ita quærere opor
teret punctum S, ut axis per centrum gravitatis tran&longs;iret, eique
da e&longs;&longs;et ratio tran&longs;ver&longs;ariorum, per quæ axis infigendus e&longs;&longs;et,
ea enim po&longs;&longs;unt centrum gravitatis compo&longs;itæ in alio puncto
con&longs;tituere.
Secunda praxis laminis poti&longs;&longs;imùm accommodata, in quibus
punctum medium &longs;atis accuratè inquiritur, ut &longs;i lamina metal
lica e&longs;&longs;et in calicem excavanda, hæc e&longs;&longs;e pote&longs;t. Impone lami
nam acutæ cu&longs;pidi cultri, aut &longs;tyli, eamque ultrò citróque
tanti&longs;per move, dum con&longs;i&longs;tat citrà periculum cadendi:
punctum enim, quod à cultri aut &longs;tyli cu&longs;pide notatur, cen
trum e&longs;t quæ&longs;itum.
Tertia praxis &longs;it iis corporibus conveniens, quæ præ&longs;tant
longitudine, qualia &longs;unt p&longs;eudocylindrica, conica, pyrami
des &c. quæ &longs;i non prædita &longs;int multâ gravitate, imponantur
funiculo brevi horizontaliter exten&longs;o, at &longs;i graviora fuerint, vel
cylindrulo vel aciei pri&longs;matis trigoni imponantur, & u&longs;que
dum in æquilibrio con&longs;i&longs;tant, promoveantur: ubi enim quie
verit corpus impo&longs;itum, ex loco contactûs innote&longs;cet vel
punctum, &longs;i in puncto &longs;e contingant, vel linea, &longs;i in lineâ, per
quam &longs;i ducatur planum à centro terræ, di&longs;tinguetur impo&longs;i
tum corpus in momenta gravitatis æqualia. Inventâ autem hu
ju&longs;modi lineâ facilè prodet &longs;e quæ&longs;itum punctum.
Quarta praxis non multùm di&longs;tat à &longs;uperiore: &longs;i nimirum
oblatum corpus impo&longs;ueris plano alicui horizontali, quod ta
men à pavimento ab&longs;it mediocri aliquo intervallo, habeat au
tem extremum marginem exactè rectum: extra &longs;uppo&longs;iti pla
ni marginem illud paulatim promove, donec eò venerit, ut &longs;i
vel minimum ulteriùs promoveretur, &longs;ponte caderet; ibíque
&longs;ecundùm rectitudinem marginis plani duc &longs;tylo lineam in cor
pore impo&longs;ito. Deinde &longs;uperficie eâdem planum tangente, &longs;i
corpus, præter longitudinem, non modicam præterea habeat
latitudinem, convertatur aliquantulum, & &longs;imili methodo in
venietur linea alia &longs;ecans priorem in puncto quæ&longs;ito, quod &longs;ci
licet re&longs;pondet centro gravitatis intra corporis &longs;oliditatem de
lite&longs;centi.
Hæc &longs;unt quæ Mechanices in&longs;tituto &longs;ufficere po&longs;&longs;int ad cen
trum gravitatis inveniendum; in molibus enim majoribus, quæ
plerumque vix differunt à pri&longs;matis, non indigemus commu-
Illud re&longs;tat, ut earum, quas at
tuli praxes, ratio, & cau&longs;æ explicentur, ex quibus clarion ha
beatur notitia eorum, quæ ad centrum gravitatis pertinent.
UT palam fiat praxibus capite &longs;uperiore allatis inveniri
punctum re&longs;pondens centro gravitatis, quod inquiritur,
indicandi &longs;unt fontes, ex quibus illæ deducuntur. Earum ita
que ratio petenda e&longs;t ex gravium naturâ, quæ extra locum &longs;ibi
debitum con&longs;tituta, in medio videlicet leviore, conantur de
or&longs;um pro viribus, ni&longs;i impediantur: quod &longs;i interpellentur
quidem, non tamen pror&longs;us de&longs;cen&longs;u prohibeantur, de&longs;cen
dunt, prout fert ob&longs;tantium impedimentorum conditio. Sic
lapis &longs;phæricus in montis clivo po&longs;itus cùm non valeat rectâ;
&longs;icut in aëre libero, deor&longs;um ferri, per planum illud inclina
tum de&longs;cendit: Sic plumbum, quod filo adnectitur laqueari, à
perpendiculo remotum de&longs;cendit circulariter. Porrò quæ de
toto ip&longs;o corpore vera e&longs;&longs;e intelligimus, ejus quoque partibus
&longs;ingulis conveniunt; cùm enim &longs;ingulæ &longs;uam habeant gravita
tem, ni&longs;i quid ob&longs;tet, de&longs;cendunt. Jam verò &longs;i contingat ita
corpus grave oppo&longs;ito extrin&longs;ecùs obice impediri, ut cunctæ
&longs;imul partes, qua&longs;i moles unà de&longs;cendere nequeant; &longs;ublato
partium nexu de&longs;cendunt, quæcunque carent impedimento:
ut &longs;i ceream candelam, aut glaciem, quam manu &longs;u&longs;tines, igni
admoveas; haud dubium, quin partes extremæ igni proximæ
lique&longs;centes, &longs;olutâ unione cum cæteris, &longs;uis nutibus deor&longs;um
latæ liberè de&longs;cendant. At &longs;i partes omnes colligatæ invicem
permaneant, eandemque figuram &longs;ervent; corpore illo &longs;u&longs;pen
&longs;o aut &longs;u&longs;tentato, fieri non pote&longs;t, ut partes aliquæ de&longs;cendant,
quin aliæ, ouæ è regione &longs;unt trans &longs;u&longs;pen&longs;ionis, aut &longs;u&longs;tenta
tionis punctum, a&longs;cendant; id autem harum gravitati re
pugnat: non igitur a&longs;cendere po&longs;&longs;unt, ni&longs;i de&longs;cendentes op
po&longs;itæ viribus ac momentis præ&longs;tent ita, ut harum gravitati
Quare &longs;i fiat corporis &longs;u&longs;pen&longs;i, aut &longs;u&longs;ten
tati con&longs;i&longs;tentia, argumentum e&longs;t æqualitatis momentorum
punctum &longs;u&longs;pen&longs;ionis, aut &longs;u&longs;tentationis hinc, & hinc u&longs;que
quaque circun&longs;tantium; &longs;i qua enim e&longs;&longs;et inæqualitas, alterutra
pars præponderaret, & ad motum incitaretur.
Sit corpus grave AB, cujus
centrum gravitatis H, linea di
rectionis HT in centrum uni
ver&longs;i producta. Si &longs;u&longs;pendatur
ex puncto C, quod e&longs;t in eadem
lineâ directionis, nece&longs;&longs;ariò con
&longs;i&longs;tit corpus horizonti paralle
lum, quia rectâ de&longs;cendere non
pote&longs;t per HT, cum in C reti
neatur; neque alterutra pars pote&longs;t de&longs;cendere, quia momen
ta partis HB, quibus deor&longs;um nititur, æqualia &longs;unt momen
tis, quibus pars HA re&longs;i&longs;tit, no elevetur; & vici&longs;&longs;im viribus
gravitatis- AH cætero qui de&longs;cen&longs;uræ reluctatur gravitas HB
pari ni&longs;u repugnans, ne attollatur; totum ergo con&longs;i&longs;tit. At &longs;i
ex M puncto &longs;u&longs;pendatur, non pote&longs;t quidem per MT per
pendicularem de&longs;cendere versùs terræ centrum, &longs;ed neque
con&longs;i&longs;tet horizonti parallelum; quia &longs;i planum intelligatur ex
terræ centro per rectam MT ductum, non dividitur corpus in
momenta æqualia, cum non tran&longs;eat per H centrum gravita
tis; igitur cum majora &longs;int momenta partis MB, quàm par
tis MA, illa præponderabit, atque de&longs;cendens circa
punctum M permanens convertetur, donec centrum gra
vitatis H &longs;it in perpendiculari MT, cui congruat recta
MO: tunc autem demum con&longs;i&longs;tet, quia planum tran&longs;iens
per MHO æqualiter di&longs;pertit momenta gravitatis; neutrâ
autem parte præponderante, utraque quie&longs;cit. Idem dicen
dum, &longs;i corpus ex I puncto &longs;u&longs;penderetur; tunc enim &longs;o
lùm fieret con&longs;i&longs;tentia, ubi in eadem directionis lineâ
e&longs;&longs;et punctum I atque H centrum gravitatis. Quod &longs;i du
plici funiculo &longs;u&longs;pendatur pondus, & illi paralleli non &longs;int,
quia neque horizonti perpendiculares, illi &longs;i producantur,
concurrent in punctum aliquod lineæ directionis, &longs;ivè &longs;upra
pondus, &longs;ivè infra, pro ratione angulorum, quos con&longs;tituunt.
trum gravitatis O, linea directio
nis IOC, &longs;i ex I &longs;u&longs;pendatur
per O, in co &longs;itu manebit; ergo
etiam, &longs;i funiculi &longs;int IH, IL,
manebit: ergo etiam, &longs;i &longs;int PH,
SL, funiculorum enim longitudo
nihil facit; Idem etiam dicendum
cum funiculi &longs;unt DH, FL; pro
ducti enim concurrunt cum linea
directionis in C, &longs;emper &longs;cilicet
perinde &longs;e habet atque, &longs;i ex I &longs;u&longs;penderetur.
Quæ verò de &longs;u&longs;pen&longs;ione dicta &longs;unt, ea, analogiâ &longs;ervatâ, de
&longs;u&longs;tentatione quoque dicta intelligantur; tunc &longs;olùm videlicet
corpus con&longs;i&longs;tere, cùm ex centro gravitatis ducta directionis
linea tran&longs;it per punctum &longs;u&longs;tentationis, quia tunc &longs;olùm æqua
lia hinc, & hinc &longs;unt momenta virtutis ad de&longs;cendendum, at
que re&longs;i&longs;tentiæ ad a&longs;cendendum: ut quando corpus aliquod
imponitur cono, vel pri&longs;ma &longs;phæræ, vel &longs;egmentum &longs;phæri
cum, plano, vel cylindrus aciei pri&longs;matis trigoni in tran&longs;ver
&longs;um; cadet enim in alterutram partem impo&longs;itum corpus, ni&longs;i
in eadem linea fuerint centrum terræ, punctum contactus, &
centrum gravitatis. Quod &longs;i corpus &longs;u&longs;tentans, atque &longs;u&longs;tenta
tum &longs;e tangant in linea, opus e&longs;t lineam illam e&longs;&longs;e in plano per
lineam directionis ducto, ut fiat æqualium momentorum con
&longs;i&longs;tentia. Quare &longs;i impo&longs;itum corpus con&longs;i&longs;tat, certi&longs;&longs;imo ar
gumento con&longs;tabit punctum, &longs;eu lineam, contactûs re&longs;pon
dere centro gravitatis. Hinc patet ratio &longs;ecundæ, & tertiæ
praxis.
In prima praxi quia facies extima, &longs;upra quam perpendicu
lum liberè movetur, e&longs;t in plano verticali, perpendiculum HC
e&longs;t parallelum lineæ directionis corporis gravis, quæ tran&longs;it
etiam per punctum &longs;u&longs;pen&longs;ionis H: planum igitur tran&longs;iens per
punctum &longs;u&longs;pen&longs;ionis H, & per perpendiculum HC, tran&longs;it
quoque per centrum gravitatis corporis. Cum verò idem pror
&longs;us dicendum &longs;it de plano tran&longs;eunte per punctum &longs;u&longs;pen&longs;io
nis R, & perpendiculum RF, illud &longs;cilicet tran&longs;ire per cen
trum gravitatis corporis; apertum e&longs;t centrum gravitatis e&longs;&longs;e in
punctum S inventum.
Quia demum, &longs;i corpus quod &longs;u&longs;tinet, & id, quod &longs;u&longs;tine
tur, in &longs;uperficie &longs;e tangant, corpus impo&longs;itum in alterutram
partem cadere non pote&longs;t (ni&longs;i fortè &longs;uppo&longs;itum planum fuerit
inclinatum) quin planum per lineam directionis ductum ita &longs;it
extra &longs;uperficiem, in qua fit contactus, ut neque illam con
tingat; con&longs;tat ratio quartæ praxis. Si namque planum ex ter
ræ centro ductum per C cen
trum gravitatis dati corporis
OS, &longs;ecet &longs;ubjectum planum,
pars corporis extra marginem
FE in aëre extans minora ha
bet momenta gravitatis, quàm
reliqua pars; hæc igitur gra
vior non pote&longs;t ab illa elevari:
ubi verò promotum corpus eò
venerit, ut planum per cen
trum gravitatis C ductum tangat extremum marginem &longs;ub
jecti plani ita, ut in eodem plano, in quo e&longs;t centrum gravi
tatis C, &longs;it etiam FE, æqualia &longs;unt gravitatis momenta par
tis CS in aëre extantis, ac CO partis plano incumbentis; &
&longs;i vel minimum ulteriùs promoveretur, pars extra planum &longs;ub
jectum extans gravior e&longs;&longs;et, adeóque de&longs;cenderet. Quare &longs;i in
corporis OS &longs;uperficie infimâ lineam de&longs;crip&longs;eris &longs;ecundùm
marginem FE, ea erit in plano tran&longs;eunte per centrum gravi
tatis. Quia verò idem contingit, &longs;i ii&longs;dem &longs;uperficiebus &longs;e con
tingentibus alium &longs;itum corpori dederis, pariterque eò u&longs;que
promoveris, ut citrà cadendi periculum promoveri ulteriùs
non po&longs;&longs;it; alia linea &longs;ecundùm marginem FE ducta erit pari
ter in plano per gravitatis centrum tran&longs;eunte, &longs;ecabitque
priorem lineam, punctum mutuæ linearum &longs;ectionis illud e&longs;&longs;e,
quod quæritur, &longs;atis liquet. Hæc e&longs;t di&longs;par philo&longs;ophandi ra
tio, &longs;i pars CO adeò longa e&longs;&longs;et, ut etiam extaret extra an
gu&longs;tias &longs;ubjecti plani; &longs;emper enim con&longs;i&longs;tit impo&longs;itum corpus,
quandiù planum per lineam directionis tran&longs;iens, aut tangit,
aut &longs;ecat &longs;ubjectum planum. Quandocunque enim linea di
rectionis non tran&longs;it per punctum, vel lineam, vel &longs;uperficiem,
&longs;u&longs;pen&longs;ione) &longs;emper in alterutram partem grave inclinatur, in
eam &longs;cilicet, in qua reperitur centrum gravitatis, cùm plura
&longs;int ex ea parte momenta gravitatis.
EX his, quæ proximè dicta &longs;unt, grave &longs;u&longs;tentatum in eam
partem inclinari, in qua e&longs;t gravitatis centrum, oritur ali
quando a&longs;cen&longs;us gravium, qui rerum naturalium ignaros in
admirationem adducit non mediocrem, &longs;i maximè tunc cor
pus de&longs;cendere intelligant, quando illud cernunt altiùs ab ho
rizonte a&longs;cendere. Sit
enim &longs;uper planum in
clinatum RN rota tantæ
latitudinis, ut po&longs;&longs;it in
plano verticali erecta
permanere, dum conver
titur; habeat autem ad
PO adnexam laminam
plumbeam cra&longs;&longs;iorem,
adeò ut totius rotæ cen
trum gravitatis &longs;it S. Jam
verò ea &longs;it plani &longs;ubjecti
inclinatio, ut rotâ illud
tangente puncto H, li
nea à terræ centro per H punctum contactûs tran&longs;iens non
tran&longs;eat per S centrum gravitatis (&longs;eu ut veriùs dicam, quia
extima &longs;uperficies rotæ cylindrica tangit planum in lineâ, pla
num ex centro terræ per lineam contactûs in H ductum non
tran&longs;eat per S) &longs;ed illud relinquat versùs &longs;uperiorem plani par
tem N; planum per rectam HO perpendicularem ductum
di&longs;tinguit rotam in momenta gravitatis inæqualia: non pote&longs;t
igitur rota in H con&longs;i&longs;tere, &longs;ed convertitur, ita ut tangat pla
num in I primùm, deinde in E, demùm in P, ubi con&longs;i&longs;tet,
trum tran&longs;ibit per P locum contactús. In hac autem conver
&longs;ione dum rotæ partes inter H & P deinceps aptantur &longs;ubjecto
plano, centrum quidem molis a&longs;cendit, &longs;ed centrum gravita
tis S de&longs;cendit. Lineam porrò SP minorem e&longs;&longs;e lineá SE, &
hanc minorem lineâ SI, & hanc lineâ SH, con&longs;tat ex prop.7.
lib.3. Eucl. &longs;i nimirum per S, & C centrum agatur diameter.
Non e&longs;t tamen cen&longs;endum quamlibet ponderis additionem
in OP &longs;atis e&longs;&longs;e, ut in quolibet plano inclinato rota a&longs;cendat;
&longs;i enim di&longs;tantia centri gravitatis à centro rotæ minor fuerit,
quàm Sinus inclinationis plani, &longs;emper de&longs;cendet; &longs;i eidem
Sinui æqualis, non a&longs;cendet; &longs;i demum eo &longs;inu major, poterit
a&longs;cendere.
Sit planum inclinatum
AB, quod in H contingat
circulum (hunc &longs;umo cir
culum, qui tran&longs;eat per
centrum tum molis tum
gravitatis rotæ) cujus cen
trum C, & ducatur recta
CH, quæ cum perpendi
culari HO faciat angu
lum CHO. Quia enim
OH producta cadit in ho
rizontem AD perpendicularis, & angulus OHA per 32.lib.1.
æqualis e&longs;t duobus internis HFA, FAH, e&longs;t autem AHC ad
contingentem factus à &longs;emidiametro rectus per 18.lib. 3. &longs;icut
& HFA e&longs;t rectus; reliquus CHO æqualis e&longs;t angulo HAF
inclinationis plani. Certum e&longs;t igitur, quòd in eam partem ro
ra convertetur, in qua fuerit centrum gravitatis. Quoniam
verò CI e&longs;t Sinus anguli CHI, po&longs;ito radio CH, e&longs;t au
tem CI minima omnium, quæ ex C puncto cadant in rectam
HO, manife&longs;tum e&longs;t, quòd, &longs;i centrum gravitatis fuerit cen
tro rotæ vicinius, ut in R, rota &longs;emper de&longs;cendet, quia cen
trum gravitatis re&longs;picit declivitatem plani: at, &longs;i fuerit
in I, a&longs;cendere non pote&longs;t, quia pars re&longs;piciens acclivita
tem plani non præponderat: &longs;i demum longiùs à centro
di&longs;titerit, ut in S, a&longs;cendere poterit, u&longs;que dum punctum S
punctum contactûs.
Ex his apertè con&longs;tat futurum, ut rota de&longs;cendat, &longs;i angulus,
quem in puncto contactûs faciunt lineæ ductæ ex centris mo
lis, & gravitatis (&longs;uppono molis centrum idem e&longs;&longs;e cum centro
rotæ, quâ rota e&longs;t) minor fuerit angulo inclinationis plani,
tunc enim centrum gravitatis re&longs;picit declivitatem plani; fu
turum autem, ut rota a&longs;cendat, &longs;i angulus ille major fuerit co
dem angulo inclinationis, quia centrum gravitatis re&longs;picit ac
clivitatem plani; futurum demùm, ut con&longs;i&longs;tat, &longs;i angulus il
le fuerit æqualis eidem angulo inclinationis plani, quia nimi
rum planum perpendiculare dividit æqualiter momenta gravi
tatis, cum tran&longs;eat per centrum gravitatis exi&longs;tens in lineâ
perpendiculari.
Hinc patet &longs;emper de&longs;cen&longs;uram rotam, &longs;i habeat centrum
gravitatis R, quia &longs;emper facit angulum, de quo dictum e&longs;t,
minorem angulo inclinationis, hoc e&longs;t angulo CHI, nam &longs;i
ducatur ad CR perpendicularis RE, & ex centro ducatur
recta CE, angulus CER e&longs;t maximus omnium, quos faciunt
lineæ ex punctis C, & R ductæ ad idem punctum circumfe
rentiæ, ut mox o&longs;tendam; atqui CER minor e&longs;t angulo CHI,
(quia ob lineas RE, IH parallelas, angulus IHC internus
per 29.lib.1. e&longs;t æqualis externo RLC, & RLC externus per
16. lib. 1. major e&longs;t interno CER, ac proinde IHC major
quàm CER) igitur quicunque angulus con&longs;titutus à rectis
exeuntibus ex C, & R minor e&longs;t angulo inclinationis; atque
adeò &longs;emper de&longs;cendet.
At &longs;i centrum gravitatis fuerit S, ductâ ad CS perpendicu
lari SM, angulus omnium maximus e&longs;t CMS: hic autem e&longs;t
æqualis externo CKI, cum IK, & SM parallelæ &longs;int con&longs;ti
tutæ; angulus verò CKI externus major e&longs;t interno CHI,
igitur angulus CMS major e&longs;t angulo CHI, hoc e&longs;t angulo
inclinationis. A&longs;cendere igitur poterit rota, quando angulus
ad contractum factus à lineis ex C, & S exeuntibus major e&longs;t
angulo inclinationis; &longs;in autem contactus fiat in co puncto, ad
quod fit angulus æqualis, con&longs;i&longs;tet; &longs;i in iis punctis, ad quæ fit
angulus minor, de&longs;cendet.
Porrò quamvis iis, qui in A&longs;tronomicarum Pro&longs;taphære&longs;eon
ad peripheriam factum à Radio circuli, & à linea perpendicu
lari in diametrum, e&longs;&longs;e maximum omnium, qui fieri po&longs;&longs;int à
Radio, & à lineâ ductâ ex eodem diametri puncto, in quod
cadebat perpendicularis; ut omnibus tamen fiat &longs;atis, non pi
gebit hîc demon&longs;trare. Sit in diametro
circuli punctum R extra centrum C, &
ad CR ducatur perpendicularis HR,
quæ producta in G, bifariam dividitur
in R: & ductis ex centro rectis CH,
CG æqualibus, &longs;unt anguli CHR,
CGR æquales, per 5. vel 8. lib.1. Fiat
angulus CER, ductis ex C & R rectis
lineis ad idem punctum E peripheriæ.
Dico angulum CER minorem e&longs;&longs;e an
gulo CHR. Ducatur enim recta EG; & erunt in I&longs;o&longs;cele
CEG æquales anguli CEG, CGE. Quia verò, per 7.lib.3.
RE major e&longs;t quàm RG, angulus RGE major e&longs;t angulo
REG, per 18.lib. 1. & ablatis æqualibus remanet REC mi
nor angulo RGC, hoc e&longs;t RHC. Similiter o&longs;tendetur angu
lum RIC minorem e&longs;&longs;e angulo RHC: ductâ enim IG, angu
li CIG, CGI &longs;unt æquales: & quoniam per 7.lib.3. RG ma
jor e&longs;t quàm RI, angulus RIG major e&longs;t angulo RGI, per
18.lib.1. &longs;i igitur ex æqualibus auferantur inæquales anguli, re
manet RIC minor, quàm RGC, hoc e&longs;t quam RHC. Ea
dem erit methodus demon&longs;trandi angulos ad puncta periphe
riæ propiora puncto H e&longs;&longs;e majores angulo CER. Ductâ enim
RD æquali ip&longs;i RE, ad punctum &longs;cilicet D æqualiter di&longs;tans à
diametro, ac di&longs;tet punctum E, & ducto radio CD, e&longs;t angu
lus CDR æqualis angulo CER. Sit autem puncto H vicinior
angulus COR, quem dico e&longs;&longs;e majorem angulo CER per
7.lib.3. & 8.lib.1. Ducta lineâ OD, anguli COD, CDO
&longs;unt æquales, quia latera CO, CD æqualia &longs;unt: at per 7.lib.3.
RO minor e&longs;t, quàm RE, hoc e&longs;t RD, igitur angulus ROD
per 18.lib.1. major e&longs;t angulo RDO, & ablatis æqualibus re
manet ROC major quam RDC, hoc e&longs;t quàm REC. Angu
li itáque recedentes à puncto H &longs;emper fiunt minores, acce
dentes verò fiunt majores.
Hoc probato con&longs;equens e&longs;t illud, quod in rotæ peripheriâ
duo &longs;unt puncta, inter quæ quodlibet punctum contingat pla
num
ductis ex centro rotæ, & ex centro gravitatis &longs;it major angulo
inclinationis; quia nimirum anguli à puncto H recedentes ad
utramque partem &longs;emper fiunt minores; ergo ad utramque e&longs;t
angulus unus æqualis angulo inclinationis, & &longs;patium inter
huju&longs;modi angulos e&longs;t quantitas peripheriæ, quæ a&longs;cendens
pote&longs;t coaptari plano inclinato: ac proinde ex horum puncto
rum di&longs;tantia definietur &longs;patium, quod pote&longs;t rota a&longs;cendens
percurrere.
Sit igitur rota, cujus centrum C, &
centrum gravitatis S: &longs;it autem CS par
tium 11, quarum CH Radius e&longs;t 16:
e&longs;t igitur CS æqualis Sinui gr. 43. 26′.
qui erit maximus angulus CIS ad peri
pheriam factus à Radio, & à lineâ IS
perpendiculari ad SC. Quare in quoli
bet plano habente minorem inclinatio
nem poterit a&longs;cendere. Ponatur plani
inclinatio gr. 15, cui æqualis &longs;it angulus CHS. Fiat igitur
ut CS 11 ad CH 16, ita Sinus anguli CHS 25882 ad
37646 Sinum Anguli CSH gr. 22. 7′; eritque angulus
SCH gr. 142. 53′. Cre&longs;cet ergo &longs;upra angulum H angulus
ad peripheriam, &longs;i ultra punctum H fiat contactus rotæ
in alio puncto viciniore puncto I, ex quo ad SC perpendi
cularis cadit; & ex I decre&longs;cit u&longs;que dum in P fiat angu
lus SPC grad. 15 æqualis angulo inclinationis. In triangu
lo itaque SPC invenitur ex ii&longs;dem datis angulus PSC
gr. 157. 53′. & angulus SCP gr. 7. 7′. qui ex angulo SCH
gr. 142. 53′ ablatus relinquit PCH gr. 135. 46′. quæ e&longs;t quan
titas arcûs HIP, quæ plano coaptatur in a&longs;cen&longs;u. Quoniam
verò quarum partium CG Radius e&longs;t 16, peripheria e&longs;t 100 1/2
earum parirer e&longs;t arcus HP ferè 38, &longs;i Radius rotæ fuerit un
ciarum pedis 16, rota a&longs;cendet in plano percurrens &longs;patium
pedum 3, & eo ampliùs. Hinc poteris aut rotæ diametrum au
gere, aut plani inclinationem minuere, &longs;i volveris rotam lon
giore &longs;patio moveri: auctâ enim rotæ diametro augetur peri-
At &longs;i
data fuerit rota (oportet non ignorari di&longs;tantiam centri gravi
tatis à centro rotæ, poterit autem primâ praxi cap.5. inve&longs;tiga
ri) certum e&longs;t illam non po&longs;&longs;e a&longs;cendere ni&longs;i per &longs;patium mi
nus longitudine &longs;emiperipheriæ; con&longs;tituto autem &longs;patio inve
nietur inclinatio plani nece&longs;&longs;aria, hac methodo. Data &longs;patij
longitudo PH reducatur ad denominationem graduum, & erit
notus angulus PCH: & quoniam anguli ad H & ad P debent
e&longs;&longs;e æquales, anguli verò in R ad verticem &longs;unt æquales, erunt
pariter æquales PCH, & PSH, qui proinde notus e&longs;t. Hujus
&longs;emi&longs;&longs;is auferatur ex recto CSI, & innote&longs;cet angulus CSH,
cum quo & duobus lateribus CS, CH invenietur per Trigo
nometriam angulus CHS æqualis angulo inclinationis plani
nece&longs;&longs;ariæ. Quod autem angulus HSI &longs;it &longs;emi&longs;&longs;is totius HSP,
hoc e&longs;t dati PCH, &longs;ic o&longs;tendo. Quia in duobus triangulis
CSP, CHS idem latus CS opponitur angulis æqualibus ad H,
& ad P, æqualia autem latera CH, & CP opponuntur angulis
quæ&longs;itis CSH, & CSP, con&longs;tat horum duorum angulorum
e&longs;&longs;e unum eundemque &longs;inum; ergo &longs;imul &longs;umpti &longs;unt æquales
duobus rectis; auferatur ex eorum &longs;ummâ unus rectus, rema
nebunt duo anguli &longs;imul CSH, ISP æquales uni recto, hoc
e&longs;t angulo ISC: auferatur communis CSH, remanebit HSI
æqualis angulo ISP: id quod oportuit demon&longs;trare.
Colligere po&longs;&longs;umus ex his, quæ hactenus explicata &longs;unt, fie
ri quidem po&longs;&longs;e, ut, &longs;i rota in plano inclinato primùm con&longs;ti
tuta exactè tangat in H, pror&longs;us con&longs;i&longs;tat; id tamen vix po&longs;&longs;e
&longs;perari, quia &longs;i in alio puncto remotiore ab I tangat, cadet, &longs;i
in puncto viciniore, a&longs;cendet. At ubi venerit in P, &longs;i ex con
cepto impetu pergat adhuc aliquantulum a&longs;cendere; centro
gravitatis S tran&longs;lato versùs plani declivitatem, & diminuto
angulo, de&longs;cendet; & ubi tran&longs;ilierit punctum P, iterùm aucto
angulo a&longs;cendet, donec omninò in P con&longs;i&longs;tat. Ubi licet
animadvertere non idem e&longs;&longs;e punctum contactus, in quo
quie&longs;ceret in plano horizontali, ac inclinato; in plano enim
horizontali quie&longs;ceret in O, ubi linea à centro rotæ C perpen
dicularis horizonti, ac tran&longs;iens per S centrum gravitatis, ter
minatur: in eo autem puncto O con&longs;i&longs;tere non po&longs;&longs;e &longs;upra pla
num inclinatum &longs;atis patet ex dictis. Porrò hæc, quæ de rotâ
râ quie&longs;cente in plano inclinato, clarius e&longs;t, quàm ut oporteat
pluribus explicare.
Unum &longs;upere&longs;&longs;e videtur o&longs;tendendum, quî verum &longs;it cen
trum gravitatis de&longs;cendere ita, ut fiat horizonti vicinius, dum
rota a&longs;cendit, & fit remotior. Id ut manife&longs;tum fiat, primò in
veniatur HS: & &longs;it ut Sinus anguli CHS gr. 15. ad &longs;inum an
guli SCH gr. 14.2. 53′. hoc e&longs;t ut 25882 ad 60344, ita CS
partium 11 ad HS 25 2/3: quæ e&longs;t altitudo centri gravitatis ante
motum. Deinde inveniatur SP; & &longs;it ut Sinus SPC gr. 15 ad
Sinum SCP gr.7. 7′ hoc e&longs;t, ut 25882 ad 12389, ita CS par
tium 11 ad SP 5 1/4, quæ in fine motus erit altitudo centri gravi
tatis &longs;upra planum inclinatum; huic autem addenda e&longs;t altitu
do, quam &longs;upra horizontem habet punctum illud plani inclinati,
in quo tanget P. Quia ergo inclinatio plani e&longs;t gr. 15, & HP
e&longs;t partium 38, tantum e&longs;t &longs;patium, quod in plano percurritur
à rota a&longs;cendente, fiat ut Radius 100000 ad 25882 Sinum an
guli inclinationis, ita 38 ad 9 4/5 altitudinem &longs;upra horizontem,
cui &longs;i addas SP 5 1/4, erit in fine motûs altitudo centri gravitatis
&longs;upra horizontem partium 15, cùm initio di&longs;taret partibus 25 2/3.
Centrum igitur gravitatis &longs;impliciter, & ab&longs;olutè de&longs;cendit,
dum rota in plano inclinato a&longs;cendit.
Po&longs;&longs;em hîc afferre aquam vi &longs;uæ gravitatis a&longs;cendentem in
cochleâ Archimedis, dum cylindrus, quem cochlea ambit,
convertitur: ab&longs;tineo tamen, quia non vacat hîc examinare,
an motus ille compo&longs;itus &longs;it ex conver&longs;ione, quâ pul&longs;u externo
agitata aqua attollatur, & ex naturali de&longs;cen&longs;u, quo per tubum
in &longs;piras &longs;inuatum de&longs;cendat; an verò quemadmodum &longs;uppo&longs;i
to cuneo reluctans pondus elevatur, vel etiam cochleâ trahitur
in plano horizontali, ita dicendum &longs;it aquam vi &longs;uæ gravitatis
in imo per&longs;i&longs;tentem à cochleâ &longs;en&longs;im &longs;ubeunte elevari &longs;imul, &
trahi, quin illa &longs;ponte &longs;ua a&longs;cendat: nam aquæ facilè tribuitur
aliquando motus, qui &longs;ubjecto corpori, cui illa in&longs;idet, conve
nit; ut liquet &longs;i ampliorem peluim ex fune &longs;u&longs;penderis, vel lu
brico in plano horizontali collocaveris, in qua &longs;it non multa
aqua in depre&longs;&longs;iore fundi parte quie&longs;cens; va&longs;e &longs;iquidem ex
improvi&longs;o vehementiùs impul&longs;o videtur aqua in oppo&longs;itam par-
veatur, quàm aqua in va&longs;e: quanquam ratione adhæ&longs;ionis aquæ
ad peluim etiam ip&longs;a motum concipiat. Quare in cen&longs;u &longs;ponte
a&longs;cendentium numeranda non videtur aqua tubo &longs;peciali cy
lindrum circumplexo elevata.
Videatur forta&longs;&longs;e aqua &longs;ponte a&longs;cen&longs;ura in tubo non æquabi
li &longs;ed conico, in plano verticali rotæ &longs;piraliter circumducto:
dum enim aqua æquilibrium &longs;uperficiei faciens in parte tubi
ampliore præponderat, convertitur rota, & illa iterum æqua
liter &longs;e librans totius molis compo&longs;itæ centrum gravitatis trans
fert extra lineam perpendicularem: &longs;i tamen ea cautio adhi
beatur, ut tanta &longs;it aquæ quantitas, quæ non planam obtineat
&longs;uperficiem &longs;ed tubi inflexione conformetur; neque ita &longs;it
&longs;piræ a&longs;cendentis ardua altitudo, ut aqua po&longs;t &longs;uperficiei libra
tionem ex ea parte ob &longs;ui paucitatem non præponderet; & præ
terea ejus figuræ &longs;it tubus, ut aqua in parte angu&longs;tiore remo
tior à perpendiculari, non ita ratione &longs;itûs augeat momenta &longs;ui
conatus deor&longs;um, ut repugnare valeat aquæ ampliorem tubi
partem occupanti. Si hæc, inquam, ob&longs;erventur (an autem
ita facile &longs;it ea ob&longs;ervare, ut quidam autumant, hic non de
finio) & centrum gravitatis transferatur extra perpendicula
rem versùs ampliorem tubi &longs;piralis partem, futurum quidem
e&longs;t, ut aqua a&longs;cendat; id tamen non e&longs;t opus centri gravitatis,
&longs;ed potius virtutis illius, qua humor &longs;e æquabiliter librat.
alia repant, alia rotentur.
QUæ capite &longs;uperiori dixi de globi aut rotæ &longs;uper planum
inclinatum con&longs;i&longs;tentiâ in puncto, in quo linea à centro
globi, aut rotæ ducta cum eâ, quæ ex centro gravitatis duci
tur, facit angulum æqualem angulo inclinationis plani, non ita
intelligi velim, qua&longs;i motus omnis deor&longs;um adimatur rotæ aut
globo cuju&longs;libet gravitatis, & in quovis plano inclinato: ibi
enim con&longs;i&longs;tentiæ, aut quietis nomine &longs;olam conver&longs;ionem
Fieri &longs;i quidem pote&longs;t, ut adeò con
tinuo lævore lubricum &longs;it planum, exactéque rotundatus globus,
ut nullam ex eminulis particulis moram recipiens deor&longs;um la
batur, volubilitate ipsâ motum nihil juvante, &longs;ed &longs;olo pondere
urgente, cum in lineâ ad horizontem perpendiculari &longs;emper
maneat centrum gravitatis, & punctum contactûs.
Neque e&longs;&longs;et diver&longs;a ratio &longs;phæræ centrum gravitatis haben
tis extra centrum molis, ac cæterorum corporum non &longs;phæri
corum: Nam gravia quæcunque in plano inclinato con&longs;tituta
tantum habent ad de&longs;cendendum momenti, ut a&longs;peritatis re
&longs;i&longs;tentiam vincant, repunt quidem, &longs;i linea directionis ab eo
rum gravitatis centro in terræ centrum ducta tran&longs;eat per con
tactum &longs;ubjecti plani, & impo&longs;iti gravis; rotantur verò, &longs;i di
rectionis linea in plani declivitatem cadat extra contactum:
&longs;ivè demùm in puncto, &longs;ivè in lineâ, &longs;ivè in &longs;uperficie con
tactus fiat. E&longs;t autem animadvertendum non e&longs;&longs;e opus, ut una
continua &longs;uperficies &longs;it, aut linea, &longs;ecundùm quam &longs;e tangant;
&longs;ed pro &longs;uperficie aut linea contactûs accipitur totum illud &longs;pa
tium, quod inter extrema contingentia rectis lineis conjuncta
intercipitur.
Sit planum inclinatum AB,
cui globus C incumbit con
tingens in puncto D. Ex cen
tro gravitatis C, quod & cen
trum molis e&longs;t ex hypothe&longs;i,
cadat linea directionis CE
perpendicularis in horizon
tem FB; quæ nece&longs;&longs;ariò ca
dit extra punctum contactûs
D; alioquin eadem linea CE
caderet ad angulos rectos &longs;u
pra planum inclinatum, & &longs;upra horizontale, id quod fieri
non pote&longs;t, cum huju&longs;modi plana non &longs;int invicem parallela.
Per D igitur punctum &longs;u&longs;tentationis ductâ GH parallelâ lineæ
directionis, &longs;i per utramque plana parallela ducantur, planum
per GH &longs;ecat &longs;phæram in partes inæqualiter graves; & idcir
co pars præponderans, in qua e&longs;t centrum gravitatis globi, mo
vetur circa punctum &longs;u&longs;tentationis D, atque adeò in gyrum
Quod &longs;i inæ
qualis fuerit &longs;phæræ &longs;ub&longs;tantia, & centrum gravitatis I in per
pendiculari GH, non de&longs;cendet &longs;phæra in gyrum acta, &longs;ed
tantùm repet, cum neutra pars præponderet.
Simili ratione parallelepipedum KL, cujus centrum gravi
tatis M, non repit; quia, cùm linea directionis MN cadat ex
tra ba&longs;im KO, quæ contingit &longs;ubjectum planum, &longs;i per extre
mam lineam KP tran&longs;eat planum PQ horizonti perpendicu
lare, dividitur parallelepipedum in duo pri&longs;mata inæqualia, &
non æquiponderantia: cum verò pri&longs;ma trapezium QLKP
præponderet pri&longs;mati trigono KOQ, quod &longs;u&longs;tinetur à ba&longs;i,
illud nece&longs;&longs;ariò de&longs;cendit, & circa lineam KP convertitur.
Contrà autem quando intra ba&longs;im contactûs, ut in cubo PR,
cujus centrum S, cadit linea directionis ST, tunc repit, & non
rotatur cubus; quia &longs;cilicet ab extrema &longs;u&longs;tentationis lineâ KP
ductum planum horizonti perpendiculare dividit cubum in
partes inæquales ita, ut pars illa, in qua e&longs;t centrum gravitatis,
& quæ à &longs;ubjecto plano tota &longs;u&longs;tinetur, præponderet, nec po&longs;
&longs;it à reliquâ parte elevari, ut circa KP convertatur.
Hinc apparet ad quantam altitudinem pertinere po&longs;&longs;it paral
lelepipedum, ut in dato plano inclinato non rotetur, &longs;ed repat:
nam ab extremâ &longs;u&longs;tentationis lineâ KP excitatum planum
horizonti perpendiculare PQ, quod bifariam in partes æqui
ponderantes dividit parallelepipedum KQ, determinat altitu
dinem maximam
repit, &longs;ed rotatur, quia linea directionis cadit extra ba&longs;im
&longs;u&longs;tentationis: in omni verò minori altitudine non rotatur, &longs;ed
repit, quia linea directionis cadit intra ba&longs;im &longs;u&longs;tentationis.
Hoc idem in corporibus cæteris, quamvis non parallelepipe
dis, ob&longs;ervandum e&longs;t, an &longs;cilicet linea directionis cadat extra
ba&longs;im &longs;u&longs;tentationis, nec ne.
Quæ tamen de cubo repente dicta &longs;unt, intelligi velim &longs;pecta
tâ per &longs;e gravium figurâ: quia per accidens fieri pote&longs;t, ut cor
pus non repat, &longs;ed rotetur, quamvis linea directionis cadat in
tra ba&longs;im, quæ planum inclinatum contingit. Nam &longs;i in motu
occurrat &longs;uper plano inclinato offendiculum aliquod, cui de
&longs;cendens corpus illidatur, fieri pote&longs;t, ut impetus ex motu con
ceptus ita promoveat centrum gravitatis in anteriora, ut linea
mum e&longs;t offendiculo, ac proinde circa illud convertatur. Hæc
autem poti&longs;&longs;imùm e&longs;t ratio, cur ex clivis de&longs;cendentes lapides,
quamquam nec orbiculares, nec admodum alti, rotentur ta
men; quia &longs;cilicet multa offendicula in clivo occurrunt, & ab
impetu per motum concepto partes &longs;uperiores promoventur
ulteriùs, inferioribus retardatis. Sic &longs;æpè ce&longs;pitantes cadimus,
quia ab offendiculo retinentur pedes, cum interim corpus re
liquum ex concepto impetu ulteriùs promoveatur, ita ut linea
directionis cadat extra ba&longs;im &longs;u&longs;tentationis.
OB&longs;ervandum e&longs;t, ait Vitruvius lib.6. cap. 11, uti omnes
&longs;tructuræ perpendiculo re&longs;pondeant, neque habeant in
ulla parte proclinationes. Nemo e&longs;t qui non intelligat præ
ceptum hoc ad ædificiorum con&longs;i&longs;tentiam pertinere; &longs;ed neque
defuerunt, qui rem &longs;ubtiliùs, quàm par &longs;it, perpendentes ina
ni timore &longs;e torquebant, ne fortè aliquando domus corrueret,
cujus parietes inter &longs;e paralleli fuerant con&longs;tituti; cùm enim
perpendicula &longs;ibi demum in terræ centro occurrant, fieri non
po&longs;&longs;e putabant, ut &longs;imul paralleli e&longs;&longs;ent parietes. Id quod Geo
metricè quidem verum e&longs;t; Phy&longs;icè tamen paralleli&longs;mus cum
perpendiculis con&longs;entit: nam &longs;i funiculos duos longitudinis
ped. 100. clavo affixos ita extendas, ut extrema eorum palmi
intervallo di&longs;tent, angulum facient acuti&longs;&longs;imum; & &longs;i lineas
duas bipedales duxeris eorum extremitatibus congruentes, vix
different à parallelis, cum intervalla jungentia utro&longs;que linea
rum terminos differant inter &longs;e &longs;olum palmi parte quinquage
&longs;ima. Longè autem majorem rationem terræ &longs;emidiameter ha
bet ad quamlibet ædificiorum altitudinem; ut proinde à paral
leli&longs;mo multo minùs recedant parietes, etiam&longs;i fuerint turrium
in&longs;tar alti&longs;&longs;imi. Ponantur enim parietes duo, aut potiùs turres,
di&longs;tare inter &longs;e pa&longs;&longs;.300; &longs;it autem parietum, vel turrium alti
tudo pa&longs;&longs;. 60, hoc e&longs;t ped.300. Con&longs;tat mihi, ut aliàs o&longs;tendi,
terrenam &longs;emidiametrum non e&longs;&longs;e minorem pa&longs;&longs;ibus Rom.
ad altitudinem 60, ita di&longs;tantia parietum, aut turrium in imo
300, ad aliud, proveniet differentia, qua di&longs;tantia turrium in
&longs;ummo vertice &longs;uperat earum di&longs;tantiam in imo pede, & erit
partium (4359/1000000) unius pa&longs;&longs;us, quæ e&longs;t minor quàm 2/5 digiti: quis
autem parallelas non dixerit turres, quæ vix uno aut altero
hordei grano di&longs;tant à paralleli&longs;mo? Quod &longs;i in tanta altitudine
atque di&longs;tantiâ di&longs;crimen hoc adeò exiguum e&longs;t, &longs;atis patet,
quid de columnarum paralleli&longs;mo dicendum &longs;it. Con&longs;tat autem
ex his ædificia in alti&longs;&longs;imis montibus con&longs;tituta habere parie
tes minùs à paralleli&longs;mo recedentes, &longs;i fuerint ad perpendicu
lum ædificati, quàm in locis depre&longs;&longs;ioribus: atque adeò, &longs;i duæ
columnæ eandem inter &longs;e po&longs;itionem &longs;ervantes de&longs;cenderent
cum &longs;ubjecto plano, ita ut alterutra columnarum illarum ad
perpendiculum de&longs;cenderet, reliqua demùm adeò inclinare
tur, ut caderet.
Sed quàm inanem &longs;ibi &longs;truant &longs;olicitudinem, qui nimis exi
guè, & exiliter ad calculos revocant &longs;tructurarum perpendicu
la, &longs;atis indicant turres inclinatæ, quæ po&longs;t aliquot &longs;ecula con
&longs;i&longs;tunt citrà ullum ruinæ periculum, quamvis illam timeant
imperiti. Duas habemus in Italiâ turres ob in&longs;ignem inclina
tionem con&longs;picuas; altera e&longs;t Bononiæ quadrata opere lateri
tio, altera Pi&longs;is rotunda ex albo marmore affabrè expolito, &
columnis 284 rite di&longs;po&longs;itis ornata. Ædificari cœpit anno
1173 Germano quodam architecto, quem ab aliis Guillel
mum, ab aliis Joannem OEnipontanum dici reperio. Rotunda
e&longs;t forma duplici muro concludente &longs;calas cochleæ in modum
ab imo ad &longs;ummum ductas: parietis cra&longs;&longs;ities e&longs;t cubitorum
6 1/3, turris altitudo cubitorum 78, ambitus in imo pede cubi
torum 80; unde colligitur diameter cubitorum ferè 25 1/2; incli
natio, &longs;eu intervallum inter ba&longs;im, & perpendiculum e&longs;t cu
bitorum 7 1/3, ut ex literis ad me inde datis habeo; quamvis
apud aliquos legerim tantùm cubitos 7, apud alios 6 1/2. Factâ
ne fuerit illa inclinatio de indu&longs;triâ, an verò &longs;ub&longs;identibus fun
damentis, incertum e&longs;t. Ego non facilè eo in illorum &longs;enten
tiam, qui id &longs;cribunt contigi&longs;&longs;e ex artificis imperitia, cui non
&longs;atis per&longs;pecta e&longs;&longs;et &longs;oli natura; tum quia fundamenta altitudi-
&longs;truendis annus &longs;olidus &longs;atis non fuit; tum quia nullam unquam
egit rimam, id quod &longs;ub&longs;idente &longs;olo rari&longs;&longs;imum e&longs;t; tum quia
potuit architectus excitari ad artis &longs;pecimen exhibendum à tur
ri Bononien&longs;i Gari&longs;endâ excitatâ anno 1110.
Turris Bononien&longs;is altitudinem habet pedum Bonon. 130;
exteriùs inclinatur ped. 9, interiùs verò ped. 1, & paulo am
plius: muri cra&longs;&longs;ities in parte infimâ e&longs;t pedum 6 1/2, in &longs;upre
ma ped. 4; cava turris ped. 7. quare lateris longitudo e&longs;t ped.
20, & ambitus, quoniam quadrata e&longs;t, ped. 80. Ex his men
&longs;uris, quas in
tulit Antonius Pauli Ma&longs;ini, turris &longs;pe
ciem exhibeo, & e&longs;t AB latus unum
ped. 20, BD inclinationis men&longs;ura
ped. 9. DC altitudo perpendicularis
ped.130; EB & AF ped. 6 1/2 cra&longs;&longs;ities
imi parietis, & CH ped. 4. cra&longs;&longs;ities
eju&longs;dem parietis EC exteriùs inclinati.
At quoniam inclinatio interior FI dici
tur e&longs;&longs;e ped.1, & paulo ampliùs, erit ID
paulo major ped.21; erecta autem ex I
perpendicularis dabit punctum G termi
num cra&longs;&longs;itiei muri AG in parte &longs;upre
mâ, & erit CG major ped. 21, cum &longs;it
æqualis ip&longs;i ID. Quare fieri non pote&longs;t,
ut KG &longs;it ped. 4; quemadmodum HC;
alioquin e&longs;&longs;et CK &longs;altem ped.25, cum
ba&longs;is AB &longs;it tantum ped.20. Hinc &longs;i li
ceat conjecturas per&longs;equi (quandoqui
dem veritatem a&longs;&longs;equi non potui, cum
non careat periculo a&longs;cen&longs;us per &longs;calas
ligneas à pluviis maximam partem cor
ruptas) exi&longs;timo AF majorem e&longs;&longs;e quàm
EB, hoc e&longs;t majorem pedibus 6 1/2, KG
verò minorem quam HC, ut turri &longs;ua
con&longs;tet Eurithmia; id quod obtineretur, &longs;i ID uno, aut alte
ro pede minor e&longs;&longs;et quàm AB, differentia enim inter ID,
& AB e&longs;&longs;et cra&longs;&longs;ities KG. Et &longs;anè memini aliquando me au-
non excedere integrum pedem. Id autem valde opportu
num accidebat, ut longè faciliùs paries AFGK &longs;uâ mole
&longs;taret: neque enim ca&longs;u inclinatam fui&longs;&longs;e turrim dicere po
teris, quam con&longs;tat prope A&longs;inellam recti&longs;&longs;imam ideò fui&longs;&longs;e
conditam, ut multo clariùs appareret inclinatio: præterquam
quod inclinatio interior minor externâ &longs;atis o&longs;tendit muros
nunquam fui&longs;&longs;e parallolos.
Porrò ut con&longs;tet ex huju&longs;modi inclinatione non magis
e&longs;&longs;e de ruinâ timendum, quàm &longs;i exactè perpendicularis e&longs;
&longs;et, examinemus, &longs;i placet, centrum gravitatis in turri Bo
nonien&longs;i; hinc enim facilis erit conjectura de cæteris. Et
primò parietis maximè inclinati &longs;ectio
verticalis illum bifariam &longs;ecans ac tran
&longs;iens per centrum gravitatis &longs;it HCBE:
cujus latera parallela HC, EB bifariam
&longs;ecta in V & R jungantur rectâ VR, cu
jus longitudo inve&longs;tiganda e&longs;t, ut in eâ
definiatur punctum S centrum gravitatis,
ac innote&longs;cat utrum perpendicularis SX,
&longs;cilicet linea directionis cadat intra ba
&longs;im EB &longs;u&longs;tentantem. Et ut à fractioni
bus minus incommodi &longs;ubeamus, liceat
a&longs;&longs;umere pedem in partes cente&longs;imas di
vi&longs;um. Cum autem EB &longs;it ped. 6 1/2, &longs;emi&longs;
&longs;is RB e&longs;t ped. 3. 25″; & quia HC e&longs;t
ped. 4, VC e&longs;t ped. 200″. Et ducatur
recta BV.
In triangulo BDC rectangulo datis BD,
inclinatione ped. 90′0′, & altitudine per
pendiculari CD ped. 130′0′, additis late
rum quadratis fit quadratum hypothenu
&longs;æ BC, quæ e&longs;t ped. 13031″. Ex datis autem lateribus BD,
& DC invenitur angulus CBD gr. 88. 33′, cui æqualis e&longs;t
inter parallelas VC, BD alternus VCB: angulus verò CBR
gr. 91. 27′.
In triangulo VCB datis lateribus VC ped. 2.0′0′, CB
ped. 130. 31″, & angulo verticali VCB gr. 88. 33′, reperitur
Ex his autem
inve&longs;tigatur VB ped. 130. 26″.
Quoniam autem angulus CBR notus erat gr. 91. 27′, &longs;i de
matur ex illo angulus VBC gr. 0. 52′. 46″. remanet VBR
gr. 90. 34′, 14″, æqualis angulo CVB alterno inter parallelas;
& nota &longs;unt latera illum con&longs;tituentia BR ped 3. 25″. & BV
ped. 130. 26″. Ex quibus datis invenitur angulus BRV gr. 88.
0′. 2″, BVR gr. 1. 25′. 44″ & ba&longs;is VR ped. 130. 326‴.
Jam verò, ex prop. 15 lib.1. Æquipond. Archimedis, divi
datur VR in S eâ ratione, ut &longs;it VS ad SR, ut duplum EB
majoris parallelarum unâ cum minore HC, ad duplum HC
unâ cum majore EB, hoc e&longs;t (quia EB e&longs;t ped. 6 1/2) & HC
ped.4.) ut 17 ad 14 1/2. Igitur ut 31 1/2 ad 14 1/2, ita VR 130. 326‴,
ad SR ped. 59. 99″. Demum ex S ducta perpendiculari SX,
quia in triangulo RXS rectangulo datur angulus SRX gr.88.
0′.. 2″. atque adeò ejus complementum RSX gr.1. 59′. 58″. &
latus SR ped. 59. 99″. invenitur latus RX ped. 209″. E&longs;t igi
tur RX linea minor, quàm RB po&longs;ita ped. 3. 25″; & idcirco
perpendicularis linea directionis SX cadit intrà ba&longs;im parie
tis EBCH.
Sed quia facturum me puto rem aliquibus gratam, &longs;i quas
inij rationes hîc exhibeam, calculi totius progre&longs;&longs;um per lo
garithmos hîc addo, ut illum po&longs;&longs;is, &longs;i placeat examinare.
Quod &longs;i paries exteriùs inclinatus etiam &longs;olitarius con&longs;i&longs;tere
po&longs;&longs;et, modò ea e&longs;&longs;et partium connexio, ut unum quid &longs;oli
dum conflarent, quia directionis linea intra ba&longs;im &longs;u&longs;tentan
tem cadit, & planum per extremam ba&longs;is lineam, & terræ cen
trum tran&longs;iens relinquit interiorem parietis partem præponde
rantem exteriori: quis po&longs;&longs;it de turris ruinâ dubitare, &longs;i eâdem
methodo deprehendat oppo&longs;iti parietis AG centrum gravita
tis e&longs;&longs;e in O, ac proinde comparatis reliquorum duorum pa
rietum centris gravitatum, totius turris centrum gravitatis e&longs;&longs;e
in intimis turris partibus? Quò igitur firmiùs &longs;ibi cohærebunt
partes turris, eò major erit inclinatio, quam obtinere pote&longs;t ci
tra cadendi periculum. Id quod pueris ip&longs;is noti&longs;&longs;imum e&longs;t,
qui turriculas inclinatas architectantur ex buxeis orbiculis,
quibus in alveolo ludunt.
Et ut res i&longs;ta plani&longs;&longs;imè o&longs;tendatur,
&longs;it &longs;upra planum inclinatum AB, pa
rallelepipedum ligneum ID ita, ut
recta CE ad horizontem perpendicu
laris tran&longs;eat per centrum gravitatis:
con&longs;tat ex dictis cap.
8. futurum e&longs;&longs;e,
ut grave ID repat, non autem rote
tur, quia pars CED non præponderat parti CEI, &longs;iqui
dem po&longs;&longs;it de&longs;cendere per planum inclinatum; quod &longs;i à lap
&longs;u impediatur, &longs;ub&longs;i&longs;tet. Jam verò intellige per C planum
FH horizontale, & adnecti pri&longs;ma trigonum CIK pa
rallelepipedo ID; utique pars CEK præponderat parti
CED, multóque minùs dubitandum erit de &longs;olidi KD rui
nâ ver&longs;us H. Quid autem aliud e&longs;t &longs;olidum KD, quam tur
ris inclinata?
Scrip&longs;eram hæc jam tum ab anno labentis &longs;æculi quinquage
&longs;imo &longs;exto; cum animum &longs;ubiit &longs;u&longs;picari, an &longs;uperiùs allatæ ex
Ma&longs;ino turris Bononien&longs;is men&longs;uræ omninò veritati re&longs;ponde
rent. Quare litteris ad P. Franci&longs;cum Mariam Grimaldum da
tis rogavi, ut pro eâ, quam ad res omnes conferre &longs;olebat, di
ligentiâ, accuratè men&longs;uras illas inquireret: hæc igitur ex ejus
re&longs;pon&longs;ione habui, quibus &longs;uperiùs dicta corrigenda &longs;unt; quæ
tamen expungere nolui, ut &longs;i lubeat, vulgarem opinionem &longs;e
qui valeas.
Extimus turris ambitus tam in imâ, quam in &longs;upremâ parte
æqualis e&longs;t, adeò ut oppo&longs;itæ facies parallelæ excurrant: &longs;in
gulorum autem laterum ad ba&longs;im latitudo e&longs;t ped. Bonon. 17.
unc. 8. murorum cra&longs;&longs;ities in imo æqualis e&longs;t; eo tantum di&longs;
crimine, quod murus, qua parte o&longs;tium patet, cra&longs;&longs;us e&longs;t ped.5.
unc.11. qui verò Septentrionem &longs;pectat, propiùs accedit ad pe
des 6. Porrò in &longs;ummâ turri murorum cra&longs;&longs;ities pariter æqualis
e&longs;t, & vix deficit à pedibus 5, quantum quidem ex a&longs;pectu à
&longs;uperiori proximæ turris A&longs;inellæ podio conjicere potuit &longs;ingu
lorum murorum lateres numerans. Areæ demum vacuæ ad ba
&longs;im latus unum e&longs;t ped. 6. alterum ped.6. unc.1.
Cum autem pluviæ per hiantem, & patulum turris verticem
deciduæ &longs;calas corruperint, nec eò veniri po&longs;&longs;it, ut demi&longs;&longs;o
perpendiculo altitudo turris inve&longs;tigetur, &longs;ub&longs;idium peten
dum fuit ex Trigonometriâ, & ex proximâ turri A&longs;inellâ, cu
jus men&longs;uræ multiplici ob&longs;ervatione innotuerant. Sit itaque
turris inclinata DC, &longs;uperioris autem podij
A&longs;inellæ altitudo EB ped.234 1/2, unde ob&longs;er
vatus e&longs;t angulus CEB gr. 18. 40′. Item in
eadem turri A&longs;inellâ patet fene&longs;tra in F, adeò
ut di&longs;tantia EF &longs;it ped.141: ibi pariter ob&longs;er
vatus e&longs;t angulus EFC gr. 51. 51′. Quare in
triangulo CEF, notum e&longs;t latus EF, & duo
anguli adjacentes, ex quibus datis colligi
tur EC di&longs;tantia ped. (117 7/12). Jam verò intelli
gantur ex C cadere duæ perpendiculares, al
tera quidem CH in planum horizontale, alte
ra verò CG in turrim A&longs;inellam; erit enim al
titudo CH æqualis altitudini GB, nam CG
e&longs;t parallela horizonti, cui turris EB perpen
dicularis in&longs;i&longs;tit. Ut igitur innote&longs;cat quæ&longs;i
ta altitudo, inveniatur in triangulo rectangu
lo CGE, ex datis latere CE ped. (117 7/12) &
angulo ob&longs;ervato CEG, gr.18.40′, latus EG
ped. (111 5/12). Jam verò &longs;i EG ped.(111 5/12) dematur
ex EB ped. 234 1/2, remanet altitudo GB, hoc e&longs;t CH,
ped. (123 1/12).
Demum ad inve&longs;tigandam turris inclinationem, applicito
ad punctum I perpendiculo ob&longs;ervatus e&longs;t angulus DIL
gr. 3. 10′.: cùm autem IL parallela &longs;it perpendiculari CH, erit
pariter angulus DCH gr.3.10′. Igitur in triangulo DCH
rectangulo ad H notum e&longs;t latus CH ped.(123 1/12), & angulus
DCH gr.3.10′, ergo & innote&longs;cit latus DH ped.6. (10/12), quæ e&longs;t
men&longs;ura inclinationis quæ&longs;itæ.
Ex his accuratioribus men&longs;uris indagemus, &longs;i placet, in
orientali pariete inclinato centrum gravitatis, & lineam di
rectionis methodo eâdem, qua &longs;uperiùs u&longs;i &longs;umus; eademque
figura &longs;ectionis verticalis re&longs;umatur. E&longs;t igitur EB ped. 6. ac
propterea RB ped. 300″; & quia HC e&longs;t ped. 5, VC e&longs;t
ped.2. 50″. BD autem e&longs;t ped. 6. unc.10, hoc e&longs;t ped.(6 10/12).
In Triangulo BDC rectangulo datis BD
ped. 6. (10/12), & altitudine perpendiculari CD
ped. (123 1/12), additis laterum quadratis fit qua
dratum hypothenu&longs;æ BC, quæ e&longs;t ped.123.27″.
Fiat igitur ut CB ped. 123. 27″, ad BD
ped. 6. 83″. ita Radius ad &longs;inum anguli BCD
gr. 3. 10′ 34″. Quare angulus reliquus CBD
gr. 86. 49′. 26″, cui æqualis e&longs;t alternus VCB
inter parallelas VC, RD; angulus autem,
qui e&longs;t deinceps, CBR gr. 93. 10′. 34′.
In
triangulo VCB datis lateribus VC ped.2-50″,
CB ped. 123. 27″, & angulo verticali VCB
gr. 86. 49′. 26″, reperitur CVB gr. 92. 0′.
36″,
& VBC. gr. 1. 9′, 58″. Ex his verò invenitur
VB ped. 122. 76″.
Jam verò in Triangulo VBR, notus e&longs;t
angulus RBV æqualis alterno CVB gr.92.
0′. 36′.
& nota &longs;unt latera RB ped. 300″, &
VB ped. 122. 76″. Quare invenitur angulus
VRB gr. 86. 35′ 43″. BVR gr. 1. 23′. 41″, & ba&longs;is VR
ped. 123. 17″.
Tum fiat ut 17 ad 16; hoc e&longs;t duplum majoris EB cum mi
nore HC, ad duplum minoris HC cum majore EB, ita VS
ad SR, & erit SR ped.59.72″. Ductâ igitur ex S centro gra-
ped. 59. 72″, & angulo VRX gr. 86, 35′, 43″, innote&longs;cit RX
ped. 3. 54″. Quare RX major e&longs;t quàm RB: & &longs;i paries ille
&longs;olitarius e&longs;&longs;et, non utique con&longs;i&longs;teret; &longs;ed quoniam reliqui
tres parietes adjecti &longs;unt, con&longs;tat ita totius molis centrum gra
vitatis e&longs;&longs;e in intima turris parte, ut linea directionis cadat in
trà turris ba&longs;im &longs;u&longs;tentantem.
Ex his di&longs;cuties timorem eorum, qui &longs;oliciti &longs;unt de obeli&longs;
corum con&longs;i&longs;tentiâ, ex inclinatione aliquâ verticis ruinam
proximam præ&longs;agientes: cum enim in huju&longs;modi molibus cen
trum gravitatis vicinius &longs;it ba&longs;i quàm vertici, &longs;i centrum incli
netur in alterutram partem &longs;patio tantùm digitali, vertex in
&longs;ignem acquiret inclinationem, con&longs;i&longs;tet tamen, quandiu linea
directionis tran&longs;ibit per ba&longs;im &longs;u&longs;tentationis. Inclinatio enim
non e&longs;t &longs;patium illud, quod inter ba&longs;im, & perpendiculum à
turris, vel obeli&longs;ci vertice demi&longs;&longs;um intercipitur (quamvis hoc
vocabulo hactenus abuti placuerit, ne à vulgo di&longs;creparem)
&longs;ed e&longs;t angulus, quem turris facit cum plano; & manente ea
dem inclinatione, intervallum illud mutari pote&longs;t pro majore,
aut minore turris longitudine. Quare quò longior e&longs;t moles in
clinata, cæteris paribus, minùs e&longs;t timendum, quia minor e&longs;t
declinatio à perpendiculari: &longs;i enim KE &longs;it pedum 100, KC
verò ped.1. angulus KEC æqualis declinationi à perpendiculo
e&longs;t gr. 0. 34. 22″. at &longs;i KE &longs;it ped. 50, & KC iterum ped. 1.
angulus KEC e&longs;t grad. 11. 32′. 13″.
Hîc autem qua&longs;i præteriens &longs;atisfaciam quærenti, cur lon
giores ha&longs;tas faciliùs, quàm breviores virgas digiti extremitate
&longs;u&longs;tineamus, quin cadant. Quia nimirum minimus angulus
declinationis à perpendiculo &longs;tatim &longs;e prodit ha&longs;tæ vertice ad
partem unam &longs;ecedente, cui &longs;tatim occurrimus ha&longs;tæ calcem
manu transferentes, ac &longs;ub vertice collocantes: verùm quia fa
cilior ha&longs;tæ con&longs;i&longs;tentia innote&longs;cit etiam, quando à &longs;uppo&longs;itâ
manu calx ejus non movetur (nam &longs;i militarem &longs;ari&longs;&longs;am terræ
perpendiculariter in&longs;i&longs;tentem con&longs;titueris, potes te &longs;emel in gy
rum contorquere, & illam qua&longs;i perpendicularem recipere, id
quod in breviore ha&longs;tâ non obtinebis) alia e&longs;t ratio petenda
primùm ex dictis, quia &longs;cilicet longior ha&longs;ta, cæteris paribus,
minùs declinat à perpendiculo, ideóque difficiliùs de&longs;cendit;
majorem percipies re&longs;i&longs;tentiam, quàm &longs;i breviorem virgam in
citares; ita aërem variis &longs;emper motibus turbatum plus etiam
impedire de&longs;cen&longs;um longioris ha&longs;tæ cen&longs;endum e&longs;t, præ&longs;ertim
&longs;i in &longs;uperiore parte aër versùs unam, in inferiore autem versùs
aliam partem moveatur: id quod in breviore virgâ non accidit,
quam modicus aër contingit, nec pote&longs;t aut adeò re&longs;i&longs;tere di
vi&longs;ioni, aut adeò diver&longs;is motibus cieri. Hinc a&longs;ta longior
tardiùs de&longs;cen&longs;um molitur, & faciliùs &longs;u&longs;tinetur, quia major
aëris dividendi quantitas, ac motus varius, magis re&longs;i&longs;tit, &
datâ æqualitate motûs minùs declinat à perpendiculo.
&longs;ubjecta planities.
POte&longs;t mons cum &longs;ubjectâ planitie, cui in&longs;i&longs;tit, dupliciter
comparari; primùm conferendo &longs;olam planitiem in ver
tice montis exi&longs;tentem cum parte &longs;ubjecti plani &longs;ibi re&longs;
pondente; deinde clivum montis comparando cum plano
horizontali. Et &longs;anè &longs;i planities in &longs;ummo montis jugo con
&longs;ideretur, certum e&longs;t illam e&longs;&longs;e plurium &longs;tructurarum ca
pacem, quàm &longs;ubjectum planum in &longs;uperficie globi ter
re&longs;tris: Quemadmodum enim &longs;uperficies &longs;phæræ majoris
plura capit ædificia, quàm minor, ita etiam &longs;phærarum
inæqualium partes &longs;imiles inæqualis &longs;unt capacitatis: Con&longs;tat
autem planitiem in &longs;ummo monte pertinere ad &longs;phæram
majorem, quàm pertineat &longs;imilis planities illi &longs;ubjecta; ac
proinde & amplior e&longs;t, & magis capax. Harum verò pla
nitierum differentia ea erit, quæ e&longs;t quadratorum di&longs;tan
tiarum à centro terræ: quòd &longs;i quadratorum huju&longs;modi
differentia exigua &longs;it & contemnenda, eo quod ad illam
quadratum &longs;emidiametri terræ habeat nimis magnam ratio
nem; planitierum pariter differentia fugiet omnem &longs;en&longs;um.
tem montis SR, in cujus vertice &longs;it pla
nities RH, cui &longs;imilis e&longs;t in &longs;uperficie
globi terreni planities SO illi parallela:
hæ autem planities &longs;imiles habent, per
20. lib.
6. duplicatam Rationem laterum
RI, SL, hoc e&longs;t, per 4. lib.
6. duplica
tam Rationis, quam habet CR ad CS.
E&longs;t igitur ut quadratum di&longs;tantiæ CR.
ad quadratum di&longs;tantiæ CS, ita plani
ties RH ad planitiem SO. Plura itaque
ædificia perpendiculariter in&longs;i&longs;tentia
po&longs;&longs;unt in planitie RH majori excitari
in montis vertice, quàm in &longs;ubjectâ
plani tie.
At &longs;i montis clivus RMOL comparetur cum &longs;ubjectâ pla
nitie SO, certum e&longs;t illum e&longs;&longs;e majorem, &longs;icuti latus RL op
po&longs;itum angulo RSL, qui non e&longs;t minor recto, majus e&longs;t la
tere SL in triangulo RSL, & RM ad SF e&longs;t ut RC ad SC:
&longs;uperficies igitur LM comprehen&longs;a &longs;ub majoribus lateribus,
& angulis non minoribus, quàm &longs;uperficies SO, major erit,
&longs;i illa per &longs;e con&longs;ideretur. Non tamen continuò major dicenda
e&longs;t capacitas, quæ plura aut ampliora recipiat ædificia; ni&longs;i
mons ad ingentem altitudinem a&longs;cendat; tunc enim perpendi
cula non &longs;unt inter &longs;e parallela, propter in&longs;ignem eorum
di&longs;tantiam. Nam &longs;i &longs;uper clivo AB
&longs;it &longs;tructura AL, cujus parietes per
pendiculares, &longs;int etiam paralleli
LB, DA, illi non magis inter &longs;e
di&longs;tant, quàm &longs;i &longs;uper plano hori
zontali NB fui&longs;&longs;ent excitati: quic
quid &longs;it, quod, &longs;icut linea AB ma
jor e&longs;t quàm NB, ita planum incli
natum majus &longs;it plano horizontali.
Non igitur plures aut ampliores &longs;tructuras recipit clivus collis,
quàm &longs;ubjectum planum horizontale. Quod verò de &longs;tructuris
dicitur, de cæteris quoque intelligendum e&longs;t, quæ perpendi
cularia in&longs;i&longs;tunt, & &longs;patium implent; at &longs;i ita &longs;e habeant, ut
homines jacere po&longs;&longs;e in clivo AB, quos non capit planum NB:
vel &longs;i in clivo &longs;e minùs invicem impediant, tunc plura huju&longs;
modi corpora in colle e&longs;&longs;e po&longs;&longs;unt quàm in planitie: &longs;i enim ra
mi arboris inferioris re&longs;pondeant trunco &longs;uperioris, certum e&longs;t
quod multò viciniores e&longs;&longs;e po&longs;&longs;unt arbores, quàm in planitie,
ubi rami &longs;e vici&longs;&longs;im impedientes majorem po&longs;tulant truncorum
di&longs;tantiam; ac proinde etiam multo plures arbores intra ea&longs;
dem parallelas erunt. Sic plures homines e&longs;&longs;e po&longs;&longs;unt in gradi
bus amphitheatri, quàm in &longs;ubjecto plano, quia graciliores,
partes &longs;uperiorum re&longs;pondent cra&longs;&longs;ioribus inferiorum, & &longs;e
minùs invicem impedientes minus relinquunt &longs;patij vacui:
quod &longs;i non homines, &longs;ed parallelepipeda, &longs;tatueres in gradi
bus, non plura &longs;tatui in iis po&longs;&longs;ent, quàm in planâ areâ gradi
bus &longs;ubjectâ.
Hæc autem ædificiorum æqualitas in clivo & in plani
tie, locum non habet ni&longs;i intra illud &longs;patium, quod inter
cipitur à perpendiculis Phy&longs;icè parallelis; &longs;tatim enim ac à
paralleli&longs;mo recedunt perpendicula, &longs;i ea fuerit altitudo, ad
quam clivus a&longs;cendens venit, ut planities parallela plano
horizontali in eâ altitudine major &longs;it, quàm &longs;imilis plani
ties depre&longs;&longs;ior, etiam plura ædificia recipiet clivus, quàm
unica planities horizontalis &longs;ubjecta. Ponamus enim per
pendicula GC, & OC jam non e&longs;&longs;e parallela, eamque e&longs;&longs;e
altitudinem KG, ut planum per G tran&longs;iens horizonti
parallelum majus &longs;it plano per O intra eadem perpendicu
la intercepto, erit quidem capacitas plani inclinati GOLF
æqualis capacitati &longs;ubjecti plani EKOL: at ulteriùs a&longs;cen
dendo capacitas FGMR non erit æqualis capacitati plani
SK continuati cum priore plano EO, &longs;ederit major, quip
pe quæ æqualis e&longs;t capacitati plani VG; e&longs;t autem pla
num VG ad planum &longs;imile SK, ut quadratum GC ad
quadratum KC: major igitur e&longs;t totius clivi ML capacitas,
quàm planitiei SO.
Et ut res apertius con&longs;tet, quandoquidem clivi alti&longs;
&longs;imorum montium, &longs;i eandem &longs;ervent inclinationem, non
&longs;unt ab imo pede ad &longs;ummum jugum æquabili, & conti
nuo ductu exten&longs;i, Sit terræ centrum H, & &longs;uperficies
tes AB, BC, CD æquales, ita ut
&longs;inguli arcus pro rectâ lineâ, & &longs;u
perficies pro plano horizontali
Phy&longs;icè u&longs;urpari po&longs;&longs;int; & tunc
&longs;olùm intelligatur mutari horizon,
quando ex A jam venerit in B,
deinde in C &c. Si igitur &longs;it pla
num inclinatum AE, ubi venerit
in E punctum perpendiculi HB
producti, non pote&longs;t rectâ progre
di, quin mutet inclinationem &longs;upra horizontem novum, ad
quem venit; quare ut &longs;ervetur &longs;imilis inclinatio, deflectit in EF,
& e&longs;t angulus HEF æqualis angulo HAE cui demum ubi ve
nerit in F, debet fieri æqualis angulus HEG. Centro autem H,
intervallis HE & HF de&longs;cribantur arcus EI, & FK. Certum
e&longs;t duarum linearum angulum con&longs;tituentium partem aliquam
extremam e&longs;&longs;e, &longs;ecundùm quam lineæ illæ non differunt, &longs;en&longs;u
judice, à parallelis; at &longs;i major pars accipiatur, jam perit paral
leli&longs;mus: Sic RA, & EB pro parallelis u&longs;urpari &longs;i po&longs;&longs;int, non
poterunt &longs;imiliter pro parallelis accipi RA, & LB: Sic LE, &
FI &longs;umuntur tanquam parallelæ citrà errorem, at non item LB,
& MC. Quare perpendicula non &longs;olùm recedunt à paralleli&longs;
mo &longs;en&longs;ibili, quia majorem angulum in centro H con&longs;tituunt,
&longs;ed etiam quia major eorum pars a&longs;&longs;umitur, in qua jam apparet
convergentia, quæ in parte minore latebat.
Cum itaque &longs;tructuræ perpendiculares in plano inclinato
occupent &longs;patium eodem modo, ac &longs;i e&longs;&longs;ent in plano horizon
tali intra ea&longs;dem parallelas, jam con&longs;tat clivi partem EF com
parandam e&longs;&longs;e cum plano EI, non autem cum plano BC; quia
in E, & I terminatur paralleli&longs;mus linearum LE, FI. E&longs;t igi
tur capacitas clivi EF æqualis capacitati EI; at capacitas EI
major e&longs;t quàm capacitas BC, ergo capacitas clivi AF major
e&longs;t, quàm capacitas planitiei AC. Eademque e&longs;to de cæteris
ratio. Hinc manife&longs;tum e&longs;t non omninò in univer&longs;um vera e&longs;&longs;e,
quæ pa&longs;&longs;im dicuntur de æquali capacitate collium, & planitiei
&longs;ubjectæ, ni&longs;i hæc certis limitibus circum&longs;cribantur; videlicet
&longs;i &longs;ermo &longs;it de iis quæ tantùm perpendiculariter in&longs;i&longs;tunt, &
convergentia adeò exigua e&longs;t, ut evane&longs;cat. Cæterùm &longs;atis
mihi videor o&longs;tendi&longs;&longs;e fieri po&longs;&longs;e, ut clivus aliquis plures
&longs;tructuras recipere po&longs;&longs;it, quàm &longs;uperficies &longs;phærica globi illi
re&longs;pondens. Si enim eadem e&longs;t &longs;emper, ut &longs;upponitur, plani
inclinatio, etiam latera turrium, vel domorum parietes æquè
invicem remoti intercipient æquales partes plani inclinati: Si
ergo &longs;tructura intercipiens &longs;emi&longs;&longs;em plani AE transferatur in
EF, æqualem partem intercipiet; at hæc minor e&longs;t &longs;emi&longs;&longs;e
ip&longs;ius EF, igitur duæ &longs;tructuræ occupantes totum planum AE,
tran&longs;latæ in EF æquale &longs;patium occupabunt, & relinquent
adhuc partem &longs;patij inanem. E&longs;&longs;e autem EF lineam majorem
linea AE patet; quia triangula AHE, EHF æquiangula
&longs;unt, & latera habent proportionalia, adeóque ut AH ad HE,
ita AE ad EF; atqui HE excedit lineam HA; igitur & EF
major e&longs;t quàm AE: ergo multo major erit &longs;uperficies ip&longs;ius
EF, quàm &longs;uperficies &longs;imilis ip&longs;ius AE. In &longs;patio igitur, quo
&longs;uperficies EF excedit &longs;uperficiem AE, poterit alia præterea
&longs;tructura excitari.
gravitatis.
DEi &longs;apientiam nunquam &longs;atis admirari po&longs;&longs;umus, quæ in
ordinandis naturæ motibus elucet; animalia enim &longs;olo
naturæ ductu adeò accuratè &longs;e ip&longs;a &longs;i&longs;tunt in lineâ directionis,
ut nemo mathematicus Geometriæ apices per&longs;crutatus po&longs;&longs;it
tam &longs;ubtiliter deprehendere, ac brevi&longs;&longs;imo temporis momento,
centrum gravitatis. Quandoquidem &longs;ive con&longs;i&longs;tentium quie
tem, &longs;ivè gradientium motum, &longs;ivè reclinantium &longs;e &longs;e inflexio
nem con&longs;ideres, miram naturæ artem intelliges, quâ præcavit,
ne corpus ingenitâ gravitate delatum præceps caderet. Id au
tem a&longs;&longs;ecuta e&longs;t motus ita di&longs;ponendo, ut linea directionis nun-
quo tamen &longs;atis con&longs;ultum e&longs;t animalis incolumitati, dum ab
anteriore pede, ubi terram attigerit, retinetur, ne ulteriùs
de&longs;cendat.
Ba&longs;is autem &longs;u&longs;tentationis non &longs;unt &longs;oli pedes, &longs;ed totum
illud &longs;patium interceptum à lineis pedum extremitates jun
gentibus; &longs;ic in quadrupedibus linea directionis debet cadere
intrà &longs;patium comprehen&longs;um lineis, quæ jungunt extrema
pedum terram contingentium, ut po&longs;&longs;it animal con&longs;i&longs;tere.
Hinc equus in po&longs;teriores pedes &longs;e erigens flexis poplitibus
reclinat &longs;e &longs;e in po&longs;teriora, & tanti&longs;per in eo &longs;itu con&longs;i&longs;tit,
dum centrum gravitatis imminet &longs;patio, quod à pedibus oc
cupatur, & ab illis intercipitur; & &longs;i extra illud &longs;patium ca
dat linea directionis, vel aver&longs;us cadit, vel iterum quatuor
pedibus in&longs;i&longs;tit. Ubi tamen ob&longs;ervandum e&longs;t ex equo & equi
te fieri unam molem compo&longs;itam unum habentem commune
centrum gravitatis: unde fit equum magis defatigari, &longs;i eques
non rectus in&longs;ideat; &longs;ed inclinatus in alterutram partem, cen
tro enim gravitatis tran&longs;lato motûs facilitas mutatur; & equite
in anteriora inclinato ac premente caput equi in po&longs;teriores
pedes erecti, centrum gravitatis in anteriora transfertur, &
occurritur periculo, ne equus aver&longs;us cadat.
Porrò dum &longs;patium à pedibus occupatum voco ba&longs;im &longs;u&longs;ten
tationis, non &longs;emper &longs;atis e&longs;t lineam directionis cadere non
extrà pedes; quia &longs;i pedes ip&longs;i &longs;olùm ex parte tangant &longs;ub
jectum corpus, ut contingit in funambulis, debet linea di
rectionis cadere in funem, cui in&longs;i&longs;tunt pedes, & &longs;i extra il
lum cadat, certa e&longs;t ruina, quia latitudo pedum non juvat.
Cum autem difficillimum &longs;it diutiùs con&longs;i&longs;tere ita, ut centrum
gravitatis &longs;emper immineat funi, ideò funambuli, vel ha&longs;tam
plumbeis laminis gravem in extremitatibus manu tenent, vel
brachiis expan&longs;is &longs;e librant, ut ha&longs;tam vel brachia extenden
tes in partem oppo&longs;itam ei, in quam gravitas inclinat, cen
trum gravitatis con&longs;tituatur in puncto, quod immineat funi
&longs;ui tentanti. Hinc oritur difficultas con&longs;i&longs;tendi, quam expe
riuntur grallatores; cum enim grallæ exiguâ &longs;ui parte tangant
terram, e&longs;t qua&longs;i linea, in qua fit &longs;u&longs;tentatio, extra quam fa
cilè cadit linea directionis: ideò tertium ge&longs;tant baculum, cui
dente intrà &longs;patium triangulare comprehen&longs;um à grallis, &
baculo.
Hîc autem maximè &longs;e prodit naturæ providentia in tam va
riâ pedum conformatione, ut ad &longs;u&longs;tentandum idonei e&longs;&longs;ent:
quadrupedibus &longs;iquidem non adeò amplos pedes tribuit, quia
ex eorum inter &longs;e di&longs;tantiâ plurimum &longs;patium intercipitur, cui
immineat centrum gravitatis: bipedibus verò latiores tribuit
pedes, quâ parte timeri potuit ca&longs;us: &longs;ic quia ex duorum cru
rum modicâ divaricatione non facilè periculum erat cadendi
in alterutrum latus, ideò humanis pedibus minorem dedit la
titudinem, quàm longitudinem; hanc verò non in æquas
di&longs;tribuit partes, &longs;ed minimam calci (præterquam in Scauris,
quos pravis fultos male talis appellat Horatius, talis &longs;cilicet
extantioribus) maximam anteriori parti conce&longs;&longs;it, ne impetu
per motum concepto tran&longs;latum centrum gravitatis in anterio
ra tran&longs;iliret ba&longs;im &longs;u&longs;tentationis. Aliquam tamen mediocrem
latitudinem pedibus conce&longs;&longs;it, ut po&longs;&longs;et homo, &longs;i res ferret, uni
tantùm pedi in&longs;i&longs;tere, & e&longs;&longs;et aliqua &longs;patij amplitudo, intrà
quam quodlibet punctum opportunum e&longs;&longs;et con&longs;i&longs;tentiæ cen
tri gravitatis. Sic aves illæ, quæ uni pedi in&longs;i&longs;tunt, cuju&longs;modi
&longs;unt grues, & ciconiæ, digitos habens longiores, quos valdè
explicant qua&longs;i in gyrum, ut amplior &longs;it ba&longs;is &longs;u&longs;tentationis; in
trà quam ut cadat linea directionis, altero pede elevato inclina
tur corpus in oppo&longs;itam partem, ut centrum gravitatis immineat
pedi &longs;u&longs;tentanti. Eandem ob cau&longs;am an&longs;eres, & anates, quæ
multâ carne abundant, & amplo &longs;unt pectore, alternâ qua
dam in dextrum, & &longs;ini&longs;trum latus inclinatione gradiuntur,
ideóque ampliores habent palmas, ut citrà cadendi periculum
centrum gravitatis faciliùs vel immineat pedi &longs;u&longs;tentanti, vel
minimùm ab eo declinet, ne majore, quàm par &longs;it, impetu
de&longs;cendens corpus & anteriori pedi incumbens, tibiæ mu&longs;cu
los, & tendines lædat. Aves verò, quæ &longs;ubtilioribus ramu&longs;cu
lis in&longs;ident non palmipedes &longs;unt, &longs;ed digitatæ (palmæ enim
avibus amphibiis ad natandum poti&longs;&longs;imum datæ videntur) ut
ramis tenaciùs inhæreant; quæ præterquàm quod exiguæ &longs;unt
gravitatis, facilè &longs;e &longs;i&longs;tunt in lineâ directionis, quæ cadat in
ramu&longs;culum, cui in&longs;i&longs;tunt, majore, vel minore angulo, quem
tantes &longs;e librant, ramumque arctè apprehentes prohibent, ne
repentino ca&longs;u circumagantur à centro gravitatis nondum im
minente ba&longs;i &longs;u&longs;tentationis.
Verùm quoniam ad aves delap&longs;us &longs;um, prætereundus non
e&longs;t u&longs;us centri gravitatis involatu; quia enim avis dum alis
aërem verberans in volatu &longs;e librat atque &longs;u&longs;pendit, ita alas
debet extendere, ut centrum gravitatis exi&longs;tat intra illud
alarum &longs;patium, in quo exercetur &longs;u&longs;tentatio; ideò &longs;i vo
luerit ad &longs;uperiora volatum dirigere, alas in anteriora ver
&longs;us caput extendit, ut centro gravitatis in po&longs;terioribus re
licto, ac deor&longs;um præponderante, caput &longs;ur&longs;um dirigatur:
contra verò, ut motum deor&longs;um dirigat, alas retrahit, ut
caput præponderet, ac deor&longs;um feratur. Hinc &longs;atis patet,
cur ubi Pavo caudæ pompam explicuerit, erecto pectore &
capite in&longs;i&longs;tat pedibus, quibus immineat centrum gravita
tis: at &longs;i caput ad anteriora inclinare voluerit, & pectus
inflectere, cogitur explicatam caudam demittere, ut &longs;yrma
te illo æquilibrium &longs;tatuat corpori, ne proruat, ut verè pro
cumberet, &longs;i pectore inclinato expan&longs;a cauda retineretur in
po&longs;itione eâdem.
Infinitum e&longs;&longs;et &longs;ingulos animalium motus per&longs;equi, in qui
bus centri gravitatis ratio habetur; &longs;atis fuerit ob&longs;erva&longs;&longs;e nos
ex declivi loco de&longs;cendentes non in&longs;i&longs;tere plantis pedum ad
angulos rectos; &longs;ed paululum in po&longs;teriora inclinari; contra
verò a&longs;cendentes jugum acclive curvari in anteriora; ut nimi
rum linea directionis cadat intrà &longs;patium, cui pedes in&longs;i&longs;tunt;
extra quod illa &longs;i caderet, nec alteri fulcro inniteremur, quod
unà cum pedibus includeret ba&longs;im &longs;u&longs;tentationis, nece&longs;&longs;ariò
nobis cadendum e&longs;&longs;et. Quòd &longs;i quis onus habens dor&longs;o impo
&longs;itum in montosâ regione iter habeat, multò magis curvari de
bet, cum a&longs;cendit, ut pedibus immineat centrum gravitatis
compo&longs;itæ ex corpore, & ex onere: quare &longs;apienti&longs;&longs;imè ru&longs;tici
aliqui in Alpibus, quæ Germaniam ab Italiá di&longs;terminant, ar
culam ex levibus a&longs;&longs;erculis, & virgulis compactam habent, cui
onera immittunt, ba&longs;is autem arculæ, quæ ge&longs;tantis corpori
adhæret, imitatur Re&longs;c Hebraicum, ita ut pars quidem dor
&longs;o, pars autem capiti incumbat: unde fit, ut centrum gravita-
que faciliùs etiam motus perficiatur, quin opus &longs;it tantâ corpo
ris inflexione. Simile quid experimur, &longs;i quis à &longs;ede &longs;urgat;
caput enim cum thorace in anteriora reclinat; pedes verò in
po&longs;teriora versùs &longs;edem retrahit, ut nimirum pedes &longs;upponan
tur centro gravitatis, quod primùm imminet parti digitis proxi
mæ, deinde corpore erecto linea directionis versùs talos rece
dit. Hinc etiam patet cur homo &longs;upinus jacens &longs;urgere non
po&longs;&longs;it, ni&longs;i retractis &longs;ub &longs;e pedibus, & thorace in anteriora pro
pul&longs;o per impetum &longs;ibi impre&longs;&longs;um. Vidi tamen non &longs;emel ho
minem, qui cum &longs;upinus jaceret, non retractis &longs;ub &longs;e pedibus
&longs;urgebat planè rectus &longs;icut &longs;tipes; ad caput autem appone
bat, vel globum tormentarium majorem, vel &longs;axum non
modicæ gravitatis; quod manu utrâque apprehen&longs;um attol
lebat, & velociter in anteriora movebat, &longs;ibique impetum
imprimebat: impetus enim impre&longs;&longs;us promovens ad ante
riora &longs;axum, & corpus ip&longs;um vincebat gravitatem corpo
ris cæteroqui ca&longs;uri; ex brachiis autem exten&longs;is &longs;axum à
corpore remotum tenentibus oriebatur, ut centrum gravi
tatis molis compo&longs;itæ longè citiùs immineret pedibus, à
quibus &longs;u&longs;tentabatur, etiam antequam planta terram at
tingeret, &longs;ed cum adhuc &longs;oli calci inniteretur. Quantum
verò impetus valeat ad vincendam oppo&longs;itam gravitatem
corporis, patet in ce&longs;pitantibus, qui naturæ ductu illico bra
chia extendunt, & in contrariam partem projiciunt, ut &longs;ci
licet impetus in oppo&longs;itam partem exæquet exce&longs;&longs;um gravita
tis, quæ ad eam partem reperitur, in quam ex ce&longs;pitatione
facta e&longs;t inclinatio.
Ex his quid in &longs;ingulis motibus dicendum &longs;it, intelli
ges; neque enim otium e&longs;t ire per &longs;ingula. Caput hoc
claudo explicatione quæ&longs;tionis, qua quæritur, quantò ma
jus &longs;patium percurrat caput quàm pedes; certum &longs;iquidem
e&longs;t hominem in lineâ directionis imminere &longs;emper terræ
centro; ac proinde &longs;i pedes ex B venerunt in C, caput ex
F in E tran&longs;latum e&longs;t per arcum FE majorem arcu BC.
Cum enim uterque arcus BC, FE &longs;ubtendatur eidem an
gulo ad centrum, &longs;unt &longs;imiles, & ut arcus BC ad totam
&longs;uam peripheriam, ita arcus FE ad &longs;uam peripheriam; &longs;unt
diametri, igitur BC ad FE, ut TB,
ad TF; atqui TF major e&longs;t quàm
TB, igitur & FE arcus major arcu
BC: ab&longs;cindatur FI, quæ ex hypo
the&longs;i intelligatur æqualis ip&longs;i BC;
e&longs;t igitur ut TB ad TF, ita FI ad
FE, & dividendo ut TB ad BF
ita FI, hoc e&longs;t BC, ad IE. Fiat ita
que ut TB &longs;emidiameter terræ mil
liar. Rom. ant.4128.pa&longs;&longs;.635. ad BF
altitudinem hominis ex. gr. ped. Rom. ant.
6. ita BC iter pe
dum mill. 500, ad IE exce&longs;&longs;um itineris capitis qui e&longs;t (726632/1000000)
unius pedis. Quòd &longs;i fiat ut terræ &longs;emidiameter ad hominis al
titudinem, ita circulus terræ maximus mill. 25941 ad exce&longs;
&longs;um itineris capitis &longs;upra iter pedum terræ ambitum percurren
tium, proveniet exce&longs;&longs;us ped. 37. unc.8. hoc e&longs;t pa&longs;&longs;.7. & pau
lò ampliùs: Quare vides in &longs;ingulis milliariis motum capitis non
habere exce&longs;&longs;um ni&longs;i partium (17429/1000000) unciæ pedis Romani anti
qui; quæ differentia &longs;en&longs;um omnem fugit.
Liceat hic ex morâ, quam in hoc Tractatu perficiendo duxi,
id utilitatis capere, quod po&longs;&longs;im pro me ip&longs;e brevi Apologiâ
re&longs;pondere, ne videar in Ageometriam lap&longs;us, cui nulla ni&longs;i ex
o&longs;citantiâ &longs;uppeteret excu&longs;atio (nam & quandoque bonus dor
mitat Homerus) & quidem tunc, cùm Mathematicas di&longs;cipli
nas in Collegio Romano publicè pro&longs;itentem maximè ocula
tum fui&longs;&longs;e oportuerat. Incidi in Magiam Naturalem P. Ga&longs;paris
Schotti part.3.lib.1. pag. 71, ubi mihi tribuit &longs;ententiam maxi
mè ab&longs;urdam, qua&longs;i in mechanicâ meâ manu&longs;criptâ (quam
&longs;cilicet anno 1653. Romæ auditoribus meis tradidi) docuerim
exce&longs;&longs;um motûs capitis &longs;upra motum pedum
cum, nimirum &longs;olum pedum &longs;ex cum dimidio, adeò ut in milliaribus
500
dine pedum &longs;ex, & terræ ambitu milliariorum
mùm attonitus, meamque o&longs;citantiam admiratus illicò anti
quàs illas meas &longs;chedulas per&longs;crutari cœpi; & nihil minus in
veniens errorem Typographo, qui pro pa&longs;&longs;ibus pedes &longs;uppo-
lum exce&longs;&longs;um pedum &longs;ex cum dimidio redargueret. Quare
contingere facile potuit, ut ille, qui tunc Romæ degebat, ex
aliquo manu&longs;cripto codice meam &longs;ententiam re&longs;cribens, ubi
men&longs;uram hanc pedibus definiebam, brevitatis ergo ad pa&longs;
&longs;us revocaverit, quam litera P notatam demùm pro pedibus &longs;it
interpretatus. Cæterùm prudens, & attentus lector me facilli
mè ab hoc errore vindicabit, &longs;i terræ ambitum mill. 21600. di
vidat per mill.500; & quotientem 43 multiplicet per (15/17) unius
pedis; deprehendet enim totum exce&longs;&longs;um pedum ferè 38, qui
excedunt pa&longs;&longs;us &longs;eptem cum dimidio. Quod &longs;i ex diametro pe
dum 34400000, & ex diametro pedum 34400012, quas ibi Au
thor ponit congruentes peripheriæ juxta Rationem 7 ad 22 con
&longs;iderentur, erit differentia circulorum pedum 38 eadem plane
cum no&longs;trâ; &longs;ed longi&longs;&longs;imè minor eâ, quam ille ibi &longs;tatuit.
Cæterùm quantus &longs;it peripheriæ majoris exce&longs;&longs;us &longs;upra mi
norem, habebitur facillimè, &longs;i majoris Radij TF, exce&longs;&longs;um
BF, &longs;tatuas tanquam circuli Radium; hujus namque circuli
peripheria e&longs;t æqualis exce&longs;&longs;ui illi. Quia enim ut minor Ra
dius TB ad majorem Radium TF, ita minor peripheria ad
majorem peripheriam, etiam convertendo & dividendo, ut
TB ad BF, ita minor peripheria ad exce&longs;&longs;um peripheriæ ma
joris, & vici&longs;&longs;im permutando ut Radius TB minor ad &longs;uam
minorem peripheriam, ita BF exce&longs;&longs;us Radij majoris ad exce&longs;
&longs;um majoris peripheriæ. Atqui exce&longs;&longs;us hic BF a&longs;&longs;umptus ut
Radius circuli habet ad &longs;uam peripheriam eandem Rationem,
quam TB Radius minor ad &longs;uam peripheriam; igitur e&longs;t ea
dem Ratio BF exce&longs;sûs Radij, ad exce&longs;&longs;um peripheriæ majo
ris, quæ e&longs;t eju&longs;dem BF ut Radij ad &longs;uam peripheriam: ergo
per 9. lib.
5. hæc peripheria æqualis e&longs;t illi exce&longs;&longs;ui periphe
riæ majoris. Cum itaque Ratio diametri ad peripheriam &longs;it ut
7 ad 22, &longs;eu ut 113 ad 355, fiat ut Radius 7 ad peripheriam
44, &longs;eu ut 113 ad 710, ita BF altitudo ped. 6. ad ped. 37.
unc. 8: qui numerus con&longs;entit cùm &longs;uperiore.
QUoniam centrum gravitatis e&longs;t in quolibet corpore
punctum illud, quod æquales gravitates circum&longs;tant,
manife&longs;tum e&longs;t non permanere idem gravitatis centrum, &longs;i
aliqua corpori additio fiat, aut detractio; neque enim manet
eadem momentorum gravitatis æqualitas circa illud punctum;
&longs;ed aliud e&longs;t punctum, per quod ducta plana dividunt totius
corporis gravitatem in momenta æqualia, & e&longs;t novum cen
trum gravitatis. Hinc patet in telluris globo, qui plurimas
mutationes &longs;ubit, corporibus gravibus ex alio in alium locum
tran&longs;latis, tolli æqualitatem partium &longs;altem in actu primo gra
vitantium, cum hæc quidem, quæ oppo&longs;itæ parti ante erat
æqualis, &longs;ubtractione nunc fiat minor, illa verò, quæ pariter
&longs;ibi oppo&longs;itæ parti proximè fuit æqualis, additione evadat ma
jor. Ex quo nece&longs;&longs;ariò colligitur mutatio centri gravitatis.
Sed quia, ut tellus &longs;uis librata ponderibus in loco &longs;ibi debi
to con&longs;i&longs;teret, debuit initio ejus centrum gravitatis congrue
re centro univer&longs;i, circa quod gravia & levia di&longs;ponuntur; id
circò dubitari pote&longs;t, utrùm mutato gravitatis centro terra mo
veri debeat, ut novum gravitatis centrum collocetur in centro
univer&longs;i. Quoniam verò huc illuc pa&longs;&longs;im tran&longs;latis corpori
bus, terra nunc in hanc, nunc in illam partem moveretur, ut
proinde qua&longs;i trepidaret; hinc factus e&longs;t quæ&longs;tioni locus, an
tellus moveatur motu trepidationis; quicquid &longs;it an motus i&longs;te
&longs;ub &longs;en&longs;um cadat, nec ne.
Terram univer&longs;am & &longs;ingulas ejus partes &longs;uâ gravitate re
pugnare, ne &longs;ur&longs;um moveantur, certum e&longs;t; at univer&longs;i cen
trum occupare, toti quidem elemento gravi&longs;&longs;imo convenit, &longs;ed
non partibus &longs;ingulis: neque enim gravitas e&longs;t appetitus &longs;ub
&longs;i&longs;tendi in centro, quem natura non &longs;atis aptè gravibus &longs;ingu
lis indidi&longs;&longs;et; cui nimirùm fieri &longs;atis non pote&longs;t, ni&longs;i corpora
&longs;e invicem penetrent; unum autem grave in centro exi&longs;tens Re&longs;tituunt &longs;e gravia in locum
&longs;uum versùs centrum pergendo, non ut ad centrum veniant;
&longs;ed ut nihil levius infra &longs;e habeant; quemadmodum & levia
versùs cælum a&longs;cendunt, non ut cælum petant, ibíque demum
quie&longs;cant, &longs;ed ne quid gravius &longs;upra &longs;e patiantur. Cæterùm
hoc ip&longs;o, quòd natura, & vacuitatem omnem eliminavit, &
corporum penetrationem pro&longs;crip&longs;it, & vim &longs;e &longs;uis locis di&longs;po
nendi corporibus indidit, &longs;atis univer&longs;i con&longs;i&longs;tentiæ & ordini
con&longs;ultum e&longs;t. Quare corpori nihil levius infra &longs;e habenti nul
lam præterea gravitationem tribuendam cen&longs;eo, præter re
&longs;i&longs;tentiam, ne &longs;ur&longs;um moveatur. Gravitas &longs;iquidem non ni&longs;i
comparatè dicitur, habitâ ratione proximi corporis, in quo
tanquam in loco exi&longs;tit id, quod grave dicitur; nam &longs;i orbis
univer&longs;us con&longs;taret unico corpore homogeneo, nihil e&longs;&longs;et aut
grave aut leve, cum nihil e&longs;&longs;et, quòd præ aliis expo&longs;ceret pro
piùs admoveri centro univer&longs;i. Cum itaque terra ad hoc uni
ver&longs;i centrum perinde &longs;e habeat, atque &longs;i corporibus levioribus
non circumfunderetur, his namque &longs;ublatis illa nec propiùs ad
univer&longs;i centrum accederet, nec longiùs ab eo recederet; ideò
pars terræ quæcumque cum reliquis comparata (ponatur hîc
tellus tota homogenea) nec gravis e&longs;t nec levis; ac proinde,
cùm nulla pars centro propior e&longs;&longs;e exigat, quàm alia, nulla
quoque e&longs;t, quæ aliam urgeat, aut premat propriè, &longs;ed omnes,
& &longs;ingulæ tantummodò repugnant, ne &longs;ur&longs;um in medium leve
transferantur.
Hinc e&longs;t quod terræ con&longs;i&longs;tentiam in loco &longs;uo, non propriè
ex libræ rationibus explicandam cen&longs;eo; quia in librâ utraque
lanx non repugnat &longs;olùm, ne attollatur, verùm etiam in aere
con&longs;tituta deor&longs;um nititur; terræ autem partes &longs;uperiores nil
infrà &longs;e levius habentes non conantur deor&longs;um. Et quemad
modum &longs;i libræ lanx utraque &longs;ubjecto plano incumberet, ea
rum con&longs;i&longs;tentia non e&longs;&longs;et æquilibrio tribuenda, quamvis
æquilibres &longs;int, &longs;ed idcircò &longs;olùm con&longs;i&longs;terent, quia infrà &longs;e
haberent corpus, quod permeare vel non exigit, vel non po
te&longs;t earum gravitas: ita terræ partes licèt adeò æqualiter &longs;int
di&longs;po&longs;itæ circa &longs;uum commune gravitatis centrum (in quo vi
res &longs;uas exererent tellure totâ in aeris locum tran&longs;latâ) ut ex illo
&longs;u&longs;pensâ tellure in æquilibrio con&longs;i&longs;terent; re tamen ipsâ non
fra &longs;e aliquid, &longs;ub quo petat exi&longs;tere, atque adeò nulla e&longs;t,
quæ deor&longs;um nitatur. Quare Poëticè &longs;olùm, non verò Philo
&longs;ophicè dictum e&longs;t.
Aëre &longs;ubjecto tam grave pendet onus.
Aer &longs;i quidem non e&longs;t &longs;ubjectus terræ, &longs;ed circumfu&longs;us; ea
namque &longs;ubjecta &longs;unt, quæ inferiora; inferiora autem, quæ
centro propiora. Terræ itaque globus nihil habet, in quod
gravitatis vires exerceat deor&longs;um conando.
Quæ cum ita &longs;int, nulla unquam continget in terrâ mutatio
atque gravium tran&longs;latio, quæ efficiat motum trepidationis.
Sit enim terræ globus AB, cujus cen
trum C &longs;it pariter centrum gravitatis:
ducto per C plano IL, hemi&longs;phærium
IAL e&longs;t æquale hemi&longs;phærio IBL;
ex quo ab&longs;ci&longs;&longs;a intelligatur portio
&longs;phærica DEB, in cujus locum &longs;uc
cedat aër. Si qua igitur pars deberet
deor&longs;um versùs C niti, non alia uti
que e&longs;&longs;et præter D & E, quæ longiùs
à centro ab&longs;unt, quàm contiguus aër
DE. At portio IDEL prævalere non
pote&longs;t hemi&longs;phærio IAL, quod deberet &longs;ur&longs;um propelli; ergo
non pote&longs;t centrum C moveri versùs A, ut punctum aliquod
inter C & K congruat centro univer&longs;i. Sed neque hemi&longs;phæ
rium IAL debet de&longs;cendere, quia nullum habet corpus leve
&longs;ibi contiguum, quod univer&longs;i centro vicinius &longs;it; non ergo
debet propellere oppo&longs;itum &longs;egmentum IDEL; cujus omnes
partes non &longs;olùm reluctantur motui, quo recedant ab univer&longs;i
centro C, &longs;ed etiam illarum aliquæ &longs;e ip&longs;æ urgent, & conan
tur versùs C. Nondum igitur terra movetur.
Quare Segmentum Sphæricum DKEB transferatur in op
po&longs;itam partem, & addatur hemi&longs;phærio &longs;uperiori etiam mons
FHG æqualis ab&longs;ci&longs;&longs;æ portioni &longs;phæricæ. Aio ne dum factam
e&longs;&longs;e mutationem, quæ ad motum telluri conciliandum &longs;ufficiat.
Quamvis enim mons ille FHG, quippe quem ambit aër le-
&longs;cendere non po&longs;&longs;e, quin totam reliquam terram impellat, eju&longs;
que re&longs;i&longs;tentiam &longs;uperet; re&longs;i&longs;tit autem primò &longs;egmentum
IDEL, cujus omnes partes magis à centro removerentur; ni
&longs;i igitur mons FHG major &longs;it &longs;egmento &longs;phærico IDEL
(vel &longs;altem non multò minor, &longs;i quidem ob majorem à centro
di&longs;tantiam augerentur momenta gravitatis, ex dictis cap.
4.)
non poterit &longs;ubjectam terram loco dimovere. Præterea etiam
hemi&longs;phærium IAL repugnat de&longs;cen&longs;ui montis FHG, quia
fieri non pote&longs;t hic motus, ni&longs;i hemi&longs;phærij partes tran&longs;iliant
planum IL, atque magis à centro recedant. Quanta igitur
gravitate præditum e&longs;&longs;e montem oporteret, qui tantam re
&longs;i&longs;tentiam &longs;uperare valeret? At nunquam fieri tantam partium
permutationem, ut id quod transfertur, &longs;it non minus &longs;emi&longs;&longs;e
hemi&longs;phærij, ut &longs;altem ratione habitâ di&longs;tantiæ à centro po&longs;
&longs;it prævalere, ita omnibus e&longs;t manife&longs;tum, ut probatione non
indigeat. Quare neque hanc gravium tran&longs;lationem motus ul
lus con&longs;equitur, quo tellus trepidare dicatur.
At, inquis, &longs;i in utrâque libræ lance &longs;int unciæ 100, & al
terutri uncia una addatur, lanx illa deprimitur, & oppo&longs;ita
elevatur; ergo exiguum pondus vim habet movendi ingens
pondus; ergo pariter mons FHG producere pote&longs;t impetum,
qui ad movendum &longs;egmentum IDEL, quantumvis gravius,
abundè &longs;ufficiat. Ego vero nego con&longs;equentiam; quia non ab
unciâ illâ additâ &longs;olâ elevatur oppo&longs;itum pondus, &longs;ed omnes
unciæ &longs;imul in medio leviore &longs;u&longs;pen&longs;æ collatis viribus deor&longs;um
conantur, atque præponderantes oppo&longs;itæ lancis pondus at
tollunt. Hoc autem nil in rem no&longs;tram facit, ubi neque mons
FHG &longs;olitariè &longs;umptus pote&longs;t &longs;ursùm propellere molem
IDEL majorem &longs;e, neque juvari pote&longs;t ab hemi&longs;phærio IAL,
quod cum nihil infrà &longs;e habeat, quod & levius &longs;it, & inter
ip&longs;um ac univer&longs;i centrum intercipiatur, neque pote&longs;t &longs;e ip&longs;um
versùs centrum urgere &longs;ecundùm aliquas &longs;ui partes ab eo remo
tiores, cum maximè partes centro proximæ valde reluctentur,
ne ab illo removeantur. Id quod in libræ lance, cui uncia fue
rit addita, reperire non poteris; totum &longs;iquidem lancis pon
dus deor&longs;um nititur.
Quod &longs;i ex librâ &longs;imilitudinem ducere placeat, petenda po-
tera in aëre libera pendeat; &longs;i enim utraque lanx plena æquali
bus ponderibus con&longs;i&longs;tat in æquilibrio, & incumbenti lanci ad
datur ponderis pars, quæ à pendulâ lance detrahatur, lances
non moventur, nec inter &longs;e mutuò confligunt ponderum gra
vitates, ni&longs;i quatenùs lanx gravior &longs;emper magis re&longs;i&longs;tit leviori,
ne ab illâ elevetur: cæterùm gravior lanx non movet leviorem,
ni&longs;i ubi demum tanto pondere prægravata fuerit, ut &longs;ubjecti
plani re&longs;i&longs;tentiam vincens illud aut frangat, aut &longs;altem depri
mat. Sic hemi&longs;phærium IAL habet rationem lancis non tan
tùm &longs;ubjecto plano incumbentis, &longs;ed, quod potius e&longs;t, &longs;uo in
loco quie&longs;centis; cui quò plus addideris ponderis, auges qui
dem re&longs;i&longs;tentiam ne &longs;ursùm versùs H propellatur, ip&longs;um verò
non conatur deor&longs;um versùs C; &longs;ed totus conatus impo&longs;ito &
adjecto monti tribuendus e&longs;&longs;et, vel (ut &longs;im maximè liberalis)
etiam exce&longs;&longs;ui illi, quo hemi&longs;phærium IAL &longs;uperat &longs;egmen
tum &longs;phæricum IDEL, qui exce&longs;&longs;us e&longs;t æqualis ip&longs;i monti,
hoc e&longs;t &longs;egmento DEB. Quare &longs;i fuerit ab&longs;ci&longs;&longs;a tertia pars
hemi&longs;phærij unius, & addatur alteri hemi&longs;phærio è regione &longs;e
cundùm diametrum, tunc ad &longs;ummum æqualis erit pars terræ
deor&longs;um nitens FMGH parti oppo&longs;itæ repugnanti IDEL; &
&longs;i velis partem FMGH remotiorem à centro magis gravitare
ita, ut ratio hujus exce&longs;sûs in gravitando po&longs;&longs;it vincere non &longs;o
lùm re&longs;i&longs;tentiam &longs;egmenti IDEL, ne &longs;ur&longs;um propellatur, &longs;ed
etiam &longs;egmenti FILG, ne &longs;ecundùm partes IL centro proxi
mas ab eo removeatur; non admodum repugnabo. Sed cum
nunquam mille&longs;ima, ne dum &longs;exta, pars terreni globi ex alio
in alium locum ex diametro oppo&longs;itum transferatur, nulla un
quam fit gravium permutatio, vi cujus tellus trepidet.
Sed unum adhuc &longs;upere&longs;t, quod per di&longs;&longs;imulantiam præ
tereundum non videtur. E&longs;to inquis, nulla fiat in tellure gra
vium tran&longs;latio, quæ tanta &longs;it, ut novum gravitatis centrum in
univer&longs;i centro con&longs;tituere valeat, ac proinde nulla &longs;it centri
terræ trepidatio: circa centrum &longs;altem nutabit tellus motu
conver&longs;ionis, validâ ventorum vi &longs;ummos montes impellente,
orbemque totum, pro variâ ip&longs;orum incur&longs;ione, modò hanc,
modò illam partem ver&longs;ante: unde forta&longs;&longs;e ortam acû magne
ticæ eodem in loco po&longs;t aliquot annos variationem &longs;u&longs;picari Cum enim tellus æqualibus circà centrum nutibus
librata permaneat, multo faciliùs omnem in partem converti
po&longs;&longs;e videtur, quàm rota ingens &longs;uo in axe &longs;u&longs;pen&longs;a: Rota &longs;ci
licet &longs;uo pondere axem premens illum, dum convertitur, te
rit; hancque affrictûs difficultatem vincat nece&longs;&longs;e e&longs;t, quod
una ex parte additur pondus, vel quæ applicatur Potentia, ut
conver&longs;ionem efficiat: tellus verò in orbem diffu&longs;a nec cen
trum premit, nec axem, cum quo ullus fiat affrictus; ac
proptereà faciliorem præbet conver&longs;ionis an&longs;am Potentiæ unam
aliquam in partem urgenti. Huju&longs;modi autem Potentia ventus
e&longs;t, non ad perpendiculum in terram incidens, &longs;ed obliquè in
præaltos &longs;altem montes incurrens; cujus viribus nihil ob&longs;tare
videtur, quin telluris globum &longs;ibi ob&longs;ecundantem inclinet;
quemadmodum, & ingentes naves, vela implens, impellit.
Huic difficultati ut me &longs;ubducam, non me in abditos magne
ti&longs;mi rece&longs;&longs;us recipio, a&longs;&longs;erendo tellurem ita arcanis nodis cæ
lo connexam, ut à &longs;ummo axium polorumque cæle&longs;tium atque
terre&longs;trium con&longs;en&longs;u divelli ac di&longs;trahi prorsùs nequeat: ne
que enim hi&longs;ce magneti&longs;mi latebris me &longs;atis protectum exi&longs;ti
marem; demptâ quippe &longs;olis Au&longs;tralibus atque Borealibus ven
tis hâc facultate tellurem convertendi, ne &longs;cilicet terre&longs;tres
poli à cæle&longs;tibus di&longs;crepent, quid prohibeat reliquos ad Orti
vum, aut Occiduum limitem pertinentes, quin &longs;uo flatu or
bem hunc volvant, adhuc &longs;upere&longs;&longs;et explicandum. Hoc qui
dem &longs;atis e&longs;&longs;e videretur ad &longs;ubmovendam &longs;u&longs;picionem illam de
acûs magneticæ variatione ob telluris conver&longs;ionem; manente
nimirum axe terre&longs;tri ita, ut cum cæle&longs;ti conveniat, aut illi
&longs;altem parallelus exi&longs;tat, nihil e&longs;t quod, etiam tellure circa
axem conversâ, magneticam declinationem commutare queat:
nam quod ad &longs;yderum a&longs;pectus &longs;pectat, parum intere&longs;t, tellus
ne? an cælum volvatur; &longs;i igitur diurna cæli conver&longs;io magne
tis declinationem non mutat, neque ad illam mutandam &longs;uffi
ceret telluris circa &longs;uum axem conver&longs;io, vi cujus alia atque
alia &longs;ydera re&longs;piceret: Præterquam quod non id temporum lap
&longs;u accideret; &longs;ed ubi ventorum impetus elangui&longs;&longs;et, illicò va
riatio illa declinationis magneticæ deprehenderetur: id quod
ab omni experimento longè abe&longs;t. Verùm adeò à no&longs;tris &longs;en
&longs;ibus &longs;ejunctæ &longs;unt magneticorum &longs;ymptomatum cau&longs;æ, ut ad
Philo&longs;ophicam &longs;cenam magneti&longs;mus.
Illud potius hìc attendendum videtur, quod montis altitu
do, atque magnitudo ad totius telluris molem Rationem habet
&longs;atis exiguam. Cum enim terræ ambitus probabiliter &longs;tatuatur,
ut aliàs o&longs;tendi, milliarium Rom.
propterea diameter &longs;it proximè mill. (9738 4/51), tota &longs;uperficies
&longs;phærica (ut pote quadrupla maximi circuli ex demon&longs;tratis
ab Archimede) e&longs;t mill. quadratorum 297. 987800 proximè.
Mons &longs;tatuatur altitudinis perpendicularis milliarium quin
que; hæc e&longs;t ad terre&longs;trem diametrum ut 1 ad 1947: ba&longs;is
montis occupet milliaria quadrata 500; hæc e&longs;t ad &longs;phæricam
totius globi &longs;uperficiem, ut 1 ad 595975. Finge jam pro mon
te granum hordei, quod promineat &longs;ecundùm &longs;uam latitudi
nem ex &longs;phærâ habente diametrum granorum 1947, hoc e&longs;t
pa&longs;&longs;uum geometricorum &longs;ex, &longs;eu pedum Rom.
culi maximi ambitus erit pedum 94 1/4: quare hujus &longs;phæræ &longs;u
perficies habet pedes quadratos 2827, hoc e&longs;t quadratas lati
tudines grani hordei paulò plures quàm 11. 579000. Igitur
grani hordei jacentis altitudo ad hujus &longs;phæræ diametrum
eandem ex hypothe&longs;i habet rationem, quam prædicti montis
altitudo ad telluris diametrum: & &longs;i decem grana &longs;ibi invicem
attigua di&longs;ponantur, ut montis ba&longs;im æmulentur, eadem erit
ratio ad &longs;uperficiem. Quamvis itaque &longs;phæra illa intelligatur
planè inanis ac levi&longs;&longs;ima &longs;olam habens &longs;uperficiem papyra
ceam, ex qua granum ordei agglutinatum promineat, an pu
tas à flatu quantumvis valido per fi&longs;tulam emi&longs;&longs;o in granum il
lud hordei incurrente convertendum e&longs;&longs;e globum papyra
ceum? Id &longs;anè ex cæteris experimentis conjicere non licet;
perinde enim e&longs;t atque &longs;i nihil promineret; neque vel mini
mùm obe&longs;t Phy&longs;icæ rotunditati. Quare neque montis altitu
do con&longs;tituta quicquam detrahet orbicularis figuræ, quod &longs;ub
Phy&longs;icam con&longs;iderationem cadat; ac proptereà nihil virium ad
tellurem convertendam obtinet ventus in montem incurrens.
Et quidem conver&longs;ionem hanc re ipsâ non fieri manife&longs;tum
e&longs;t; &longs;i quidem cum nulla vincenda e&longs;&longs;et gravitas, quæ longiùs
à centro gravium recederet, vel quæ axem tereret, facillima
videretur e&longs;&longs;e globi totius conver&longs;io circa centrum, non &longs;olùm
mediocribus ventis flantibus. Hi autem aliquando diuturni
&longs;unt; cuju&longs;modi poti&longs;&longs;imum &longs;unt Ete&longs;iæ, quibus maritimi cur
&longs;us celeres, & certi diriguntur. Tot igitur dierum &longs;patio, ven
to oppo&longs;itos montes vehementiùs urgente, non modica fieret
terreni globi inclinatio; ac propterea non eadem demum per
maneret eodem in loco Poli &longs;uprà Horizontem altitudo, quo
ties ab alterutro cardine Au&longs;trali Boreali ve, aut à &longs;ol&longs;titiali
Brumali-ve limite tam ortivo quàm occiduo ventus &longs;piraret, at
que multarum ædium facies non eandem ampliùs re&longs;picerent
cæli plagam; quare & &longs;cietherica Horologia quantumvis ac
curatè &longs;emel de&longs;cripta po&longs;t non adeò multas temporum inclina
tiones toto ferè cælo di&longs;creparent; aliis enim, atque aliis &longs;ub
inde flantibus ventis, varia oriretur orbis conver&longs;io, atque alia
planorum cum circulis horariis &longs;ectio, quæ de&longs;criptis lineis non
congrueret. Hujus autem mutationis nullum in toto terra
rum orbe ve&longs;tigium apparet, ni&longs;i fortè fabulas liceat com
mini&longs;ci.
Quòd &longs;i conver&longs;ionem hanc non omninò circa centrum
quamcumque in partem fieri, &longs;ed tantummodo circa axem,
dixeris, ut argumenti vim effugias; Quid illud e&longs;t, quod ita
terre&longs;trem axem cum cæle&longs;ti colligatum velit, ut tamen ter
re&longs;tres meridianos à primâ mundi molitione con&longs;titutos tem
poris lap&longs;u cum cæle&longs;tibus meridianis non convenire permit
tat? Sed & aliud profectò, nec illud quidem leve, incommo
dum &longs;ubeas nece&longs;&longs;e e&longs;t; dum enim conver&longs;ionem ad&longs;truis ab
ortu in occa&longs;um, & vici&longs;&longs;im ab occa&longs;u in ortum, fieri poterit,
ut po&longs;t aliquot annos non planè &longs;pernenda conver&longs;io facta fue
rit, ac proinde temporum numeratio cælo non re&longs;pondeat.
Nam &longs;i ab ortu in occa&longs;um ex.
gr. proce&longs;&longs;erit tellus, minus tem
poris numerabitur quàm pro ratione cæle&longs;tium motuum; ut
contigi&longs;&longs;e fertur navi cui à Victoriâ nomen inditum e&longs;t, in ex
peditione Magellanicâ; cum &longs;cilicet po&longs;t totius orbis ambitum
redux in Hi&longs;palen&longs;em portum, ex quo ante tres annos &longs;olve
rat, intraret, tunc primùm ob&longs;ervarunt &longs;e à rectâ temporis nu
meratione defeci&longs;&longs;e die uno; quippe qui cum juxta diurnam
cæli conver&longs;ionem ab ortu in occa&longs;um iter in&longs;titui&longs;&longs;ent, ju&longs;to
tardiùs &longs;emper &longs;ol illis occiderat, exiguo quidem &longs;ingulis die-
cis illis acce&longs;&longs;ionibus in integrum diem excreverat. Contra ve
rò accideret, &longs;i ab occa&longs;u in ortum &longs;emper navigaretur; ju&longs;to
enim breviores e&longs;&longs;ent dies, ac propterea eorum numerus ac
cre&longs;ceret. Hæc autem in temporum numeratione incon&longs;tan
tia, &longs;i ventorum impetu tellus modò in ortum, modò in occa
&longs;um converteretur, quantam perturbationem inveheret in
A&longs;tronomiam? Neque tibi quicquam &longs;uffragari exi&longs;times, &longs;i
ex varia ventorum oppo&longs;itas in plagas &longs;ivè &longs;imul, &longs;ivè &longs;ubinde,
&longs;pirantium commutatione conver&longs;iones illas compen&longs;ari dixe
ris: id enim ad incertum revocat omnes A&longs;tronomorum calcu
los, ubi meridianorum circulorum &longs;ectiones &longs;tabiles non perma
neant; cum ad orbem totum inclinandum, ut tu quidem au
tumas, &longs;atis &longs;it, &longs;i unâ aliquâ in regione ventus montes impel
lat; quî verò certus &longs;im factam ab Arge&longs;te telluris conver&longs;io
nem in ortum, æquatam demum fui&longs;&longs;e à Vulturno, aut ab
Euro-Au&longs;tro?
Verùm quàm infirmæ &longs;int validi&longs;&longs;imorum ventorum vires ad
globum hunc terraqueum inclinandum, expendamus, etiam&longs;i
montium perpendicula non quinque tantùm milliaribus defini
ta velis, &longs;ed multò altiora. Statue in ingenti lacu compo&longs;itam
ex trabibus aliquot ratem, quam in littore &longs;tans facilè funiculo
modereris: Tùm ratem aliam paris quidem latitudinis, &longs;ed cen
tuplò longiorem, compone: Poteris-ne hanc funiculo eodem,
ac labore non majori, trahere perinde atque priorem? Negabis
utique, quamvis enim utraque lacui &longs;tagnanti innatet, nec
vincenda &longs;it alterutrius gravitas, ut à centro gravium magis re
cedat; licet utraque parem in motu ab aquâ dividendâ re&longs;i&longs;ten
tiam inveniat (eju&longs;dem quippe &longs;unt latitudinis &longs;olâ di&longs;crepan
tes longitudine, & æqualis e&longs;t utriu&longs;que immer&longs;io propter ean
dem &longs;ingularum trabium molem, atque &longs;pecificam gravitatem)
quia tamen di&longs;par e&longs;t ratium magnitudo, & impetu extrin&longs;e
cùs accepto utraque eget, ut moveatur, palàm e&longs;t majore im
petu opus e&longs;&longs;e, ut ratis major trahatur, ac propterea po&longs;&longs;e hanc
adeò augeri, ut impetus ad illam movendam nece&longs;&longs;arius exce
dat vires Potentiæ ratem minorem funiculo moderantis. Ita
planè e&longs;t. Sed jam animum transfer ad in&longs;titutam di&longs;putatio
nem, ut di&longs;picias, undè irrep&longs;erit dubitatio hæc de telluris
rit rota &longs;uo in axe &longs;u&longs;pen&longs;a, quæ levi negotio, nec valido im
pul&longs;u, volvitur. Rota &longs;iquidem tota deor&longs;um gravitat, ac
proptereà axem premit; quia autem in axe &longs;u&longs;penditur, fieri
non pote&longs;t, ut pars altera de&longs;cendat, quin oppo&longs;ita a&longs;cendat.
Quandiu conatus ad de&longs;cendendum æqualis e&longs;t re&longs;i&longs;tentiæ ad
a&longs;cendendum, rota quie&longs;cit; nec volvitur, ni&longs;i alterutri parti
fiat acce&longs;&longs;io Potentiæ, quæ pariter de&longs;cen&longs;um juvet, vel quia
ip&longs;a quoquè deor&longs;um conatur cum parte de&longs;cendente, vel quia
&longs;ur&longs;um nitens partem alteram elevat, oppo&longs;itamque deprimet
&longs;uapte naturâ de&longs;cendentem. Non tamen huju&longs;modi rotæ &longs;u&longs;
pen&longs;æ conver&longs;io tribuenda e&longs;t &longs;oli Potentiæ; &longs;ed pars rotæ de
&longs;cendens atque Potentia collatis viribus elevant partem rotæ
a&longs;cendentem, eíque impetum imprimunt. At in telluris circa
&longs;uum centrum, vel axem, conver&longs;ione nihil ade&longs;&longs;et, quod Po
tentiam juvaret; quia nulla e&longs;t pars, quæ deor&longs;um conetur,
aut &longs;ur&longs;um, ut po&longs;&longs;it oppo&longs;itæ parti impetum aliquem impri
mere; nulla etenim pars in huju&longs;modi conver&longs;ione ad centrum
gravium accederet, aut ab illo recederet. Totus igitur impe
tus à vento imprimendus e&longs;&longs;et toti telluris globo, ut à &longs;uâ, quæ
&longs;ecundùm naturam e&longs;t, quiete dimoveretur. Atqui globi ter
raquei ea e&longs;t moles, ut contineat milliaria cubica proximè
48670. 200000 (omnis nimirum &longs;phæra æqualis e&longs;t cono, cu
jus altitudo par e&longs;t Radio &longs;phæræ, ba&longs;is autem æqualis &longs;uperfi
ciei &longs;phæræ, ex dictis verò paulò &longs;uperiùs, & &longs;uperficies & Ra
dius globi hujus innote&longs;cit) nullus igitur adeò vehemens e&longs;t
ventus, qui tantæ moli impetum imprimere valeat; nullus &longs;i
quidem excogitari pote&longs;t ventus, qui globum marmoreum, aut
etiam ex argillâ, in planitie æqui&longs;&longs;imâ con&longs;titutum, &longs;i mille
pa&longs;&longs;us Geometricos in diametro numeret, convolvere valeat.
Adde in telluris conver&longs;ione, &longs;i illa fieret, quò vehementior
e&longs;&longs;et ventus in montem incurrens, validior e&longs;&longs;et re&longs;i&longs;tentia aëris
à reliquis montibus dividendi; &longs;ed & multorum ingentium
fluminum contrariam in partem labentium impetus ob&longs;i&longs;teret,
ne tellus vento flanti ob&longs;ecundaret. Quod &longs;i hæc levis e&longs;&longs;e mo
menti dixeris ad ob&longs;i&longs;tendum, levis pariter momenti e&longs;&longs;e ven
torum impetum, nece&longs;&longs;e e&longs;t, fatearis: neque hic arduum e&longs;&longs;et
ventorum atque fluminum vires invicem conferre, aquarum-
randum e&longs;t: &longs;atisfuerit monui&longs;&longs;e non mediocrem intercedere
analogiam inter aquarum guttas in rivulos primùm, deinde in
majores rivos, ac demum in torrentem concurrentes, atque
terræ expirationes in ventum congregatas, quæ multum vi
rium obtinent, &longs;i plurimæ in unum coëant, quemadmodum
& aquis contingit.
inclinato.
PLanum inclinatum dicitur planum quodcumque non tran
&longs;it per centrum gravium & levium, hoc e&longs;t per centrum
univer&longs;i; huju&longs;modi &longs;iquidem planum non cadit ad angulos
æquales in &longs;phæricam terræ &longs;uperficiem. Hinc etiam planum
horizonti parallelum reipsâ e&longs;t inclinatum, ni&longs;i adeò exiguum
&longs;it ac breve, ut puncti vicem obtineat, &longs;i cum terreni globi &longs;u
perficie conferatur. Sit univer&longs;i
centrum A, plana BA, & CA &longs;unt
verticalia & perpendicularia, qui
bus &longs;i corpus aliquod grave appli
cueris, illud non impedietur, quin
per &longs;uam directionis lineam de&longs;cen
dat. At verò tam planum BC, quam
planum CD inclinata &longs;unt, nec cor
pus grave illis impo&longs;itum pote&longs;t
rectâ &longs;ecundùm directionis lineam
de&longs;cendere, &longs;ed ab illâ declinare co
gitur plano ob&longs;i&longs;tente. Sunt autem anguli inclinationis ABC,
ACD. Quod &longs;i planum parallelum horizonti ita exiguum &longs;it,
ut à &longs;phæricâ &longs;uperficie, quam tangit, non recedat; tunc in
quacumque ejus parte con&longs;tituatur corpus grave, perinde e&longs;t,
atque &longs;i in puncto D collocatum concipiatur. Sin autem ita à
modum &longs;i e&longs;&longs;et planum DF, illud e&longs;t inclinatum, & fit angulus
DFA inclinationis. Ubi ob&longs;ervandum e&longs;t non eandem e&longs;&longs;e
&longs;ingularum plani partium inclinationem; angulus enim in
clinationis AEC major e&longs;t inclinatione ABC, per 16.
lib.
1. & &longs;imiliter AFD maior e&longs;t angulo ACD. Quare
&longs;tatim atque ea e&longs;t puncti E à puncto B di&longs;tantia, ut an
gulus à perpendiculis in centro A factus contemni non po&longs;
&longs;it, alia e&longs;t etiam phy&longs;icè inclinatio, & corporis eju&longs;dem
gravitatio mutatur.
Quoniam verò corpus grave plano inclinato impo&longs;itum ita
aëre circumfunditur, ut petat infrà illum de&longs;cendere, & re
&longs;i&longs;tat, ne &longs;ur&longs;um moveatur; ideò gravitare dicitur.
Sed cavendum e&longs;t, ne ex vocabulorum &longs;imilitudine er
ror &longs;ubrepat: quandoquidem aliud e&longs;t
inclinato,
aërem corpus grave, putà, lapis, gravitat in quocunque
plano etiam perpendiculari, non tamen gravitat in pla
num perpendiculare, nulla&longs;que vires &longs;uæ gravitatis con
tra illud exercet, quamvis in eo exi&longs;tens, & re&longs;i&longs;tat &longs;ur
&longs;um trahenti, & conetur, ut vincat vires retinentis, ac
quicquid moram infert, & impedimentum motui. In pla
no itaque inclinato exi&longs;tens corpus grave (&longs;ubjectum pla
num &longs;upponitur optimè lævigatum, nec motui officiens
partium prominularum a&longs;peritate) gravitat quidem, &longs;ed mi
nùs quàm in plano perpendiculari, & pro variâ planorum
inclinatione, varia pariter e&longs;t gravitatio, ut quotidiana nos
docet experientia. Quâ igitur ratione gravitatio minuatur,
hîc e&longs;t examinandum; capite &longs;equenti gravitatio in Planum
inclinatum explicabitur.
Cogno&longs;citur autem gravitatio ex re&longs;i&longs;tentiâ, quâ corpus
repugnat contra vires illud retinentis, ne deor&longs;um feratur,
aut &longs;ur&longs;um trahentis; neque enim alio ni&longs;u gravia gravi
tant, quàm quo re&longs;i&longs;tunt impedienti motum gravitati
convenientem. Et quidem experimento aliquo pote&longs;t gra
vitationis varietas inve&longs;tigari; &longs;i nimirum planum BO ex
ligno, aut marmore accuratè lævigetur, & extremitati B
adnectatur orbiculus D facillimè circa axem ver&longs;atilis, pon-
rotulæ, & adnectatur funi
culus per D tran&longs;iens, ex
cujus extremo pendeat lanx
E, cui pondera immitti po&longs;
&longs;int: pro variâ enim plani
BO inclinatione etiam pon
dera in lance mutare opor
tebit, ut pondus A &longs;u&longs;ti
neatur, & plura erunt, quò magis ad perpendiculare accedet
planum BO. Verùm quia nunquam carere poteris &longs;u&longs;picione,
an corporum affrictus aliquid afferat impedimenti; ideò &longs;eclu
&longs;is omnibus, quæ extrin&longs;ecùs accidere po&longs;&longs;unt, re&longs;i&longs;tentiam ex
&longs;olâ gravitate ortam opus e&longs;t con&longs;iderare.
Re&longs;i&longs;tentia verò omnis re&longs;pondet violentiæ, quam patitur
id quod re&longs;i&longs;tit; minori etenim conatu minorem vim illatam
propul&longs;are &longs;tudet natura, quæ validiùs ob&longs;i&longs;tit majori violen
tiæ: id quod ita rationi e&longs;t con&longs;onum, & obviis experimentis
manife&longs;tum, ut in hoc demon&longs;trando &longs;upervacaneum &longs;it im
morari. Con&longs;tituantur itaque duo
æqualis ponderis corpora in D &
in C; &longs;ingulis alligetur funiculus,
qui per B tran&longs;eat, & &longs;ur&longs;um tra
hantur &longs;imul ita, ut æqualiter mo
veantur. Ab&longs;olutâ motûs particu
lâ, corpus alterum ex D a&longs;cendit
in H in plano perpendiculari; al
terum in plano inclinato ex C ve
nit in E, & CE linea æqualis e&longs;t
lineæ motûs DH. Non eandem
tamen utrumque grave &longs;ubiit vio
lentiam; nam motus DH fuit &longs;impliciter, & ab&longs;olutè violen
tus; at motus CE eatenus &longs;olùm gravitati adver&longs;atur, quate
nus a&longs;cendit; a&longs;cen&longs;um autem metitur linea DG, quam ab
&longs;cindit EG horizonti parallela. Hîc &longs;cilicet planum DC in
tellige horizontale nihil à &longs;phæricá &longs;uperficie di&longs;crepans, ut
communiter contingit: quòd &longs;i non ita &longs;e haberet; &longs;ed e&longs;&longs;et
ampli&longs;&longs;imum planum, men&longs;ura violentiæ illatæ ponderi in C
KC & OE. E&longs;t itaque gravitatio in plano perpendiculari ad
gravitationem in plano inclinato, ut re&longs;i&longs;tentia ad a&longs;cenden
dum in uno ad re&longs;i&longs;tentiam ad a&longs;cendendum in alio; re&longs;i&longs;tentiæ
autem &longs;unt, ut violentia, quam corpora &longs;ubeunt in motu; vio
lentia demum e&longs;t ut HD ad GD, hoc e&longs;t per 7. lib.
5. ut CE
ad DG. Sed ut CE ad DG, ita EB ad GB, per 2. lib.
6. &
ut BE, ad BG ita BC ad BD, per 4. lib.
6. igitur gravitatio
in perpendiculari ad gravitationem in inclinato e&longs;t ut BC ad
BD, hoc e&longs;t ut Secans anguli inclinationis ad Radium.
Quæ autem de totis DH, & CE lineis dicta &longs;unt, de &longs;ingu
lis earum particulis æqualibus dicta intelligantur; ductis quip
pe parallelis horizonti, eadem e&longs;t omnium Ratio: hîc namque
&longs;upponimus planum BC non adeò magnum e&longs;&longs;e, ut &longs;ingula
ejus puncta cum diver&longs;is horizontibus comparanda &longs;int, omnes
&longs;iquidem perpendiculares lineæ directionis non qua&longs;i conver
gentes, &longs;ed phy&longs;icè parallelæ accipiuntur. Quòd &longs;i tam lon
gum e&longs;&longs;et planum, ut phy&longs;icè mutatus intelligeretur angulus
inclinationis, non eadem e&longs;&longs;et Ratio gravitationis in toto, ac in
partibus: &longs;ed mutato angulo inclinationis mutaretur utique
ejus Secans; ac proinde inæqualium Secantium Ratio ad eum
dem Radium inæqualis, gravitationum pariter inæqualem ra
tionem o&longs;tenderet.
Quod &longs;i a&longs;cendentium per vim extrin&longs;ecùs illatam corporum
re&longs;i&longs;tentiam atque gravitationem metimur ex violentiâ, quam
pro planorum varietate &longs;ubeunt; eorum pariter in de&longs;cendendo
efficacitatem ex ip&longs;o de&longs;cen&longs;u argui æquum e&longs;&longs;et, datâ motûs
in diver&longs;is planis æqualitate. Sed quia de&longs;cen&longs;us naturæ pro
pen&longs;ioni congruit, fieri non pote&longs;t, ut in alio atque alio plano
æquales &longs;int motus i&longs;ochroni; tardior enim e&longs;t, qui in plano in
clinato perficitur, neque, &longs;i æqualis ponderis corpora de&longs;cen
dant ex H & E, quando illud ad D pervenit, hoc pote&longs;t attin
gere punctum C: ideò non ex de&longs;cen&longs;u gravitationem metiri
oportet, cum motus æquales non habeantur: ni&longs;i fortè ea&longs;dem
movendi vires tribuas gravitati non impeditæ in perpendicula
ri, ac impeditæ in plano inclinato. Qua propter gravitationis
momenta ad de&longs;cendendum non aliunde meliùs æ&longs;timantur,
quàm ex repugnantiâ ad a&longs;cendendum: &longs;ic enim vulgari argu-
tumque iis ad de&longs;cendendum virium tribuimus, quantum re
&longs;i&longs;tunt, ne ab oppo&longs;itâ libræ lance deor&longs;um conante eleventur.
Eadem igitur e&longs;t gravitationis Ratio, &longs;eu propen&longs;ionis ad de
&longs;cendendum, quæ e&longs;t re&longs;i&longs;tentiæ ad a&longs;cendendum: Cum verò
re&longs;i&longs;tentiam in plano inclinato ad re&longs;i&longs;tentiam in perpendicu
lari o&longs;ten&longs;um &longs;it e&longs;&longs;e, ut Radius ad Secantem anguli inclinatio
nis, hoc e&longs;t ut BD ad BC, erit pariter vis de&longs;cendendi in
plano BC ad vim de&longs;cendendi in plano BD, reciprocè ut BD
ad BC.
Eadem ratione in plano CD &longs;uperficiem globi tangente,
gravitatio in CD ad gravitationem in perpendiculari CA e&longs;t
ut CD ad CA; e&longs;t enim CA Secans anguli inclinationis
DCA. Si enim ducatur KF Tangens, triangula CKF,
CDA &longs;unt &longs;imilia, angulus enim ad C communis e&longs;t, & am
bo rectangula ad D & K; quare ut CK ad CF, ita CD ad
CA; &longs;ed gravitatio in CF ad gravitationem in CK e&longs;t reci
procè ut CK ad CF: igitur gravitatio in plano inclinato CD
globum tangente, ad gravitationem in perpendiculari CA, e&longs;t
ut CD ad CA.
Hinc e&longs;t quod in planis horizontalibus, quæ ut plurimum
habemus, corpora non de&longs;cendant, aut moveantur: quia ni
mirum à puncto, in quo grave &longs;tatuitur, ex. gr. F, ductæ li
neæ FA perpendicularis & FD Tangens faciunt angulum
DFA inclinationis adeò magnum, ut Radius ad ejus &longs;ecan
tem penè infinitam non habeat &longs;en&longs;u perceptibilem Rationem,
vel &longs;altem non tantam, ut gravitatio, quæ ratione inclinatio
nis plani congruit corpori, non elidatur à re&longs;i&longs;tentiâ, quæ ori
tur ex corporum a&longs;peritate. Quare &longs;ublatâ, aut potiùs impeditâ,
gravitatione corpus quie&longs;cit in plano horizontali.
Et hæc e&longs;t ratio, cur violentiam determinans, quam grave
a&longs;cendens patitur, a&longs;&longs;ump&longs;erim in perpendiculari BA par
tem GD, quam ab&longs;cindit parallela horizonti; hæc enim
men&longs;ura phy&longs;icè non di&longs;crepat à verâ men&longs;urâ, quæ a&longs;&longs;umen
da e&longs;&longs;et, &longs;i mente concipias rectam lineam DC tangere circu
lum, cujus &longs;emidiameter &longs;it millecuplo major. Men&longs;ura &longs;i qui
dem a&longs;censûs petenda e&longs;t ex exce&longs;&longs;u, quo perpendicularis EA
&longs;uperat perpendicularem AC; illo enim intervallo, quo magis
rece&longs;&longs;it à centro, a&longs;cendit.
Ex quo fit quod, &longs;i planum inclinatum BC cum perpendi
culari CA faceret angulum acutum ACB, corpus ex C u&longs;que
in L (in quod punctum cadit perpendicularis AL) de&longs;cende
ret, quia &longs;emper magis ad centrum accederet: ex L autem in E
a&longs;cenderet, & a&longs;cen&longs;um metiretur exce&longs;&longs;us perpendiculi EA
&longs;uprà perpendiculum LA. Quare ut ex C a&longs;cenderet, debe
ret e&longs;&longs;e planum inclinatum IC, quod cum CA faceret angu
lum ICA &longs;altem rectum. Ubi ex occa&longs;ione licet ob&longs;ervare
po&longs;&longs;e dari duos montes, qui cum valle intermediâ planitiem
unam con&longs;tituant; &longs;i nimirum montium vertices e&longs;&longs;ent E, & C,
ex quibus in imam vallem L de&longs;cenderetur: & aqua per mon
tium venas de&longs;cendens in L po&longs;&longs;et fontem aut lacum creare.
Re autem ipsâ &longs;emper contingit angulum BCA e&longs;&longs;e obtu&longs;um
vel non minorem recto. Ponatur enim terræ &longs;emidiameter DA
1000, & planum DC: (e&longs;&longs;et autem planum DC longius
milliar.4.) erit angulus DAC, gr. 0. 3′. 26′; atque adeò DCA
gr. 89. 56′. 34″.
Jam verò &longs;it CD ad DB ut 100 ad 87; erit
angulus BCD gr.4.1. 1′. 23″: quare totus BCA gr.130. 57′.
57′.
Nunc &longs;i libeat comparare perpendiculum EA cum perpendi
culo GA, &longs;tatue GD &longs;emi&longs;&longs;em totius BD; e&longs;t igitur & GE
&longs;emi&longs;&longs;is ip&longs;ius DC: Quare GE e&longs;t partium 50, quarum GA e&longs;t
100043 1/2: addantur quadrata GE 2500 & GA 10008701892 1/4,
& &longs;ummæ radix quadrata (100043 102543/200086) major verâ e&longs;t EA, quæ
non excedit perpendicularem GA 100043 1/2 ni&longs;i particulis (2500/400172).
Quoniam autem DAC angulus inventus e&longs;t grad. 0. 3′. 26′;
eju&longs;que Secans AC e&longs;t partium (100000 5017/100000), quarum AD
po&longs;ita e&longs;t 100000; di&longs;crimen inter AC, & AE &longs;uperiùs in
ventam, e&longs;t partium (43 46227/100000), quæ e&longs;t proximè eadem men&longs;u
ra, ac DG po&longs;ita partium 43 1/2. Quod &longs;i in plani inclinati lon
gitudine
ta con&longs;tit ita e&longs;t, pote&longs;t citrà errorem a&longs;&longs;umi tanquam men&longs;ura
a&longs;censûs pars perpendiculi BA intecepta ab horizontali DC,
& parallelâ EG, &longs;atis patet id multò magis licere in planorum
longitudinibus minorem Rationem habentibus ad eandem ter
ræ &longs;emidiametrum. Manet itaque con&longs;tituta regula gravitatio
nis, videlicet gravitationem in plano inclinato ad gravitationem
in perpendiculari e&longs;&longs;e, ut e&longs;t Radius ad &longs;ecantem anguli incli
nationis.
Quamvis verò in partibus inferioribus plani inclinati &longs;it &longs;em
per major angulus inclinationis, quàm in &longs;uperioribus, & pro
inde minor &longs;it Ratio, quam habet Radius ad &longs;ecantem anguli
majoris, ac ea, quam idem Radius habet ad &longs;ecantem anguli
minoris: non tamen ea e&longs;t gravitationis differentia, cujus ratio
habenda &longs;it; cum enim adeò exiguus &longs;it angulus BAC, ejus
quantitas di&longs;tribuitur per omnes inclinationis angulos, qui
fiunt in punctis intermediis inter B & C; atque adeò contem
nendum e&longs;t in praxi di&longs;crimen illud, quod oritur ex alio atque
alio inclinationis angulo in codem plano. Quod &longs;i in&longs;ignis e&longs;&longs;et
Rationum varietas, notabilis quoque e&longs;&longs;et gravitationis diver
&longs;itas idem enim contingeret, ac &longs;i non idem e&longs;&longs;et planum. Sed
hoc communiter non accidit.
Ex his illud manife&longs;tâ con&longs;ecutione conficitur, quod &longs;i duo
plana inclinata inter &longs;e comparentur, eju&longs;dem corporis gravita
tiones in illis &longs;unt reciproce ut Secantes angulorum inclinatio
nis: hoc e&longs;t, &longs;i fuerint duo plana inclinata BS, BC, gravitatio
in BS ad gravitationem in BC e&longs;t ut BC ad BS. Quia enim
gravitatio in BC ad gravitationem in BD e&longs;t ut BD ad BC;
& gravitatio in BD ad gravitationem in BS e&longs;t ut BS ad BD,
igitur ex æqualitate, per 23. lib.5. gravitatio in BC ad gravi
tationem in BS e&longs;t ut BS ad BC.
Hinc prætereà fit, ut, &longs;i gravia in planis con&longs;tituta habeant
Rationem eandem, quam &longs;ecantes angulorum inclinationis ha
bent inter &longs;e vel ad Radium, eorum gravitatione, &longs;int æquales.
Sit ad horizontalem, SC per
pendicularis BD, & inclina
tæ BS, BC, per quas lineas
ducta intelligantur plana, &
in planis gravia diver&longs;a, & ut
BD ad BC ita pondus O ad
pondus M, & ut BD ad BS
ita pondus O ad pondus N.
Dico ponderum M, O, N, gravitationes in &longs;uis planis e&longs;&longs;e
æquales.
perpendiculari BD &longs;unt ut
pendiculari BD, ad gravitationem O in eadem perpendiculari,
e&longs;t ut M ad O, hoc e&longs;t ut BC ad BD; &longs;ed gravitatio M in per-
BC, e&longs;t pariter ut BC ad BD; igitur per 11. lib.
5. gravita
tio M in perpendiculari ad gravitationem O in perpendiculari
e&longs;t, ut gravitatio M in perpendiculari BD ad gravitationem
M in inclinatâ BC; igitur per 14. lib.
5. gravitatio O in per
pendiculari BD æqualis e&longs;t gravitationi M in inclinatâ BC.
Eâdem methodo o&longs;tenditur æqualem e&longs;&longs;e gravitationem N in
inclinatâ BS, gravitationi O in perpendiculari BD. Quare
gravitationes M & N æquales inter &longs;e &longs;unt, cum æquales &longs;int
gravitationi O.
Con&longs;tat itaque ii&longs;dem viribus retineri po&longs;&longs;e, aut &longs;ur&longs;um trahi,
majus pondus in plano inclinato, quàm in perpendiculari, ea
dem enim e&longs;t illorum gravitatio, ut o&longs;tendi; vires autem reti
nentis aut trahentis debent gravitationi corporis proportione
re&longs;pondere. Quare datis viribus, quæ po&longs;&longs;int datum pondus O
&longs;u&longs;tinere in perpendiculari BD, cogno&longs;ci pote&longs;t gravitas pon
deris quod eædem vires &longs;u&longs;tinere valebunt in dato plano BC in
clinato: &longs;i nimirùm fiat ut Radius ad &longs;ecantem anguli datæ in
clinationis, ita datum pondus O ad pondus M quæ&longs;itum. De
tur O lib.
15. & angulus DBC gr. 36. Fiat ut radius 10000000
ad &longs;ecantem 12360680, ita lib.
15. ad lib.
18 1/2; quod e&longs;t pon
dus M æquè gravitans in plano BC cum pondere O in per
pendiculari. Contra verò dato pondere M &longs;u&longs;tinendo ii&longs;dem
viribus, quibus &longs;u&longs;tinetur O in perpendiculari, invenietur in
clinatio plani: &longs;i fiat ut pondus O lib.
15. ad pondus M datum
lib.
50, ita Radius 10000000 ad 333.33333.&longs;ecantem anguli in
clinationis DBC gr. 72. 32′. 32″.
Demum dato pondere & pla
ni inclinatione nota fiet potentia, &longs;i ut Secans datæ inclinatio
nis ad Radium, ita fiat datum pondus ad aliud pondus, quod
potentia valet &longs;u&longs;tinere in perpendiculari. Sit enim DBC
gr. 36, & M lib.
50. Erit ut Secans 12360680 ad Radium
10000000, ita M lib.
50 ad pondus O ferè lib.40 1/2, quod po&longs;&longs;it
à potentia in aere libero &longs;u&longs;tineri. Quare potentia &longs;u&longs;tinens
pondus in plano inclinato e&longs;t ad pondus, ut Radius ad Secan
tem anguli inclinationis; & potentia potens movere cum &longs;it ma
jor potentiâ &longs;u&longs;tinente, etiam majorem habet Rationem quàm
habeat Radius ad Secantem. Id quod intelligitur ex vi præcisè
gravitationis; quicquid inferat di&longs;criminis partium conflictus.
COn&longs;tituta Ratione gravitationis in plano inclinato, deter
minatis &longs;cilicet momentis, quæ ad de&longs;cendendum obtinet
corpus grave exi&longs;tens in plano inclinato, &longs;upere&longs;t explicanda
gravitatio, quam idem corpus exercet in planum inclinatum
illud urgendo, atque deor&longs;um premendo. Certum e&longs;t autem
planum verticale &longs;eu perpendiculare nullo pacto urgeri à cor
pore gravi, quod liberè de&longs;cendere pote&longs;t per &longs;uam directionis
lineam, quæ cum non occurrat plano verticali, nullum ab eo
recipit impedimentum. Quare corporis gravitas vires totas
exercet, aut de&longs;cendendo, aut repugnando contra retinentem,
qui non plus adhibere debet conatûs in retinendo, etiam &longs;i pla
num verticale amoveatur: atque adeò nihil omninò gravitat in
planum verticale. Contra verò in planum horizontale, quam
maximè gravitant corpora; eò quod directionis lineâ in illud
incurrente ad angulos rectos, motus omnis impeditur, &
cunctas gravitatis vires deor&longs;um contendentes ita &longs;ubjectum
planum excipit, ut nihil reliquum &longs;it virium, quas vel minimo
motu exerceat. Hinc &longs;i corporis in plano horizontali jacentis
an&longs;am teneas, nihil tibi pror&longs;us e&longs;t laborandum, nec quicquam
percipis ponderis; at &longs;ubmoto plano lacertis omnibus e&longs;t con
tendendum, ut illud retineas; tota enim gravitatio cum reti
nente luctatur, quæ planum &longs;u&longs;tinens urgebat. In hoc itaque
planum verticale cum horizontali comparatur, quod cum ver
ticale nihil impediat motum, corpus in plano verticali omninò
gravitat, &longs;ed in illud non gravitat: cum autem horizontale
pror&longs;us impediat motum, corpus in plano horizontali nihil gra
vitat, &longs;ed in illud totam &longs;uam gravitationem exercet. Eædem
igitur vires, quæ ad de&longs;cendendum in plano verticali impen
derentur, in urgendo plano horizontali in&longs;umuntur.
Quæ cum ita &longs;int, &longs;atis con&longs;tat corpora gravia ita in pla
no inclinato gravitare, & obtinere momenta ad de&longs;cenden-
ludque urgeant.
Id verò fieri non pote&longs;t ni&longs;i pro ratione impedimenti & mo
ræ, quam &longs;ubjectum planum motui infert &longs;u&longs;tinendo corpora
gravia; quæ proinde &longs;ibi relicta à directionis lineâ declinant,
motúmque deflectunt. Porrò in plano inclinato quantum &longs;ub
&longs;it impedimenti, &longs;tatim apparet, ac innote&longs;cit, quantum reli
quum &longs;it virium ad de&longs;cendendum; vires enim, quæ reliquæ
&longs;unt, &longs;i adjiciantur viribus impeditis, totam virium omnium
&longs;ummam conflare debent. Atqui ex &longs;uperiori capite notæ &longs;unt
vires, quibus corpus gravitat in plano inclinato; igitur quæ e&longs;t
differentia gravitationis in plano inclinato, à gravitatione in
plano verticali, quod & perpendiculare, ea e&longs;t men&longs;ura im
pedimenti, quod à &longs;ubjecto plano infertur motui; atque
adeò gravitationis corporis in planum.
Cum itaque o&longs;ten&longs;um fuerit
in plano BS ad gravitationem in plano BD
e&longs;&longs;e reciprocè ut BD ad BS, hoc e&longs;t, ut Ra
dius ad &longs;ecantem anguli inclinationis cum
verticali, hoc e&longs;t ut BV ad BS, patet vires
non impeditas ad vires impeditas e&longs;&longs;e ut
BV ad VS, quandoquidem totas gravita
tis vires refert BS. In planum igitur inclinatum BS gravitatio
e&longs;t ut VS, quæ in planum horizontale e&longs;&longs;et &longs;ecundùm totas
vires ut BS. Quare gravitatio in planum horizontale ad gra
vitationem in planum inclinatum e&longs;t ut Secans BS ad exce&longs;
&longs;um Secantis &longs;upra Radium, VS; &longs;eu, quod in idem recidit, &longs;i
gravitatio in plano inclinato ad gravitationem in verticali po
natur ut Sinus complementi anguli inclinationis ad Radium,
ita BR Radius ad DR Sinum ver&longs;um anguli inclinationis. Id
autem, quod de plano BS dictum e&longs;t, de plano quoque BC, &
cæteris quibu&longs;cunque dictum intelligatur; cum enim gravita
tio in plano inclinato BC ad gravitationem in perpendiculari
&longs;it ut BD, hoc e&longs;t BX, ad BC, erit gravitatio in planum ho
rizontale ad gravitationem in inclinatum ut BC ad XC, hoc
e&longs;t ut BT ad DT. Quare gravitatio in planum BS ad gravi
tationem in planum BC, e&longs;t ut DR Sinus ver&longs;us inclinationis
DBS, ad DT Sinum ver&longs;um inclinationis DBC; a&longs;&longs;umptis
neæ &longs;pectentur, non e&longs;t Ratio ut DR ad DT, &longs;ed ut OT
ad DT; neque enim idem e&longs;t Radius BS & BC; ac proinde
OT major e&longs;t, quàm DR, &longs;icuti Radius BI major e&longs;t Radio
BS; vel a&longs;&longs;umpto eodem Radio BD, Ratio e&longs;t ut VS ad XC,
exce&longs;&longs;us &longs;ecantium &longs;upra Radium.
Id verò ex dictis &longs;ub finem capitis &longs;uperioris videtur mani
fe&longs;tum: nam &longs;i in plano BC retinetur pondus lib.
50. ii&longs;dem
viribus, quibus in perpendiculari &longs;u&longs;penderentur lib.
40 1/2, pa
tet à plano &longs;u&longs;tineri lib.9 1/2; ac proinde grave, quod habet gra
vitatem totam ut 100, in plano BC gravitabit ut 81, & urge
bit ut 19 &longs;ubjectum planum.
Ex his fieri pote&longs;t &longs;atis quæ
renti, cur &longs;u&longs;tinens columnam
OR plus gravitatis percipiat,
quàm qui &longs;u&longs;tinet columnam
SR: quia nimirum, qui &longs;u&longs;tinet,
e&longs;t pars plani inclinati, in quo ja
cens concipitur columna: quan
do igitur e&longs;t pars plani habentis
inclinationem LOR, gravitas,
quæ &longs;u&longs;tinetur à &longs;ubjecto plano, &longs;e habet ad totam gravitatem
ut Sinus Ver&longs;us anguli LOR ad Radium; Quando autem e&longs;t
pars plani habentis inclinationem VSR, gravitatio in &longs;ub
jectum planum &longs;u&longs;tinens e&longs;t ad totam gravitationem ut Sinus
Ver&longs;us anguli VSR ad eumdem Radium. Atqui Sinus Ver&longs;us
anguli VSR minoris minor e&longs;t Sinu Ver&longs;o anguli LOR ma
joris; igitur minor e&longs;t gravitatio SR, quam OR. Verum qui
dem e&longs;t illud, quod &longs;i in R aliquo obice prohibeatur, ne de
&longs;cendat; variatâ inclinatione, quo fit minor &longs;u&longs;tinentis labor,
eò augetur magis conatus potentiæ in R detinentis columnam,
ne juxta plani inclinationem de&longs;cendat. Hinc &longs;i duo &longs;int co
lumnam inclinatam deferentes, qui illam in R &longs;u&longs;tinet, plus
&longs;ubit laboris, quàm qui in O, aut S: quia præter gravitatio
nem, quam percipit tanquam pars plani inclinati SR aut OR,
debet præterea retinere columnam proclivem ad de&longs;cen&longs;um
propter plani inclinationem; ideò cùm &longs;calas, aut montis cli
vum con&longs;cendunt, qui in &longs;uperiore loco e&longs;t, minimum &longs;ubit Huc etiam revocari po&longs;&longs;e videtur ratio, ob quam in
elevando pontes illos ver&longs;atiles, qui arcium portis opponuntur,
initio major percipiatur difficultas, &longs;ed demùm facillimè ele
ventur. Verùm id ex dicendis inferiùs clariùs con&longs;tabit; neque
enim omnium gravium, quocunque &longs;e tandem modo habeant,
eadem e&longs;t ratio; cum animum diligenter advertere oporteat, ut
innote&longs;cat planum inclinatum, in quo &longs;uam gravitationem
exercent, & habent vires ad de&longs;cendendum.
Non e&longs;t autem per di&longs;&longs;imulantiam prætereunda difficultas,
quæ face&longs;&longs;ere po&longs;&longs;et aliquid negotij, & gravitationis Rationem
con&longs;titutam convellere videretur. E&longs;t &longs;iquidem certum apud
omnes mechanicos, tam ubi de libra, quàm ubi de vecte &longs;ermo
e&longs;t, aliam &longs;ervari Rationem quàm Sinuum Ver&longs;orum in mo
mento potentiæ, aut ponderis determinando. Sit vectis, aut
libræ brachium EC, hypomochlion
&longs;eu centrum C; attollatur in H, aut
in D; omnes con&longs;entiunt momentum
potentiæ aut ponderis in E ad mo
mentum in H, e&longs;&longs;e ut HC ad IC,
ad momentum verò in D e&longs;&longs;e ut DC
ad FC. E&longs;t igitur, inquis, gravitatio
in planum DC ad gravitationem in
planum horizontale EC, ut FC ad DC; in planum verò HC,
ut IC ad HC, hoc e&longs;t ut Sinus Rectus anguli inclinationis ad
Radium.
Priùs verò, quàm me ab hac difficultate expediam, o&longs;tendo
non &longs;atis aptè gravitationem in planum inclinatum de&longs;umi po&longs;
&longs;e ex Sinu Recto anguli inclinationis. Quandoquidem vis de
&longs;cendendi in plano DC ad
ut DF ad DC, igitur &longs;i gravitatio in
tationem
ac tota vis gravitandi, ubi nullum e&longs;t impedimentum, e&longs;t DC;
igitur DC, & DF plus FC, æquales &longs;unt, contra 20.lib.1.Eucl.
Neque hic liceat ad æqualitatem potentiarum confugere, ut
&longs;icut per 47. lib.
1. Eucl. linea DC pote&longs;t quadrata linearum
DF, FC, ita vis totius gravitatis æqualis gravitationibus in
plano inclinato & in planum inclinatum eandem &longs;ervet pro
portionem laterum trianguli DFC, adeò ut totam gravitatem
inclinato Radius, Tangens verò gravitationem in planum in
clinatum. Si enim Quadratum DC æquale e&longs;t quadratis DF,
& FC &longs;imul &longs;umptis, non tamen linea DC æqualis e&longs;t aggre
gato linearum DF & FC: neque eadem e&longs;t inter lineas DF
& DC Ratio, quæ inter earum quadrata; &longs;ed e&longs;t &longs;ub duplica
tâ quadratorum: Quare cum gravitatio in plano inclinato DC
ad gravitationem in perpendiculari, non &longs;it ut quadratum DF
ad quadratum DC; &longs;ed ut linea DF ad lineam DC, fru&longs;trà ad
quadrata confugimus, quorum nulla hîc habetur ratio.
In eo itaque æquivocatio con&longs;i&longs;tit, quod pondus in D con&longs;ti
tutum, & applicatum brachio DC concipitur e&longs;&longs;e in plano in
clinato DC, contra quàm res e&longs;t: in eo &longs;iquidem plano intel
ligendum e&longs;t, in quo ad motum determinatur; illud autem e&longs;t
planum DG, quod tangit circulum ED; & &longs;ic deinceps, pro
ut diver&longs;a circuli puncta à diver&longs;is planis contingi po&longs;&longs;unt.
Quare in D momentum ad de&longs;cendendum per DG ad totam
gravitationem e&longs;t ut DF ad DG, hoc e&longs;t ut FC ad CD, per
8. lib.6. hoc e&longs;t ut FC ad EC. E&longs;t igitur brachium libræ &longs;eu
vectis CD, &longs;u&longs;tinens pondus &longs;eu potentiam D, quæ cum ha
beat vires univer&longs;as ut EC, gravitationis autem momenta ha
beat &longs;olùm ut FC, impeditur à &longs;u&longs;tinente ut FE; e&longs;t autem
EF Sinus Ver&longs;us anguli FCD, hoc e&longs;t anguli inclinationis
FDG. Quare gravitatio ponderis contrà &longs;ubjectum corpus,
quod impedit motum perpendicularem, ad totam gravitatio
nem e&longs;t, ut Sinus Ver&longs;us anguli inclinationis plani, per quod
fieri pote&longs;t motus, ad Radium.
Hinc vides valdè di&longs;parem e&longs;&longs;e rationem gravitationis in
&longs;u&longs;tinendo corpore inclinato, &longs;i illud liberè moveri po&longs;&longs;it, ac &longs;i
circa centrum perfici debeat motus. Nam &longs;i DC &longs;it columna,
aut pons ver&longs;atilis, retineaturque in C, jam punctum C vicem
obtinens &longs;ubjecti plani, illiu&longs;que munere fungens, &longs;u&longs;tinet
ponderis partem EF, reliqua FC, quæ e&longs;t men&longs;ura momenti
ad de&longs;cendendum, debet &longs;u&longs;tineri à potentia motum impe
diente per DG. Sin autem per DC planum columna moveri
po&longs;&longs;it rectâ & de&longs;cendere, vis de&longs;cendendi ad totam gravitatio
nem e&longs;t ut DF ad DC, gravitatio autem contra &longs;u&longs;tinentem
e&longs;t ad totam gravitationem ut Sinus Ver&longs;us anguli inclinationis
clinatum, habet rationem plani inclinati. Neque id mirum vi
deri debet, quandoquidem plurimum refert, an per planum
DG an verò per DC &longs;it determinatio ad motum, & quâ ra
tione &longs;u&longs;tinens opponatur virtuti motivæ: quare cùm diversâ
ratione opponatur motui circa centrum C, ac motui per pla
num DC, etiam di&longs;par erit in &longs;u&longs;tinendo difficultas.
Ex his, quæ tùm hoc, tùm &longs;uperiori capite di&longs;putata &longs;unt,
habes quid funambulis re&longs;pondeas volatum mentiri meditanti
bus, cum pectore in&longs;i&longs;tentes intento funi, diductis cruribus &
exten&longs;is brachiis, corpus æqualibus momentis librant, séque
ex editâ turri in depre&longs;&longs;iorem locum præcipites dant; &longs;i fortè,
ut noverint, quàm &longs;olidus e&longs;&longs;e debeat ac validus funis, quo iis
utendum e&longs;t, quærant, quantis momentis corpus urgeat &longs;ub
jectum funem. Datâ enim turris altitudi
ne BD, & depre&longs;&longs;ioris loci, in quem de
&longs;cendendum e&longs;t, di&longs;tantiâ DC, collectí&longs;
que in &longs;ummam harum quadratis, Radix
&longs;ummæ dabit BC funis longitudinem; ex
quâ &longs;i auferatur BX turris altitudini BD
æqualis, erit BC divi&longs;a in X juxtà Ratio
nem momentorum, quæ corporis gravitas
exercet in plano inclinato, & in planum
inclinatum. Sic po&longs;itâ BD ped. 150, & DC ped. 200, BC e&longs;t
ped. 250: ex quâ &longs;i auferatur BD, erit BX 150, & XC 100.
Statue autem totius gravitatis corporis funambuli momenta
220; hæc dividantur in duas partes, quarum major &longs;it &longs;e&longs;qui
altera minoris, &longs;icut BX inventa e&longs;t ip&longs;ius XC &longs;e&longs;quialtera, &
erunt momenta quidem ad de&longs;cendendum in plano inclinato
132, momenta verò gravitationis in planum inclinatum, hoc
e&longs;t in &longs;ubjectum funem, 88. Hæc tamen intelligenda &longs;unt eâ
factâ hypothe&longs;i, quòd funis rectâ intentus permaneret: cæte
rùm cum & &longs;uopte pondere, & &longs;ub impo&longs;iti corporis mole &longs;ub
&longs;idat, atque inflectatur, præ&longs;ertim circà medium, &longs;atis apparet
adhuc majorem &longs;ubjecti plani inclinationem æ&longs;timandam e&longs;&longs;e,
quàm quæ ex altitudine DB & di&longs;tantiâ DC inferatur, quin
& illam pro diversâ ab extremitatibus di&longs;tantiâ &longs;ubinde muta
ri, ac proinde validiori fune opus e&longs;&longs;e.
&longs;uspen&longs;orum.
COn&longs;ideratâ corporum gravitatione tùm in plano inclinato,
tùm in planum inclinatum, con&longs;equens e&longs;t, ut ad eorum
dem gravitationem, &longs;i ex fune &longs;u&longs;pendantur, gradum facia
mus; hæc enim illi valdè affinis e&longs;t &longs;peculatio: id quod facilè
intelligat, qui&longs;quis animum advertere voluerit, remque totam
penitiùs intro&longs;picere. Ex his &longs;i quidem, quæ hactenus di&longs;puta
ta &longs;unt, lux, opinor, non modica ad hanc, quam examinan
dam &longs;u&longs;cipimus quæ&longs;tionem, derivabitur.
Pendeat ex clavo C ad perpen
diculum globus ferreus A, quem
&longs;uppo&longs;itum planum horizontale
BD ita exactè contingat, ut nihil
de funiculi CA intentione remit
tatur. Satis apparet &longs;ubjecto pla
no BD non incumbere globum A,
&longs;ed omnia &longs;uæ gravitationis, qua
deor&longs;um nititur, momenta exer
cere contrà clavum C, ex quo &longs;u&longs;pen&longs;us ad perpendiculum
pendet. Quod &longs;i aut clavus C, nemine funem retinente, revel
leretur, aut funis CA præcideretur, jam tota vis de&longs;cendendi,
quæ corpori A ine&longs;t, urgeret &longs;ubjectum planum BD; nec ta
men in motum erumperet globus, quia planum BD; pari u&longs;que
quaque ad perpendiculum inclinatione libratur, atque adeò
motui pror&longs;us ob&longs;i&longs;tit.
Jam verò &longs;i globum A pariter ex perpendiculo CA penden
tem contingat planum aliud non quidem horizontale, &longs;ed in
clinatum EF, manife&longs;tum e&longs;t totam pariter gravitationem
exerceri contra clavum C retinentem, planumque contingens
bus, ut in inclinato plano EF ad de&longs;cen&longs;um pronus contra &longs;ub
jectum planum nitatur, à quo cogitur, ut in motu à recto, quod
ad univer&longs;i centrum e&longs;t, itinere deflectat.
Quod &longs;i planum inclinatum EF ita &longs;u&longs;pen&longs;o globo A &longs;ubji
ciatur, ut recta linea centrum gravitatis A, & punctum &longs;u&longs;
pen&longs;ionis H conjungens parallela &longs;it lineæ EF, quam in plano
inclinato de&longs;cendens globus percurreret; momenta quidem
gravitationis, quæ in eo plano obtineret globus ad de&longs;cenden
dum, exercebit adversùs clavum retinentem in H, &longs;ubjectum
verò planum EF perinde urgebitur, atque &longs;i nullo retinente li
bera e&longs;&longs;et globo de&longs;cendendi facultas: vis enim, quâ prohibe
tur globus, ne moveatur &longs;ecundùm rectam lineam, ut con&longs;tat,
opponitur de&longs;cen&longs;ui in plano inclinato; ejus autem directio
AH non opponitur nitenti in planum, cui parallela e&longs;t.
Contra verò &longs;i globus in plano inclinato con&longs;titutus retinea
tur &longs;ecundùm rectam lineam, quæ ad perpendiculum cadit in
&longs;ubjectum planum EF, nimirum &longs;ecundùm lineam LO, im
peditur quidem, ne contra planum nitatur; &longs;ed vis i&longs;ta &longs;ic reti
nens nullâ ratione adver&longs;atur motui in plano inclinato, quin
ii&longs;dem gravitatis momentis de&longs;cendat globus in eo plano; &longs;i
quidem retinentis directio LO maneat &longs;emper adversùs illud
planum perpendicularis. Nam &longs;i potentia retinens &longs;ecundùm
eam directionem agat, ut neque congruat perpendiculari LO,
neque parallelæ HA, ob&longs;i&longs;tet gravitationi corporis &longs;ivè in pla
no inclinato, &longs;ivè in planum inclinatum pro ratione anguli,
quem retinentis directio inter perpendicularem LO, & paral
lelam HA interjecta, con&longs;tituet cum plano inclinato. Quæ
enim inter LO & CA fuerit, elidet omnem corporis conatum
adversùs planum, à quo illud avellit; non autem omnem eum,
qui in plano inclinato deor&longs;um rapit. Quæ verò fuerit inter
CA & HA, tollet quidem de&longs;cen&longs;um in plano EF inclinato;
&longs;ed non omninò prohibebit, quin &longs;ubjectum planum, cui aliqua
tenus nititur, urgeat. Id quod facilè intelligas, &longs;i plana &longs;ubjecta
BD horizontale, & EF inclinatum ex maximè flexili mate
ria, puta, papyro, concipias; in quâlibet enim &longs;u&longs;pen&longs;ione
inter C, & L, planum BD horizontale flectetur ex pondere,
non autem inclinatum EF: contrà verò in omni &longs;u&longs;pen&longs;ione
rizontale BD, quia nimirum inclinatum EF prohibet, ne recta
HA ad perpendiculum accedens verticalis fiat.
Unum hic præterea con&longs;iderandum venit, quod &longs;uperiori
capite &longs;ubindicatum fuit; &longs;i videlicet non ex flexili fune deor
&longs;um pendeat globus, &longs;ed rigido bacillo circà axem inferiùs po
&longs;itum ver&longs;atili adnectatur &longs;upe
riùs. Sic rectus bacillus AB, cujus
extremitas altera adnexum ha
beat globum B, altera &longs;it circà
axem A ver&longs;atilis. Satis aperta
conjectura e&longs;t bacillum AB vi
cem &longs;ubire plam, cui innitatur
globus in B, qui proinde prohi
betur, tùm ne ad perpendiculum
cadat per BD, tùm ne per BA
delabatur: linea igitur plani, per quod moliri motum poterit
globus B, nulla alia congruentiùs a&longs;&longs;ignari queat præter BC,
quæ cum bacillo BA rectum angulum con&longs;tituit. Perindè igi
tur in motum incitabitur, atque &longs;i in plano e&longs;&longs;et, cujus inclina
tio angulum efficeret æqualem angulo elevationis bacilli &longs;uprà
planum horizontale GA. Cum enim recta BD producta ca
dens in planum horizontale, angulum BSA Rectum efficiat,
reliqui duo &longs;imul SAB, ABS, Recto ABC æquales &longs;unt; &
communi ABS dempto, &longs;upere&longs;t SAB elevationis angulus
æqualis angulo SBC inclinationis plani. Quare ductâ Tan
gente DE, erit BE Secans anguli inclinationis, BD verò Ra
dius: ac proptereà ad de&longs;cendendum in huju&longs;modi plano BC
momenta, ad totam gravitatem in perpendiculo BD, erunt ut
Radius BD ad Secantem BE, juxta ea, quæ cap.
13. hujus lib.
demon&longs;travimus.
Quia tamen in motu globus ex bacilli conver&longs;ione circà
axem A non pote&longs;t percurrere rectam BC, &longs;ed ita retinetur à
bacillo, cui adnectitur, ut de&longs;cendat in F, jam in alio plano
minorem inclinationem habente con&longs;titutus intelligitur, nimi
rùm in plano FG, quod cum perpendiculo FL efficit angulum
inclinationis GFL æqualem angulo LAF elevationis: id quod
eâdem planè methodo, ac &longs;uperiùs factum e&longs;t, demon&longs;tratur.
in alio atque alio plano inclinato con&longs;tituitur, ita alia atque alia
obtinere gravitatis momenta: in B &longs;iquidem gravitat ut BD ad
BE, in F verò ut HF ad FI. Cum igitur Radius utrobique ex
hypothe&longs;i æqualis &longs;it, videlicet DB, & HF, major autem &longs;it
BE Secans majoris anguli DBE, quàm FI Secans minoris an
guli HFI, con&longs;tat ex 8. lib.
5. majorem Rationem e&longs;&longs;e HF ad
FI minorem, quàm DB ad BE majorem, atque adeò globum
magis in F quàm in B gravitare, ut deor&longs;um moveatur, atque
adeò minùs etiam conniti contrà planum, in quo e&longs;t, videlicet
adversùs bacillum FA, magis verò adversùs bacillum BA.
Ex his attentè perpen&longs;is facilis e&longs;t tran&longs;itus ad &longs;u&longs;pen&longs;orum
corporum gravitationem inve&longs;tigandam. Sit enim jam non in
feriùs, &longs;ed &longs;uperiùs po&longs;itus
Axis A, circa quem ver&longs;a
tilis e&longs;t funiculus AB, cui
globus B adnectitur. Con
&longs;tat &longs;anè non ad perpendi
culum BD cadere po&longs;&longs;e
globum B; &longs;ed à recto deor
&longs;um tramite deflectere, fu
niculo &longs;cilicet AB eum re
tinente, quemadmodum ri
gidus bacillus OB eum ali
quatenùs &longs;u&longs;tineret. Quia autem bacillo OB &longs;u&longs;tinente, vis
de&longs;cendendi ea e&longs;&longs;et, quæ per planum inclinatum BC, eadem
pariter e&longs;t funiculo retinente; videlicet per planum BC, in
quod recta AB ad rectos angulos incidit. Momenta igitur gra
vitatis in eo plano inclinato, ad gravitatis momenta &longs;i corpus
liberè de&longs;cenderet, in eâ &longs;unt Ratione, quæ e&longs;t DB ad BE;
hoc e&longs;t DO ad OB per 8. lib.6. hoc e&longs;t KB ad BA per 4.lib.6.
Haud di&longs;pari methodo ratiocinantes o&longs;tendemus globi in F
con&longs;tituti momenta ad gravitandum e&longs;&longs;e perinde, atque &longs;i e&longs;
&longs;et in plano inclinato FI, in quod ad rectos angulos cadit fu
niculus AF; ac proinde gravitatio in F, &longs;i de&longs;cendendi vis
præcisè &longs;pectetur, ad gravitationem globi liberi, e&longs;t ut HF
ad FI, hoc e&longs;t, ut GF ad FA.
Ex quo apertiùs liquet, quàm ut in eo explicando diutiùs
vitatis corporis &longs;u&longs;pen&longs;i, pro ut major aut minor e&longs;t angulus
declinationis à perpendiculo AG, haud aliter quàm &longs;i in aliis
atque aliis planis inclinatis con&longs;titueretur; quo enim minor e&longs;t
declinationis angulus GAF, eò major e&longs;t angulus inclinatio
nis plani, quippe qui e&longs;t illius complementum. Con&longs;tat &longs;i qui
dem angulos GAF, GFA &longs;imul, e&longs;&longs;e æquales tùm Recto
AFI, tùm Recto GFH; ac proinde dempto communi GFI,
remanet HFI angulus inclinationis plani æqualis angulo
GFA, qui e&longs;t complementum anguli declinationis GAF.
Quare quò declinationis angulus major e&longs;t, eò minus e&longs;t
complementum, ac propterea e&longs;t minor angulus inclinationis
plani: in plano autem minùs inclinato majora &longs;unt gravitatis
momenta. Quò igitur corpus &longs;u&longs;pen&longs;um magis à perpendiculo
removetur, eò majora percipiuntur gravitatis momenta, ma
jorque vis requiritur in eo, qui motum prohibere voluerit, ut
& ip&longs;a experientia unicuique facilè demon&longs;trat, & ratio evin
cit; cum enim AB & AF æquales &longs;int, major e&longs;t Ratio KB
ad BA, quàm GF ad FA per 8. lib.
5. e&longs;t nimirum KB major,
& GF minor.
Quoniam verò quò major e&longs;t gravitatio in plano inclinato,
minor e&longs;t in planum inclinatum; hoc ip&longs;o, quod facto declina
tionis angulo GAB majore, quàm GAF, major e&longs;t ad de&longs;cen
dendum propen&longs;io, minor e&longs;t conatus adversùs axem A reti
nentem. Id quod manife&longs;to etiam experimento deprehen
des, &longs;i ob&longs;ervaveris minùs intentum e&longs;&longs;e funiculum AB,
quàm AF.
Hinc & illud &longs;atis dilucidè apparet, quod longitudinis
funiculi non exigua ratio habenda e&longs;t; ex eâ &longs;cilicet pen
det, quod in plano magis aut minùs inclinato con&longs;titutum
cen&longs;eatur corpus grave &longs;u&longs;pen&longs;um. Si enim globus F ex fu
niculo AF pendeat, declinationis angulus e&longs;t GAF: at
verò &longs;i funiculus, quo &longs;u&longs;penditur, &longs;it MF, angulum de
clinationis facit GMF, qui cum externus &longs;it, major e&longs;t
interno MAF per 16. lib.
1. ac propterea minor e&longs;t incli
natio plani FN facientis cum rectâ MF angulum Rectum,
quàm &longs;it inclinatio plani FI, cui perpendicularis e&longs;t recta
AF. Plus igitur momenti ad gravitandum habet glo-
longiore AF.
Quæ cum ita &longs;int, haud &longs;anè incongrua &longs;e nobis offert me
thodus pondus ex depre&longs;&longs;iore in altiorem locum transferendi;
&longs;i videlicet id curemus, ut ex &longs;atis valido & longiore fune &longs;u&longs;
pendatur; &longs;ublato etenim partium attritu, qui fieret, &longs;i per pla
num raptaretur pondus, minore virium jacturâ trahi pote&longs;t.
Sit corpus grave ubi A, quod at
tollere oporteat, & in &longs;uperiorem
locum RS transferre. Si ex C brevio
ri fune &longs;u&longs;pendatur, trahere illud po
terit u&longs;que in R, quicumque facto de
clinationis angulo ACR pote&longs;t illud
cum aliquo virium exce&longs;&longs;u retinere,
& ob&longs;i&longs;tere gravitatis momentis, quæ
obtinet in R. At &longs;i ex longiore fune
DA pendeat, idem corpus A trahi
poterit, & retineri in S, ne deor&longs;um labatur, & quidem mino
re conatu; facto enim declinationis angulo ADS minore,
quàm ACR, in S pariter minùs gravitat quàm in R. Angu
lum autem ADS minorem e&longs;&longs;e angulo ACR con&longs;tat, &longs;i rectæ
AR, AS ducantur: nam CA, CR æqualia &longs;unt latera ex hy
pothe&longs;i, item DA, DS æqualia; e&longs;t &longs;cilicet idem funiculus,
qui primum perpendicularis cadit, deinde à perpendiculo re
movetur: in Triangulo I&longs;o&longs;cele CAR anguli ad ba&longs;im AR
æquales &longs;unt per 5. lib.
1. item in triangulo I&longs;o&longs;cele DAS an
guli ad ba&longs;im AS æquales inter &longs;e &longs;unt. Porrò angulus DAS
major e&longs;t angulo CAR; ergo & reliquus DSA major reliquo
CRA. Cum itaque tres anguli utriu&longs;que trianguli &longs;int æquales
duobus Rectis per 32. lib.
1. &longs;i ex &longs;ummâ duorum Rectorum au
ferantur duo majores anguli DAS, DSA, relinquitur ADS
minor, quàm &longs;i ex eâdem duorum Rectorum &longs;ummâ auferan
tur duo minores CAR, CRA, hoc e&longs;t minor quàm ACR.
Ut autem clariùs innote&longs;cat, quænam &longs;it gravitationum Ratio
pro funiculi longitudine, &longs;it corpus grave in R: & primùm
quidem ex C pendeat funiculo breviore CR, deinde ex D lon
giore funiculo DR: qui&longs;quis retineat corpus in R con&longs;titu
tum, atque de&longs;cen&longs;u prohibeat, faciliùs retinebit, cum ex D,
major e&longs;t angulo XDR per 16. lib.
1. Verùm qua Ratione, in
quis, vires, quas in utroque ca&longs;u retinens exerit, di&longs;criminan
tur? utique &longs;ecundùm Reciprocam funiculorum Rationem co
natur ob&longs;i&longs;tens corporis propen&longs;ioni ad de&longs;cen&longs;um; quæ enim
Ratio gravitationum corporis, ea e&longs;t virium gravitationibus
repugnantium: comparatà autem corporis in R con&longs;tituti gra
vitatione, &longs;i ex C pendeat, cum eju&longs;dem ibidem po&longs;iti gravita
tione, &longs;i pendeat ex D, e&longs;t reciprocè ut DR ad CR; igitur
& vires retinentis corpus ex C pendens &longs;unt ut DR, retinen
tis verò idem corpus ex D pendens &longs;unt ut CR. Id quod hinc
conficitur, quia corpus in &longs;u&longs;pen&longs;ione, po&longs;itionem habens CR,
gravitat ut XR ad RC, po&longs;itionem verò habens DR gravitat
ut XR ad RD; duæ autem Rationes XR ad RC, & XR ad
RD &longs;unt reciprocè ut RD ad RC. Quotie&longs;cumque enim duæ
&longs;unt Rationes, quarum idem e&longs;t Antecedens terminus, & di
ver&longs;us Con&longs;equens, eæ &longs;unt reciprocè ut con&longs;equentes.
Quòd &longs;i quis Rationes inter &longs;e comparare non a&longs;&longs;uetus de
hoc ambigeret, an Rationes eumdem vel æqualem anteceden
tem terminum habentes &longs;int reciprocè ut Con&longs;equentes, facilè
intelliget, &longs;i animadvertat Rationes eumdem Con&longs;equentem
terminum habentes e&longs;&longs;e inter &longs;e directè, ut antecedentes.
Quemcumque enim interrogaveris, quæ &longs;it Ratio 2/7 ad 6/7 illicò
re&longs;pondebit e&longs;&longs;e &longs;ubtriplam, &longs;ecunda &longs;cilicet ter continet pri
mam, ut con&longs;tat &longs;i ter po&longs;itam Rationem 2/7 in &longs;ummam colligas;
neque enim
triplicatam; Ratio &longs;iquidem 2/7 e&longs;t &longs;ubtriplicata Rationis (8/343). Si
igitur pariter quæras, quænam &longs;it Ratio 7/2 ad 7/6 rectè re&longs;ponde
bit eam e&longs;&longs;e triplam, hoc e&longs;t reciprocè ut 6 ad 2: id quod ma
nife&longs;tè apparebit, &longs;i illas ad denominationem eandem, hoc e&longs;t
ad eumdem Con&longs;equentem terminum reduxeris, &longs;unt nimirum
ut (42/12) ad (14/12), hoc e&longs;t ut 6 ad 2.
Ex quibus obiter patet methodus exponendi per lineas pro
portionem duarum Rationum etiam numeris non explicabi
lium; &longs;i videlicet fiat ut Antecedens &longs;ecundæ Rationis ad &longs;uum
Con&longs;equentem, ita Antecedens datus primæ Rationis ad alium
novum Con&longs;equentem; erit enim prima Ratio data ad &longs;ecun-
ad datum Con&longs;equentem primæ Rationis: aut etiam &longs;i fiat ut
Con&longs;equens &longs;ecundæ Rationis ad &longs;uum Antecedentem, ita con
&longs;equens primæ Rationis ad alium novum Antecedentem; erit
enim prima ratio data ad &longs;ecundam Rationem datam, directè
ut datus Antecedens primæ Rationis ad novum Antecedentem.
Con&longs;ideratâ hactenus unicâ & &longs;implici corporis gravis &longs;u&longs;
pen&longs;ione, gradum facere oportet ad gravitationis rationes in
ve&longs;tigandas, &longs;i duplex fuerit &longs;u&longs;pen&longs;io. Sit enim globus A tùm
ex B, tùm ex C &longs;u&longs;pen&longs;us fu
niculis BA & CA. Haud du
bium quin tota corporis gravi
tas ex B & C pendeat; &longs;ed quâ
Ratione &longs;ingulæ vires eidem
gravitati ob&longs;i&longs;tant, de hoc po
te&longs;t ambigi. Verùm ni&longs;i mea
mihi nimiùm blanditur opi
nio, ex dictis facilis videtur
explicatio. Corpus &longs;iquidem
ex duplici fune &longs;u&longs;pen&longs;um ita
con&longs;titutum e&longs;t, ut alterutro
fune præci&longs;o ex reliquo pen
deat, & de&longs;cendens moveatur
circà punctum, cui alligatur
funis. Quare unu&longs;qui&longs;que ob&longs;i&longs;tit momentis, quibus ex altero
gravitat; nimirum funiculus CA retinens globum, ne de&longs;cen
dat, repugnat momentis gravitatis, quibus globus A &longs;e ip&longs;e
deor&longs;um urget circa punctum B ex fune BA: Contrà verò fu
niculus BA eundem globum retinet, ne circa punctum C ex
funiculo CA moveatur de&longs;cendens, atque adcò ob&longs;i&longs;tit, mo
mentis gravitatis ad de&longs;cendendum circà idem punctum C. At
qui momenta de&longs;cendendi ex fune BA ad gravitatem in per
pendiculo &longs;unt ut DA ad AB, & ex fune CA &longs;unt ut EA ad
AC, ex his, quæ &longs;uperiùs di&longs;putata &longs;unt. Sunt igitur duæ Ra
tiones DA ad AB, & EA ad AC.
Quare fiat angulus DAF æqualis angulo EAC, & e&longs;t trian
gulum DAF ob angulorum æqualitatem &longs;imile triangulo
EAC; ac propterea per 4. lib.
6. ut EA ad AC, ita DA ad Ergo vis de&longs;cendendi ex CA e&longs;t ut DA ad AF, & vis
de&longs;cendendi ex BA e&longs;t ut DA ad AB: igitur duæ hæ Ratio
nes &longs;unt reciprocè ut BA ad AF; atque adeò B quidem reti
nens, ne de&longs;cendat ex CA, exerit vires ut BA; C verò reti
nens, ne de&longs;cendat ex BA, adhibet conatum ut FA; & quæ
componitur ex BA, AF, totum gravitatis momentum, quod
corpori &longs;u&longs;pen&longs;o ine&longs;t, repræ&longs;entat. Momentum, inquam,
gravitatis potiùs, quàm gravitatem totam; totius &longs;i quidem
gravitatis nomine vires ip&longs;as de&longs;cendendi intelligimus, quas
corpus grave obtinet &longs;ibi prorsùs relictum &longs;eclu&longs;o quolibet im
pedimento, à quo certam de&longs;cendendi regulam accipiat: Mo
menti autem vocabulo ip&longs;as de&longs;cendendi vires &longs;ignificamus
non per &longs;e & &longs;olitariè acceptas; &longs;ed quatenus ex corporis po&longs;i
tione, cæterorumque quæ circum&longs;tant, ad majorem aut mino
rem motùs velocitatem determinatur. Con&longs;iderato itaque ni&longs;u
corporis A ad de&longs;cendendum & cùm perpendicularis e&longs;t funi
culus BD, & cum declinat BA, Ratio momentorum e&longs;t ut
BA ad AD. Similiter momentum ex perpendiculari CE ad
momentum ex declinante CA e&longs;t ut CA ad AE, hoc e&longs;t ut
FA ad AD: e&longs;t igitur corporis A ex duplici funiculo BA, CA
pendentis totum gravitandi momentum, quod ex lineis BA,
AF componitur.
Hic autem hæ&longs;itantem videre mihi videor non neminem ex
iis, quæ dicebantur, colligentem corpus A primùm ex decli
nante BA æquè ac ex perpendiculari BD gravitare; deinde
plus ad de&longs;cendendum momenti obtinere, &longs;i ex duobus funi
culis, quàm &longs;i ex unico pendeat. Si enim angulus declinatio
nis DBA &longs;it gr. 22. 12′; e&longs;t DA &longs;inus dati anguli ad radium
BA ut 37784 ad 100000: & &longs;i angulus declinationis ECA
&longs;it gr. 54. 35, e&longs;t EA &longs;inus dati anguli ad Radium CA ut
81496 ad 100000. At ex con&longs;tructione triangulum DAF &longs;i
mile e&longs;t triangulo EAC; igitur DA ad AF e&longs;t ut 81496 ad
100000. E&longs;t autem DA in particulis Radij BA partium 37784;
igitur &longs;i fiat ut 81496 ad 100000, ita 37784, ad aliud, erit AF
earumdem particularum 46363, quarum BA e&longs;t 100000. Qua
re compo&longs;ita BA, AF momenta &longs;unt 146363, cum tamen
momentum in perpendiculari AD &longs;it tantum 100000. Cum
verò dictum &longs;it B clavum re&longs;i&longs;tere ponderi A ut BA, C autem
declinat BA à perpendiculo: Atqui etiam in perpendiculo BD
retinet ut 100000, igitur idem e&longs;t ponderis tùm ex BD, tùm
ex BA momentum; id quod e&longs;t ab&longs;urdum.
Sed & illud prætereà ex dictis con&longs;equi videtur, quod eju&longs;
dem corporis majus &longs;it momentum, &longs;i ex duobus funiculis, quàm
&longs;i ex unico pendeat. Fiat enim angulus DBH æqualis angulo
declinationis ECA, & a&longs;&longs;umptâ BH æquali ip&longs;i BA, ducatur
ad BD perpendicularis HI: erit utique triangulum BHI &longs;imi
le triangulo CAE, ac propterea ut EA ad AC, ita IH ad
HB, hoc e&longs;t ad AB. Sunt igitur duæ Rationes eundem Con
&longs;equentem terminum habentes, atque adeò inter &longs;e in ratione
Antecedentium, ac proinde cùm vis de&longs;cendendi ex BA &longs;it ut
DA ad AB, & vis de&longs;cendendi ex CA &longs;it ut IH ad AB, vires
de&longs;cendendi invicem comparatæ &longs;unt ut DA ad IH, totum
que momentum componitur ex DA 37784, & IH 81496.
Quare momentum quod in perpendiculari, &longs;i unico funiculo
penderet ex BD, e&longs;&longs;et 100000, pendente corpore A ex duo
bus funiculis BA, CA, fit majus, &longs;cilicet 119280. ut quid igi
tur ex pluribus funiculis illud &longs;u&longs;pendere oportuit?
Quibus difficultatibus ut fiat &longs;atis, & id, quod inquirimus,
enucleatiùs explicetur, illud ob&longs;ervo, quod funiculus BA &longs;i
præcisè &longs;pectetur, quatenus ex eo corpus grave pendet, retinet
globum A, ne rectâ de&longs;cendat per lineam ip&longs;i BD parallelam,
&longs;ed cogit illum deflectere in motu: quare adversùs clavum B,
globus A exercet ea momenta, quæ exerceret in planum incli
natum, cui BA ad rectos angulos in&longs;i&longs;teret. At &longs;i globus ex alio
prætereà funiculo CA pendeat, idem funiculus BA re&longs;i&longs;tit
etiam momentis illis, quibus globus A de&longs;cenderet in plano in
clinato, cui CA ad rectos angulos in&longs;i&longs;teret, quæ momenta (ut
&longs;ummum) &longs;unt ad BA radium ut 81496. Momenta verò qui
bus urgeret planum inclinatum perpendiculare ad BA, &longs;unt, ex
dictis &longs;uperiori capite, ut Sinus Ver&longs;us anguli inclinationis pla
ni; inclinatio autem plani, ut paulò &longs;uperiùs hoc eodem capite
demon&longs;travimus, e&longs;t complementum anguli declinationis
DBA. Quare differentia inter DA 37784 &longs;inum rectum an
guli declinationis, & radium BA 100000, cum &longs;it Sinus Ver
&longs;us anguli inclinationis plani, &longs;unt momenta 62216 addenda
bus funiculus BA repugnat, &longs;i pondus pendeat etiam ex CA;
cum tamen &longs;i ex ip&longs;o tantùm funiculo BA penderet, & aliquis
e&longs;&longs;et præcisè obluctans viribus ad de&longs;cendendum, idem funicu
lus BA re&longs;i&longs;teret &longs;olùm momentis 62216.
Eâdem methodo deprehendes funiculum CA, &longs;i ex eo &longs;olo
globus pendeat, retinere momenta 18504: at &longs;i etiam ex BA
globus pendeat, additis momentis 37784, tota momentorum
&longs;umma e&longs;t 56288. Jam &longs;ummam hanc priori 143712 adde, &
erit tota momentorum &longs;umma 200000: perinde atque &longs;i corpo
ris gravitas fui&longs;&longs;et duplicata. Id quod deprehendes, quo&longs;cum
que demùm declinationis angulos &longs;tatueris &longs;ivè majores, &longs;ivè
minores; &longs;emper enim eandem &longs;ummam momentorum om
nium invenies 200000: & funiculus minoris declinationis plus
momentorum &longs;u&longs;tinebit, tùm quia Sinus Ver&longs;us majoris incli
nationis plani major e&longs;t, tum quia Sinus Rectus alterius anguli
declinationis majoris item major e&longs;t.
Hæc tamen ut veritati congruant, ita &longs;olùm accipienda &longs;unt,
ut momenta &longs;ingula ex utrâque funiculorum declinatione orta
particulatim &longs;umantur: pondus &longs;cilicet ex utroque &longs;u&longs;pen&longs;um
perinde hactenus con&longs;ideratum e&longs;t, ac &longs;i momenta ip&longs;a de&longs;cen
dendi in diver&longs;as partes abeuntia momentum quoddam ex
utri&longs;que temperatum non con&longs;tituerent; re autem ipsa quod ex
iis componitur momentum, non ex ip&longs;orum momentorum ad
ditione conflatur, &longs;ed ex ip&longs;is temperatur. Si enim mobile &longs;it
ubi A, impetum verò cum tali
directione habeat, quâ deferri
po&longs;&longs;it æquabiliter per rectam
AB, alio autem impetu feratur
æquabiliter directum in C, no
tum omnibus e&longs;t motum, qui ex
AB & AC componitur, non fieri ex earum additione, &longs;ed tem
perari in lineam AD, quæ dimetiens e&longs;t parallelogrammi, quod
ex earumdem linearum AB, AC longitudine, ac mutuâ incli
natione formam de&longs;umit. Quâ in re plurimum intere&longs;t, quam
invicem habeant inclinationem directiones motuum in diver&longs;a
abeuntium; quò enim acutiorem angulum con&longs;tituunt, eò lon
giùs provehitur mobile, ut AB, AC in acutum angulum
angulus, eò etiam brevius e&longs;t iter ip&longs;ius mobilis, ut contingit,
&longs;i ex B directum per rectas BA, BD ad obtu&longs;um angulum
con&longs;titutas moveatur, &longs;i&longs;titur enim in C, & brevior e&longs;t diame
ter BC quàm AD, ut ex 24. lib.
1. &longs;atis manife&longs;tum e&longs;t geo
metris, & ip&longs;a motuum natura po&longs;tulat; qui nimirum &longs;ibi in
vicem magis adver&longs;antur, magi&longs;que in diver&longs;a abeunt, &longs;e ma
gis elidunt, id quod fit ex angulo obtu&longs;o DBA; qui verò mi
nùs in diver&longs;a abeunt, id quod fit ex angulo acuto CAB, &longs;e pa
riter minùs elidunt.
Sint itaque, ut priùs, funiculi BA, CA, ex quibus A pon
dus &longs;u&longs;penditur: ducatur ad BA perpendicularis AR, & e&longs;t
planum inclinatum, in quo de&longs;cendendi momentum e&longs;t ut
DA; &longs;imiliter ad CA perpendicularis AG ducatur referens
planum inclinatum, in quo de&longs;cendendi momentum e&longs;t AE.
Sumatur igitur AR quidem ip&longs;i AD æqualis, AG verò ip&longs;i
AE pariter æqualis, &longs;i funiculi BA, & CA æquales fuerint;
&longs;in autem inæquales &longs;int, fiat angulus DBH æqualis angulo
declinationis ECA, & &longs;umptâ BH æquali ip&longs;i BA, duca
tur ad BD perpendicularis HI, eritque ut EA ad AC,
ita IH ad HB, hoc e&longs;t ad AB; ac propterea ip&longs;i IH, quæ
refert momentum AE, &longs;umatur AG æqualis. Ex quo fit cor
pus A &longs;u&longs;pen&longs;um hâc ratione momenta de&longs;cendendi habe
re in diver&longs;as partes abeuntia AR, AG: perfecto igitur paral
lelogrammo ARNG, ex duobus illis momentis temperatur
momentum AN.
Ip&longs;ius autem AN longitudinem inve&longs;tigare non e&longs;t diffici
le; cum enim noti &longs;upponantur anguli declinationum DBA,
ECA, angulus RAG conflatur ex eorum complementis,
quippe qui æqualis e&longs;t duobus angulis inclinationis planorum
AR, & AG. Porrò ex hypothe&longs;i &longs;unt angulus DBA gr. 22.
12′, & angulus ECA gr. 54. 35′: jungantur &longs;imul, & eorum
&longs;umma gr. 76. 47′ auferatur ex gr. 180, ut re&longs;iduum gr. 103.
13′ &longs;it angulus RAG, cui æqualis e&longs;t oppo&longs;itus RNG; ac
proinde notus e&longs;t angulus G, qui e&longs;t &longs;uo oppo&longs;ito R æqualis,
uterque &longs;cilicet gr. 76. 47′ quæ e&longs;t &longs;umma angulorum decli
nationis. Sunt igitur in triangulo AGN nota latera AG,
GN (e&longs;t enim ex 34. lib.
1. GN oppo&longs;ito lateri AR æquale)
te&longs;cit tertium latus AN. Quare cum latus AG &longs;it ex &longs;upe
riùs con&longs;titutis 81496, & GN, hoc e&longs;t AR, 37784, fiat ut
laterum AG, GN &longs;umma 119280 ad eorumdem differen
tiam 43712, ita &longs;emi&longs;ummæ angulorum ad ba&longs;im, hoc e&longs;t
gr. 51. 36 1/2 Tangens 126205 ad 46249 Tangentem gr. 24. 49′ 2/5
differentiæ infra, vel &longs;upra eandem &longs;emi&longs;ummam. E&longs;t igitur
angulus GAN gr. 26. 47′ (3/10). In triangulo itaque AGN noti
&longs;unt duo anguli A, & G, ac latus GN angulo A oppo&longs;i
tum; igitur ut anguli A gr. 26. 47′ (3/10) Sinus 45070 ad anguli G
gr. 76. 47′ Sinum 97351, ita latus GN 37784 ad latus AN
81613.
Ex quibus apparet de&longs;cendendi momentum, quod compo
nitur ex momentis in planis inclinatis, non e&longs;&longs;e 119280 ex eo
rum &longs;ummâ, &longs;ed ita temperari, ut longè minus &longs;it, videlicet &longs;o
lùm 81613.
Methodo eâdem operantes deprehendemus ponderis in H
con&longs;tituti, ac ex funiculis BH, CH &longs;u&longs;pen&longs;i momentum ita
componi ex momento HI bis &longs;umpto (&longs;i quidem anguli decli
nationum DBH, ECH & funiculi æquales &longs;int) ut in unum
ex utroque nimirum HI & HO temperatum HS coale&longs;cat.
Unde con&longs;tabit quò majores fuérint declinationum anguli, eò
longiorem futuram lineam HS, atque adeò etiam majus mo
mentum de&longs;cendendi; plana &longs;iquidem inclinata acutiorem
angulum con&longs;tituunt. Quam momentorum varietatem pau
lò inferiùs manife&longs;to experimento comprobabimus: ubi con&longs;ta
bit pondus hâc ratione &longs;u&longs;pen&longs;um ex duobus funiculis plus ha
bere aliquando momenti ad de&longs;cendendum, quàm in perpen
diculari &longs;u&longs;pen&longs;ione.
Quemadmodum verò de momentis de&longs;cendendi in planis
inclinatis ratiocinati &longs;umus, ita pariter in unum coale&longs;cere di
cenda &longs;unt momenta, quibus funiculi pondus retinentes ip&longs;um
quodammodo avellere conantur à plano inclinato, ne illud ur
geat; hæc enim pariter momenta in diver&longs;a abeunt &longs;ecun
dùm ip&longs;am funiculorum directionem. Sunt autem momenta
illa Sinus Ver&longs;i angulorum inclinationis planorum; qui haben
tur, &longs;i Sinus Recti complementorum, hoc e&longs;t angulorum de-
clinationis funiculorum, de
mantur ex Radio. Itaque ex
BA auferatur BF ip&longs;i DA
æqualis, & e&longs;t FA Sinus Ver
&longs;us anguli inclinationis: po&longs;ita
e&longs;t autem declinatio DBA
gr.22. 12′, igitur FA e&longs;t parti
cularum 62216; & declinatio
ECA gr. 54. 35′; igitur factâ
CG æquali ip&longs;i AE, remanet
GA particularum 18504, quarum CA e&longs;t 100000. Quare ut
habeantur particulæ eju&longs;dem rationis cum particulis AF, fiat
ut CA ad AG, ita BA ad AH, & e&longs;t AH particularum 18504
homologarum particulis AF. Perficiatur parallelogrammum
AHIF; & quia funiculus CA retrahit à plano inclinato juxta
momentum ac directionem HA, funiculus verò BA retrahit à
plano inclinato &longs;ecundùm momentum ac directionem FA, di
rectionibus in diver&longs;a abeuntibus, temperatur ex his momentis
momentum AI diameter parallelogrammi.
Porrò in diametri AI inve&longs;tigatione methodus e&longs;t eadem,
quâ paulò antè utebamur: Cum enim tres anguli BAD, BAC,
CAE &longs;int duobus Rectis æquales, anguli verò BAD, CAE
noti &longs;int, quippe complementa angulorum declinationis DBA,
ECA, innote&longs;cit reliquus FAH, qui æqualis e&longs;t &longs;ummæ an
gulorum declinationis. E&longs;t igitur FAH gr.76.47′, ac proinde
angulus AFI gr.103.13′ notus e&longs;t, unâ cum lateribus FA 62216
& FI 18504. Fiat igitur ut laterum &longs;umma 80720 ad eorum
dem differentiam 43712, ita angulorum ad ba&longs;im AI &longs;emi&longs;um
mæ gr. 38. 23′1/2. Tangens 79235 ad 42907 Tangentem dif
ferentiæ infra vel &longs;upra eandem &longs;emi&longs;ummam, hoc e&longs;t gr. 23.
13′.1/2 dempta igitur hæc differentia ex &longs;emi&longs;&longs;ummâ gr.38.23′ 1/2,
reliquum facit angulum FAI gr.15.10′. Fiat demùm ut anguli
FAI gr.15.10′. Sinus 26163 ad anguli AFI gr. 103. 13′.
hoc e&longs;t
ad &longs;upplementi gr.76.47′. Sinum 97351, ita latus FI 18504
ad ba&longs;im AI 68852.
Inventa itaque momenta compo&longs;ita tùm in planis inclinatis,
tùm in plana inclinata, dividantur juxta Rationem momento-
tribuendum &longs;it in pondere retinendo. Momentum de&longs;cenden
di compo&longs;itum inventum e&longs;t &longs;u&longs;periùs 81613, &longs;implicia &longs;unt
81496, & 37784. Fiat ut igitur ut &longs;implicium momentorum
&longs;umma 119280 ad eorum alterutrum, puta ad 37784, ita mo
mentum compo&longs;itum 81613 ad aliud, & provenit 25852 pars
illius momenti pertinens ad funiculum CA, qui retinet pon
dus; cujus vis de&longs;cendendi e&longs;t DA 37784. Reliqua autem mo
menti 81613 pars 55761 pertinet ad funiculum BA retinentem
pondus, cujus vis de&longs;cendendi e&longs;t EA 81496. Pari ratione fiat
ut Sinuum Ver&longs;orum angulorum inclinationis &longs;implicium
62216, atque 18504 &longs;umma 80720 ad eorum alterutrum, pu
ta ad 18504, ita momentum compo&longs;itum inventum 68852 ad
aliud, & provenit pro minori 15783, pro majori verò 53069.
Quare funiculus BA minorem habens declinationem, & plus
&longs;u&longs;tinet in &longs;uo plano magis inclinato, cui perpendicularis e&longs;t,
nimirum ut 53069, & plus retinet in plano reliquo minùs in
clinato, nimirum ut 55761: contra verò funiculus CA, & mi
nus &longs;u&longs;tinet, &longs;cilicet ut 15783, & minus retinet &longs;cilicet ut
25852. Funiculus itaque BA exercet vires ut 108830, & fu
niculus CA ut 41635, & totum corporis &longs;u&longs;pen&longs;i momentum
e&longs;t 150465.
Non &longs;ola autem momenta de&longs;cendendi in planis inclinatis
con&longs;iderari oportere, &longs;ed & ea, quæ e&longs;&longs;ent adversùs plana
ip&longs;a inclinata, uti dictum e&longs;t, ex eo apertè conficitur, quòd
ubi funiculi concurrerent ad acuti&longs;&longs;imum angulum, vix quic
quam virium in retinendo pondere exercere opus e&longs;&longs;et; te
nui&longs;&longs;imum quippe, e&longs;&longs;et momentum, quod ex parvis mo
mentis per acuti&longs;&longs;imorum angulorum Sinus Rectos definitis
componeretur: &longs;i verò nihil præterea momenti addendum e&longs;
&longs;et; à magnâ gravitatione, quæ in perpendiculari e&longs;t, ad ferè
nullam tran&longs;itus e&longs;&longs;et, facta vel modicâ à perpendiculo decli
natione; atque adeò vix intenti e&longs;&longs;e deberent funiculi: id quod
manife&longs;to experimento adver&longs;atur.
Illud po&longs;tremò hîc o&longs;tendendum &longs;upere&longs;t, plus &longs;cilicet in
e&longs;&longs;e po&longs;&longs;e momenti ad de&longs;cendendum corpori ex duobus funi
culis invicem inclinatis &longs;u&longs;pen&longs;o, quàm &longs;i ex unico ad per
pendiculum pendeat. Orbiculo circà &longs;uum axem C ver&longs;atili,
ac &longs;ecundùm extremam
oram excavato, in&longs;eratur
funiculus AFB, ex quo
æqualia hinc, & hinc
pondera A, & B pen
deant: nullus planè &longs;e
quitur motus, quia utrum
que ex perpendiculo pen
det, & quantâ vi alterum conatur deor&longs;um, pari nu&longs;u alterum
repugnat, ne elevetur. Quærenti igitur, quantum momenti
pondus B habeat ad de&longs;cendendum, utique re&longs;pondebis omni
nò par e&longs;&longs;e momento ponderis A. Jam verò &longs;it funiculus AFD,
qui in D religetur, & ponderi A &longs;umatur æquale pondus E,
vel potiùs ip&longs;um B transferatur in E, & funiculo AFD ad
nectatur in H; ut &longs;int qua&longs;i duo funiculi DH, FH. Quæro
quantum ad de&longs;cendendum momenti habeat pondus E, hoc e&longs;t
pondus B in H tran&longs;latum, quod e&longs;t æquale ponderi A: &longs;i tan
tumdem habet momenti, quantum pondus A, planè manebit
immotum, intento funiculo FD; at &longs;i E de&longs;cendens cogat
a&longs;cendere pondus A, utique plus momenti habet quàm A, hoc
e&longs;t, plu&longs;quàm B perpendiculariter pendens. Id quod re ipsâ
contingit; & quidem tàm certo experimento, ut non &longs;olùm
pondus E prævaleat ponderi A, &longs;i &longs;it ei æquale, verùm etiam &longs;i
minus &longs;it eodem pondere A. Non igitur hoc ab&longs;urdum e&longs;t,
quod con&longs;titutam à nobis momentorum hypothe&longs;im con&longs;equa
tur, &longs;ed potiùs ip&longs;i naturæ no&longs;tra con&longs;entit hypothe&longs;is, cui ro
bur adjicit experientia; nec ex eo capite perperam philo&longs;opha
ti videmur, quòd in perpendiculo minus momenti, quàm ex
duplici funiculo &longs;u&longs;pen&longs;um pondus habere dicendum &longs;it.
Ex his, quæ de corpore ex binis funiculis &longs;u&longs;pen&longs;o hactenus
di&longs;putata &longs;unt, non difficilis erit conjectura eorum, quæ dicen
da &longs;int, &longs;i ex tribus aut quatuor &longs;u&longs;pendatur, &longs;ivè illi immedia
tè adnectantur ip&longs;i ponderi, &longs;ivè funiculus unus demum in plu
ra capita dividatur, ex quibus fiat &longs;u&longs;pen&longs;io: neque enim his
diutiùs ad nau&longs;eam immorandum cen&longs;eo.
PRoxima e&longs;t iis, quæ hactenus di&longs;putata &longs;unt, præ&longs;ens in
ve&longs;tigatio gravitationis corporum, &longs;ive nisûs, quo motui
re&longs;i&longs;tunt, cùm obliquè in plano aliquo trahuntur, aut elevan
tur: &longs;icut enim toto conatu repugnant elevanti ad perpendicu
lum, & ab&longs;trahenti à plano, cui in&longs;ident, ita pro majori, aut
minori obliquitate tractionis aut elevationis magis etiam, aut
minùs, ob&longs;i&longs;tere experimur. Et primùm quidem &longs;uper plano
inclinato AB duo pondera
pror&longs;us æqualia, & &longs;imilia
intelligantur po&longs;ita in B
& C, atque linea CE &longs;it
horizonti BE perpendicu
laris, ac pondus C filo DC
ad perpendiculum &longs;u&longs;pen
datur, ita tamen, ut con
tingat planum in C, & &longs;it
recta DE. Item ex D
puncto ducatur filum DB,
ut &longs;ur&longs;um trahatur B pon
dus incumbens plano in
clinato, dum pariter pon
dus C &longs;ur&longs;um rectâ trahi
tur, & à plano avellitur: horum autem funiculorum trahatur
ex D pars æqualis. Quando igitur C venerit in V, æquali men
&longs;urâ BP multatum intelligitur filum DB, & remanet longi
tudo DP, hoc e&longs;t DO; pondus enim, cum filum in D trahe
retur, ex B venit in O. Ductâ itaque lineâ ON horizonti pa
rallelâ, erit EN altitudo perpendicularis, ad quam a&longs;cendit
pondus B in plano inclinato interea, dum pondus C venit in V,
aut E venit in M, e&longs;t enim EM a&longs;&longs;umpta ip&longs;i CV æqualis.
Quare cum pondus B obliquè trahitur &longs;uper planum inclina-
culari elevatione avellitur.
Hoc tamen ita intelligendum e&longs;t, ut ob&longs;ervetur alia e&longs;&longs;e
momenta, cùm tractionis linea parallela e&longs;t ip&longs;i plano inclina
to, ac cùm in planum inclinatum cadit obliqua, ut hîc li
nea DB. Si enim in plano inclinato &longs;umatur BR æqualis
perpendiculari EM, gravitatio per rectam BC, &longs;eu per li
neam eidem parallelam, ad gravitationem in perpendiculo
CE e&longs;t reciprocè ut EC ad BC, &longs;eu ut ES ad BR aut EM,
ex &longs;uperiùs dictis cap.13. At verò cum tractio obliqua e&longs;t,
gravitatio e&longs;t ut EN ad EM, &longs;ivè ut BO ad BX: punctum
autem O altius e&longs;t puncto R, ac proptereà in huju&longs;modi
obliquâ tractione plus violentiæ infertur ponderi, quàm in
tractione parallelâ, plus enim a&longs;cendit. Porrò lineam BO
longiorem e&longs;&longs;e lineâ BR e&longs;t manife&longs;tum; &longs;iquidem duo la
tera DO, OB per 20. lib.1. majora &longs;unt reliquo DB: e&longs;t
autem ex hypothe&longs;i DP ip&longs;i DO æqualis, ergo reliqua
BP minor e&longs;t, quàm BO: &longs;ed & ip&longs;i BP, hoc e&longs;t ip&longs;i
EM, æqualis a&longs;&longs;umpta e&longs;t BR; igitur BR minor e&longs;t quàm
BO. Id quod etiam hinc con&longs;tat, quia in triangulo I&longs;o
&longs;cele DOP angulus OPB infra ba&longs;im major e&longs;t recto,
cum &longs;it deinceps angulo DPO ad ba&longs;im acuto; ergo per
19.lib.1. latus BO majus e&longs;t latere BP, hoc e&longs;t BR; igi
tur etiam EN major e&longs;t quàm ES, & plus difficultatis
percipitur in obliquâ hâc tractione, quàm in tractione pa
rallelà.
Similiter intelligatur pondus C elevatum fui&longs;&longs;e ex D
(quod punctum D concipiatur multò altius, quàm in præ
&longs;enti &longs;chemate) ad perpendiculum altitudine æquali ip&longs;i ET,
pondus verò B æquali tractione funiculi veni&longs;&longs;e ex B in G,
demptâ &longs;cilicet longitudine BF ip&longs;i ET æquali, atque
adeò DF, DG æquales &longs;unt: ip&longs;i autem ET æqualis &longs;u
matur BI; quæ &longs;imili ratione demon&longs;tratur brevior, quàm
BG: ex quo pariter &longs;it hîc etiam ad majorem altitudi
nem perpendicularem EH elevari, quàm &longs;i tractio pa
rallela fui&longs;&longs;et plano inclinato, & elevatio ad altitudi
nem EL.
Ex his manife&longs;tum e&longs;t plus virium requiri ad trahendum
quas, quàm per lineam plani inclinati BC, aut illi paral
lelam: dum enim per obliquas illas lineas fit tractio, pon
dus quidem non omninò ab&longs;trahitur à plano, &longs;icut in tractio
ne perpendiculari, &longs;ed nec omninò incumbit plano, &longs;i
cut in tractione parallelâ ip&longs;i plano; ac propterea, quò ma
gis tractio ad perpendicularem accedit, eò majorem inve
nit in pondere re&longs;i&longs;tentiam. Patet autem altitudinum per
pendicularium EH, EL differentiam HL majorem e&longs;&longs;e,
quàm &longs;it altitudinum perpendicularium EN, ES differen
tia NS. Comparatis enim triangulis i&longs;o&longs;celibus DPO,
DFG, anguli ad ba&longs;im PO majores &longs;unt angulis ad ba&longs;im
FG, quia angulus PDO minor e&longs;t angulo FDG: ergo
angulus BPO, qui e&longs;t infra ba&longs;im, minor e&longs;t angulo
BFG infra ba&longs;im. Fiat igitur ip&longs;i BPO æqualis angulus
BFK, ac proinde K cadit inter puncta I & G. Sunt ergo
triangula BPO, BFK habentia angulum ad B communem
æquiangula, & &longs;imilia, ac per 4. lib.6. ut PB, hoc e&longs;t BR,
ad BO, ita FB, hoc e&longs;t BI, ad BK; & invertendo, ac
dividendo, & iterùm invertendo ut BR ad RO, ita BI
ad IK. Atqui IG major e&longs;t quam IK, ergo per 8.lib.5.
Ratio BI ad IG minor e&longs;t Ratione BI ad IK, hoc e&longs;t BR
ad RO. Cum itaque per 2. lib.6. ut BR ad RO, ita ES
ad SN; & ut BI ad IG, ita EL ad LH, major e&longs;t Ra
tio ES ad SN, quàm EL ad LH, & permutando major
e&longs;t Ratio ES ad EL, quàm SN ad LH; e&longs;t autem ES
minor quàm EL, ergo etiam SN multò minor e&longs;t quàm
LH; ac proinde quo magis à perpendiculari recedet obli
qua tractio, momentum ponderis magis accedit ad momen
tum eju&longs;dem in plano inclinato per tractionem parallelam,
hoc e&longs;t, minore differentiâ hoc excedit. Momentum igitur
perpendicularis tractionis ad momentum obliquæ tractionis
minorem Rationem habet, quàm ad momentum tractionis pa
rallelæ plano inclinato.
Ex his ob&longs;ervare e&longs;t aliquod paradoxum, pondus &longs;cilicet obli
quâ hâc elevatione tractum plus moveri, quàm potentiam tra
hentem; hæc enim movetur &longs;ecundùm men&longs;uram funiculi
tracti, hoc e&longs;t BP &longs;eu BR illi æqualis, o&longs;ten&longs;um e&longs;t autem Id quod etiam manife&longs;tum e&longs;t,
&longs;i tractio obliqua non ab&longs;trahat pondus à plano, &longs;ed qua&longs;i il
lud adversùs planum trahat.
Sit enim planum AB, &longs;uper
quo globus C, & funiculus
obliquus DC; ex D autem
pendeat ad perpendiculum
æquale pondus E. Uterque fu
niculus pariter trahatur, &
cum E venerit in F, æqualis
pars CG decedit funiculo
DC; remanet autem longitu
do DG æqualis longitudini
DH, & centrum globi C ve
nit in H. Dico CH motum
globi majorem e&longs;&longs;e &longs;upra CG
motum potentiæ trahentis.
Ducatur enim recta GH; e&longs;t
I&longs;o&longs;celes DGH, ergo angulus HGC infra ba&longs;im major e&longs;t
recto; ergo CH per 19.lib.1. major e&longs;t quàm CG. Ip&longs;i autem
CH æqualem e&longs;&longs;e di&longs;tantiam contactuum RS manife&longs;tum
e&longs;t, quia ex centris H & C rectæ cadunt in S & R ad angu
los rectos, atque adeò &longs;unt parallelæ: &longs;unt æquales CR & HS,
ut pote Radij eju&longs;dem globi; igitur per 33.lib.1. CH, & RS
æquales &longs;unt & parallelæ. Quare &longs;ivè centrum &longs;pectetur, &longs;ivè
puncta contactuum, perinde e&longs;t; &longs;emper enim major e&longs;t glo
bi motus motu potentiæ trahentis; & quia RS major e&longs;t quàm
CG, hoc e&longs;t quàm motus, qui fieret in ip&longs;o plano inclinato
tractione parallelâ, hinc e&longs;t quod huju&longs;modi obliquâ tractio
ne ad majorem altitudinem perpendicularem pari tempore tra
hitur, majorémque proptereà violentiam &longs;ubiens majoribus
indiget viribus, quàm &longs;i tractione parallelâ elevaretur.
Sed jam trahatur iterum funiculus ita, ut ip&longs;i CG primæ
tractioni æqualis &longs;it &longs;ecunda tractio HL; & crit centrum globi
in M, & æquales DM, DL. Anguli MDH, HDC &longs;i di
cantur æquales, etiam per 3.lib.6. ut MD ad DC ita MH
ad HC: e&longs;t igitur MH minor quàm HC, major tamen quàm
HL, quia &longs;ubten&longs;a e&longs;t angulo MLH obtu&longs;o, ut pote infra ba-Atqui ex hypothe&longs;i anguli MDL, HDG
&longs;unt æquales; ergo I&longs;o&longs;celium anguli infra ba&longs;es, hoc e&longs;t MLH,
HGC &longs;unt æquales: angulus autem externus MHL major e&longs;t
interno HCD, hoc e&longs;t HCG, per 16.lib.1. igitur reliquus
HML minor e&longs;t reliquo CHG. Itaque in duobus triangulis,
angulis CGH, HLM ex hypothe&longs;i o&longs;ten&longs;is æqualibus &longs;ub
tenditur illi quidem majus latus CH, huic verò minus HM,
& angulis inæqualibus CHG majori, HML minori æquale
latus CG, HL: id quod omninò ab&longs;urdum e&longs;&longs;e con&longs;tat ex
doctrinâ & Canone Sinuum; &longs;ubten&longs;æ &longs;iquidem inæquales an
gulorum æqualium &longs;unt in circulis inæqualibus, major in majori
circulo, minor in minori, in quibus utique fieri non pote&longs;t, ut
angulorum inæqualium &longs;ubten&longs;æ &longs;int æquales. Non igitur fieri
pote&longs;t ut factá &longs;ecunda tractione HL æquali priori CG, angu
lus MDH æqualis &longs;it angulo HDC; alioquin triangulum
HLM (cujus ba&longs;is HM ex hypothe&longs;i arguitur minor ba&longs;e
CH, quæ tamen &longs;unt angulis ad G & L æqualibus &longs;ubten&longs;æ)
e&longs;&longs;et in circulo minore, quàm &longs;it circulus, in quo e&longs;&longs;et triangu
lum CGH; in circulo autem minore, angulo minori HML
&longs;ubten&longs;a HL e&longs;&longs;et æqualis ip&longs;i CG &longs;ubten&longs;æ angulo majori
CHG in circulo majore.
Quod &longs;i dicatur angulus MDH minor, quàm HDC, ergo
angulus MLH infra ba&longs;im minor e&longs;t angulo HGC infra ba
&longs;im: atqui angulus MHL externus major e&longs;t
igitur reliquus angulus LMH vel e&longs;t æqualis angulo GHC,
vel illo minor, vel illo major. Sit æqualis: quoniam æqualibus
lineis CG, HL &longs;ubtenduntur, &longs;unt in circulis æqualibus; ergo
cùm angulus MHL major &longs;it angulo HCG, etiam oppo&longs;itum
latus ML majus e&longs;t quàm HG: ergo I&longs;o&longs;celes MDL habens
angulum minorem &longs;ub brevioribu lateribus habet majorem
ba&longs;im, & I&longs;o&longs;celes HDG habens angulum majorem &longs;ub late
ribus
&longs;urdum
LMH, GHC &longs;int æquales, &longs;i MDH minor e&longs;t quàm HDC.
Quandoquidem igitur LMH, GHC non &longs;unt æquales, dica
tur angulus LMH minor quàm GHC, & quia æqualibus li
neis HL, CG &longs;ubtenduntur, triangulum HLM e&longs;t in circulo
majore, triangulum verò CHG in minore. Cum autem angu-
ten&longs;a illius, ut potè in circulo majori, &longs;cilicet ML major e&longs;t
quàm HG &longs;ubten&longs;a anguli minoris in circulo minori: atque
hinc idem quod priùs, &longs;equitur ab&longs;urdum angulum verticalem
MDL, ex hypothe&longs;i minorem, & brevioribus lateribus com
prehen&longs;um ba&longs;im habere majorem, quàm &longs;it ba&longs;is anguli verti
calis HDG majoris &longs;ub lateribus longioribus.
Sed neque dici pote&longs;t angulus HML major quàm CHG;
quia, &longs;i MDL minor e&longs;t quàm HDG, angulus DML ad ba
&longs;im I&longs;o&longs;celis major e&longs;t quàm DHG pariter ad ba&longs;im; ergo &longs;i
DML majori addatur major HML, & DHG minori adda
tur minor CHG, erit totus DMH major toto angulo DHC,
internus &longs;cilicet major externo, contra 16.lib.1. Si igitur an
gulus HML comparatus cum angulo CHG non pote&longs;t e&longs;&longs;e
æqualis, neque minor, neque major, factâ hypothe&longs;i anguli
MDL minoris quàm HDC, nece&longs;&longs;ariâ con&longs;ecutione confici
tur angulum MDL non e&longs;&longs;e minorem angulo HDG.
Cum itaque angulus MDL neque æqualis, neque minor &longs;it
angulo HDG, &longs;equitur quod &longs;it major: igitur & angulus in
fra ba&longs;im MLH major e&longs;t angulo HGC; item angulus MHL
major e&longs;t quàm HCG; ergo HML reliquus minor e&longs;t reliquo
CHG: at i&longs;tis æquales lineæ HL, CG &longs;ubtenduntur, igitur
triangulum HML e&longs;t in majore circulo, ac proinde angulo
MLH majori, quàm CGH, etiam majus latus &longs;ubtenditur:
quapropter MH, hoc e&longs;t SN, illi parallela & æqualis, major
e&longs;t quàm CH, hoc e&longs;t RS: atque adeò ad majorem altitudi
nem elevatur per SN, quàm per RS factâ æquali tractione, &longs;eu
æquali motu potentiæ trahentis. Ex quo & manife&longs;tum e&longs;t pro
majori obliquitate & rece&longs;&longs;u tractionis à paralleli&longs;mo cum pla
no inclinato etiam trahenti difficultatem augeri.
Facilè ex dictis colliges, quanto laboris compendio Romæ
altioribus rotis in&longs;truantur birota (antiquis Ci&longs;ia dicebantur)
adeò ut unicus equus temoni applicitus, illumque &longs;ubjecto pla
no proximè parallelum &longs;ervans, dum clivum a&longs;cendit, ingentia
pondera trahat, quibus &longs;anè par non e&longs;&longs;et, &longs;i rotarum axis mi
nùs à &longs;ubjecto plano di&longs;taret, & equitractio e&longs;&longs;et obliqua &longs;ur
&longs;um: quamvis, ut aliàs &longs;uo loco explicabitur, ip&longs;a rotarum am
plitudo plurimum conferat. Similiter in navium tractione, quæ
quid juvare funis longitudinem, ut &longs;cilicet minùs obliqua &longs;it
tractio, ex dictis confirmatur: quamvis enim tractiones in plano
inclinato con&longs;ideraverimus, ut gravium elevationem expende
remus, aliquid etiam facit obliquitas tractionis in plano horizon
tali, cuju&longs;modi e&longs;t aqua, cui navis innatat; pars &longs;iquidem de
mer&longs;a ob&longs;tantem undam repellere debet; nec planè inutile e&longs;t,
&longs;ecundùm quam lineam dirigatur motus potentiæ trahentis, vi
cujus impedimentum &longs;uperandum e&longs;t.
Hactenus nobis de tractione &longs;ermo fuit, quæ motum inferens
non ni&longs;i &longs;patiis, per quæ motus e&longs;t, determinari potuit. Quo
niam verò in obliquis tractionibus non eandem &longs;emper analo
giam &longs;ervari, quæ in parallelâ tractione eadem perpetuò e&longs;t, de
prehendimus, inquirendum &longs;upere&longs;t, quæ demum Ratio mo
mentorum &longs;it pro &longs;ingulis obliquitatibus, ut con&longs;tet, quibus vi
ribus retineri po&longs;&longs;it, ne in proclive labatur pondus, etiam&longs;i vires
ad illud ulteriùs elevandum non &longs;uppetant. Quamquam autem
pondera qua&longs;i molis expertia unico puncto expre&longs;&longs;imus in plano
ip&longs;o inclinato, ut in 1.fig.hujus cap.
re tamen verâ centrum gra
vitatis attendendum e&longs;t, ut in 2. &longs;chemate, quod utique di&longs;tat à
plano, cui corpus grave incumbit: hujus verò di&longs;tantiam nulla
certior men&longs;ura definit, quàm linea ex eo cadens in &longs;ubjectum
planum ad angulos rectos, hæc quippe omnium brevi&longs;&longs;ima e&longs;t.
Sit igitur planum inclinatum AB,
cui impo&longs;itus globus centrum ha
bet gravitatis C, & contingit pla
num in D; ac propterea etiam, quæ
à centro ad contactum ducitur
recta CD, di&longs;tantiam determinat,
cum &longs;it plano perpendicularis ex
18.lib.3. Jam recta CE parallela
plano ducatur, & &longs;it linea &longs;u&longs;pen
&longs;ionis, quam claritatis gratiâ paral
lelam vocemus: & per D punctum,
in quod cadit linea di&longs;tantiæ cen
tri gravitatis tran&longs;eat perpendicu
laris horizonti linea FD quæ in G
&longs;ecat lineam CE. Con&longs;tat trian-
rallela e&longs;t lineæ AS pariter perpendiculari ad horizontem, an
guli SAB, ADG alterni æquales &longs;unt per 27.lib.1. Et quo
niam angulus CDA ex con&longs;tructione e&longs;t rectus, complemen
tum CDG æquale e&longs;t angulo complementi ABS; anguli verò
DCG, BSA &longs;unt recti, hic quidem ex hypothe&longs;i, ille autem
propter linearum CE, DA paralleli&longs;mum: igitur reliquus
CGD reliquo BAS æqualis e&longs;t; ac proptereà per 4. lib.
6. ut
BA ad AS, ita DG ad GC. Quoniam itaque, &longs;i pondus in
plano inclinato ad pondus in perpendiculari &longs;it ut inclinata BA
ad perpendicularem AS, eorum momenta æqualia &longs;unt, &
æquiponderant, etiam globus æqualia ad de&longs;cendendum habet
momenta, ac potentia habeat vires ad retinendum in parallelâ
EC, &longs;i globi gravitas ad potentiam retinendum &longs;it ut DG ad
GC. Verum quidem e&longs;t globum non per lineam FD, &longs;ed per
CT à centro gravitatis perpendicularem horizonti deor&longs;um ni
ti: Sed quia CT ip&longs;i FD parallela e&longs;t, triangulum CTD
triangulo DGC &longs;imile e&longs;t & æquale; atque adeò parùm in
tere&longs;t, utrùm lineis DG, GC, an verò lineis CT, TD eadem
Ratio exponatur.
Sed jam retineatur globus per rectam CH; utique perinde &longs;e
cundùm eam directionem &longs;e habet, atque &longs;i e&longs;&longs;et planum HCK;
globus enim &longs;u&longs;tinetur per lineam DC, & retinetur ex H, ac
proinde &longs;ecundùm
quam &longs;ubjecti plani inclinatio ob&longs;taret, ne &longs;ecundùm rectam
HCK procederet, &longs;i &longs;ibi dimitteretur, & alia atque alia plana
con&longs;tituerentur. Planum itaque illud HC declinat à perpen
diculari, cum quâ con&longs;tituit angulum CID æqualem externo
KCT propter paralleli&longs;mum perpendicularium FD, CT per
27. lib.
1. qui utique CID minor e&longs;t externo CGD per 16.
lib.
1. & quidem differentia anguli ICG per 32.lib.1. Fiat
ergo angulus BAP æqualis angulo CIG; quia BAS o&longs;ten&longs;us
e&longs;t æqualis ip&longs;i CGD, remanet PAS æqualis angulo ICG.
Quare BPA externus æqualis e&longs;t duobus internis, &longs;cilicet recto
PSA, & acuto SAP, per 32.lib.1. igitur idem angulus BPA
æqualis e&longs;t toti angulo DCI. Sunt itaque æquiangula & &longs;imi
lia duo triangula BAP & DIC, atque per 4.lib.6. ut BA ad
AP, ita DI ad IC. Atqui pondera &longs;uper BA & AP, quæ &longs;int
æqualium momentorum e&longs;t globus, & potentia retinens per
HC, &longs;i globus ad potentiam &longs;it ut DI ad IC, hoc e&longs;t ut CN
ad ND, &longs;i ex D intelligatur exire DN parallela ip&longs;i HC.
Eâdem ratione &longs;i linea obliqua, per quam globus retinetur,
&longs;it infra parallelam CE, ut &longs;i &longs;it CX, o&longs;tendetur globi gravita
tem ad potentiam retinentem e&longs;&longs;e ut DQ ad QC, e&longs;t enim
qua&longs;i planum inclinatum faciens cum perpendiculari angulum
DQC majorem interno DGC, hoc e&longs;t majorem angulo BAS
illi æquali. Fiat igitur angulo DQC æqualis angulus BAY:
& quia ABY æqualis e&longs;t angulo CDQ, ut &longs;uperiùs dictum
e&longs;t, triangula BAY, DQC &longs;unt æquiangula & &longs;imilia, ac per
4.lib.6. ut BA ad AX, ita DQ ad QC: ergo quia pondera &longs;u
per BA, & AY, quæ &longs;int in Ratione BA ad AY, æquiponde
rant, etiam globi & potentiæ retinentis momenta æqualia &longs;unt,
&longs;i fuerint ut DQ ad QC.
Hic autem tria ob&longs;ervanda occurrunt.
Primum e&longs;t, quòd
Rationes prædictæ momentorum potentiæ retinentis compara
tæ ad pondus idem, quamvis pro diversâ obliquitate aliis atque
aliis lineis explicentur DQ ad QC, & DG ad GC, DI ad
IC, omnes tamen exponuntur comparatè ad eandem BA in
triangulo BAY; in quo ip&longs;æ quoque inter &longs;e invicem compara
ri po&longs;&longs;unt. Secundum e&longs;t, quòd &longs;i obliquitas tàm &longs;upra, quàm
infra parallelam CE æqualis &longs;it, hoc e&longs;t angulus ICG æqualis
&longs;it angulo GCQ, momenta potentiæ retinentis in H & X
æqualia &longs;unt; inter &longs;e &longs;iquidem &longs;unt ut AP, & AY, quæ lineæ
æquales &longs;unt; nam anguli PAS, YAS æquales &longs;unt ex hypo
the&longs;i, & con&longs;tructione, anguli autem ad S &longs;unt recti & latus
AS e&longs;t utrique triangulo commune; ergo etiam per 26.lib.1.la
tera AP & AY æqualia &longs;unt. Tertium e&longs;t, quòd in lineá CE
parallelâ minus virium exigitur ad retinendum globum, quàm
in cæteris: nam & linea AS vires potentiæ repræ&longs;entans om
nium minima e&longs;t, utpote perpendicularis.
Ex his & illud colligitur, quod &longs;i linea, &longs;ecundùm quam
pondus retinetur in plano inclinato, &longs;it parallela horizonti,
eadem e&longs;t philo&longs;ophandi methodus. Si enim &longs;uper plano in
clinato AB &longs;it pondus tangens in C, cujus gravitatis centrum
&longs;it D, & linea retentionis DE horizonti parallela, ducatur
CF perpendicularis horizonti; & Ratio
ponderis ad vires retinentes erunt ut CF
ad FD. Fiat enim angulus BAH æqua
lis angulo CFD, qui utique e&longs;t rectus,
cum DE ex hypothe&longs;i &longs;it horizonti pa
rallela, FC verò perpendicularis: ergo
&longs;uper AB, AH æquiponderant pondera,
quæ &longs;int ut AB ad AH; paria igitur &longs;unt
momenta, &longs;i pondus ad vires potentiæ re
tinentis in eâdem Ratione &longs;it ut AB ad AH, hoc e&longs;t ut CF ad
FD. Quia enim BAH angulus e&longs;t rectus per 8.lib.6. e&longs;t ut
BA ad AH, ita BG ad GA; e&longs;t autem BG ad GA ut CF ad
FD; quia nimirum FC perpendicularis horizonti e&longs;t paralle
la ip&longs;i AG, & anguli BAG, FCA alterni &longs;unt æquales per
27.lib.1. DCA verò e&longs;t rectus ex hypothe&longs;i; igitur & DCF
complementum recti æquale e&longs;t angulo ABG: utrumque
triangulum e&longs;t rectangulum; ergo ut BG ad GA, ita CF
ad FD.
Hinc apparet fieri po&longs;&longs;e, ut ad retinendum pondus in tali &longs;i
tu aliquando plus virium requiratur, quàm ad &longs;u&longs;tinendum il
lud in perpendiculari; quando videlicet ex inclinatione plani
AB con&longs;equitur lineam CF minorem e&longs;&longs;e quàm FD: immò
cre&longs;cit retinendi difficultas, &longs;i adhuc retentio fiat per lineam
inferiorem horizontali DE, quæ cum perpendiculari CF con
&longs;tituat angulum DIC obtu&longs;um; cum enim cre&longs;ceret linea DI
&longs;upra DF, & IC decre&longs;ceret infra FC, e&longs;&longs;et minor Ratio pon
deris in perpendiculo ad potentiam obliquè retinentem,
quæ proinde major e&longs;&longs;e deberet, ut fieret momentorum æqua
litas.
Concipe autem &longs;ublatum triangulum totum BAH, & DC
e&longs;&longs;e columnam, quæ in eodem &longs;itu inclinata retineri debeat:
jam &longs;atis con&longs;tat ex dictis, quâ ratione di&longs;poni oporteat funes,
ut qui funium extremitates tenent, minus laboris impendant.
Non e&longs;t tamen eadem funis retinentis, & fulcri &longs;u&longs;tentantis
ratio: in &longs;upponendis enim fulcris illud poti&longs;&longs;imùm attenditur,
quòd fulcrum ip&longs;um integrum permaneat, citrà &longs;ci&longs;&longs;ionis aut
fractionis periculum; id quod habetur, quò magis perpendicu
lari ad horizontem &longs;itui proximum collocatur; parùm &longs;cilicet
&longs;imus de fulcri ip&longs;ius firmitate. Cæterùm &longs;i tu ip&longs;e fu&longs;tem
manu tenens cogaris inclinatam columnam &longs;u&longs;tinere, punctum
autem &longs;u&longs;tentationis, cui fulcrum applicatur, magis à &longs;ub
jecto plano di&longs;tet, vel &longs;altem non minùs, quàm centrum gra
vitatis columnæ, experieris minori conatu opus e&longs;&longs;e, &longs;i ful
crum axi columnæ perpendiculare &longs;it, qui &longs;itus re&longs;pondet re
tentioni parallelæ plano inclinato, majorem verò adhiben
dum e&longs;&longs;e conatum, &longs;i fulcrum cum eodem axe acutum aut ob
tu&longs;um angulum con&longs;tituat; id quod obliquis elevationibus
re&longs;pondet.
Quòd &longs;i infra centrum gravitatis applicetur fulcrum, jam
con&longs;tat hoc ita e&longs;&longs;e collocandum, ut ei idem centrum im
mineat, alioquin aut columna corruet, aut multis viri
bus tibi contendendum erit, ut illam &longs;u&longs;tentes à lap&longs;u; &longs;i
tamen ea &longs;it complexio tùm inclinationis, tùm obicis co
lumnæ pedem retinentis, ne excurrat, aut elevetur, tùm po
&longs;itionis fulcri, ut aliquatenus &longs;u&longs;tineri columna po&longs;&longs;it, ne pror
sùs ruat.
Sed quoniam hîc columnæ mentio incidit, præ&longs;tat ele
vationes corporum, quæ non tota elevantur, &longs;ed eorum
altera extremitas &longs;ubjecto alicui fulcro aut plano innititur,
altera elevatur aut &longs;u&longs;penditur, con&longs;iderare: neque enim hîc
reputanda &longs;unt momenta gravitatis perinde, ac &longs;i totum cor
pus elevaretur aut &longs;u&longs;penderetur, quemadmodum paulò an
te dicebatur; immò verè longè minora &longs;unt pro ratione
di&longs;tantiæ à centro gravitatis, ut ex inferiùs dicendis, ubi de
æquilibrio, atque de vecte &longs;ermo erit, con&longs;tabit. Cavendum
autem plurimum e&longs;t ab æquivocationibus, quæ obrepere
po&longs;&longs;unt, ni&longs;i animum advertas ad gravitatem, &longs;ivè per totam
longitudinem, quæ movetur, aut ad motum incitari pote&longs;t,
diffu&longs;am, &longs;ivè qua&longs;i in unum punctum ibi collectam, ubi ele
vans applicatur, ut in vecte, aut librâ; hinc enim non mo
dica momentorum inæqualitas oritur. Nam &longs;i puncto appli
cationis re&longs;pondeat centrum gravitatis, multò majores ad
elevandum, aut &longs;u&longs;pendendum corpus requiruntur vires,
quàm &longs;i centrum gravitatis à puncto applicationis aliquo in
tervallo &longs;ejungatur.
Hinc &longs;i &longs;it pri&longs;ma AB ho
rizontaliter collocatum, eju&longs;
que extremitas A innitatur
apici pyramidis, altera verò
extremitas B &longs;u&longs;pendatur per
pendiculari funiculo CB, vel
&longs;u&longs;tentetur &longs;uppo&longs;ito ad
pendiculum
liter res &longs;e habet, & pares requiruntur vires tam in &longs;u&longs;penden
te CB, quàm in &longs;u&longs;tentante DB: hæ tamen vires non pares
e&longs;&longs;e debent toti ponderi pri&longs;matis; &longs;ed quia centrum gravita
tis E ab utroque extremo æqualiter di&longs;tare &longs;upponitur, &longs;e
mi&longs;&longs;is tantùm gravitatis percipitur in B. Quod &longs;i in codem
horizontali &longs;itu retineatur pri&longs;ma &longs;ivè à &longs;u&longs;pendente obliquo
IB, &longs;ivè ab obliquo &longs;u&longs;tentante OB, utique retinentis, aut
&longs;u&longs;tentantis vires æquipollere debent viribus retinentis aut
&longs;u&longs;tentantis ad perpendiculum CB aut DB. Quemadmo
dum igitur pondera illa &longs;uper BO & BD æquiponderant,
quæ &longs;unt ut BO ad BD, ita vires, quæ &longs;ecundùm ea&longs;dem
lineas ac directiones æqualem effectum præ&longs;tare debent; in
eâdem Ratione BO ad BD e&longs;&longs;e oportet: Vires ergo retinen
tis BI obliqui ad vires retinentis CB ad perpendiculum &longs;unt
ut BO ad BD, hoc e&longs;t, ductâ parallelâ CI, ut IB ad CB,
propter triangulorum OBD, CBI &longs;imilitudinem.
Ut autem non hîc perperam nos philo&longs;ophari innote&longs;cat,
finge &longs;ublatam ex A pyramidem, & con&longs;titutam in G ita,
ut ex B ad perpendiculum dependeat pondus aliquod æqui
librium efficiens cum pri&longs;mate: quo perpendiculari pondere
&longs;ublato, ut pri&longs;ma horizontale permaneat, certum e&longs;t &longs;uper
plano inclinato BO requiri pondus, quod ad pondus per
pendiculare ex BD &longs;it ut BO ad BD: igitur &longs;i loco pon
deris applicentur &longs;ecundùm eandem rectam lineam BO vires
alicujus viventis, à quo retineatur pri&longs;ma in eodem &longs;itu ho
rizontali, &longs;atis apparet conatum debere e&longs;&longs;e ut BO ad cona
tum, qui &longs;ecundùm perpendicularem requireretur ut BD.
Sicut itaque conatus deor&longs;um trahens, cum fulcrum e&longs;t in
G citrà centrum gravitatis E, ex inclinatione lineæ, &longs;ecun
dùm quam fit, de&longs;umitur, ita etiam conatus &longs;u&longs;pendens IB,
gravitatis E, de&longs;umendus e&longs;t pariter ex inclinatione lineæ, &longs;e
cundùm quam applicatur pri&longs;mati, comparatè ad conatum per
pendicularem CB, vel DB, habita &longs;emper ratione di&longs;tantiæ
fulcri à centro gravitatis.
Ne quid verò dubitationis
&longs;uper&longs;it, utrum OB deor&longs;um,
& IB &longs;ur&longs;um trahentium pa
res &longs;int vires &longs;ecundùm ean
dem rectam lineam OI, &longs;int
rotulæ duæ H & F circa &longs;uum
axem ver&longs;atiles infixæ extre
mitatibus regulæ, aut tigilli,
& ex funiculo rotularum ca
vitatibus in&longs;erto dependeant
æqualia pondera L & G. Hæc
pondera &longs;ibi vici&longs;&longs;im æquipon
derare manife&longs;tum e&longs;t, quem
cumque tandem &longs;itum &longs;ivè
perpendicularem, &longs;ivè incli
natum, habeat regula, aut ti
gillus, cui rotulæ infixæ &longs;unt. Sit libræ jugum AB æqualiter
in E divi&longs;um, circa quod punctum &longs;tabile moveri queat, &
in A adnectatur funiculo HF: ex B autem dependeat pondus
D æquale ponderi G, &longs;ed ita obliquè di&longs;po&longs;itum, ut linea BO
parallela &longs;it lineæ AF. Submove pondus L, remanent G
& D, quorum neutrum prævalere pote&longs;t; &longs;unt enim æqualia
inter &longs;e, & per lineas &longs;imiliter inclinatas AF, BO agunt. Re
pone pondus L, & amove pondus G, item removeatur pon
dus D, & &longs;ur&longs;um ponatur æquale C; aio libræ jugum AB
adhuc retinere eumdem &longs;itum; quia &longs;cilicet pondera C & D
vici&longs;&longs;im æquiponderabant, &longs;icut etiam G & L: igitur quantum
virium habebat pondus D ad æquiponderandum ip&longs;i G, tan
tumdem virium habet pondus C ad æquiponderandum ponde
ri L, hoc e&longs;t eidem ponderi G. Sivè igitur in &longs;uperiori &longs;che
mate con&longs;iderentur vires deor&longs;um trahentes aut &longs;u&longs;tentantes
OB, &longs;ive retinentes IB, perinde e&longs;t, & æqualium momento
rum cen&longs;endæ &longs;unt.
Non jam horizontale &longs;it
pri&longs;ma AB, &longs;ed inclinatum,
& puncto A &longs;tabili innixum:
momenta ad de&longs;cendendum,
ac proinde repugnantia ad
a&longs;cendendum, ut &longs;uperiùs in
nuimus cap.14; æ&longs;timanda
&longs;unt in plano DC inclinato,
quod cum AB angulos facit
rectos, & cum horizonte AE
concurrit in puncto E. Ducatur per B perpendicularis ad ho
rizontem FH, & ex H ad BE perpendicularis HO. Momen
ta gravitatis pri&longs;matis in perpendiculari ad momenta eju&longs;dem
in inclinatà &longs;unt reciprocè ut inclinata EB ad perpendicula
rem BH, hoc e&longs;t per 8.lib.6. ut HB ad BO, &longs;ive (ductâ ex D
&longs;uper DB inclinatam perpendiculari DG &longs;ecante rectam HF
in F) ut BF ad BD, propter &longs;imilitudinem triangulorum OBH,
DBF. Vires ergo retinentes in D ad vires retinentes in F &longs;unt
ut DB ad BF.
Retineatur pri&longs;ma &longs;ecundùm obliquam GB, quæ producta
u&longs;que ad Horizontalem concurrat in L. Iterum ex L ad DE
cadat ad angulos rectos LC, quæ perpendicularem FH &longs;ecabit
in I: e&longs;t autem IC parallela ip&longs;i HO; ac propterea per 4.lib.6.
ut HB ad BO, ita IB ad BC, & per 11.lib.5. ut IB ad BC,
ita BF ad BD. Ad retinendum igitur pri&longs;ma in eodem &longs;itu in
clinationis BAE per obliquam GB, vires æquipollentes viri
bus retinentibus in perpendiculari FB e&longs;&longs;e oportet ut BL ad
BI, quemadmodum retinentes per rectam DB &longs;unt ut BC.
Quare datâ corporis inclinatione, cujus gravitas retinenda e&longs;t
in eodem &longs;itu, &longs;umatur eju&longs;dem axis tran&longs;iens per gravitatis
centrum, & ad axis extremitatem mobilem ducatur ip&longs;i axi per
pendicularis DB, in quâ a&longs;&longs;umpto quolibet puncto D, ducatur
prædicto axi parallela DG, quæ &longs;ecans lineas qua&longs;libet obli
quas, & perpendicularem ad Horizontem, dabit omnium obli
quarum &longs;u&longs;pen&longs;ionum Rationem: Sic recta DG &longs;ecans perpen
dicularem FB & obliquam GB determinat Rationem virium in
utrâque &longs;u&longs;pen&longs;ione, ut &longs;cilicet &longs;int in Ratione BF ad BG, &
&longs;ic de reliquis.
Quòd &longs;i in gradibus data &longs;it inclinatio pri&longs;matis, & funiculi
oblique &longs;u&longs;pendenti declinatio a perpendiculo, &longs;tatim ex tabu
lis Sinuum, aut etiam Secantium, apparebit Ratio quæ&longs;ita li
nearum: angulus enim, quem perpendicularis ad axem facit
cum perpendiculari ad Horizontem, æqualis e&longs;t angulo incli
nationis pri&longs;inatis; angulo &longs;iquidem BAE inclinationis pri&longs;ma
tis, æqualis e&longs;t angulus EBH per 8.lib.6. ac proptereà etiam
ex 15.lib.1. qui illi e&longs;t ad verticem DBF. Hinc &longs;i inclinatio
nis angulus &longs;it gr. 36. DB ad BF erit ut Radius ad Secantem
gr. 36. vel ut Sinus gr.54. complementi gr.36. ad Radium. At
angulus, quem facit linea obliquæ &longs;u&longs;pen&longs;ionis cum perpendi
culari ad horizontem tran&longs;eunte per pri&longs;matis punctum; in quo
&longs;u&longs;penditur, e&longs;t æqualis angulo, quem eadem &longs;u&longs;pen&longs;ionis li
nea facit cum perpendiculo tran&longs;eunte per aliud extremum
eju&longs;dem lineæ &longs;u&longs;pen&longs;ionis, cui applicatur potentia retinens:
duæ enim perpendiculares prædictæ &longs;unt inter &longs;e parallelæ, &
linea &longs;u&longs;pen&longs;ionis in eas incidens alternos angulos facit æquales
per 27.lib.1. Si igitur GB à &longs;uo perpendiculo, quod ex G in
horizontem cadat, declinat gr.25. etiam FBG e&longs;t gr.25. To
tus igitur angulus DBG e&longs;t aggregatum anguli inclinationis
pri&longs;matis, & anguli declinationis funiculi &longs;u&longs;pendentis: igitur
DBG e&longs;t gr.61, & po&longs;itâ DB ut Radio, erit BG Secans gr.61.
Vel &longs;i comparanda &longs;it BG cum BF, qui angulus GFB ex
ternus per 32.lib.1. æqualis e&longs;t duobus internis oppo&longs;itis tran
guli DBF, erit GFB gr.126; at FBG e&longs;t gr.25, igitur FGB
e&longs;t gr.29. Quare BF ad BG e&longs;t ut Sinus gr. 29. ad Sinum
gr.126, hoc e&longs;t &longs;upplementi gr.54. Apparet ex his primò minimas vires exerceri, &longs;i linea reten
tionis cadat ad perpendiculum in axem corporis elevati cum in
clinatione; quia &longs;cilicet cum in D &longs;it angulus rectus, recta BD
e&longs;t omnium linearum ex B puncto excuntium, & in rectam
DG cadentium minima: quò autem major fuerit obliquitas,
eò etiam majores vires requiri, quia longiores &longs;unt Secantes
angulorum majorum in B po&longs;ito Radio BD.
Secundò fieri pote&longs;t, ut pare: vires requirantur, &longs;i linea re
tentionis faciat cùm axe corporis elevati angulum acutum, ac
&longs;i faciat cùm eodem angulum obtu&longs;um, ut &longs;i fuerit recta MB;
ip&longs;a enim pariter opponitur angulo recto BDM, ac proinde
qui pote&longs;t e&longs;&longs;e æqualis angulo DBF, vel DBG; quo ca&longs;u
etiam ip&longs;a BM æqualis erit ip&longs;i BF aut BG. Ex quo ulteriàs
&longs;equitur, &longs;i à retinente obliquè fiat tractio elevando magis ac
magis pri&longs;ma &longs;ic inclinatum, mutari &longs;ubinde momenta: hoc ta
men intercedit di&longs;crimen, quod trahentis linea initio applicata,
ut angulum faciat acutum cum axe pri&longs;matis, in ipsâ tractione
&longs;emper majorem facit cum ip&longs;o axe angulum, donec veniat ad
angulum rectum con&longs;tituendum, ut &longs;i MB traheretur, donec
coincidat cùm DB, quæ pariter moveri intelligatur: contrà
verò trahentis linea applicata, ut cum axe faciat angulum ob
tu&longs;um, in ipsâ tractione magis adhuc obtu&longs;um angulum con&longs;ti
tuit, donec tractionis linea (&longs;i tamen fieri id po&longs;&longs;it) in unam
rectam lineam cum axe pri&longs;matis conveniat. Quare in primâ
illâ tractione minuitur conatus, in hac &longs;ecunda augetur.
INNOTUIT, opinor, quantum ad præ&longs;ens in&longs;titu
tum &longs;atis e&longs;&longs;e po&longs;&longs;it, centrum gravitatis ex iis, quæ
libro &longs;uperiore dicta &longs;unt: nunc propiùs ad ip&longs;am
machinalem &longs;cientiam accedendum, quam Mecha
nicam dicimus. Hæc Geometriæ &longs;ubjicitur; neque
enim, ut illa, puram corporum quantitatem moli&longs;que exten
&longs;ionem ab&longs;tractè con&longs;iderat, &longs;ed quatenus gravitati illigatam
aut levitati; nihil tamen &longs;olicita de ipsâ corporum materie, au
reáne &longs;it, an lapidea. Quamvis autem ea quoque Statices pars,
quam Hydro&longs;taticen indigitamus, &longs;e pariter in corporum gra
vitate con&longs;iderandâ exerceat, aliam tamen &longs;ibi contemplatio
nem a&longs;&longs;umit; motum &longs;iquidem corporum &longs;ingulorum naturæ
congruentem, pro humorum, in quos incurrunt, diver&longs;itate,
poti&longs;&longs;imùm &longs;peculatur: Mechanice verò eatenus &longs;olùm ingeni
tam corporibus propen&longs;ionem in motum aut quietem explorat,
ut earum facultati per&longs;pectæ vim po&longs;&longs;it opportunâ in&longs;trumento
rum machinatione inferre. Quapropter ut certâ methodo ma
chinas oneribus movendis pares con&longs;truere valeamus, motus
machinalis cau&longs;as antè cognitas habere nece&longs;&longs;e e&longs;t, quàm ma
chinas ip&longs;as aggrediamur. His porrò jactis fundamentis ope
ro&longs;um non erit inædificare, & machinarum &longs;ingularum vires,
&longs;ivè &longs;implices illæ &longs;int, &longs;ivè compo&longs;itæ, exponere: adeò ut iis
ritè intellectis, quæ hoc &longs;ecundo libro di&longs;putabuntur, vix quie
quam in reliquo opere &longs;uper&longs;it difficultatis.
FInis, quò demum unaquæque actio refertur, primus animo
concipitur, præ&longs;tituiturque, & idonea ad agendum &longs;ub&longs;i
dia, quæ deligenda &longs;unt, moderatur. Hinc ille primus nobis
in hâc contemplatione occurrit; quem &longs;cilicet ad finem ma
chinæ in&longs;tituantur, in&longs;truantúrque, con&longs;iderandum; ut ad
hanc qua&longs;i regulam cæteræ cau&longs;æ dirigantur, & formentur.
Fortè dixerit qui&longs;piam magnificè, eo con&longs;ilio machinas à no
bis excogitatas, ut naturam arte vincamus; quemadmodum
enim &longs;cribit Antipho Poeta apud Ari&longs;totelem in quæ&longs;t.Mechan.
&longs;ub initium, Sed hic plani&longs;
&longs;imè philo&longs;ophandi locus e&longs;t, non gloriandi in&longs;olentiùs. Quare
fatendum e&longs;t apertè, adhiberi machinas in &longs;ub&longs;idium infirmi
tatis; ut quod virium imbecillitas onus loco movere, aut omni
nò, aut ni&longs;i ægerrimè &longs;ola nequiret, illud demum facilè, quò
libuerit, aut trahat, aut impellat, aut etiam expellat quantum
vis reluctans, &longs;i machina accedat.
Dupliciter autem in&longs;ita corporibus gravitas ob&longs;i&longs;tit moventi,
&longs;i ab alio in alium locum transferenda fuerit: di&longs;paribus enim
momentis mora infertur motui, &longs;i hic fluido in corpore ac &longs;e
quaci, puta in aëre aut aquâ, perficiatur, ac &longs;i &longs;uprà &longs;olidam
con&longs;i&longs;tentemque planitiem raptetur moles, &longs;ive Horizonti pa
rallela jaceat planities, &longs;ive molli aut arduâ inclinatione eriga
tur in clivum. Et quidem &longs;i &longs;olidum in corpus non incumbat
onus, &longs;ed in aëre &longs;u&longs;pen&longs;um pendeat, ac &longs;ur&longs;um trahere opor
teat, certos ad calculos revocari gravitatis momenta poterunt,
quibus machina proportione re&longs;pondeat: nam quamvis aër aëri
præ&longs;tet tenuitate, non ea tamen e&longs;t in levitatibus differentia, ut
hinc in gravium corporum momentis di&longs;&longs;imilitudo notabilis
oriatur. Quare &longs;icut laberetur turpiter, qui machinam &longs;axo ab
imo mari ad &longs;ummam &longs;uperficiem elevando parem in&longs;trueret, &longs;i
nullâ factâ virium acce&longs;&longs;ione illud in aërem extrahi po&longs;&longs;e &longs;ibi
aërem, qui pro di&longs;pari ejus levitate modum machinæ &longs;tatueret;
in materiâ etenim, ex quâ machina componitur, nullus e&longs;t
huic minutæ &longs;ubtilitati locus, quæ aciem omnem fugit, ni&longs;i
cum veritas in di&longs;putatione limatur. Id quod de eâ pariter
gravitationis inæqualitate dictum velim, quæ ex inæquali à cen
tro gravium di&longs;tantiâ ortum habet, ut lib.1. cap.
4. di&longs;putatum
e&longs;t: Quia in tantulo Spatio, in quo nos labor no&longs;ter exercet,
illa momentorum exuperantia &longs;ub &longs;en&longs;um non cadit. Quo cir
ca &longs;atis &longs;upérque habemus, quòd moventis vires ac molis mo
vendæ pondus reputantes ita inter &longs;e conferamus, ut virium
imbecillitas adhibitâ machinâ convale&longs;cat, & repugnanti one
ris gravitati non re&longs;i&longs;tat modò, &longs;ed & præ&longs;tare po&longs;&longs;it, nullâ aut
loci aut aëris habitâ ratione.
Verùm quàm facile e&longs;t corporis gravitatem cùm ex mate
riæ &longs;pecie, tùm ex molis magnitudine inve&longs;tigare; tàm mul
tis difficultatibus impedita res e&longs;t, &longs;i examinandum &longs;it,
quantùm ex mutuo corporum &longs;e contingentium tritu retardetur
motus: non enim qui&longs;quis pendulum in aere majoris campanæ
malleum pote&longs;t à perpendiculo dimovere, earum e&longs;t virium, ut
illum pariter in terrâ jacentem propellere valeat: & decennis
puer arrepto fune illigatam cymbam, modicè fluctuante &longs;alo,
ad &longs;e trahit; quam vix, aut ne vix quidem, robu&longs;tioris lacerti
vir dimoveat, ubi areno&longs;o vado in&longs;ederit: cum tamen eadem aut
ligneæ cymbæ aut ferreo malleo gravitas innata permaneat.
E&longs;t autem tùm &longs;ubjecti corporis con&longs;i&longs;tentis, tùm impo&longs;iti one
ris movendi &longs;uperficies &longs;pectanda, quatenus &longs;e contingunt:
Nam &longs;i lapideum globum pondo 100 in planitie con&longs;titutum
non rotare modo, &longs;ed & rectâ urgere po&longs;&longs;is, non itidem cubum
pondere parem & materiâ &longs;imilem æquali facilitate urgebis;
quia &longs;cilicet globus tenui&longs;&longs;imâ &longs;ui parte &longs;uppo&longs;itam planitiem
contingens minus invenit impedimenti ex proximè &longs;ubjecti
corporis a&longs;peritate, quæ prominulas impo&longs;iti globi particulas re
moretur; at cubus longè pluribus &longs;ui partibus plano adhæret, at
que adeò multiplicatá partium hujus in illius partes incurren
tium re&longs;i&longs;tentiâ, augeri quoque movendi
Quoniam verò obtineri nequit, ut corporum &longs;e contingen
tium &longs;uperficies &longs;int continuo lævore lubricæ, earum autem
veniens &longs;ub certam legem non cadit; &longs;ed quantum conjectura
a&longs;&longs;equi valemus, illa potius ex antiquis experimentis æ&longs;timanda
videtur, quàm mathematicis ratiocinationibus indaganda. In
hoc uno nimirùm facem præferre pote&longs;t Geometria, ut &longs;i reli
qua pror&longs;us paria &longs;int, nec alia &longs;it quàm molis aut figuræ di&longs;&longs;i
militudo, quantum ex hoc capite movendi difficultas augea
tur, minuaturve, innnote&longs;cat: cæterùm plenè atque perfectè
explicare, quantum re&longs;i&longs;tentiæ ex a&longs;perarum &longs;uperficierum
conflictione oriatur, quis ni&longs;i temerè conetur?
Po&longs;teriori huic malo, quod &longs;uperficierum aliqua a&longs;peritas
creat, occurritur, &longs;i pingui &longs;equacíque materiâ oblitæ lubri
cæ fiant: Sic Automatis, rotarum &longs;e &longs;e mutuá collabellatione
mordentium conver&longs;ione, horas indicantibus velocitas conci
liatur, &longs;i quis denticulos oleo leviter perungat: &longs;ic plau&longs;trorum
tarditatem, equorumque laborem, ut imminuant aurigæ, axes
rotarúmque modiolos axungiâ illinunt; & cæmentarij majora
&longs;axa attollentes, trochleæ orbiculis &longs;apone perfricatis, quærunt
laboris compendium. Hinc Am&longs;terodami pa&longs;&longs;im ob&longs;ervatur
lubricas fieri trahas cerui&longs;iæ doliis, &longs;imilíve pondere, onu&longs;tas;
cum enim equus non procul abe&longs;t à ponte, in quem a&longs;cenden
dum e&longs;t, is, qui equum agit, centonem unguine delibutum
currenti trahæ &longs;ub&longs;ternit, ut expre&longs;&longs;us ex centone pinguis hu
mor inficiat duo illa longiora tigna, quibus traha in&longs;i&longs;tit, ac
proinde lubrica machina faciliùs raptetur per vias lateribus
&longs;tratas. Sic Dio lib.50. de Augu&longs;to loquens.
mes ex mari exteriore per murum in &longs;inum tran&longs;tuli&longs;&longs;e, & loco Pa
langum, per quos ducerentur, tergoribus animalium recens cæ&longs;orum
loco inunctis u&longs;um,lib.13.v.444.
Atque recens cæ&longs;i tergo prolap&longs;a juvenci,
Æquorcem rota ducebat per gramina puppim.
Verùm nec frequens e&longs;&longs;e pote&longs;t, nec commodum, remedium
hoc ex pingui liquore petitum; illud certius erit ad imminuen
dam moram ex tritu corporum ortam, quod ea &longs;e invicem
quàm minimùm contingant. Quoniam verò deducendi one
ris &longs;uperficiem amplam mutare &longs;æpè nequimus, aut illud rap
tandum trahæ imponimus, quæ non ni&longs;i tigillis duobus læviga-
jus rotæ &longs;olum calcantes dum convertuntur, axem tantum
modo terunt, compendio &longs;anè mirabili; nam dum rotæ modio
lus axem &longs;emel terit, pedes circiter viginti provehitur onus, aut
demum &longs;ublato corporum mutuo tritu cylindros, vel &longs;cytalas
illi &longs;ubjicimus, ut nihil noceat &longs;oli a&longs;peritas, ni&longs;i quatenus hæc
cylindrorum vel &longs;cytalarum conver&longs;ionem remoratur.
Huc &longs;pectat id, quod non &longs;ine voluptate ob&longs;ervare aliouan
do contigit Bononiæ. Tres erant viri nec admodum robu&longs;ti,
qui ut aliquot ingentes &longs;accos farinâ plenos in domum infer
rent, paratum habuerunt axem binis rotulis circiter &longs;e&longs;quipal
maribus in&longs;tructum; axi jungebatur cra&longs;&longs;iu&longs;culus temo &longs;acco
rum longitudinem vix &longs;uperans. Erecto &longs;acco machinulam ap
plicabant, tùm &longs;accum pariter cum temone reclinabant, & ne
temoni incumbens juxtà longitudinem &longs;accus in alterutram
partem inclinaretur, duo hinc & hinc retinebant pariter, ac
propellebant, ut tertium arrepto temone trahentem labore le
varent: Hâc ratione alium atque alium &longs;accum tenui&longs;&longs;imo la
bore in domum importarunt; erectoque iterum temone delap
&longs;us e&longs;t ex machinulâ &longs;accus, &longs;tetitque erectus.
Ex his itaque con&longs;tat in machinâ in&longs;truendâ non &longs;olùm in
genitæ corpori movendo gravitatis rationem habendam e&longs;&longs;e;
&longs;ed & plani, &longs;uper quo illud deducendum e&longs;t, jacens-ne &longs;it?
an erectum?
læve, an a&longs;perum?
amplâ, an tenui &longs;uperficie
contingat? hinc &longs;i quidem varia re&longs;i&longs;tentiæ momenta exur
gunt. Illud tamen plerumque contingit, quod &longs;i attollendo ad
perpendiculum oneri par fuerit machina, illa pariter &longs;ufficiat
ad onus idem &longs;uper plano horizontali, aut inclinato deducen
dum: vix enim fieri pote&longs;t (ni&longs;i &longs;umma &longs;it &longs;uperficierum &longs;e
contingentium a&longs;peritas) ut quantum re&longs;i&longs;tentiæ demitur à
plano &longs;u&longs;tinente, tantumdem addatur ex mutuo prominentium
particularum conflictu.
Quamquam & ip&longs;a a&longs;peritas facit aliquod laboris compen
dium: nam licèt continens ac perpetuus non &longs;it motus, &longs;ed al
ternâ quiete interruptus &longs;uper arduo clivo, modico tamen co
natu prohibetur moles, ne prolap&longs;a &longs;i&longs;ipheum crect laborem;
quia a&longs;pera &longs;uper&longs;icies motui ob&longs;i&longs;tens efficit ne corporis gravi
tas deor&longs;um conetur pro plani inclinatione. Satis igitur fuerit
perpendiculum &longs;u&longs;tollendo cæteroqui impares vires &longs;ufficiant:
qui enim valuerit, adhibitâ machinâ, molem attollere, poterit
illam pariter, eju&longs;dem machinæ ope, in plano quocunque tra
here aut propellere; &longs;i maximè cylindri aut rotæ ei &longs;ubji
ciantur.
Hîc autem fortè nec à præ&longs;enti in&longs;tituto alienum, nec lecto
ri injucundum accidat, &longs;i quæ, aliquando commini&longs;ci placuit,
&longs;ubjiciam, cum narrantem quendam audirem de campaná in
gentis ponderis facillimè agitatâ &longs;ubjectis æneis rotulis, quæ
demum longo ævo confectæ di&longs;&longs;ipatæ fuere; &longs;ed quonam artifi
cio, quóve ordine di&longs;po&longs;itæ fui&longs;&longs;ent, enarrare omninò non
poterat. Quare mecum ip&longs;e reputans, quî fieri id potui&longs;&longs;et, in
eam incidi &longs;ententiam, ut exi&longs;timarem gravi&longs;&longs;imam campanam
potui&longs;&longs;e facilè pul&longs;ari, imminutâ re&longs;i&longs;tentiâ, quæ oritur ex mu
tuo fulcri, & axis tritu. Sint
enim binæ rotulæ B & C ex
ære &longs;olido, quarum diameter
&longs;it in aliquâ Ratione multiplici
ad diametrum axis, cui cam
pana innititur. Axis autem &longs;e
midiameter &longs;it AE, rotulæ ve
rò BE in ratione duplâ; ergo
& periphæriæ &longs;unt in eâdem Ratione: dum igitur punctum I
in H perficit quadrantem, convertit pariter rotulam; cujus pe
ripheriæ &longs;emiquadranti coæquatur. Quare &longs;i rotula infixa e&longs;&longs;et
axi, cujus &longs;emidiameter BG e&longs;&longs;et æqualis &longs;emidiametro AE,
fieret affrictus cum octante peripheriæ axis rotulæ B; &longs;ed quia
etiam in rotulâ C fieret æqualis affrictus cum eju&longs;dem axe, jam
nihil ferè emolumenti haberetur, quia totus affrictus æquè e&longs;
&longs;et, ac &longs;i quadrans EO in fulcro &longs;tabili & cavo converteretur:
& potiùs laboris in agitandâ campanâ compendium e&longs;&longs;et, &longs;i ro
tulæ fixæ hærerent, axis &longs;i quidem cylindricus cum &longs;it, &longs;ubjectas
rotulas in lineâ tangeret modico &longs;cilicet tritu; rotularum autem
axes concavis earum partibus congruunt in &longs;uperficie, quæ te
ritur, dum rotulæ convertuntur: ni&longs;i fortè cylindrica axis
BG &longs;uperficies convexa paulò minor e&longs;&longs;et concavâ rotulæ
&longs;uperficie, eæque propterea &longs;ecundùm lineam &longs;e continge-
contingit.
Verum non e&longs;t nece&longs;&longs;e rotulis B & C tàm &longs;olidos axes dare;
nam &longs;i axis AE toti campanæ oneri ferendo par e&longs;t, bini æqua
les axes duplici ponderi re&longs;i&longs;tunt: &longs;atis igitur e&longs;&longs;et, &longs;i axes &longs;in
guli B & C, oneris &longs;emi&longs;&longs;em &longs;u&longs;tinerent. Cum verò cylindro
rum re&longs;i&longs;tentiæ, ne frangantur, &longs;int in triplicatâ Ratione &longs;ua
rum diametrorum, &longs;ufficeret inter &longs;emidiametrum AE, & ejus
&longs;emi&longs;&longs;em duas medias proportione continuâ reperire, quæ enim
proxime minor e&longs;&longs;et ipsá AE, e&longs;&longs;et &longs;ufficiens &longs;emidiameter cy
lindri &longs;ubduplam habentis &longs;oliditatem ac re&longs;i&longs;tentiam. Sed
adhuc minor requiritur &longs;emidiameter, quia onus axes rotula
rum B & C obliquè premit; ex quo fit campanæ gravitationem
in axes illos e&longs;&longs;e &longs;ecundùm lineas AB, AC, non autem juxtà
perpendiculum AD: igitur ut AD ad AB, ita reciprocè gra
vitatio &longs;uper AB ad gravitationem &longs;uper AD: atqui gravita
tio in alterutrum axium, ut &longs;ummum &longs;ubdupla e&longs;t totius gra
vitationis; ergo gravitatio &longs;uper BA minor e&longs;t &longs;ubduplâ. Quâ
autem Ratione minor &longs;it con&longs;tat. Cum enim detur tùm &longs;emi
diameter AE, tùm etiam BE, nota e&longs;t tota BA, & BD, pari
ter, ip&longs;i BE æqualis, nota e&longs;t; igitur ex 47 lib.
1. etiam AD
innote&longs;cit, cujus &longs;cilicet quadratum habetur, &longs;i ex BA quadra
to dematur quadraturm BD.
Cum itaque, ex hypothe&longs;i, BA &longs;it 3, cujus quadratum 9, &
BD 2, cujus quadratum 4, remanet quadratum 5, eju&longs;que Ra.
dix 2. 23″. e&longs;t recta DA: gravitatio igitur &longs;uper BA ad totam
campanæ &longs;uper utrumque axem B, & C, gravitationem e&longs;t
223 ad 600′. Quoniam verò &longs;olidorum &longs;imilium re&longs;i&longs;tentia
e&longs;t in triplicatâ Ratione laterum homologorum (in cylindris
autem diametrorum ratio habetur) quærantur duo medij pro
portionales numeri inter 600″ & 223″. Id quod a&longs;&longs;equeris, &longs;i
cuju&longs;libet extremi quadratum ducas in alium extremum, pro
ducti enim Radix cubica e&longs;t terminus proximus illi numero,
cujus quadratum a&longs;&longs;ump&longs;i&longs;ti. Primi igitur 600 quadratum
360000 duc in 223, & producti 80280000, Radix cubica e&longs;t
431 1/3 proximè: alterius verò extremi 223 quadratum 49729
ductum in 600 dat 29837400, cujus Radix cubica 310 proxi
mè e&longs;t alter medius. Sunt igitur quatuor numeri 600. 431 1/3.
223 continuè proportionales proximè, &longs;pretis fractiuncu
lis. Quare &longs;i &longs;iat ut 600′ ad 431′, ita &longs;emidiameter AE ad BN,
erit hæc &longs;emidiameter quæ&longs;ita &longs;ufficienter re&longs;i&longs;tens.
Quoniam itaque BE dupla e&longs;t ip&longs;ius AE, & AE ad BN
facta e&longs;t ut 600 ad 431, erit BE ad BN ut 1200 ad 431; & &longs;e
cundùm hane eandem Rationem &longs;e habebunt &longs;emiquadrantes
ab illis de&longs;eripti. Atqui octans peripheriæ ex Radio BE æqua
lis e&longs;t quadranti ex Radio. AE; igitur quadrans EO ad &longs;emi
quadrantem ex Radio BN e&longs;t pariter ut 1200 ad 431: Qui igi
tur affrictus axis campanæ cum fulcro &longs;tabili & cavo e&longs;&longs;et 1200,
rotulæ B cum &longs;uo axe e&longs;t 431, cui æqualis e&longs;t alterius rotulæ C
affictus cum &longs;uo axe; ac proinde &longs;ubjectis rotulis, quarum dia
meter &longs;it tantum dupla diametri axis. campanæ, affrictus e&longs;t ut
862, ad affrictum qui e&longs;&longs;et ut 1200. Si itaque rotularum dia
meter ad campanæ axem. non tantùm dupla, &longs;ed vel tripla,
vel quadrupla &longs;it, multò minor erit affrictus, majorque in agi
tanda campanâ facilitas.
Quamvis autem i&longs;tâ con&longs;imilivè diligentiâ indu&longs;triâque plu
rimum imminui po&longs;&longs;it particularum conflictus, quæ &longs;e vici&longs;&longs;im
terentes moram atque impedimentum motui inferrent; non illa
tamen ex eo propriè veréque dicitur motio machinalis, quòd
in&longs;trumento atque apparatu aliquo perficiatur, ni&longs;i, &longs;pectatâ
dumtaxat oneris gravitate, potentia illi movendo cæteroqui im
par, &longs;ub&longs;idium &longs;ibi comparet ex machinâ. Machina autem non
idem e&longs;t, &longs;i plenè atque perfectè interpretari velis, ac in&longs;tru
mentum; licet enim machina omnis in&longs;trumentum &longs;it, non ta
men in&longs;trumentum quodlibet machinæ vocabulum continuò
&longs;ortitur, &longs;i motionem aliquatenùs juvet; &longs;ed illud prætereà ef
ficiat nece&longs;&longs;e e&longs;t, quod ejus ope naturalem ac in&longs;itam vim cor
poris loco dimovendi &longs;uperet vis minor extrin&longs;ecùs adhibita.
Cum ergò onus hærere in &longs;alebrâ, non ex in&longs;itâ vi, &longs;ed ex proxi
mi etiam atque continentis corporis a&longs;peritate proveniat, &
in&longs;trumenta, quibus hoc tantummodo impedimentum tollitur,
idem planè efficiant, quod pinguis humor lubricum parans iter;
neque hæc machinæ magis dici po&longs;&longs;unt, quàm centones ungui
ne delibuti, &longs;i ritè &longs;ub&longs;ternantur, neque motus propterea inter
machinales numerandus videtur, quorum hîc cau&longs;as ye&longs;tigare
nobis propo&longs;itum e&longs;t. Quamquam negandum non &longs;it hæc pari-
machinis, præcipuo nimirum mechanices &longs;copo. affinia &longs;unt;
etiam&longs;i ad illas non velut &longs;ubjectæ partes ad genus revocentur:
& in&longs;trumentis huju&longs;modi &longs;i machinæ appellationem tribuere
placuerit, non admodum de nomine di&longs;putabo; res enim hîc
&longs;pectatur, non verba penduntur.
Sed neque hîc di&longs;putare velim, utrùm in motuum machina
lium cen&longs;um irrepant, an verò iis ritè annumerandi &longs;int motus
illi, quos &longs;ur&longs;um deor&longs;um, ultrò citróque perficiendos eatenus
expeditè, nec exiguo laboris compendio, molimur, quatenus
eos intervallis ita di&longs;tinguimus, ut nos quidem corpus deprima
mus, ut adducamus, ab alio verò extollatur, aut reducatur: in
his &longs;iquidem &longs;æpè nihil e&longs;t, quod no&longs;tram imminuat operam,
&longs;i motiones &longs;ingulæ attendantur; quamquam motui univer&longs;o
adjumentum importat continens illa conatûs no&longs;tri, alienique
&longs;ub&longs;idij, vici&longs;&longs;itudo. Hinc &longs;i quis
ad contundendam in æneo morta
rio A contumacem aliquam mate
riam graviore pi&longs;tillo ferreo opus
habeat, haud dubium quin ei mul
tâ lacertorum vi contendendum
&longs;it, ut illum extollat; cumque ope
ro&longs;ius multo &longs;it inflexum corpus
erigere, quàm erectum inclinare,
multóque mole&longs;tius brachia tanto
pondere pre gravata attollere, quàm
eorum gravitati ob&longs;ecundando de
primere, &longs;atis con&longs;tat, quantum &longs;i
bi laboris detractum eat, &longs;i &longs;uperio
re in loco tran&longs;ver&longs;um tigillum
CD circa axem E ver&longs;atilem &longs;tatuat, paribú&longs;que intervallis
hinc ex C pendeat fune &longs;u&longs;pen&longs;us pi&longs;tillus B, hinc verò in D
plumbea ma&longs;&longs;a adnectatur, quâ ita pi&longs;tillus præponderetur, ut,
nemine hunc retinente aut deprimente, illa aliquanto gravior
in &longs;ubjectum prodeuntis è pariete tigni caput G recidens &longs;pon
te &longs;ub&longs;idat. Omnis &longs;cilicet extollendi pi&longs;tilli labore &longs;ublato,
vel &longs;olum brachiorum pondus pi&longs;tillo additum &longs;atis e&longs;&longs;e ali
quando poterit ad leviu&longs;culè tundendam materiam, licebitque Id quod
pariter continget, &longs;i operâ unâ opus duplex efficere placuerit;
nam &longs;i ex D plumbeæ ma&longs;&longs;æ loco alius pendeat æque, ac plum
bum, gravis pi&longs;tillus, pondere præpollens elevabit pi&longs;tillum B,
aliámque vici&longs;&longs;im in altero &longs;ubjecto mortario conteret mate
riam &longs;ponte &longs;uâ cadens: cumque pi&longs;tillorum gravitates non ad
modum inter &longs;e di&longs;pares &longs;int, neque multum laboris eum &longs;ubi
re nece&longs;&longs;e erit, cui pi&longs;tillum B deprimendi munus incumbit.
Quâ in re, &longs;i motus univer&longs;us ita tribuatur in partes, ut tun
dentis quidem motiones &longs;ingulæ &longs;eor&longs;im &longs;pectentur, non ille
profectò &longs;e juvari &longs;entit, quippe quem, præter vires ad commi
nuendam materiam nece&longs;&longs;arias, conatum quoque adhibere
oportet ad vincendam præponderantis plumbi, aut pi&longs;tilli gra
vitatem. Cæterùm &longs;i totius motûs, qui Ar&longs;i pariter con&longs;tat ac
The&longs;i, habeatur ratio, in&longs;iciari nemo poterit, minus multo la
boris impendi, quàm &longs;i hæc omnia &longs;ublata intelligantur. Qua
re nec incongruum pror&longs;us videatur motûs machinalis voca
bulum, cum ver&longs;atilis tigillus CD ad libræ Rationes manife&longs;tò
revocetur, quam certè ex machinarum albo nemo expungit, ni
&longs;i qui &longs;olas quinque facultates, & quæ ex his componuntur, ma
chinas indigitare voluerit, & libram ad vectem referri po&longs;&longs;e
pernegarit.
Nec di&longs;&longs;imilis ineunda videtur dicendi ratio, &longs;i quid alternis
ciendum motibus &longs;ic di&longs;ponitur, ut, cum primùm quidem mo
vetur, corpus aliud vi flectatur, quod po&longs;tmodum facultate
ela&longs;ticâ, &longs;e re&longs;tituens illud vici&longs;&longs;im moveat; quemadmodum
pa&longs;&longs;im in eorum officinis videre e&longs;t, qui rudes arborum, aut
elephantini dentis particulas in toreumata elaborant: primùm
enim artifex pede &longs;ubjectum vectem premens, toreuma in gy
rum ducit, ha&longs;tulámque &longs;uperiore in loco po&longs;itam pariter in
flectit; quæ &longs;ibi mox &longs;uam reparans rectitudinem, funiculum
que cylindrulo ver&longs;atili circumplicatum retrahens, illud iterum
&longs;ua per ve&longs;tigia ver&longs;at, ut accuratè exqui&longs;itéque tornetur. Sic
aliquid &longs;ubtiliter ac delicatè &longs;ecturus, ut &longs;errulam rectâ addu
cas, reducá&longs;que, operæ tantùm &longs;emi&longs;&longs;em tibi re&longs;ervans, arcum
intentum ex adver&longs;o &longs;tatuito, ac medio nervo &longs;errulam alliga
to; hac enim adductâ magis flectetur arcus, qui &longs;e &longs;e mox re&longs;ti
tuens illam vici&longs;&longs;im reducet.
Hæc &longs;anè laboris in movendo compendia ex ela&longs;mate, vel ex
anti&longs;acomate petita, quemadmodum & ea, quæ mutuum cor
porum tritum atque conflictum minuunt, ut pote Mechanico
artificio con&longs;tituta, eumdemque in finem ac machinæ, quibus
hoc nomen præcipuè tribuitur, videlicet in infirmæ potentiæ
&longs;ub&longs;idium excogitata, e&longs;to illis primas deferant, non tamen
omninò rejicerem, &longs;i in machinarum cen&longs;u prodirent, ii&longs;que
&longs;e peterent ad&longs;cribi. Triplicem enim in &longs;peciem tribui po&longs;&longs;e vi
detur univer&longs;um machinarum genus: Prima eas complectitur
facultates, quarum ope motui facilitas conciliatur, quocum
que tandem ex capite &longs;ivè tantummodo ex in&longs;itâ in corporibus
gravitate, &longs;ivè non ex eâ dumtaxat, &longs;ed ex partium a&longs;peritate
movendi difficultas con&longs;urgat. Altera e&longs;t, quæ mutuam qui
dem corporum &longs;e contingentium conflictionem minuit, &longs;ed ad
vincendam oneris gravitatem ip&longs;i potentiæ momenta non addit.
Tertia demùm eatenus per &longs;e, quia talis e&longs;t, moventem juvat,
quatenus ejus operam alternam efficit, cum tamen neque gra
vitatem vincat, neque quod ex partium triru impedimentum
oritur, extenuet, ni&longs;i cum alterutra, aut utraque &longs;uperiori &longs;pe
cie, amico fœdere copuletur. Alternam autem operam appel
lo, cum in motu ex duplici motione compo&longs;ito alterutram effi
cit potentia, &longs;ivè illæ &longs;ibi invicem adver&longs;antes &longs;uccedant, ut
Ar&longs;is ac The&longs;is, Adductio atque Reductio, &longs;ivè in unam tem
perentur, ut cum premere &longs;imul oportet ac agitare: &longs;ic plana
vitra expolientes in &longs;pecula, inter ip&longs;a, & lacunar bacillum in
flectunt, qui &longs;e re&longs;tituere tentans vi ela&longs;ticâ, &longs;peculum validè,
quantum opus e&longs;t, admovet atque applicat ad &longs;ubjectum pla
num, adeò ut ad artificem à pre&longs;&longs;u immunem nil aliud &longs;pectet,
quàm &longs;peculum urgere, retrahere, contorquere. Verùm ta
met&longs;i de his omnibus in hac tractione pa&longs;&longs;im &longs;e offeret dicendi
locus, primus tamen di&longs;putationis no&longs;træ &longs;copus erit prima illa
&longs;pecies, ip&longs;æ nimirum facultates, quarum poti&longs;&longs;imum momen
ta expendimus, cum motûs machinalis cau&longs;as inquirimus.
explicatur.
QUicquid movetur, qualecumque e&longs;t, cau&longs;am habeat mo
ventem nece&longs;&longs;e e&longs;t, ut hoc quidem &longs;ponte &longs;uâ, illud ve
rò alienâ vi ex alio in alium locum migret. Suopte ingenio mo
ventur tùm corpora gravia aut levia, ut &longs;i extrà præ&longs;criptum
&longs;ibi à naturâ locum con&longs;tituta fuerint, &longs;uo quæque ordine di&longs;
ponantur; tùm rara aut den&longs;a, ut &longs;i per vim hæc extenuata fue
rint, illa concreverint, naturæ &longs;tatum &longs;ibi reparent; tùm ani
mantia, quibus cum à naturâ tributum &longs;it, ut &longs;e, vitam, cor
pu&longs;que tueantur, &longs;timulos admovet appetitus, ut ea declinent,
quæ nocitura videantur, omniaque, quæ &longs;int ad vivendum ne
ce&longs;&longs;aria, acquirant, & parent. Vi extrin&longs;ecus impre&longs;sâ locum
mutant, quæcumque in motu non &longs;erviunt naturæ, &longs;ed alieno
reguntur arbitrio; ut iis contingit, quæ raptantur, pelluntur, in
gyrum ducuntur, projiciuntur, & hujus generis motibus
cientur.
Quoniam verò gravium, & levium celeritatem naturâ ur
gente incitari, jaculorum autem, ac mi&longs;&longs;ilium, motum u&longs;que
eò &longs;en&longs;im langue&longs;cere, ut planè deficiat, ob&longs;ervamus; etiam&longs;i
moventi naturæ, quæ ex Philo&longs;ophi decretis &longs;ub&longs;tantia e&longs;t, mo
tûs originem ultimam tribuamus, jure tamen optimo aliquid
naturæ ip&longs;i ac motui, interjectum agno&longs;cimus (Impetum no
minamus) cujus intentionem ac remi&longs;&longs;ionem velocitas ac tar
ditas con&longs;equatur. Cum enim eadem de&longs;cendentis lapidis na
tura per&longs;everet, nec illa in &longs;uâ pote&longs;tate &longs;it, aut optione delatâ,
ut eligat utrum velit, motum arbitrio &longs;uo incitare, aut remit
tere valeat; qui fieri po&longs;&longs;it, ut de&longs;cendens velocitatem augeat,
ni&longs;i ei, quem primùm produxit, alium atque alium momentis
&longs;ingulis impetum adjiciat? Illud certè extrà omnem controver
&longs;iam po&longs;itum videtur, naturam gravem &longs;ponte &longs;uâ non a&longs;cen-
jectam rupem delap&longs;um re&longs;ilire cogit, aut &longs;ibi relictum plum
bum ex fune &longs;u&longs;pen&longs;um ultrà perpendiculum, naturá repugnan
te, &longs;ur&longs;um provehit, & eò quidem altiùs, quò ex altiore loco
globulus aut plumbum deciderunt? ni&longs;i quia conceptus naturâ
procurante impetus pergit motum efficere, ipsâ etiam naturâ
quantum pote&longs;t, ob&longs;i&longs;tente. Quòd &longs;i corpus alienâ vi longiùs
emi&longs;&longs;um moveatur, extrin&longs;ecùs impetum imprimi nece&longs;&longs;e e&longs;t:
quem &longs;anè non concipit, ubi primùm à projiciente &longs;ejunctum
fuerit; nihil enim prode&longs;&longs;et ad longiorem lapidis jactum fun
dam iterum ac tertiò circumducere, ni&longs;i alium atque alium im
petum lapis conciperet, quandiù funditori adhærens unâ cum
ip&longs;o movetur.
Quæcumque igitur moventur, impetum habent, quo ferun
tur; cui &longs;atis probabili conjectura, proxima vis motum efficien
di tribuenda videtur. Id quod in projectis quidem, ii&longs;que om
nibus, quæ naturâ repugnante moventur, ita manife&longs;tum e&longs;t,
ut id pluribus demon&longs;trare non oporteat; nulla &longs;iquidem ade&longs;t
in&longs;ita motûs cau&longs;a; ab impetu igitur illo extrin&longs;ecùs impre&longs;&longs;o
motum effici nece&longs;&longs;e e&longs;t. At in cæteris, quibus &longs;e movendi
principium ine&longs;t, neme jure negaverit aut in motu impetum
acquiri, aut velocitatis incrementum ex impetus acce&longs;&longs;ione ori
ri: quî enim fieret, ut excurrentes objectam fo&longs;&longs;am ampliore
&longs;altu tran&longs;ilirent faciliùs, quàm nullo præcedente cur&longs;u, &longs;i in
cur&longs;u ip&longs;o conceptus impetus non augeretur? Jam verò &longs;i &longs;e
cundo temporis momento incitatur magis motus, quàm primo,
urgente &longs;cilicet etiam impetu, quem corpus priore motu acqui
&longs;ivit; hic utique impetus, quem nunc gignere non pote&longs;t
prior motus, cum perierit, extitit pariter cum priore motu:
natura igitur movens priore momento & motum effecit & im
petum. Atqui impetum ex eorum &longs;altem genere e&longs;&longs;e, quæ mo
tum efficiant, con&longs;tat ex velociore motu po&longs;terioribus momen
tis, naturâ pror&longs;us immutatâ, factoque impetûs incremento:
contrà verò motu, quâ motus e&longs;t, impetum non augeri &longs;atis
indicant mi&longs;&longs;ilia, quorum velocitas, dum moventur, &longs;en&longs;im
elangue&longs;cit. Igitur & priore illo temporis momento non mo
tus impetum; &longs;ed impetus motum proximè effecit; impetum
autem procreavit innata movendi vis; cui id circo motio tri-
& ad motum efficiendum natura de&longs;tinavit. Quid?
quòd mo
tui per &longs;e, quia ex alio in alium locum continuata migratio e&longs;t,
efficientiam ægrè tribuere po&longs;&longs;umus: quippe qui, cum in
fluxione con&longs;i&longs;tat, ita ut locus loco, &longs;eu potius, ut &longs;cholæ lo
quuntur, Ubicatio Ubicationi, priori &longs;cilicet pereunti &longs;uccedat
po&longs;terior æquè fugax, inferioris notæ cen&longs;endus e&longs;t quàm im
petus naturâ &longs;uâ aliquandiù permanens: labentia enim &longs;tanti
bus deteriora e&longs;&longs;e, cæteris paribus, quis neget? effectum au
tem causâ præ&longs;tabiliorem e&longs;&longs;e non po&longs;&longs;e ip&longs;a originis notio &longs;ua
det, ne quid effectus habeat, quod non acceperit, aut aliquid
cau&longs;a dederit, quo ip&longs;a careret. Non igitur impetum motus,
&longs;ed motum impetus efficit.
Porrò cum definitas ad agendum vires unaquæque cau&longs;a ob
tineat, certa e&longs;t impetûs men&longs;ura, quæ cum innatâ movendi
facultate ita adæquatur, ut eo qua&longs;i termino circum&longs;cripta cen
&longs;enda &longs;it potentia movens, nec unquam validiore conatu po&longs;&longs;it
&longs;e ip&longs;a urgere; &longs;i tamen omnem impetum antecedente motu a&longs;
&longs;umptum mente &longs;ecernas. Et quidem omne animal (quippe
cui ine&longs;t appetitio & declinatio naturalis ejus, quod naturæ ac
commodatum e&longs;t, aut infen&longs;um) non &longs;emper univer&longs;am illam
impetûs men&longs;uram exequitur, &longs;ed ut vult, ita utitur motu &longs;ui
corporis, quem aucto aut diminuto impetu modò intendit, mo
dò remittit, pro ut interiore motu, rerumque appetitu &longs;imula
tur. Contrà verò inanimum non &longs;uo arbitrio motûs intentio
nem moderatur, &longs;ed naturæ juribus ob&longs;equens nihil prætermit
tit impetûs, & quantum eniti pote&longs;t, opportunum in locum, &longs;i
bique à naturâ con&longs;titutum, contendit. Cave tamen exi&longs;times
parem e&longs;&longs;e lapidis eju&longs;dem, & in aëre, & in aquâ de&longs;cendentis
impetum: natura &longs;cilicet ex medio dividendo, in quo perficien
dus e&longs;t motus, metitur impetûs modum.
Sed quoniam non pauca &longs;unt, quæ motui &longs;æpè adver&longs;antur,
hinc e&longs;t non &longs;emper eandem e&longs;&longs;e corporis &longs;e moventis velocita
tem, quamvis pari impetu producto connitatur: deteritur nimi
rum tantum impetus, quantum &longs;atis e&longs;t ad impedimentum &longs;ub
movendum. Sivè enim objectum corpus propellendum &longs;it, &longs;ivè
medij particulæ locum ægrè dantes divellendæ aut compri
mendæ &longs;int, &longs;ivè connexam molem pariter rapi oporteat, &longs;ivè
in alium locum migret præter naturam, irritus reddatur corpo
ris in motum propen&longs;i conatus; &longs;atis con&longs;tat illud motu agitan
dum e&longs;&longs;e exteriùs: atque adeò quantum impetus illi imprimi
tur oppo&longs;itæ propen&longs;ioni æquale, motui tantumdem &longs;ub
trahitur.
In iis &longs;anè, quæ alienâ vi extrin&longs;ecùs moventur, quia infi
nitè progredi non licet, aliqua demum origo deprehenditur,
cui naturalis &longs;it motus: natura &longs;iquidem vis e&longs;t ciens motus in
corporibus nece&longs;&longs;arios; ita tamen certis tenetur legibus uni
ver&longs;itatis rerum concinnitatem &longs;pectantibus, ut ne ab iis di&longs;ce
dat, &longs;ingularibus corporibus vim aliquam inferri permittat, ubi
adver&longs;is propen&longs;ionibus inter &longs;e confligentibus validior præ&longs;tat
imbecilliori. Sic quia nefas e&longs;t aut corpora inanitatibus inter
jectis conci&longs;a hiare, aut unum in proximi corporis locum, ni&longs;i
eo recedente, penetrare, aut diverticula flexione&longs;que in motu
&longs;ponte quærere; ideò & liquor in longiore &longs;iphonis, aut &longs;piri
talis diabetis, crure de&longs;cendens continuum liquorem in brevio
re crure a&longs;cendere cogit, totumque ex va&longs;e demum exhaurit; &
rapidè lap&longs;us torrens &longs;axa rapit, objecta&longs;que moles disjicit; &
ad perpendiculum cadens lapis &longs;ubjectum vitrum comminuit,
&longs;uique ve&longs;tigium in terrâ validiùs pre&longs;sâ relinquit. Verùm il
lud firmum ac perpetuum e&longs;t, quòd ubi plus violentiæ opus e&longs;t,
parem conatum languidior motus con&longs;equitur. Id quod in
&longs;iphone ABC ob&longs;ervare in promptu e&longs;t, ex
cujus o&longs;culo C inæqualis aquæ copia de
fluit paribus temporis intervallis: quò enim
magis aquæ &longs;uperficies in va&longs;e deprimitur,
eò lentiùs aqua ex &longs;iphone dilabitur:
quamvis &longs;cilicet aquæ crus BC implentis
pares &longs;int &longs;emper ad de&longs;cendendum vires, &longs;i
nihil, aut &longs;altem non inæqualiter, repugnet,
aquæ tamen crus BD brevius, & BI longius, & BA adhuc
longius implentis di&longs;par e&longs;t in a&longs;cen&longs;u repugnantia; ac pro
pterea cum earumdem virium BC minor &longs;it Ratio ad majorem
re&longs;i&longs;tentiam BI, quàm ad minorem BD, languidior quoque
motus e&longs;t de&longs;cendentis aquæ ex BC, cùm graviorem aquam
BI, quàm cùm minùs gravem BD &longs;ursùm trahere oportet. At
& aquæ in va&longs;e &longs;uperficies I paribus ab&longs;int ab Horizonte inter
vallis, aquam ideò hærere, nec amplius ex E fluere con&longs;tat,
quia aquæ BE ad de&longs;cendendum propen&longs;ionem, par aquæ BI
repugnantia, ne a&longs;cendat, elidit. Quòd &longs;i demum aquam in
va&longs;e imminuas, ut ejus &longs;uperficies paulò infra I, atque adeò
infra E o&longs;culum deprimatur, non jam aqua hæret in E, &longs;ed &longs;ua
per ve&longs;tigia in EB remeare cogitur, præponderatâ nimirum
majore gravitate aquæ implentis crus paulo longiùs quàm BI,
atque adeò quàm BE, quod illi ex hypothe&longs;i con&longs;tituimus
æquale; tantóque velociùs ab aquâ interioris cruris raperetur
exterior, quantò depre&longs;&longs;ior facta fui&longs;&longs;et in va&longs;e aquæ &longs;uper
ficies.
Hinc itaque fit, ut pro variâ corporis motui ob&longs;i&longs;tentis re
pugnantiâ modò plus, modò minus impetûs reliquum &longs;it, quo
motû, celeritas aut tarditas perficiatur. Et &longs;i tanta &longs;it eorum
omnium, quæ motui moram inferunt, ob&longs;i&longs;tentia, ut ad eam
vincendam plus impetûs nece&longs;&longs;e &longs;it, quàm pro potentiæ facul
tate, tunc nullus efficitur motus, quo corpus ex loco in locum
transferatur, &longs;ed aliqua ex peregrino impetu fit partium com
pre&longs;&longs;io, aut di&longs;tractio; neque enim omnes corporis particulæ
homogeneæ &longs;unt, aut ita compactæ citrà omnes poros, ut nul
la tenuiorum particularum compre&longs;&longs;io aut di&longs;tractio con&longs;equi
po&longs;&longs;it. Quod &longs;i ea &longs;it corporis per vim movendi natura aut po&longs;i
tio, ut nullum planè &longs;ivè lationis, &longs;ivè rotationis, &longs;ivè vibratio
nis, &longs;ivè con&longs;tipationis, &longs;ivè dilatationis motum concipere po&longs;
&longs;it, aut violento in &longs;tatu permanere languido illo impetu, quem
vis extrin&longs;eca efficere valeret, nullum quoque impetum reci
pit; quippe qui idcircò imprimeretur, ut motum præter natu
ram efficeret, aut ut naturalem motum retunderet, aut etiam
pror&longs;us impediret. Quemadmodum enim &longs;i corporis alicujus
&longs;pecificam gravitatem in aquâ mutari non po&longs;&longs;e con&longs;tet, infer
re continuò licet, corpus idem neque raritatem neque den&longs;ita
tem in aquâ a&longs;&longs;ùmere po&longs;&longs;e; ex his &longs;iquidem &longs;pecificæ gravita
tis mutatio oriretur: ita pariter ubi nihil haberi pote&longs;t eorum,
quæ impetum extrin&longs;ecùs impre&longs;&longs;um nece&longs;&longs;ariò con&longs;equuntur,
impetum quoque abe&longs;&longs;e non immeritò conjectamus.
Si quis tamen animum diligentiùs adverrat, manife&longs;tò de-
vendum &longs;it, minùs verò, &longs;i tardiùs: &longs;ic ferreæ an&longs;æ cubiculi
o&longs;tio infixæ magnetem armatum applicui, & &longs;iquidem paulò
velociùs magnetem traherem, disjungebatur ab ansâ; at len
tiùs trahentem &longs;ub&longs;equebatur o&longs;tium, magnetis &longs;cilicet vim
non &longs;uperans, ubi lentè res peragebatur.
An non oneri, quod potentia præ &longs;ui tenuitate propellere
non po&longs;&longs;e videtur, motus, qui momentis &longs;ingulis &longs;en&longs;um om
nem fugiat, conciliari pote&longs;t, adeò ut, &longs;i illa quidem con&longs;tan
ter urgeat, elap&longs;o demùm longo temporis intervallo appareat?
Sic incumbentem glebam tenerrimus na&longs;centis frugis caulicu
lus tandem di&longs;cutit; duri&longs;&longs;ima marmora &longs;cindens caprificus lo
co movet; & ædificia &longs;ub&longs;edi&longs;&longs;e, ac inæquabile &longs;olum pre&longs;&longs;i&longs;&longs;e,
rimæ demùm loquuntur. Tota igitur corporis, quod præter
naturam movendum e&longs;t, repugnantia metienda e&longs;t, quâ ex
principio ip&longs;o motum detrectante, quâ ex motûs celeritate, aut
tarditate: adeò ut pro variâ horum connexione di&longs;par movendi
difficultas oriatur.
Ex quo fit impetu eodem moveri celeriùs po&longs;&longs;e corpus, quod
minorem &longs;ubit violentiam, tardiùs verò, cui vis major infer
tur, &, &longs;i eadem &longs;it reciprocè Ratio tarditatis ad velocitatem,
quæ e&longs;t minoris violentiæ ad majorem violentiam, parem fore
utrobique movendi difficultatem, cùm par &longs;it repugnantia, quæ
ex motûs tùm &longs;pecie, tùm intentione componitur. Si enim mo
les aliquâ tantâ vi raptetur, ut, quo tempore decies arteria pul
&longs;um edit, pa&longs;&longs;um unum conficiat; quantum virium adhiberi
oporteat, ut paribus temporis momentis ad tres pa&longs;&longs;us eadem
moles promoveatur? utique, &longs;i cætera omnia paria &longs;int, triplo
majorem conatum adhibendum concedes, inten&longs;ione exten
&longs;ionem compen&longs;ante: nam quemadmodum iterùm ac tertiò re
petendus fui&longs;&longs;et prior ille conatus ad æquale &longs;emper &longs;patium pa
ri tarditate percurrendum; ita quamvis conatui conatus non
&longs;uccedat, triplici tamen conatu opus erit, ut tempore eodem
motus ille triplo major perficiatur. Nonnè & agricolæ terram
&longs;ubigentes fo&longs;&longs;ione glebarum, tam multiplices adhibent operas,
quàm breviori tempore opus ab&longs;olvere meditantur? Eò igitur
magis re&longs;i&longs;tit corpus motui, quò celeriùs agitandum e&longs;t; con
trà verò minùs repugnat, quò tardiùs.
Quare &longs;i duo &longs;int corpora, quorum alterum alteri præ&longs;tet
triplo majori gravitate, atque hæc pari celeritate attollenda &longs;int,
di&longs;parem exigunt conatum pro gravitatis Ratione: &longs;i par &longs;it eo
rum gravitas, motus autem alterius reliquo triplo velocior e&longs;&longs;e
debeat, inæqualem pariter exigunt conatum, &longs;ed pro ratione
velocitatis: &longs;i demùm & di&longs;par &longs;it gravitas, & inæqualis velo
citas, eam e&longs;&longs;e con&longs;tat repugnantiam, quæ tùm ex gravitate,
tùm ex velocitate componitur; atque adeò &longs;i corpus alterum
triplo gravius triplo etiam velociùs movendum e&longs;&longs;et, noncuplex
e&longs;&longs;et ejus repugnantia; &longs;in autem triplo levius triplo majori
velocitate quàm corpus triplo gravius, moveretur, par e&longs;&longs;et eo
rum ob&longs;i&longs;tentia, paremque conatum exigerent.
Hinc &longs;atis apertè con&longs;tat, datâ tum re&longs;i&longs;tentiarum, tum velo
citatum Ratione, &longs;i gravitas altera nota &longs;it, reliquam facilè inno
te&longs;cere: &longs;i nimirùm nota gravitas per &longs;uam velocitatem ducatur,
& in datâ Ratione re&longs;i&longs;tentiarum reperiatur huic producto ter
minus homologus; quo per ignotæ gravitatis velocitatem da
tam divi&longs;o, prodibit Quotiens index quæ&longs;itæ gravitatis. Sint
duo corpora inæqualia, & ad ea movenda requiratur conatus
in Ratione &longs;e&longs;quialterâ, motus autem eorum &longs;int ut 7 ad 8, &
illud quod minùs re&longs;i&longs;tit, moveturque velocitate ut 7, numeret
gravitatis libras 4. Reliqui corporis validiùs re&longs;i&longs;tentis, cujus
velocitas e&longs;t ut 8, gravitas &longs;ic invenietur.
Libræ 4 ducantur per numerum &longs;uæ velocitatis 7, & fit 28.
Quia igitur re&longs;i&longs;tentiæ &longs;unt, ut 2 ad 3 ex hypothe&longs;i, & unius
corporis re&longs;i&longs;tentiâ, quæ ex gravitate & motûs velocitate com
ponitur, e&longs;t 28, fiat ut 2 ad 3, ita 28 ad aliud, & erit 42 re
&longs;i&longs;tentia alterius corporis compo&longs;ita ex ejus velocitate & gravi
tate. Atqui velocitas nota e&longs;t 8; igitur divisâ totâ re&longs;i&longs;tentiâ
42 per 8; prodibit quotiens 5 1/4 index quæ&longs;itæ gravitatis. Quare
ad movendas libras 5 1/4 velocitate ut 8, requiritur conatus &longs;e&longs;
quialter conatûs nece&longs;&longs;arij ad movendas libras 4 velocitate
ut 7. Eadem e&longs;to de reliquis ac &longs;imilibus conjectura.
Ex his præterea manife&longs;tum e&longs;t corporis per vim dimovendi
re&longs;i&longs;tentiam ex &longs;olâ naturâ, & principio in&longs;ito, quod motui re
pugnat, ab&longs;olutè definiri non po&longs;&longs;e; motum &longs;i quidem ab omni
prorsùs celeritatis aut tarditatis men&longs;urâ &longs;ejungere non po&longs;&longs;u
mus; idcircò non ni&longs;i habitâ ratione celeritatis, aut tarditatis,
poterit. Quare & impetus à facultate movendi principium ha
bente productus major &longs;it nece&longs;&longs;e e&longs;t, quàm dimoti corporis
repugnantia; quæ varia prorsùs cùm &longs;it, nunc quidem majo
rem, nunc verò minorem impetum exigit, ut ab eo vincatur;
nam &longs;i pares confligerent vires, à neutrâ parte &longs;taret victoria.
Quod autem ad ip&longs;am motûs originem &longs;pectat, ea, quæ vi
vunt, ab iis, quæ vitâ omnino carent, &longs;ecernenda &longs;unt: hæc
enim (&longs;cilicet non viventia) propterea motum expetunt, ut
violentiam, quam &longs;ubeunt, excutiant, nec unquam à loco, &longs;eu
&longs;tatu, &longs;ecundùm naturam opportuno &longs;ponte recedunt; quem
admodum eunti per &longs;ingula con&longs;tabit. Sic gravibus & levibus
&longs;uis in locis quietem natura indixit, non motum; nec deor
&longs;um conantur aut &longs;ur&longs;um, ni&longs;i alieno in loco, hoc e&longs;t, in me
dio di&longs;pari gravitate aut levitate prædito con&longs;titutâ: &longs;ic quæ
cumque ela&longs;ticâ facultate pollent, motum non moliuntur, ni&longs;i
cum &longs;ibi naturalem partium figuram, &longs;itumque reparare opor
tet. At motum, cujus origo vita e&longs;t, natura perficit, etiam&longs;i
nulla præce&longs;&longs;erit violentia: &longs;ic &longs;tirpes dum augentur, & cre&longs;
cunt, earum particulæ locum mutant; &longs;ic vitali facultate in
fluentibus per nervos in
males vocant, intenduntur mu&longs;culi, motu&longs;que membrorum con
&longs;equitur: quamvis ante motum nec &longs;tirpis particulæ, nec anima
lis membra vim
Quæcunque igitur ob id ip&longs;um in motum prona &longs;unt, quia
vim patiuntur, impetum illicò concipiunt, ac vis iis illata e&longs;t,
quo naturalem locum, &longs;eu &longs;tatum, recipere valeant, licèt &longs;æpè
irrito conatu, ni&longs;i quatenùs adver&longs;o hoc impetu illatam ab ob
&longs;i&longs;tente violentiam retundunt, vim aliquam illi vici&longs;&longs;im infe
rentes. Sic onera bajulorum humeros, quibus &longs;u&longs;tinentur,
premunt, aut penduli brachij; ex quo &longs;u&longs;penduntur, mu&longs;cu
los ac ligamenta fatigant: id quod pariter in corpore inanimo
cernere licet; quemadmodum enim ex diuturnâ prementis
deor&longs;um ponderis, ac mu&longs;culorum &longs;ursùm urgentium luctâ,
di&longs;&longs;ipatis &longs;piritibus, la&longs;&longs;itudo in animali oritur, ita pariter &longs;ub
jectum a&longs;&longs;erem longâ temporis morâ pondus curvat, aut etiam
demùm frangit, & funem, ex quo pendet, non intendit &longs;olùm, &longs;ed
etiam tandem aliquando corrupto particularum nexu disjicit.
Quo id autem pacto contingat, explicare opero&longs;um non fue
rit funiculi texturam con&longs;ideranti; ex tenui&longs;&longs;imis &longs;cilicet linei
aut cannabini corticis longâ maceratione, & plurimâ tun&longs;ione
extenuati particulis in &longs;piram contortis filum cohæret; ex filis
autem plu&longs;culis in &longs;piram pariter contortis funiculus, & pluri
bus funiculis cra&longs;&longs;iores rudentes conflantur: quod &longs;i di&longs;&longs;olvatur
omnis &longs;pira, non cohærent funiculi aut fili partes. Spira di&longs;
&longs;olvitur factâ in contrarium revolutione; quò autem laxioribus
gyris flectitur, eò faciliùs villi &longs;inguli ex cæteris, quibus im
plicantur, extrahuntur; & uno ab aliorum communione &longs;e
juncto, amplitudo &longs;patij faciliorem exitum proximis relinquit:
ex quo fit faciliùs &longs;emper ac faciliùs po&longs;&longs;e funiculum frangi;
filo enim uno rupto, aut extracto, facilior e&longs;t in contrarium re
volutio, & &longs;pira fit amplior, ac reliqua fila faciliùs extrahun
tur. Ob&longs;ervamus autem non rarò appen&longs;um ex funiculo pon
dus aliquandiu in gyrum contorqueri; dum &longs;cilicet &longs;uâ gravi
tate deor&longs;um connitens intendit funiculum, contorta fila in
contrarium revolvuntur. Sed &, quamvis nulla fieret in con
trarium revolutio, &longs;atis con&longs;tat ex illâ inten&longs;ione funiculum
di&longs;trahi, ac produci; atque adeò &longs;piram laxiorem fieri, paula
timque unum aut alterum villum educi, locumque fieri vapo
ribus, qui proximum villum corrumpentes faciliori &longs;ci&longs;&longs;ioni pa
rant, atque adeò, &longs;erpente lue, demùm non tot integri &longs;uper
&longs;unt villi, qui po&longs;&longs;int ponderis gravitati ob&longs;i&longs;tere, quin dif
fringantur. Ex quo &longs;atis apparet &longs;u&longs;pen&longs;um pondus, licèt non
omninò de&longs;cendat, impetum tamen concipere, quo retinenti
repugnat, & vim aliquam vici&longs;&longs;im infert.
Nec ab&longs;imili ratione in reliquis vim patientibus contingere
ob&longs;ervabimus, ea &longs;cilicet moliri illicò naturalis &longs;tatûs repara
tionem, aliquidque efficere, licèt tenui&longs;&longs;imum, quod demum
appareat, ubi temporis morâ augmentum ceperit. Sic ha&longs;tam
per vim inflexam &longs;i continuò dimittas, illa &longs;e&longs;e re&longs;tituit, facul
tate ela&longs;ticâ; at &longs;i dies aliquot, aut etiam diutiù per vim &longs;i
nuata perman&longs;erit, &longs;ibi dimi&longs;&longs;a antiquam rectitudinem non re
parat; elanguit nimirùm facultas ela&longs;tica, quæ ex violentâ par
ticularum compre&longs;&longs;ione aut di&longs;tractione oriebatur. Cùm enim
primùm ha&longs;ta flectitur, particulæ concavam curvaturæ partem
re&longs;picientes comprimuntur, contra verò, quæ convexam re&longs;pi-
&longs;unt, dum vim illicò prorsùs excutere conantur, con&longs;pirant, ut
pri&longs;tinam ha&longs;tæ rectitudinem moliantur: Quod &longs;i id non li
cuerit, hæ quidem aliam ex angu&longs;tiis evadendi, quâ facilior
patet via, rationem tentant, ita ut demùm &longs;ubtili&longs;&longs;imas in ru
gas cri&longs;pentur, illæ verò &longs;e&longs;e ad angu&longs;tiora &longs;patia &longs;en&longs;im reci
pientes mutuum nexum &longs;olvunt, tenui&longs;&longs;imo&longs;que poros relin
quunt, aut &longs;i qui priùs interjecti fuerint, ampliùs hiare per
mittunt. Id quod ubi jam contigerit, fru&longs;trà &longs;ubmoves, quæ
admoveras impedimenta; & &longs;pontè curvaturam ha&longs;ta &longs;ervat,
ni&longs;i fortè particulis omnibus adhuc per tempus non licuerit
vim totam excutere; tunc enim &longs;e &longs;e languidiùs re&longs;tituunt, pro
ratione reliquæ violentiæ. Hinc patet arcum, quò fuerit con
tentus atque adductus vehementiùs, remitti aliquando, & ma
nualium tormentorum rotas interdum laxari oportere, ne vis
ela&longs;tica languidior facta minùs utilis fiat.
Ex his igitur paulò enucleatiùs explicatis, in quibus longio
re temporis fluxu motum aliquem tardi&longs;&longs;imum contigi&longs;&longs;e, at
que adeò etiam impetum jam tum ab initio &longs;tatim fui&longs;&longs;e pro
ductum con&longs;tat, conjecturam in reliquis capio, & ab iis impe
tum concipi &longs;tatuo, quæ aut loco naturali dimota, aut incon
gruam partium po&longs;itionem nacta id repetunt, quod natura exi
git. Motus autem non pro impetûs tantum, &longs;ed & pro re
&longs;i&longs;tentiæ modo con&longs;equitur.
UT impetûs natura, quam inquirimus, explicatiùs atque
di&longs;tinctiùs innote&longs;cat, ex quo pariter, quæ corpora, quâ
ve ratione, impetum re&longs;puant, intelligamus, hîc nobis e&longs;t
ve&longs;tigandum, quâ ratione conceptum &longs;emel impetum abji
ciant: hinc nimirum in uberiorem ip&longs;ius re&longs;i&longs;tentiæ notitiam
venientes ad explicandam motûs machinalis cau&longs;am propiùs
accedemus.
Et &longs;anè conceptum impetum, naturâ &longs;uâ, nec flabilem &longs;em
per permanere, nec ad unicum temporis punctum durare, &longs;a
tis con&longs;tat: &longs;ivè enim &longs;pontè profluat ex naturâ debitum &longs;ibi
locum quærente, &longs;ivè alienâ vi impre&longs;&longs;us &longs;uo loco corpus ex
trudat, perpetuus e&longs;&longs;e nequit; omnis &longs;cilicet motus terminum
habeat nece&longs;&longs;e e&longs;t; nam &longs;i violentus quidem e&longs;t, perennis uti
que non e&longs;t; &longs;in autem naturalis, quem violentus præce&longs;&longs;erit,
certis definitur terminis; à loco enim, in quo quietem natura
indixit, corpus infinito intervallo non abe&longs;t, ac proinde ubi
eum attigerit, demùm conquie&longs;cet, nec impetu perpetuo opus
erit, cùm motum ce&longs;&longs;are oporteat. Sed neque temporis mo
mento circum&longs;cribi impetum &longs;ivè in naturali motu acqui&longs;itum,
&longs;ive in violento impre&longs;&longs;um, plura &longs;unt, quæ palam faciunt: ut
enim reliqua &longs;ileam nullæ e&longs;&longs;ent funependulorum o&longs;cillatio
nes, nullus emi&longs;&longs;æ &longs;agittæ motus, &longs;i conceptus impetus illicò
periret.
In duo autem veluti genera tribuendus e&longs;t Impetus ex natu
râ dimanans; alius Innatus, &longs;eu qua&longs;i in&longs;itus, alius Acqui&longs;itus
dicitur, Innatum, &longs;eu qua&longs;i in&longs;itum, voco, non quem corpus
jugiter obtineat, &longs;ive &longs;uo in loco, &longs;ive in alieno quie&longs;cat; &longs;ed
eum, qui facultati &longs;e movendi præcisè re&longs;pondet, nullo facto
per continuam adjectionem incremento: quandiù enim corpus
ita &longs;imili &longs;ecundùm gravitatem corpore circumfunditur, ut na
turali in loco con&longs;i&longs;tere dicendum &longs;it, quare conetur motum:
conatum autem hîc ab impetu non di&longs;tinguo: &longs;atis igitur citrà
quemlibet impetum &longs;uo &longs;e tutatur in loco per hoc, quod eá fa
cultate &longs;it præditum, quæ in contrariam partem conniti valeat
illicò, ac vis inferri cæperit. Hinc nullum aquæ impetum tri
buo intrà aquam con&longs;i&longs;tenti; &longs;ed tunc &longs;olùm cùm &longs;itula plena
è lacu extrahitur, ea aquæ pars impetum habet, quæ &longs;uprà &longs;ub
jectam lacûs &longs;uperficiem aere circumfuia motum expetit, quo
&longs;uum repetat locum repugnans &longs;u&longs;tinenti. Impetum hunc, qui
naturali &longs;e movendi facultati re&longs;pondet, & e&longs;t ip&longs;a gravitatio,
&longs;eu naturalis ad de&longs;cen&longs;um propen&longs;io, Innatum voco, & is e&longs;t,
cui extrin&longs;eca cau&longs;a repugnat motum impediens. Quòd &longs;i &longs;u&longs;
pen&longs;um corpus &longs;ibi relinquatur, ita &longs;uum in locum contendit,
ut vis naturalis æquè &longs;emper ad agendum applicata, nec impe
dita, momentis &longs;ingulis novum impetum acquirat, qui propterea
ficit: &longs;apienti &longs;anè naturæ in&longs;tituto; nam &longs;i corpora per &longs;e ip&longs;a
ac &longs;uâ &longs;ponte mota non accelerarent; &longs;ed naturalis motus pla
ne æquabilis e&longs;&longs;et, tardè nimis locum &longs;uum con&longs;equerentur;
atque adeò augendus continuò fuit impetus, ut & motus in
crementum acciperet: at &longs;i innatus impetus valdè
corpora nonni&longs;i ægerrimè aliò transferri, aut alieno in loco re
tineri pro animalium, & hominis utilitate po&longs;&longs;ent; finge &longs;cili
cet animo tibiam tanto impetu innato repugnare, ne attollatur,
quanto impetu in aëre ex 200 pa&longs;&longs;uum altitudine de&longs;cenderet;
quanto id tibi e&longs;&longs;et incommodo? Quare peropportunum acci
dit, ut vehemens non e&longs;&longs;et &longs;ingularum particularum impetus
innatus, qui tamen ubi motum efficeret, novâ acce&longs;&longs;ione po&longs;
&longs;et augeri.
Quod ad impetum Innatum &longs;pectat, quem à gravitatione
ipsá & proxima motus exigentia non &longs;ejungo, utique fru&longs;trà
e&longs;&longs;et, &longs;i omni pror&longs;us effectu careret; impetus autem motum
aut efficit, aut &longs;altem exigit: propterea illum &longs;tatim perire au
tumo, ac fuerit corpus in loco &longs;uo: Id quod hoc deprehendes
experimento. Scrobem defo&longs;sâ humo altè excavato; &longs;itulam
aquæ pienam, & noti ponderis, intrà illam &longs;u&longs;pendito; tùm
aquam in &longs;crobem tantâ copiâ derivato; ut &longs;itulan u&longs;quequa
que circumplectatur: illicò evane&longs;cet totius aquæ priùs in &longs;itu
lâ gravitantis pondus, quin & &longs;itula ip&longs;a pro gravitatum &longs;ecun
dùm &longs;peciem di&longs;&longs;imilitudine levior apparebit, ut ex Hydro&longs;ta
ticis con&longs;tat. Periit ergo innatus impetus, quo aqua &longs;itulam
replens de&longs;cen&longs;um moliebatur.
At impetum Acqui&longs;itum non continuò perire, ac eò ventum
fuerit, ubi quie&longs;cendum e&longs;&longs;et, hinc &longs;altem di&longs;ces, quod
ligneum globum aquæ cæteroqui innataturum &longs;i in &longs;ublime at
tollas, & ex illâ altitudine cadere permittas, infrà aquæ &longs;uper
ficiem de&longs;cendere, ac penitùs immergi videbis; quamquam
po&longs;tea emergat, & ubi aliquoties &longs;ub&longs;ultaverit, demùm pro
gravitatum aquæ, & ligni di&longs;paritate emer&longs;us quie&longs;cat. Quæ
&longs;anè immer&longs;io, ni&longs;i Acqui&longs;itus impetus adhuc duraret, omninò
non contingeret. Verùm nihil rem per &longs;e &longs;atis ab&longs;tru&longs;am æquè
in lucem evocat, ac funependulorum motus; plumbum enim
ex filo &longs;u&longs;pen&longs;um, & à perpendiculo dimotum, ita de&longs;cendens
quidem) minori tempore a&longs;cendens de&longs;cribat. Cui autem, re
pugnante plumbi gravitate à naturá in&longs;itâ, tribuatur a&longs;cen&longs;us,
ni&longs;i impetui acqui&longs;ito dum de&longs;cenderet, adhuc po&longs;t de&longs;cen&longs;um
duranti? Quemadmodum verò in de&longs;cen&longs;u po&longs;teriores motûs
partes prioribus velociores &longs;unt, factâ nimirum novi impetûs
acce&longs;&longs;ione, ita ex oppo&longs;ito a&longs;cen&longs;us ex celeritate in tarditatem
de&longs;init, factâ acqui&longs;iti impetûs dece&longs;&longs;ione continuâ, donec ita
elanguerit, ut gravitas ip&longs;a &longs;uperet, & iterum de&longs;cendens al
ternas vibrationes efficiat. Perit igitur Acqui&longs;itus impetus non
totus &longs;imul; &longs;ed &longs;en&longs;im extenuatur; idque non aliâ ratione,
quàm quâ proportione impeditur motus, quocumque tandem
ex capite impedimenta oriantur. Cum enim impetus contra
rium impetum non habeat, &longs;i præci&longs;a quidem impetûs natura
&longs;pectetur (quippe qui unus & idem contrariorum motuum ori
go e&longs;t, ut ex funependulis ultrò citróque &longs;ponte vibratis & ex
pilâ lu&longs;oriâ deor&longs;um cadente, ac vi concepti impetûs &longs;ur&longs;um
re&longs;iliente, con&longs;tat) reliquum e&longs;t, ut pereat pro ratione eorum,
quæ aut motui corporis ob&longs;i&longs;tunt, aut illud aliò quoquomodo
dirigunt.
Præ&longs;tat autem hîc funependuli
motum paulò attentiùs con&longs;iderare.
Sit plumbeus globulus B filo AB
connexus clavo in A. Si globulo li
ceret, quâ impetus innatus urget viâ,
de&longs;cendere, utique rectam BC per
curreret; &longs;ed funiculo retinente co
gitur arcum BK de&longs;cribere, adeò ut
&longs;emper in alio & alio plano inclinato
con&longs;titutus, alia, & alia habeat gra
vitatis momenta, ut lib.
1. cap.
15 explicatum e&longs;t; hæc autem
&longs;unt pro Ratione Sinuum angulorum declinationis à perpendi
culo AK. Quare totum momentum, quod in B e&longs;&longs;et ut AB,
&longs;ingulis momentis in de&longs;cen&longs;u libero per rectam BC paribus
&longs;altem incrementis augeretur (Quicquid &longs;it an etiam pro Ra
tione duplicatâ temporum, de quo alias di&longs;putabimus) &longs;ed
cum à rectitudine deflectat, cum venerit in D, non additur
momentum ut EF, &longs;ed ut ED; &longs;imiliter in G momentum non Augetur igitur impetus in de&longs;cen&longs;u
BK non omninò pro Ratione
durat, &longs;ed pro Ratione momentorum gravitatis, quæ &longs;ubinde
obtinet minora & minora; pars
gravitate producti deteritur in intendendo filo, quo retinetur.
Quapropter ubi in K venerit per arcum BK, non tantum ha
bet impetûs, quantum &longs;i per lineam perpendicularem arcui
BK æqualem de&longs;cendi&longs;&longs;et; in motu enim ad perpendiculum
cum nihil retineat aut impediat, totus impetus ad de&longs;cen&longs;um
urget velociùs, quàm ubi repugnat aliquid. Ex quo fit quod,
cùm arcus BK ad Radium AB, hoc e&longs;t ad BC æqualem
proximè ut 11 ad 7, ex Cyclometricis, multò plus temporis in
percurrendo arcu BK, quàm in rectâ BC, in&longs;umitur; tardiùs
&longs;cilicet movetur quàm in perpendiculari, quæ ad BC e&longs;&longs;et ut
11 ad 7. manente itaque, quamdiu corpus naturâ urgente mo-
vetur, impetu acqui&longs;ito, qui re&longs;i&longs;tentiam excedit, in fine
de&longs;censûs in K totus impetus e&longs;t ut aggregatum omnium Si
nuum Quadrantis: at in perpendiculari BC in fine de&longs;censùs
in C e&longs;&longs;et ut aggregatum omnium parallelarum ip&longs;i AB in
Quadrato AC; ac propterea (in re Phy&longs;icâ &longs;i liceat cum geo
metrizantibus per Indivi&longs;ibilia ratiocinari) erit impetus per ar
cum BK acqui&longs;itus ad impetum per rectam BC acqui&longs;tum ut
Quadrans ABK ad Quadratum AC, hoc e&longs;t ut 11 ad 14, ex
iis quæ in Cyclometriâ demon&longs;trantur.
Quoniam verò ubi ad perpendiculum AK globulus de&longs;cen
dens venerit, nihil objicitur, quod motum pror&longs;ùs impediar,
quin ad ea&longs;dem partes pergat ferri ex præconcepti impetûs di
rectione, non &longs;i&longs;tit in perpendiculo; &longs;ed ulteriùs pergens a&longs;cen
dit, nec ni&longs;i per arcum circà centrum A, funiculo &longs;cilicet reti
nente. Sed jam repugnat a&longs;cen&longs;ui gravitas plumbi, non qui
dem quantum in perpendiculo KA, verùm pro ratione Sinuum
angulorum declinationis; qui cum &longs;emper a&longs;cendendo cre&longs;
cant, major e&longs;t etiam momentorum gravitatis Ratio nitentium
contrà impetum de&longs;cendendo acqui&longs;itum. Quare tantum abe&longs;t,
ut novus &longs;ingulis temporis punctis impetus &longs;ur&longs;um directus pro
ducatur, ut potius ex eo tantumdem dematur, quanta e&longs;t
a&longs;cendentis plumbi repugnantia. Hinc e&longs;t a&longs;cen&longs;um initio ve
lociorem e&longs;&longs;e, quia adhuc multus e&longs;t impetus acqui&longs;itus, & pro
atque adeò motus efficitur celerior: quia verò diminuto &longs;en&longs;im
impetu, & auctis
nationis
Ratio, tardior &longs;equitur motus, & plus acqui&longs;iti impetûs perit, do
nec demùm pror&longs;us evanuerit, & &longs;uperante gravitate globulus
iterum de&longs;cendat. Quamvis autem &longs;i po&longs;itio &longs;ola &longs;pectetur, ii&longs;
dem Reciproce gradibus minui videatur impetus, quibus fuit
auctus, totidemque momentis temporis, ita ut quantum po&longs;tre
mo temporis puncto acce&longs;&longs;it, tantumdem primo decedat, adhuc
tamen aliqua e&longs;t ob&longs;i&longs;tentiæ appendicula ex aëre dividendo, ac
propterea paulo ampliùs extenuatur impetus acqui&longs;itus, quàm
pro Ratione incrementi Sinuum declinationis: quò autem ve
locior e&longs;t motus, magis etiam aër dividendus comprimitur,
den&longs;atú&longs;que plus ob&longs;i&longs;tit quàm rarus; quòd &longs;i medium non fue
rit compre&longs;&longs;ionis capax, &longs;altem æquali tempore plures medij
partes &longs;cinduntur, quàm in motu tardiori, ac propterea etiam
multiplex e&longs;t medij re&longs;i&longs;tentia: Ex quo fit arcum a&longs;censûs pau
lò minorem &longs;emper e&longs;&longs;e arcu de&longs;censûs, &, cum vici&longs;&longs;im glo
bus remaneat ex humiliore loco ac priùs de&longs;cendens, brevio
rem pariter &longs;ecundi a&longs;censûs arcum perfici, atque ita deinceps,
ut &longs;ervatâ eâ in motu &longs;emper minori reciprocando con&longs;tantiâ
demum quie&longs;cat in perpendiculo.
At, inquis, dura magis ob&longs;i&longs;tunt corpori, ejú&longs;que motum
validiùs impediunt, quàm mollia, quæ dum &longs;e comprimi pa
tiuntur, & loco pauli&longs;per cedunt, motui aliquantulùm & ex
parte ob&longs;ecundant: &longs;i igitur pro Ratione impedimenti debili
tatur acqui&longs;itus impetus, minus detrahitur impetûs corpori,
quod ex alto decidens à &longs;ub&longs;tratis paleis excipitur, quàm &longs;i ad
&longs;axum allideretur; vehementiùs igitur à luto quàm à &longs;àxo re
flecteretur, contrà quàm docet experientia.
Fateor eburneum globum &longs;egniùs re&longs;ilire delap&longs;um in gle
bam humore perfu&longs;am, quàm in marmor; non tamen his con
&longs;equens e&longs;t, ut impetûs acqui&longs;iti diminutioni alius &longs;tatuendus
&longs;it modus, quàm ex impedimento: ubi enim globus cadens ex
timam &longs;ubjecti corporis &longs;uperficiem attigerit, non quie&longs;cit, &longs;ed
pergit moveri, aut deor&longs;um comprimendo corpus molle, aut
illicò &longs;ursùm reflexum à duro. Ita autem à corpore molli ex-
tum; ac proinde quò magis cedit &longs;ubjectum corpus, eò diutiùs
movetur globus cum ip&longs;o, vel intrà ip&longs;um; atque interea plus
impetûs perit: quid igitur mirum, &longs;i languidiùs po&longs;tea re&longs;iliat,
cum exigua impetûs portio reliqua &longs;it? Quòd &longs;i
jectum corpus, impetu nondum debilitato reflecteretur vali
diùs. Hinc fieri pote&longs;t adeò molle e&longs;&longs;e &longs;ubjectum corpus, ut
dum illud penetrat decidens globus, tantum impetûs deper
dat, ut, quod reliquum fit, non &longs;atis &longs;it ad vincendam in&longs;itam
globo gravitatem, qui propterea neque re&longs;ilire valeat. Quam
vis itaque corpus molle minùs ob&longs;i&longs;tat quàm durum, diutiùs
tamen re&longs;i&longs;tit; & per aliquot momenta aliquoties diminutus
impetus minore men&longs;urâ, eò decrementi venire pote&longs;t, ut ma
gis imminutus demum fuerit, quàm &longs;i unico momento magis
ob&longs;titi&longs;&longs;et corpus durum. Cæterùm paribus momentis plus pe
rit impetûs ex alli&longs;ione ad corpus durum, quàm ad molle, quip
pe quod magis opponitur motui. Porrò huic rei explicandæ
&longs;imilitudo aliqua peti po&longs;&longs;et ex luce, cui &longs;anè &longs;i contingat per
medium diaphanum quidem, &longs;ed den&longs;um, pergere, languidiùs
multò reflectitur à &longs;peculo, in quod incurrit, &longs;i den&longs;ioris me
dij longior fuerit tractus, quàm &longs;i brevior, perinde atque eò
minùs reflectitur corpus, quò molliori magi&longs;que &longs;ub&longs;identi cor
pori occurrit. &longs;ed quoniam quæ de luce dicenda e&longs;&longs;ent, fortè ob
&longs;curiora acciderent, ab huju&longs;modi &longs;imilitudine
Sed ex illud e&longs;t in durorum corporum colli&longs;ione ob&longs;ervan
dum, quod aliqua particularum compre&longs;&longs;io aliquando contin
git &longs;ivè in alterutro, &longs;ivè in utróque, quæ &longs;e facultate ela&longs;ticâ
re&longs;tituentes motum reflexum juvant: id autem manife&longs;to ex
perimento con&longs;tat in pilâ ex gummi, ut vocant, Indico, quæ
ad terram eli&longs;a frequenti&longs;&longs;imè &longs;ub&longs;ultat; at ubi in corpus molle
incidit, neque hujus neque illius partes violentam compre&longs;&longs;io
nem &longs;ubeunt, quam &longs;e&longs;e re&longs;tituentes excutere debeant. Sic &
pilá in &longs;phæri&longs;terio ludentes &longs;atis nôrunt eam validiùs reflecti
objecto recticulo, quàm ligneo batillo; intenti &longs;cilicet nervi ex
contortis &longs;iccati&longs;que animalium inte&longs;tinis reticulum con&longs;tituen
tes cùm pilæ ictum excipiunt, flectuntur quidem aliquantu
lum; &longs;ed illicò &longs;ibi pri&longs;tinam rectitudinem reparantes pilam ex
cutiunt (id quod ligneo ba&longs;tillo non contingit) novoque hoc
majorem: quòd &longs;i in reticulo flaccidi, & remi&longs;&longs;i &longs;int nervi, lan
guidè pila reflectitur.
Ad quandam autem reflexionis &longs;peciem pertinere cen&longs;enda
e&longs;t concu&longs;&longs;io, &longs;ive vibratio, aliquarum &longs;altem corporis partium,
ubi totum ex reliquo impetu re&longs;ilire nequit: &longs;ic corpus ita at
tollens, ut &longs;ummis pedibus innitaris, po&longs;tmodum recidens in
talos, eò validiorem partium concu&longs;&longs;ionem percipies, quò ve
lociùs recides. Simile quid etiam in inanimis contingere ratio
&longs;uadet, neque enim ita &longs;emper &longs;olida aut pror&longs;us homogenea
tota moles e&longs;t, ut nullæ omninò partes concuti valeant: quin
etiam alli&longs;i corporis partes, &longs;i non adeò tenaci vinculo inter &longs;e
cohæreant, ex reliquo impetu aliæ aliò di&longs;tractæ de&longs;iliunt.
Hinc, docente naturâ, ex alto de&longs;ilientes ubi terram pedi
bus attigerint, genua antror&longs;um inflectunt, qua&longs;i calcaneis in
&longs;e&longs;&longs;uri, ne conceptus ex &longs;altu impetus &longs;uperiorem corporis par
tem deor&longs;um validiùs urgens &longs;ubjectas tibias, & genua ita pre
mat, ut inde divi&longs;io aliqua membrorum, aut o&longs;&longs;ium luxatio, aut
nervorum &longs;eu tendinum nimia di&longs;ten&longs;io dolorem gignat: hoc
autem valet illa genuum inflexio ad extenuandum impetum,
quod & flexili mollitiâ &longs;ub&longs;idens terra uligino&longs;a, &longs;i quando la
pis in eam ex alto deciderit. Sic Atlas Sinicus pag.
123. in XI.
Provinciâ Fokion, ubi &longs;ermo e&longs;t de flumine Min, quod vio
lento cur&longs;u per &longs;axa volvitur, ait naves, quibus ibi navigatur,
ex diverbio vocari
&longs;i&longs;tentibus con&longs;tent a&longs;&longs;eribus, imò ne clavis quidem compaginatis;
&longs;ed vimine quodam lenti&longs;&longs;imo; unde tamet&longs;i in &longs;axa impingat na
vis, &longs;apè tamen minimè rumpitur, quia vix re&longs;i&longs;tit. Et pag.127.
de catadupis aquarum in flumine per quod ad Jenping naviga
tur loquens ait.
f actionis incurrant periculum, &longs;citè præmittunt nautæ aliquot &longs;tra
minis fa&longs;ces, ad quos navis leviùs impingat, ac tran&longs;eat.
Jam verò ad impetum extrin&longs;ecùs impre&longs;&longs;um mentem ocu
lo&longs;que intendente, non illum &longs;emper momento perire animad
vertimus, aut illicò, ac externus agitator ce&longs;&longs;at.
Unde enim tit, ut concitato navigio, cùm vela nautæ con
traxerunt, aut remiges inhibuerunt, retineat tamen ip&longs;a navis
motum & cur&longs;um &longs;uum, intermi&longs;&longs;o ventorum incur&longs;u, pulsúve ni&longs;i quia navis, etiam nullo impellente, vi impre&longs;sâ
urgetur. Quid rhedam cur&longs;u procedente faciliùs quàm initiò
promovet, equis licet languidius connitentibus? curve onus
aliquod ingens protrudentes, aut trahentes hoc maximè ca
vent, ne contentionem illam quies interrumpat, experientiâ
&longs;atis edocti incitatum &longs;emel minori labore propelli, quàm com
moveri quie&longs;cens? ni&longs;i quia reliquus ex priore motu impetus
adhuc per&longs;everans po&longs;teriorem motum juvat. Hoc tamen tria
hæc differunt, quòd onus, ce&longs;&longs;antibus iis, qui protrudebant,
con&longs;i&longs;tit illicò (ni&longs;i fortè volubilitatem habens, aut &longs;ubjectis
cylindris innixum, adhuc modicum quid volvi aut progredi
pergat) rheda currentes equo, &longs;ubita funium abruptione dis
junctos &longs;equitur ad pa&longs;&longs;us aliquot non adeò multos pro viæ
æquabilitate præcedenti&longs;que velocitatis ratione; navigium verò
&longs;ubmi&longs;&longs;is antennis, remi&longs;que ce&longs;&longs;atione torpentibus aliquandiu,
intervallo non &longs;anè contemnendo, provehitur. Oneris &longs;cilicet
motui, cui volubilitatem neque ars, neque natura dederit, im
pedimento e&longs;t ip&longs;a extremitas a&longs;pera &longs;ubjectam planitiem &longs;ale
bris quandóque non carentem contingens, gravita&longs;que ita va
lidè premens, ut major futurus e&longs;&longs;et partium tritus, quàm pro
impetûs modo, qui reliquus e&longs;&longs;et, &longs;uperari po&longs;&longs;et: Id quod cur
renti rhedæ idcircò non contingere planum e&longs;t, quia licèt
nihilo levior &longs;it quàm onus protru&longs;um, minùs tamen rotarum
modioli leniter cum axibus confligentes motum retardant. At
navis &longs;ponte &longs;uâ innatans, ventorum incur&longs;ione, remorúmve
pul&longs;u diutiùs acta, vix, aut fortè ne vix quidem, mole &longs;uâ re
luctatur, ni&longs;i quatenus diffindenda e&longs;t aqua; nec &longs;inè multo fa
cilitatis compendio, prior &longs;iquidem unda, quam prora impel
lens excitat, aliam ante &longs;e urget ad ea&longs;dem partes: propterea
impre&longs;&longs;us navi impetus modicum nactus impedimentum diù
durat, illámque promovet. Quare idem de impetu extrin&longs;ecùs
a&longs;&longs;umpto dicendum e&longs;t, quod de acqui&longs;ito; nimirùm minui pro
Ratione eorum, quæ in&longs;tituto motui ob&longs;i&longs;tunt, aut etiam pror
sùs perire.
Præter ea autem quæ utrique motui tùm naturali, tùm vio
lento æquè opponuntur, (cuju&longs;modi e&longs;t medium dividendum,
objecti corporis occur&longs;us, aut contingentis tritus atque con
flictus, retinaculum, quod certo limite motum definiat, & alia
quæ ex inhærente corpori gravitate oritur, &longs;ive illi innatus im
petus, &longs;ive acqui&longs;itus modum &longs;tatuat. Neque id &longs;impliciter
tantùm, &longs;ed comparatè con&longs;iderandum e&longs;t, quam &longs;cilicet in
plagam impul&longs;us motum dirigat, & quatenus gravitatis pro
pen&longs;ioni opponatur. Quemadmodum enim qui in pilâ aroma
ta pin&longs;unt, nihil repugnantem, quin & impul&longs;ui ob&longs;ecundan
tem, experiuntur pi&longs;tilli gravitatem deprimentes; contrà verò
attollentes fatigat eadem gravitas directò deor&longs;um urgens; me
dium autem quiddam tenet in ob&longs;i&longs;tendo, &longs;i motio tran&longs;ver&longs;a
contingat; &longs;icut experiri licet, &longs;i ex funiculo pendens idem
pi&longs;tillus à perpendiculo dimoveatur; minore enim conatu opus
e&longs;t: ita quò minùs in oppo&longs;itam gravitati plagam dirigitur im
pul&longs;us, eò etiam diutiùs per&longs;everat minus habens impedimenti.
Hinc e&longs;t quod gravitas æquabiliter toto corpore fu&longs;a &longs;i aut ex
centro &longs;u&longs;pendatur, aut coni apici in&longs;i&longs;tat, levi negotio, ac &longs;a
tis diù, in gyrum convertitur; innatum videlicet gravitatis im
petum vis ip&longs;a &longs;u&longs;pendens aut &longs;u&longs;tentans elidit; nihil verò im
pul&longs;um remoratur præter aut funiculi &longs;u&longs;pendentis &longs;piras paulò
&longs;pi&longs;&longs;iores, aut tritum cum &longs;ubjecto cono, aëri&longs;que dividendi
re&longs;i&longs;tentiam; quæ tamen &longs;i tollatur in corpore orbiculari circà
centrum commoto, etiam longior fit conver&longs;io. Sic ferream
&longs;agittam palmarem cra&longs;&longs;iu&longs;culam in&longs;tar acûs magneticæ in
æquilibrio con&longs;titutam levi&longs;&longs;imo impul&longs;u ac diuti&longs;&longs;imè in gy
rum agi ob&longs;ervavi; vix enim acuti&longs;&longs;imum verticem, cui innite
batur, terebat, & aëris intrà eumdem gyrum circumducti mo
dica erat re&longs;i&longs;tentia. Id autem multo luculentiùs apparet in
verticillo, cujus axem perpolito alveolo in&longs;i&longs;tentem extremo
pollice ac indice leviter comprimens, ac paulò celeriùs vertens,
eò diuturniori vertigine contorqueri videbis, quò pauciores
minore&longs;que offenderit in &longs;ubjectâ tabulâ a&longs;peritates, ad quas al
li&longs;us paululùm inclinetur, aut aliò reflectatur.
Quòd &longs;i magnetis polo ritè armato chalybeum axiculum
congruo verticulo in&longs;tructum admoveris, ut planè à magnete
&longs;u&longs;pendatur, tùm &longs;ummis digitis opportunè axem terentibus
vertiginem ei delicatè ac molliter conciliaveris, miraculi loco
tibi erit tàm diuturna conver&longs;io; quippe cui non &longs;ubjectialveoli
a&longs;peritates &longs;altitare cogentes, non gravitas ip&longs;a premens, tritum-
&longs;i&longs;tunt, motúmve aliquatenus impedientes impre&longs;&longs;um impe
tam imminuunt; &longs;ed magnetico radio &longs;u&longs;pen&longs;us intra &longs;e perpe
tuò volvitur lævi&longs;&longs;imum chalybem magnetis polo adhærentem
leni&longs;&longs;imè terens.
Illud etiam in motu, qui ab extrin&longs;eco provenit, con&longs;ide
randum e&longs;t, quòd contingere pote&longs;t duos ade&longs;&longs;e motores, qui
corporis motum in diver&longs;as partes dirigant: quare alter alteri
ob&longs;i&longs;tit, & motus ex duplici directione compo&longs;itus is e&longs;t, qui
non re&longs;pondeat men&longs;uræ duplicis illius impetûs, &longs;i &longs;inguli in
tegrè accipiantur. Con&longs;tat enim, &longs;i æquabili & æquali cona
tu urgeant corpus, moveri aut per diametrum Quadrati, &longs;i di
rectiones &longs;int ad angulum rectum con&longs;titutæ; aut per Diago
nalem lineam Rhombi, &longs;i directiones obliquæ &longs;int: &longs;i verò
æquabiles quidem &longs;int, &longs;ed inæquales conatus, per diametrum
Rectanguli aut Rhomboidis moveri, pro ut ad rectum aut obli
quum angulum directiones &longs;ibi invicem re&longs;pondent. Semper
autem minor e&longs;t motus quàm pro duorum illorum impul&longs;uum
ratione; diameter &longs;iquidem brevior e&longs;t aggregato duorum
adjacentium laterum. Quòd &longs;i æquabiles non &longs;int impetus,
vel &longs;altem alter æquabilis &longs;it, alter acceleratus aut retardatus,
linea curva de&longs;cribitur; quæ pariter minor e&longs;t duabus rectis,
quæ vi &longs;ingulorum impetuum de&longs;criberentur; ab illis &longs;i qui
dem continetur.
Hîc tamen advertendus animus e&longs;t, & ob&longs;ervare oportet
æquabilem impul&longs;um (&longs;i continuus &longs;it, nec morulis inter
ruptus) e&longs;&longs;e non po&longs;&longs;e, ni&longs;i ab animali &longs;emper æqualiter conan
te efficiatur; quia gravium de&longs;cen&longs;us naturaliter acceleratur;
ela&longs;mata verò dum &longs;e re&longs;tituunt, &longs;emper languidiùs &longs;ingulis
momentis conantur, &longs;i quidem virtus ela&longs;tica con&longs;ideretur:
quamquàm po&longs;teriore momento quod e&longs;t reliquum prioris im
petûs, inten&longs;ionem efficit additum po&longs;teriori licèt remi&longs;&longs;o.
Vix igitur contingere pote&longs;t motum unum à duplici impetu
extrin&longs;ecùs impre&longs;&longs;o fieri per lineam rectam ni&longs;i corpus à du
plici motore æquabiliter urgeatur.
Cum itaque impetus acqui&longs;itus, aut aliundè impre&longs;&longs;us, &longs;it
qualitas propter motum in&longs;tituta, quæ non ni&longs;i in motu pro
ducitur, ita pariter ni&longs;i in motu, & cum motu non con&longs;erva-Quare &longs;i corpus eò deveniat, ut nullo pror&longs;us pacto agi
tari queat, aut interiore motu cieri, quo momento impeditur
motus, ne &longs;it, eo momento impetus perit, ce&longs;&longs;ante videlicet
causa effectiva ab ejus con&longs;ervatione eo ip&longs;o quod ce&longs;&longs;at finis,
propter quem impetus e&longs;t. Quod &longs;i impedimentum occurrat
non prorsùs motum tollens (ut &longs;i globus in plano horizontali
rotatus veniat ad planum inclinatum, per quod ex concepto
impetu a&longs;cendat) tunc pro ratione impedimenti extenuatur
impetus, donec tandem pereat.
comparetur.
MOtus omnis nec in oppo&longs;itas, nec in diver&longs;as plagas, &longs;ed
per certam lineam dirigitur; unico quippe in loco, non
in pluribus, eodem temporis puncto e&longs;&longs;e pote&longs;t corpus.
Nihil igitur motui moram & impedimentum inferre pote&longs;t,
ni&longs;i directò aut obliquè illi &longs;ecundùm eam lineam, per quam
in&longs;tituendus e&longs;&longs;et, antè, ponè, ad dextram, ad lævam, &longs;ur&longs;um,
deor&longs;um opponatur. Si enim duo corpora eádem pergerent viâ,
& maximâ velocitatis, aut tarditatis con&longs;piratione con&longs;entirent,
tunc neque po&longs;terius ab eo quod antè e&longs;t, traheretur, neque
prius à po&longs;teriore urgeretur, neque alterum alteri impedimen
to e&longs;&longs;et. Hinc manife&longs;tum e&longs;t non po&longs;&longs;e impedimentum &longs;upe
rari, quin ei vis aliqua inferatur.
Rem porrò univer&longs;am duas in partes tribuere po&longs;&longs;umus, ut
duplex Re&longs;i&longs;tentiæ genus &longs;tatuatur; Formalem alteram, alte
ram Activam &longs;cholæ vocarent. Corpus enim, quod ob&longs;tat, aut
retinet, &longs;i motum prorsùs nullum conetur in&longs;tituto aut de&longs;ti
nato motui adver&longs;antem, re&longs;i&longs;tit quidem, &longs;ed Formaliter; nihil
&longs;cilicet efficit, quo repugnet, &longs;ed &longs;uo tantùm &longs;e tutatur in loco:
Sin autem & contrà nitatur, aut retrahat, jam non ob&longs;i&longs;tit &longs;o
lùm, ne loco per vim dimoveatur; &longs;ed etiam impetum in con-
proptereà Activè re&longs;i&longs;tit. Huic autem verbo, cum
cimus, &longs;ubjecta notio e&longs;t, in causá e&longs;&longs;e ne motus fiat, aut &longs;al
tem non ea velocitate, quæ virtuti movendi non impeditæ cæ
teroqui re&longs;ponderet. Sic paries, in quem incurris, tibi re&longs;i&longs;tit
Formaliter, ne procedas, & aqua &longs;tagnans, cui collo tenus im
mergeris, progredienti re&longs;i&longs;tit Formaliter, ne velociter, &longs;icut
intra aërem movearis pro ratione impetus, quo conaris progre
di: qui verò occurrens te repellit, ut &longs;i coneris contra ictum
fluvij, non Formaliter tantùm, &longs;ed etiam Activè re&longs;i&longs;tit; non
&longs;olùm enim ob&longs;tat, quia ejus in locum &longs;uccedere non potes,
ni&longs;i cum loco dimoveas, &longs;ed etiam tibi adver&longs;um impetum im
primit, ut te loco extrudat.
Cum itaque impedimenta motûs externo impetu &longs;ubmoven
da &longs;int, virtus autem movendi certa &longs;it ac definita, con&longs;tat vi
res omnes, quæ in corpore promovendo, &longs;i nihil ob&longs;taret, exer
cerentur, duas in partes di&longs;trahi, ad movendum &longs;cilicet cor
pus, & ad tollenda impedimenta, Concipit igitur impetum,
qui motum efficiat, & ob&longs;tanti corpori impetum imprimit, ut
loco cedat. Quid igitur mirum, &longs;i di&longs;tractis viribus languidior
&longs;equatur metus? Quia verò quò majori velocitate corpus
ob&longs;tans propellendum e&longs;t, aut trahendum, majori quoque im
petu impre&longs;&longs;o opus habet, palàm e&longs;t majorem quoque in pro
pellente, aut &longs;ecum rapiente, impetum requiri, ut majorem re
&longs;i&longs;tentiam vincens &longs;e ip&longs;um pariter moveat.
Hic autem quid monui&longs;&longs;e oporteat vim re&longs;i&longs;tendi &longs;uperan
dam e&longs;&longs;e à virtute movendi? quis enim ambigat, an, &longs;i pares
illæ fuerint, nullus futurus &longs;it motus? Quòd &longs;i impedimentum
prorsùs immotum adversùs conantem per&longs;tat, nullum pariter
recipit impetum; qui &longs;cilicet, etiam &longs;i priùs fui&longs;&longs;et, motu ce&longs;
&longs;ante periret. Hinc in animali defatigatio membrorum oritur,
quando prorsùs in irritum conatus cadit; impetus enim, quem
concipit, ut æqualem motum imprimeret impedimento, &longs;i hoc
&longs;uperari po&longs;&longs;et, in animali ip&longs;o motum aliquem efficit, &longs;ed quia
progredi vetatur ab o&longs;tante aut retinente impedimento, impe
tus ille non totius animalis motum ulteriùs promovet; &longs;ed mem
brorum partes alias comprimit, alias di&longs;tendit, unde & dolor
aliquis, & la&longs;&longs;itudo provenit. At &longs;i corpus, cui motus debetur,
tes, ex arbitrio temperare, & quia &longs;olidum e&longs;t ac durum, nul
lam pati compre&longs;&longs;ionem aut di&longs;tentionem partium po&longs;&longs;it, &longs;icut
& corpus ob&longs;tans aut retinens compre&longs;&longs;ionem omnem aut
di&longs;tentionem re&longs;puit; tunc nullum concipit aut imprimit im
petum præter innatam gravitationem, aut levitationem, cùm
per vim in loco non debito detineatur. Ex hoc conjecturam ca
pere licet de eo, quod contingit, quando virtute movendi re
&longs;i&longs;tentiam vincente impedimentum &longs;ubmovetur; impediri vi
delicet, ne producatur motus, juxta re&longs;i&longs;tentiæ modum atque
men&longs;uram; quæ &longs;icuti non quâlibet minimâ vi &longs;uperari pote&longs;t,
ita majori cedit.
Verùm quonam id pacto contingat, ut explicare conemur,
illud ob&longs;erva, quòd &longs;i corpus idem quadruplo velociùs moveri
debeat, ac moveretur priùs certâ impetûs men&longs;urâ, utique qua
druplo majorem impetum exigit, ut pro impetûs inten&longs;ione
aut remi&longs;&longs;ione velocior aut tardior &longs;equatur motus. At &longs;i cor
pus aliud movendum quadruplo gravius exhibeatur, in hoc im
petus ille quadruplex &longs;ubquadruplam efficiet inten&longs;ionem, ac
propterea etiam motum habebit tardiorem, &longs;i cætera &longs;int paria,
pro impetûs inten&longs;ione. Si cætera, inquam, &longs;int paria; &longs;æpè
enim aër, aut aqua plus velociori motui re&longs;i&longs;tunt, quàm tardio
ri, & moles major efficit, ut non omninò velocitas inten&longs;ioni
impetûs re&longs;pondeat. Hæc tamen nunc mente &longs;ecernamus, per
inde atque &longs;i nihil officerent motui.
Quoniam igitur motus ab omni velocitatis aut tarditatis men
&longs;urâ &longs;ejungi nequit, finge corpus per vim movendum huju&longs;
modi e&longs;&longs;e, ut &longs;pectatâ mole &longs;eu materiâ, ac &longs;pecificâ gravitate,
ad percurrendum &longs;patium pa&longs;&longs;uum 100 unius horæ quadrante,
indigeret impetu, cujus inten&longs;io e&longs;&longs;et particularum 4 in &longs;ingu
lis corporis movendi partibus: molem autem, exempli gratiâ,
di&longs;tinctam concipe in particulas 100 minimas. Quare &longs;pectatâ
tùm exten&longs;ione tùm inten&longs;ione impetûs, nece&longs;&longs;e e&longs;t illi à mo
tore imprimi impetûs particulas 400. Quòd &longs;i corporis per vim
movendi moles ac materia e&longs;&longs;et quadruplex alterius, &longs;i nimi
rum ratione materiæ exten&longs;ionis particulas haberet 400, jam
impetus idem &longs;ubquadruplam efficeret inten&longs;ionem, & &longs;ingulæ
impetûs particulæ &longs;ingulis corporis particulis ine&longs;&longs;ent; atque
tis: partamen utrobique e&longs;&longs;et, illud quidem velociùs, hoc tar
diùs movendi difficultas, cum in utroque particulas 400 impe
tûs produci oporteret; utriu&longs;que enim impetûs exten&longs;iones &
inten&longs;iones e&longs;&longs;ent Reciprocè in eadem Ratione. In corpore
itaque, ex quo motus originem ducit, tanta vis movendi ine&longs;&longs;e
debet, ut & corpori impedienti, quod &longs;ubmovetur, congruen
tem motui impetum imprimat, hoc e&longs;t particulas 400, & ip&longs;um
&longs;e pariter promoveat: nihil enim accepto extrin&longs;ecùs impetu
agitatur à motore prorsùs immoto, ut eunti per &longs;ingula patebit.
Jam verò quoniam idem corpus modò remi&longs;&longs;iùs, modò con
citatiùs moveri pro impetûs inten&longs;ione videmus, probabilis
conjectura e&longs;t in iis, quæ non &longs;uo arbitrio, &longs;ed naturæ reguntur
imperio, totum impetum produci, qui virtuti efficiendi re&longs;pon
det: hæc autem in impedimento, cujus re&longs;i&longs;tentia vincitur,
impetum eâ inten&longs;ionis men&longs;urâ imprimit, quæ illi motûs ve
locitatem conciliet ip&longs;ius corporis moventis velocitati con
gruentem, adeò ut movendi facultas totas &longs;uas vires exerat
partim impetum imprimens &longs;ubmovendo impedimento, partim
motum efficiens in ip&longs;o corpore: ex quo fit quod eò remi&longs;&longs;iorem
motum in &longs;e motor efficiat, quò major &longs;ecundùm inten&longs;ionem
impetus impeditur ab impedimento. Sic plumbeus globus bili
bri, &longs;i, funiculo excavatæ volubilis orbiculi curvaturæ in&longs;erto,
connectatur cum globulo &longs;ubduplæ gravitatis, non eá veloci
tate de&longs;cendit, qua de&longs;cenderet &longs;ibi relictus ab&longs;que ullâ appen
dice; velociùs tamen movetur, quàm &longs;i e&longs;&longs;et globuli adjuncti
tantùm &longs;e&longs;quialter; quia &longs;cilicet ut ad æqualem velocitatem
temperentur motus tùm impedimenti &longs;ur&longs;um, tùm corporis mo
ventis deor&longs;um, minor inten&longs;ivè impetus impediendus e&longs;t à glo
bulo &longs;ubduplo quàm à &longs;ub&longs;e&longs;quialtero; ac propterea major e&longs;t
&longs;ecundùm inten&longs;ionem reliquus impetus motum efficiens con
citatiorem.
Quòd autem à globo de&longs;cendente imprimatur impetus glo
bulo, quem &longs;ur&longs;um trahit, hinc con&longs;tat, quod &longs;i globulus ille
non &longs;it admodum gravis, tùm demum &longs;ub&longs;ilit, ubi globus ve
lociter de&longs;cendens &longs;ubjectum planum attigerit: quid enim il
lum &longs;ub&longs;ilire cogeret quie&longs;cente jam globo, à quo trahebatur,
ni&longs;i adhuc aliquid impre&longs;&longs;i impetûs remaneret? At quòd im-
ille concepit, proximè efficiatur, hinc &longs;ibi &longs;uadent plures, quia
ex alterâ parte impetum ab impetu produci po&longs;&longs;e manife&longs;tum
videtur ex percu&longs;&longs;ionibus projectorum, ut cùm globus pro
jectus in quie&longs;centem globum impactus illum trudit; ex alterâ
cau&longs;am proximam effectui homogeneam congruenter naturæ
&longs;tatuimus; &longs;ic enim & calorem in nobis à calore potiùs quàm à
&longs;ub&longs;tantiâ ignis proximè produci exi&longs;timamus. Sed quid de
percu&longs;&longs;ionum impetu dicendum &longs;it, &longs;uo loco con&longs;tabit inferiùs.
Motoris demùm velocitatem inten&longs;ioni impetûs concepti
non re&longs;pondere experimur, cum valdè conantes ut onus rapte
mus; parùm progredimur; at &longs;i funis ex improvi&longs;o abrumpa
tur, illicò corruimus, impetu &longs;cilicet concepto motum validiùs
efficiente, ubi de&longs;ierit impetum oneri, quod raptabatur, im
primere.
Hinc fit quòd, &longs;i ea fuerit corporum di&longs;po&longs;itio, ut impedi
mentum tardè &longs;ubmovendum &longs;it, ac proinde remi&longs;&longs;iore impetu
opus habeat, qui &longs;ibi imprimatur; corpus verò, cui motus
omnis tribuitur, non æquali tarditate cum impedimento ferri
nece&longs;&longs;e &longs;it, &longs;ed velociùs præ illo moveri po&longs;&longs;it, hoc &longs;anè eò mi
nùs habet re&longs;i&longs;tentiæ, quò minorem in intentione impetûs men
&longs;uram impedimento eidem imprimere debet, ut illud &longs;ubmo
veatur. Contrà verò &longs;i ita fuerint di&longs;po&longs;ita, ut impedimentum
velociùs præ ip&longs;o motore moveri oporteat, multò magis re&longs;i&longs;tit,
quàm &longs;i pariter moverentur, plus enim impetûs imprimendum
e&longs;t, ut motus con&longs;equatur.
Hactenùs re&longs;i&longs;tentiam poti&longs;&longs;imùm Formalem, impedimento
nihil in adver&longs;um conante, contemplati &longs;umus; jam ad Acti
vam tran&longs;eamus, cum &longs;cilicet duo corpora invicem aut omni
nò, aut ex parte repugnant, quia motum in diver&longs;as aut oppo
&longs;itas plagas directum moliuntur. In medio va&longs;e aquâ pleno &longs;ta
tuatur lignea tabella cra&longs;&longs;iu&longs;cula, eique lapis imponatur: dum
illa conatur a&longs;cendere, hic de&longs;cendere, &longs;e invicem urgent; &longs;ed
cum &longs;e vici&longs;&longs;im permeare nequeant, &longs;i paribus quidem viribus
confligant, &longs;ine motu con&longs;i&longs;tunt; &longs;in autem imparibus, aut
ambo a&longs;cendunt, aut ambo de&longs;cendunt, pro ut &longs;ive tabellæ le
vitas, &longs;ive lapidis gravitas oppo&longs;itam vicerit. Quod &longs;i lapis ta
bellæ non impo&longs;itus, &longs;ed &longs;uppo&longs;itus, arctè tamen connexus
cem urgent, &longs;ed vici&longs;&longs;im retrahunt, quandiù vinculum non
revellatur, aut rumpatur. Hic verò &longs;ubdubitet qui&longs;piam,
utrùm corpora, quæ contrario ni&longs;u reluctantur, &longs;ibi vici&longs;
&longs;im impetum imprimant, nec ne, aut æqualem, &longs;i pares fue
rint vires, aut, &longs;i impares, inæqualem: Quando enim ob vi
rium æqualitatem utrumque corpus con&longs;i&longs;tit, codem pacto
quies &longs;equitur, &longs;i unumquodque &longs;uam gravitationem aut levi
tationem &longs;ervans nihil alteri imprimat, ac &longs;i lignea tabella levi
tans partem impetûs &longs;ur&longs;um directi conferat impo&longs;ito lapidi, à
quo gravitante vici&longs;&longs;im recipiat tantumdem impetús deor&longs;um
directi; ex quo fiat, ut lapis habens concepti ac innati impe
tûs deor&longs;um directi vires æquales viribus impetûs &longs;ur&longs;um di
recti con&longs;i&longs;tat, idemque in ligneâ tabellâ contingat. Cùm ve
rò inæquales fuerint vires, id quod validius e&longs;t, eodem modo
&longs;uperat, &longs;ive nihil contrarij impetûs ab infirmiore oppo&longs;ito re
cipiat, &longs;ed minorem motum vi &longs;ui impetûs producat pro ratio
ne virium, quibus &longs;uperat; &longs;ivè partem impetûs contrarij reci
piat, quæ proprij impetûs vires attenuet.
Quotidianum e&longs;t hujus æqualitatis aut inæqualitatis experi
mentum in iis, quæ innatant humori; hæc enim humori im
po&longs;ita, quia in aëre gravitant, de&longs;cendunt; pars verò immer&longs;a
levitat in humore; prægravata tamen à reliquâ parte extante
deor&longs;um adhuc urgetur, donec inter partem immer&longs;am & ex
tantem fiat æquilibrium, & tantumdem pars immer&longs;a levitet in
humore, ac extans gravitat in aëre. Sic ma&longs;&longs;a plumbea argento
vivo impo&longs;ita de&longs;cendit, donec molis plumbeæ pars (2/13) extet; e&longs;t
enim &longs;pecifica plumbi gravitas ad &longs;pecificam mercurij gravita
tem ut 11 ad 13. levitat itaque plumbum in mercurio ut 2, gra
vitat in aëre ut 11; igitur plumbeæ ma&longs;&longs;æ partes 11 levitantes
&longs;ingulæ ut 2 parem habent conatum &longs;ur&longs;um, ac partes 2 gra
vitantes &longs;ingulæ ut 11 conantur deor&longs;um. Quòd &longs;i ita depri
meretur plumbum, ut ejus partes 12 immergerentur, & una
extaret; jam unica pars gravitans ut 11 vinceretur à partibus
12 levitantibus &longs;ingulis ut 2, ac propterea adhuc pars una
emergeret: quemad modum &longs;i quatuor partes extarent, & no
vem immergerentur, harum levitas 18 ab illarum gravitate 44
vinceretur, ideóque adhuc duæ immergerentur.
Jam &longs;i dixeris à partis immer&longs;æ levitantis momentis 18 impe
diri momenta 18 partis extantis gravitantis, adeò ut &longs;uper&longs;int
tantùm vires juxtà exce&longs;&longs;um gravitatis, &longs;cilicet momentorum
26, juxta quem exce&longs;&longs;um impetum imprimat parti immer&longs;æ, ut
deprimatur, tunc autem cum paria &longs;uerint levitatis atque gra
vitatis momenta, jam non invicem agere, &longs;ed &longs;e vici&longs;&longs;im impe
dire, probabilior forta&longs;&longs;e videatur alicui philo&longs;ophandi ratio
hîc, ubi directè &longs;ibi invicem adver&longs;antur directiones; alteruter
enim aut neuter impetus movet oppo&longs;itum corpus. Verùm
quoniam ubi lineæ directionum motûs non &longs;unt in directum
po&longs;itæ; &longs;ed inclinationem habent, motus mixtus, qui &longs;equitur,
ex utroque impetu unum motum temperari indicat, in eam fe
ror &longs;ententiam, ut exi&longs;timem duo corpora obliquè &longs;ibi invicem
repugnantia vici&longs;&longs;im imprimere, & recipere impetum in diver
&longs;as plaga directum pro modo virtutis uniu&longs;cuju&longs;que, adeò ut
&longs;i paria &longs;int momenta, medius planè inter utramque directio
nem &longs;equatur motus, &longs;i di&longs;paria, &longs;equatur pro modo exce&longs;sûs.
Fieri autem hane mutuam impetûs communicationem hinc
apparet, quòd &longs;i duo corpora, quorum virtus movendi ut AB
& AC, inloco, ubi A, con&longs;ti
tuta moveri cœperint, alterum
quidem, quod ad dexteram e&longs;t,
cum directione AB, alterum
verò, quod ad &longs;ini&longs;tram, cum di
rectione AC, ita &longs;e impediunt,
ut quod ad lævam e&longs;t, urgeat reliquum, ne per rectam AB proce
dat; hoc verò quod ad
AC incedat; &longs;ed propellat ita, ut ambo habeant directionem
mixtam AD. Hæc autem lineæ AD cum major &longs;it &longs;ingulis
lateribus AB, AC in rectangulo, aut rhomboide, ut quadra
to, aut rhombo, cavè nè putes &longs;ingulis corporibus &longs;upra pro
prium impetûs modum factam e&longs;&longs;e aliquam ab externo impetu
virium acce&longs;&longs;ionem: quî enim fieri po&longs;&longs;it, ut corpus nullo re
pugnante po&longs;&longs;it certo tempore percurrere lineam AB, dimi
nutis verò impetûs viribus ex re&longs;i&longs;tentià, pari tempore longio
rem lineam AD percurrat? An quia recipiat à corpore re
pugnante impetum, cujus acce&longs;&longs;ione augeatur proprius impe
tus, qui reliquus e&longs;t? At &longs;i propter virium æqualitatem percur-
ro recipit impetús, quantum tribuit: igitur non e&longs;t major vis
impetus, quàm &longs;i nihil repugnaret: ex quo fit neque motum ve
lociorem e&longs;&longs;e po&longs;&longs;e, ut pari tempore diametrum percurrant,
quo &longs;ingula de&longs;criberent latus Quadrati.
Non igitur ex illà mutuá impetus in diversâ directi commu
nicatione fit in &longs;ingulis corporibus impetûs inten&longs;io major (&longs;i
propriè loquendum &longs;it, habent enim impetus illi, conceptus
&longs;cilicet, & impre&longs;&longs;us, directionem diver&longs;am) quàm ferat pro
pria &longs;ingulorum virtus: id autem poti&longs;&longs;imùm con&longs;tat, quando
pora enim in motu breviorem Rhombi aut Rhomboidis
trum
Finge itaque corpus, quod percurreret AB, nullo impedi
mento prohiberi, quin moveatur eádem velocitate per AD;
utique &longs;olùm æquale &longs;patium AI decurreret, impediret tamen,
ne aliud corpus habens directionem AC, illique perpetuò
adhærens, decurreret juxta &longs;uam directionem &longs;patium æquale
ip&longs;i AC; &longs;ed tantùm EI, hoc e&longs;t Sinum anguli BAD loco
Tangentis eju&longs;dem anguli, po&longs;ito Radio AI.
Finge iterum alterum corpus habens directionem AC eâ
dem velocitate moveri per AD; utique non ni&longs;i &longs;patium AF,
ip&longs;i AC æquale, motu dimetiretur, prohiberetque, ne reli
quum corpus habens directionem AB, illique perpetuò adhæ
rens, progrederetur ni&longs;i in F, hoc e&longs;t &longs;patio æquali ip&longs;i BD;
&longs;ed versùs B non procederet ni&longs;i juxta men&longs;uram AG mino
rem ipsâ AC. Atqui utrumque &longs;uam habet directionem, &
non per AD, &longs;eque vici&longs;&longs;im impediunt; igitur dum &longs;imul mo
ventur, neque &longs;ub&longs;i&longs;tunt in F, neque veniunt in I; &longs;ed medio
loco con&longs;i&longs;tunt, puta in O.
Dixeris forta&longs;&longs;e AO æqualem ip&longs;i AE ita, ut &longs;it &longs;icut DB
ad BA, ita IE ad EA, hoc e&longs;t ad AO, aut AO e&longs;&longs;e medio
loco proportionalem inter AF & AI, hoc e&longs;t inter AC & AB
men&longs;uras virium impetûs &longs;ingulorum corporum. Hoc tamen
&longs;ecundo loco propo&longs;itum non facilè admi&longs;erim, quia ubi æqua
les &longs;unt virtutes movendi, medio loco proportionalis e&longs;t æqua
lis &longs;ingulis extremis, ac propterea utrumque corpus impeditum
æque velociter moveretur, ac non impeditum. Primum verò,
enim potior ulla apparet ratio, cur ad in&longs;tituendam analogiam
a&longs;&longs;umatur potiùs IE, quàm quælibet alia minor linea cadens
inter G & E. Ego autem libentiùs profiteor me ne&longs;cire, quà
Ratione analogia hæc in&longs;tituatur, quam aliquid certi divinan
do &longs;tatuere.
Verùm quamvis non utrumque corpus velociùs moveatur
quàm pro &longs;uâ virtute, alterum tamen quod urgetur, &longs;eu rapitur
à validiori, pote&longs;t, factâ impetûs acce&longs;&longs;ione, plus &longs;patij percur
rere, quàm pro &longs;uis viribus: impeditur &longs;iquidem motus non ab
&longs;olutè, &longs;ed juxtà eam directionem. Hinc fit corpus habens di
rectionem & velocitatem AC minorem velocitate AB promo
veri ultrà punctum F in linea mixti motûs AD.
At inquis: an &longs;i nautæ remis incumbant, veli&longs;que obliquis
ventum excipiant, tardior erit motus, quàm &longs;i navis vel à &longs;olis
remigibus, vel à &longs;olo vento impelleretur? contrarium &longs;anè vi
detur experientia evincere. Verùm &longs;i rem attentiùs con&longs;ideres,
aliam planè e&longs;&longs;e rationem deprehendes, cum duo corpora &longs;e
moventia vici&longs;&longs;im &longs;e impediunt, aliam cùm unum à duplici ex
trin&longs;eco impetu in diver&longs;a directo impellitur: de illis hactenùs
&longs;ermo fuit, neque ulla ratio &longs;uadere pote&longs;t velocius à tardiore
incitari, quamquam tardius à velociore urgeatur, ut dictum e&longs;t.
At &longs;i unum corpus à duobus æqualis aut inæqualis virtutis
impetum recipiat, utique magis inten&longs;us, vel &longs;i inten&longs;ionem
propriè dictam neges, certè major e&longs;t impetus, quàm &longs;i ab al
terutro tantùm reciperet impetum: quare nil mirum, &longs;i ea mo
tûs velocitas con&longs;equatur, quæ utrumque impetum &longs;ingillatim
&longs;umptum vincat, quamvis utroque &longs;imul &longs;umpto minor &longs;it, quia
habent directiones oppo&longs;itas, ut alibi explicabitur. Hinc e&longs;t
navim velociùs agi velis remi&longs;que, quàm &longs;i aut &longs;olâ ventorum
vi, aut &longs;olâ remigum ope propelleretur, & cymbam, dum &longs;e
cundo flumine rapitur, &longs;imulque remis ad alteram ripam im
pellitur, velociùs moveri, quàm aut in &longs;tagno eâdem remigum
operâ, aut à flumine ce&longs;&longs;antibus remis ageretur. Quemadmo
dum enim neque ventus remos impellit, neque ab his ventus
impellitur, ita neque &longs;e vici&longs;&longs;im immediatè impediunt, aut &longs;ibi
mutuò repugnant; atque adeò non e&longs;t hîc eadem philo&longs;ophan
di ratio, ac cum duo corpora &longs;ibi invicem immediatè re&longs;i&longs;tunt,
virtutis men&longs;uram motum concipiat.
Ex his quæ hactenùs dicta &longs;unt, illud &longs;atis con&longs;tare videtur,
quòd animal eatenùs in motu difficultatem ac re&longs;i&longs;tentiam per
cipit, quatenùs multum impetûs concipere debet, ex quo mu&longs;
culorum contentio oritur, neque tamen ea &longs;equitur motûs ve
locitas, quæ tanto impetui re&longs;ponderet, dum &longs;ubmovendo im
pedimento maximam virium partem impendit impetum impri
mens: unde fit plurimum influentis &longs;piritûs animalis ab&longs;umi in
tàm diuturnâ, vel tàm validâ mu&longs;culorum contentione, ac
proinde la&longs;&longs;itudinem &longs;equi, atque aliquando etiam contento
rum mu&longs;culorum dolorem, cum id non contingat &longs;ine aliquâ
partium compre&longs;&longs;ione aut di&longs;tentione. Quò igitur velociùs
moveri pote&longs;t animal pro ratione concepti impetûs, eò mino
rem percipit in &longs;ubmovendo impedimento difficultatem; &
quidem maximè &longs;i alternâ contentionis ac remi&longs;&longs;ionis mu&longs;cu
lorum vici&longs;&longs;itudine labor mite&longs;cat.
Curio&longs;iùs autem inquirenti, quam Rationem habeat motoris
impetus ad impetum corpori, quod movetur, quatenus move
tur, impre&longs;&longs;um, ut aliquatenus &longs;atisfaciam, a&longs;&longs;ero ut minimum
duplam e&longs;&longs;e, non quidem inten&longs;ivè, aut exten&longs;ivè &longs;ed enti
tativè. Quatenùs, inquam, movetur, hoc e&longs;t quatenus vinci
tur ejus re&longs;i&longs;tentia: cæterùm potentia movens in &longs;e producit, &
in mobili æqualem impetum; &longs;ed quemadmodum ubi calor fri
gori permi&longs;cetur illud vincens, non percipitur ni&longs;i quatenus
excedit vim frigoris, ita impetus oneri impre&longs;&longs;us eatenus mo
vet, quatenùs eju&longs;dem re&longs;i&longs;tentiam &longs;uperat: Hunc autem ex
ce&longs;&longs;um &longs;ubduplum impetûs motoris &longs;atis probabili conjecturâ
affirmo. Illud enim hoc mihi &longs;uadet, quòd motoris virtutem
metitur exce&longs;&longs;us impetûs, quem ille habet &longs;uprà impedimenti
re&longs;i&longs;tentiam: re&longs;i&longs;tentiæ autem modus, ut &longs;æpiùs dictum e&longs;t,
ex velocitate motûs, quæ concilianda e&longs;t gravitati corporis &longs;ub
movendi, de&longs;umitur; hoc enim ideò re&longs;i&longs;tit partibus ex gr.100
impetûs, quia &longs;i &longs;olùm fuerint 100 partes impetûs, fieri non po
te&longs;t ut moveatur tantâ velocitate, &longs;ed pluribus impetûs parti
bus indiget: exce&longs;&longs;us igitur virtutis motoris æqualis e&longs;t ut mi
nimum re&longs;i&longs;tentiæ mobilis; atque adeò tota virtus motoris, hoc
e&longs;t impetus ab eo conceptus, æquivalet tùm re&longs;i&longs;tentiæ mobi-
principio motûs eju&longs;dem mobilis: atqui motus hic æqualis e&longs;t
motui, cui illud re&longs;i&longs;tit, totus igitur impetus motoris duplus e&longs;t
impetùs, qui motum efficit in mobili, quatenus movetur.
Hinc e&longs;t eodem conatu motoris di&longs;parem effici motum, &longs;i
potentia æqualiter moveatur cum mobili, ut con&longs;tat: quia ni
mirum impetus mobili impre&longs;&longs;us inæqualem habet inten&longs;io
nem, quamvis entitativè æqualis &longs;it. Si enim tota motoris vir
tus &longs;it 20, & decem impetûs particulas re&longs;i&longs;tentiam &longs;uperantes
mobili imprimat, in quo inten&longs;io fiat ut 1, in mobili gravitatis
&longs;e&longs;quialteræ, particulæ eædem decem impetûs inten&longs;ionem ef
ficiunt ut 2/3; quare & hujus motus erit &longs;ub&longs;e&longs;qui alter, ac pro
inde motor, qui æqualiter cum mobili movetur, etiam tardio
rem habet motum, quàm cùm motum priori mobili conci
liabat.
Patet igitur ex his nunquam fieri po&longs;&longs;e, ut corpus grave mi
noris aut æqualis virtutis alterum moveat ita, ut planè in velo
citate con&longs;entiant; illud enim corpus minùs aut æquè grave
concipere non pote&longs;t impetum, qui & &longs;ibi ad motum &longs;ufficiat,
& alteri impetum imprimat: finge &longs;cilicet animo fui&longs;&longs;e impe
tum impre&longs;&longs;um corpori æquè vel magis gravi; hîc utique cum
non excedat re&longs;i&longs;tentiam mobilis, nullum efficere pote&longs;t mo
tum; igitur neque impre&longs;&longs;us fuit impetus, ne &longs;it omninò inuti
lis. Quòd &longs;i eâ ratione di&longs;ponantur ut motor velociùs moveri
po&longs;&longs;it quàm mobile, jam fieri pote&longs;t, ut à minore majus movea
tur: nam &longs;i motor certâ quâdam velocitate movere po&longs;&longs;it pon
dus unius libræ motu &longs;ibi æquali, eodem conatu & eádem ve
locitate &longs;e movens movebit pondus centum librarum, &longs;i hoc ita
&longs;it di&longs;po&longs;itum, ut centuplo tardiùs moveatur: quia nimirum
idem entitativè impetus in hoc pondere centuplo remi&longs;&longs;ior,
quàm in pondere unius libræ, &longs;ufficit ad motum centuplo tar
diorem. Motus &longs;iquidem centum librarum &longs;ubcentuplus in ve
locitate, æqualis e&longs;t motui unius libræ centuplo in velocitate;
&longs;i enim libra percurrit centum &longs;patij digitos &longs;ibi &longs;uccedentes in
longitudine, pari tempore centum libræ percurrunt quidem
unicum digitum longitudinis &longs;patij, centum tamen &longs;patia digi
talia percurrunt, &longs;ingulæ &longs;cilicet libræ digitum.
POtentiam oneri movendo cæteroqui imparem præ&longs;tare po&longs;
&longs;e, &longs;i machina adhibeatur, quotidiano experimento di&longs;ci
mus; adeò ut ip&longs;a unica pluribus potentiis machinâ de&longs;titutis
virtute æqualis &longs;it, & quæ pondus &longs;olitarium ac &longs;implex loco
pror&longs;us movere non poterat, ubi &longs;e ad machinam applicuerit,
jam non ponderi tantùm, &longs;ed & machinæ motum conciliet.
Quid ergo illud &longs;it, ex quo huju&longs;modi virium incrementum
oritur, hîc perve&longs;tigandum e&longs;t; & ad illud cau&longs;æ genus revo
catur, quam Scholæ Formalem appellant; e&longs;t &longs;cilicet ratio, per
quam fit, ut &longs;it, atque dicatur Machina: hoc autem incremen
tum virium, ut ex dicendis con&longs;tabit, ex machinæ figurâ pen
det &longs;ecundùm quam potentiæ, & ponderis motus &longs;ibi invicem
pro ratâ portione re&longs;pondent.
A machinâ quâ machina e&longs;t, potentiæ moventis vires non
augeri certum e&longs;t; nihil enim illi interioris virtutis impertitur,
& quâ machina e&longs;t, ab omni innatâ gravitate &longs;ejuncta intelli
gitur: vectis &longs;iquidem, ferreus &longs;it, &longs;ive ligneus, machinæ ra
tionem non immutat, &longs;i &longs;ola intercedat materiæ gravioris aut
levioris di&longs;paritas.
Quòd &longs;i faciliùs ferreo vecte tricubitali deor&longs;um premens at
tollas &longs;axum, quàm &longs;i ligneo vecte pariter tricubitali utaris
(quia nimirum ferreus vectis habet &longs;ibi adnexam ex gravi ma
teriâ, quâ con&longs;tat, potentiam, quæ deor&longs;um urgendo te juvat,
ut &longs;axum attollatur,) id planè e&longs;&longs;e extra vectis naturam, quâ
vectis e&longs;t, manife&longs;tum erit, &longs;i non deor&longs;um, &longs;ed &longs;ur&longs;um, aut à
lævâ in dextram connitendum &longs;it, ut duo connexa disjungas;
tunc enim ferrei vectis gravitas &longs;u&longs;tentanda laborem potiùs
creabit, quàm ut præ &longs;imili ligneo vecte motum hunc facilio
rem reddat. Quare præter Mechanicæ facultatis in&longs;titutum
machinis accidit, ut gravitate &longs;uâ potentiæ moventis vires ad
augeant, non quidem illam immutando, facto interiore virtu-
viribus agat, con&longs;ociando.
Sed & illud animadvertendum e&longs;t, vix unquam fieri po&longs;&longs;e,
ut potentia movens nihil pror&longs;us impedimenti à machina reci
piat: &longs;ivè enim machinæ ip&longs;ius pars aliqua gravis elevanda e&longs;t;
&longs;ivè membrorum, in quæ machina di&longs;tribuitur, invicem con
fligentium, &longs;eque vici&longs;&longs;im terentium a&longs;peritas ob&longs;i&longs;tit; &longs;ive mo
tus (ut machinæ ip&longs;i, cui applicatur potentia, ob&longs;ecundet) à
&longs;uâ directione inflectitur; &longs;ivè quid huju&longs;modi intercedit, quod
aliquid de motûs velocitate imminuat, quæ cæteroqui concep
tum potentiæ ab omni machinâ ab&longs;olutæ impetum con&longs;equere
tur. Ex his tamen aliqua &longs;unt, quæ ita motui potentiæ offi
ciunt, ut ad retinendum onus juvent; hujus &longs;iquidem gravitas
minùs adversùs potentiam valet, &longs;i & ip&longs;um, quia machinæ il
ligatum à recto in centrum gravium tramite deflectere, vel
mutuum partium &longs;e terentium conflictum vincere cogatur, ut
vim potentiæ inferat. Verùm hæc, quamvis, ubi res ad praxim
deducitur, per incuriam di&longs;&longs;imulanda non &longs;int, &longs;ub &longs;taticam
con&longs;iderationem hîc non cadunt, ubi machinarum vires ex
penduntur; harum enim figura perindè attenditur, atque &longs;i
nihil adjumenti, nihil detrimenti ex materiâ accederet.
Ad rem itaque propiùs accedentibus recolenda &longs;unt ea, quæ
in &longs;uperioribus hujus libri capitibus di&longs;putata &longs;unt, proximam
videlicet motûs effectricem cau&longs;am impetum e&longs;&longs;e &longs;ive ab inte
riore virtute manantem in iis, quæ &longs;ponte &longs;uâ moventur, &longs;ivè
extrin&longs;ecùs aliunde impre&longs;&longs;um iis, quæ naturâ repugnante per
vim cientur: ex cujus impetûs inten&longs;ione, quatenùs omnem
re&longs;i&longs;tentiam &longs;uperat, motuum velocitas oritur: nunquam autem
à velocitate aut tarditate motum &longs;ejungi po&longs;&longs;e certum e&longs;t, quip
pe qui nec &longs;inè &longs;patio per quod decurratur, nec &longs;inè partium
&longs;ibi certâ lege &longs;uccedentium continuatione ac &longs;erie intelli
gi pote&longs;t. Quare & re&longs;i&longs;tentiæ momenta tùm ex corporis
movendi gravitate, tùm ex velocitate componi &longs;æpiùs innui
mus, ut hinc innote&longs;cat fieri facilè po&longs;&longs;e, ut, &longs;icut eju&longs;dem
gravitatis re&longs;i&longs;tentia inæqualis e&longs;t, &longs;i velocitate inæquali mo
venda &longs;it, & gravitatum inæqualium di&longs;paria &longs;unt re&longs;i&longs;tentiæ
momenta, &longs;i Ratio, quæ ex gravitatum & velocitatum Ratio
nibus componitur, &longs;it Ratio Inæqualitatis, quia gravior velo-
par &longs;it re&longs;i&longs;tentia, &longs;i quæ inter gravitates intercedit Ratio, ea
dem reciprocè inter velocitates inveniatur. Quemadmodum
enim quæcumque calori adver&longs;antur, vehementiorem quidem
validi&longs;&longs;imè re&longs;puunt, tenui&longs;&longs;imum verò facillimè admittunt;
haud di&longs;pari ratione pondera, &longs;i velociùs incitare velis, im
pensiùs reluctantur, minimo ac tardi&longs;&longs;imo motui levi&longs;&longs;imè ob
&longs;i&longs;tunt.
Quoniam igitur naturâ definitum e&longs;t, quantam gravitatem,
quantáque velocitate, pro certâ impre&longs;&longs;i impetûs men&longs;urâ, mo
vere po&longs;&longs;it Potentia concepto impetu, qui pro ratâ portione
re&longs;pondeat impetui quem illa oneri imprimit, ut Potentia, &
onus æquali velocitate moveantur; &longs;atis con&longs;tat eandem impe
tús men&longs;uram parem e&longs;&longs;e movendo oneri graviori, &longs;i quá Ra
tione po&longs;terior hæc gravitas priorem gravitatem vincit, eâdem
Reciprocè Ratione prioris velocitas po&longs;terioris tarditatem &longs;u
peret; utrobique &longs;cilicet par e&longs;t re&longs;i&longs;tentia, ac proinde ab eâ
dem potentiâ vinci pote&longs;t. Cùm enim ea, quæ &longs;imul æqualiter
moventur, æquali impetu ferantur; &longs;i Potentia tàm tardè mo
veretur ac pondus per machinam, indigeret impetu ex. gr. &longs;ub
quintuplo ejus quo illa movetur quintuplo velociùs ac ip&longs;um
Pondus. Verùm impetus hîc &longs;ubquintuplus ineptus e&longs;&longs;et ad
oneris re&longs;i&longs;tentiam quintuplo ferè majorem vincendam; &longs;ed &longs;o
lum &longs;uperare po&longs;&longs;et ac movere 1/5 ponderis. Quinque igitur im
petus huic æquales po&longs;&longs;unt totam re&longs;i&longs;tentiam &longs;uperare. Cum
itaque in motu quintuplo velociori Potentiæ &longs;it verè impetus
quintuplus, poterit etiam elevare pondus, quod e&longs;t quintuplo
majus, quàm &longs;it 1/5 ip&longs;ius. Verùm híc ubi de motûs velocitate
&longs;ermo e&longs;t, non is quidem ab&longs;olutè accipiendus e&longs;t; &longs;ed quâ
parte gravium naturæ repugnat: &longs;i enim plumbeus globus
A ex C dependeat funiculo CA, & circà ver&longs;atilem or
biculum B &longs;tabili axi infixum ducatur filum connectens
globos A & D, cottum quidem e&longs;t globum A, &longs;i u&longs;que ad
B perveniat, tantumdem &longs;patij in arcu AB percurrere,
non tamen tantumdem a&longs;cendere, quantum globus D &longs;e
cundùm rectam BD de&longs;cendit; &longs;ed a&longs;cen&longs;um metitur AE,
nimirum Sinus Ver&longs;us arcûs AB, qui minor e&longs;t codem
arcu (arcus &longs;iquidem major e&longs;t rectâ AB lineâ ip&longs;um &longs;ub-
tendente, quæ oppo&longs;ita
recto angulo E major e&longs;t
quàm trianguli ba&longs;is AE)
ac propterea re&longs;i&longs;tentiæ
momenta non ea &longs;unt, quæ
ex velocitate motûs AB,
&longs;ed AE, & ipsâ globi A
gravitate componuntur. Ex
quo fit globum D quam
vis minorem po&longs;&longs;e globo A
graviori præ&longs;tare, ac illum
ad certam altitudinem ele
vare, ut cuilibet experiri
licet, cum tamen illi a&longs;cen
&longs;um &longs;uo de&longs;cen&longs;ui æqualem
nullatenùs conciliare po&longs;&longs;it.
Quòd &longs;i idem globus A ex breviore funiculo HA dependeat,
experimento con&longs;tat opus e&longs;&longs;e globo D gravitatem addere, ut
valeat illum per arcum AF elevare ad eandem altitudinem
AE: magis quippè laborio&longs;um e&longs;t breviore motu AF, quàm
longiore motu AB ad eandem altitudinem a&longs;cendere; atque
adeò plus virium in D requiritur, ut globo A majorem impetum
imprimat, ex cujus inten&longs;ione plus &longs;ingulis temporis momentis
a&longs;cendat in hoc po&longs;teriore motu, quàm in priore. Ne tamen
motui globi D tribue men&longs;uram arcûs AB &longs;ed rectæ AB.
Sicut autem ubi potentiæ & oneris æquales e&longs;&longs;e debent mo
tus, potentiæ vires gravitate oneris majores e&longs;&longs;e oportet, ut vim
illi inferant; ita pariter ubi potentia & onus in motuum velo
citate di&longs;&longs;entiunt, & illa quidem velociùs, hoc tardiùs move
tur, nece&longs;&longs;e e&longs;t majorem e&longs;&longs;e Rationem Potentiæ ad Onus (licet
illa minor &longs;it onere) quàm &longs;it Ratio tarditatis hujus ad illius
velocitatem; ut &longs;cilicet ratio Potentiæ ad onus, quæ ex mo
tuum, & virium Rationibus componitur, &longs;it Ratio majoris inæ
qualitatis. Sit ex.
gr. Ratio motûs Potentiæ ad motum Oneris
ut 3 ad 2; &longs;i Ratio virium potentiæ ab&longs;olutè &longs;umptæ ad gravi
tatem oneris &longs;it Reciprocè ut 2 ad 3, Ratio ex his Rationibus
compo&longs;ita e&longs;t Æqualitatis, &longs;cilicet 1 ad 1, & motus nullus &longs;e
quitur; multò minùs &longs;i fuerit Ratio minor quàm 2 ad 3; prove-
Ratione 2 ad 3. Sit ex hypothe&longs;i Ratio 4 ad 5; jam Ratio com
po&longs;ita ex Rationibus 3 ad 2, & 4 ad 5, e&longs;t Ratio 6 ad 5 majoris
Inæqualitatis.
Neque hoc ita dictum intelligas, qua&longs;i motus ip&longs;e Potentiæ,
eju&longs;que velocitas, efficiendi vim haberet; &longs;ed ex ipsá majore
potentiæ velocitate innote&longs;cit impetum, qui radix e&longs;t motûs,
minus invenire impedimenti ex onere, quod minùs re&longs;i&longs;tit, eo
quòd tardiùs movendum e&longs;t, quàm &longs;i æqualem velocitatis gra
dum cum potentiâ &longs;ortiri deberet. Quare licèt potentia minor
&longs;it, ac pauciores entitativè particulas impetús producere valeat,
quàm potentia major, &longs;atis in aperto e&longs;t fieri po&longs;&longs;e, ut potentia
major majorem inveniens re&longs;i&longs;tentiam nequeat impetum im
primere, ac movere onus, quod movebitur à minore potentiâ,
&longs;i onus idem minùs re&longs;i&longs;tat, cum &longs;it tardiùs movendum: impe
tus enim à minore potentiâ oneri impre&longs;&longs;us &longs;atis e&longs;t ad vincen
dam minorem hanc re&longs;i&longs;tentiam; cum tamen potentia major
non &longs;atis habeat virtutis, ut eam impetús men&longs;uram oneri im
primat, quæ majorem illius re&longs;i&longs;tentiam &longs;uperaret.
In eo igitur totum Mechanices artificium con&longs;i&longs;tit, ut &longs;ua
in&longs;trumenta ita di&longs;ponat, loci&longs;que congruis ita Potentiam, &
Onus collocet, ut Potentiæ motus velocior &longs;it præ motu Oneris:
tùm horum motuum Ratione attentè per&longs;pectâ definies, quæ
nam Potentia datum Onus movere, vel quodnam Onus à datâ
Potentiâ moveri queat; &longs;i nimirum Potentiæ vires ad oneris
gravitatem majorem habeant Rationem, quàm &longs;it Ratio motùs
Oneris ad motum Potentiæ. Neque enim Machina aut Poten
tiæ vires auget, aut oneris gravitatem minuit, &longs;ed Ponderis re
&longs;i&longs;tentiam ad Potentiæ virtutem accommodat.
Phy&longs;ica autem cau&longs;a hæc e&longs;t, quia impetus à Potentiâ pro
ductus, qui in onere minori movendo æque velociter cum po
tentiâ
movendo minorem quidem habet inten&longs;ionem, &longs;ed quæ &longs;atis e&longs;t
pro minore Fac enim oneris particulas graves e&longs;&longs;e 20,
illique à
tûs, quibus vincitur Oneris re&longs;i&longs;tentia: inten&longs;io in &longs;ingulis par
ticulis gravitatis e&longs;t particularum impetûs 5, juxtà quam inten
&longs;ionis men&longs;uram &longs;equitur motus æque velox Potentiæ & oneris,
&longs;um. Sit adhuc eadem Potentia; &longs;ed offeratur Onus, cujus
particulæ gravitatis &longs;int non jam 20; &longs;ed 50: Potentiæ virtus e&longs;t
eadem; quapropter non ni&longs;i re&longs;i&longs;tentiam vincere pote&longs;t, cui
vincendæ &longs;ufficiant particulæ 100 impetus; hæ autem in One
re graviore ut 50 efficerent &longs;olùm inten&longs;ionem ut 2: Non igitur
Potentia & onus æquè veloci motu, qui re&longs;pondeat inten&longs;ioni
ut quinque, &longs;icuti priùs, moveri poterunt; &longs;ed ut onus moveri
po&longs;&longs;it, impetúmque à potentiâ recipere, opus e&longs;t ita illud col
locare, ut quò magis Ratione gravitati re&longs;i&longs;tit; eò minùs ra
tione tarditatis motûs re&longs;i&longs;tat, &longs;eque eâ ratione temperent duæ
hæ re&longs;i&longs;tentiæ, ut una confletur re&longs;i&longs;tentia non major illâ, quæ
oriebatur ex onere gravi ut 20 æqualiter movendo: id quod
fiet, &longs;i motus Potentiæ, quatenùs machinæ applicatur, ad mo
tum oneris &longs;it ut 5 ad 2 in Reciprocâ Ratione inten&longs;ionum im
petûs producti. Quare motus Potentiæ ad motum oneris e&longs;t
duplus &longs;e&longs;quialter, quemadmodum po&longs;terior hæc oneris gravi
tas ut 50 e&longs;t prioris gravitatis ut 20 dupla &longs;e&longs;quialtera: atque
hinc manife&longs;tum e&longs;t particulas gravitatis 50 re&longs;i&longs;tentes ut 2 ra
tione motûs comparati cum motu potentiæ, requirere particu
las 100 impetûs, quemadmodum particulæ gravitatis 20 re
&longs;i&longs;tentes ut 5 ratione motûs comparati cum motu eju&longs;dem Po
tentiæ requirunt particulas 100 impetûs. Quid igitur mirum, &longs;i
potentia eadem eodem conatu movet onus ut 50 velocitate ut 2,
quo conatu movet onus ut 20 velocitate ut 5?
Servatur itaque perpetua quædam ju&longs;titia inter potentiæ vi
res, oneris gravitatem, &longs;patia motuum, ac tempora; quò enim
decre&longs;cunt potentiæ vires, aut oneris gravitas augetur, eò bre
viora &longs;unt &longs;patia, & longiora tempora motuum ip&longs;ius oneris;
&longs;ed ampliora &longs;patia motuum potentiæ debilioris, quæ præ one
re velociùs movetur. Hinc dato onere graviori &longs;ubmovendo,
aut potentiam augeri, aut, &longs;i illa immutata permaneat, oneris
motum imminui, &longs;eu potentiæ motum augeri nece&longs;&longs;e e&longs;t: Te
nui enim potentiâ ingens pondus citò moveri non pote&longs;t.
Formalem igitur Machinæ Rationem, quâ Machina e&longs;t, in eo
&longs;itam e&longs;&longs;e deprehendimus, quòd ea figura &longs;it, quæ potentiæ,
& oneris motibus legem ita &longs;tatuat, ut Potentia velociter, Pon
dus lentè moveatur; &longs;ic enim fit, ut minor oneris re&longs;i&longs;tentia vir-
ne re&longs;pondeat. Satis igitur erit, ubi &longs;ingularum machinarum
vires expendendæ erunt motuum inire rationes, qui ex machi
næ agitatione oriuntur: nam &longs;i Potentia præ Onere velociùs
moveatur, operæ pretium faciet Machinator; modò non adeò
tenuis &longs;it motuum Ratio, ut quicquid utilitatis ex machinæ fi
gurà accedit, deferatur ex partium &longs;e terentium conflictu; nam
perinde e&longs;&longs;et, ac &longs;i oneri gravitas adderetur.
Ex his liquet à non paucis plus operæ labori&longs;que con&longs;ump
tum, quàm par e&longs;&longs;et, ut Ari&longs;toteli adhærerent in referendis
machinarum viribus in circuli naturam planè admirandam:
miraculorum omnium e&longs;&longs;e principium. Ea igitur quæ circà libram fiunt,
ad circulum referuntur, quæ verò circa vectem, ad ip&longs;am libram;
alia autem ferè omnia, quæ circa mechanicas &longs;unt motiones, ad
vectem. Ni&longs;i enim fucum veritati faciamus, quæ demum mi
racula ita circulum à reliquo figurarum vulgo &longs;ecernunt, ut in
cum admiratio omnis corrivata confluat, nec ni&longs;i hinc in cæte
ras derivetur? An quòd linea eadem, quâ circuli ambitus de
finitur, omnis latitudinis expers, cava pariter atque convexa
amico fœdere copulat, quæ &longs;ibi invicem repugnant? Cavum
&longs;i quidem à convexo, quæ recto interjecto di&longs;criminantur, per
inde di&longs;&longs;idere cen&longs;emus, atque minus à majori, inter quæ &longs;ibi
adver&longs;antia id, quod æquale e&longs;t, intercedit. At hæc ita vulga
ria &longs;unt, ut non Hyperbolæ &longs;olùm, ac Parabolæ, aut Nicome
dis Conchoidi, aut Archimedis Spiralibus, aut Dino&longs;trati
Quadratici, cæteri&longs;que omnibus extrà Geometricas leges cur
vis lineis communia &longs;int; verùm etiam in angulo quocumque
rectilineo facilè ab omnibus ob&longs;erventur; cum lineæ rectæ, qui
bus inclinatis angulus con&longs;tituitur, hinc quidem &longs;ibi mutuis
nutibus annuere, hinc verò abnuere videantur; quibus oppo
&longs;itis nutibus media pariter interjacet directa po&longs;itio, omni in
clinatione &longs;ubmotâ.
An ipsâ na&longs;centis Circuli exordia admiratione non carent,
quòd æquè ex Radij eju&longs;dem in centro &longs;ub&longs;i&longs;tentis quiete, ac
circumlati motu oriatur? Sed quid hæc in circulo potiùs &longs;u&longs;
piciamus, quàm in Helice, cui gene&longs;is haud di&longs;par contingit?
Quòd &longs;i circulo primas ideò deferendas exi&longs;timemus, quòd
unde &longs;ump&longs;it exordium; & circumacta, quæ ex adver&longs;o
&longs;unt, partes oppo&longs;itis cieat motibus, ita ut progredientibus
&longs;upremis infimæ regrediantur, & in ima detrudantur &longs;i
ni&longs;træ, dextris in altiora provectis: Quid Ellip&longs;im præjudi
cio repellimus? cum & hæc unico limite cavo pariter atque
convexo in &longs;e&longs;e redeunte circum&longs;cripta in contrarias partes
incitetur; nec à rectâ tantummodo lineâ alternis auctà cre
mentis, imminutáque decrementis altero terminorum quie&longs;
cente, &longs;ed etiam (quod verè miraculo proximum e&longs;t)
utroque extremo flexilis lineæ in binis Ellip&longs;eos umbilicis
defixo ab illâ in alios, atque alios angulos &longs;inuata de&longs;
cribatur.
At, inquis, in circulo &longs;emidiametri partes codem im
pellente circà centrum agitatæ ita di&longs;pari velocitate ferun
tur, ut earum tarditas aut concitatio intervallo, quo &longs;in
gulæ à centro ab&longs;unt, &longs;it analoga. Verùm & hoc Ellip&longs;i,
ac plano Helicoidi aliquatenùs pro &longs;uo modulo commune
e&longs;t; &longs;emidiametri enim circumactæ puncta à centro remo
tiora velociùs feruntur. Partes autem quie&longs;centi centro pro
piores cunctabundas moveri, naturæ pro viribus oppo&longs;ita
di&longs;terminantis in&longs;tituto con&longs;entaneum e&longs;&longs;e nemo non videt,
qui tarditatem interjici videt quietem inter, ac motûs ve
locitatem. Quare &longs;apienti&longs;&longs;imo con&longs;ilio factum, ut eorum,
quæ firmo nexu invicem &longs;olidata &longs;ub&longs;i&longs;tunt, vel particu
læ omnes æquis pa&longs;&longs;ibus moveantur, vel &longs;i qua moræ di&longs;
pendium &longs;ubeat, finitimarum velocitas, &longs;ervatâ aliquâ vi
cinitatis analogiâ minuatur: ne &longs;cilicet &longs;olutâ compage di&longs;
&longs;iliant.
Quæ verò ad explicandum, cur ea, quæ centro propiora
&longs;unt, tardiùs in gyrum contorqueantur, Author illius libri
Quæ&longs;t. mechan. commini&longs;citur de duplici motu, naturali vi
delicet, ac præter naturam, quibus feratur ea, quæ circu
lum de&longs;cribit linea (qua&longs;i breviorem lineam vis major à tra
hente centro illata magis à naturali motu, qui &longs;ecundùm
Tangentem e&longs;t, deflecteret) ea &longs;unt, quæ facillimè cor
ruant, & minimè cum Ari&longs;totelis doctriná cohæreant, qui
lib.
1. de Cælo. &longs;umma 4. circularem motum & &longs;implicem, &
enim,12;
perfectorum e&longs;t, recta verò linea nulla. Quis ergo in circulo
motus præter naturam? 8.
quod corpus &longs;implex, quod natum e&longs;t ferri circulari motu &longs;ecun
dùm &longs;uam ip&longs;ius naturam. Ea certè quibus in&longs;ita e&longs;t in mo
tum propen&longs;io, in gyrum aguntur, ut &longs;ydera; aut &longs;altem mo
tu in &longs;e recurrente circulum æmulantur, ut ex cerebri & cor
dis &longs;y&longs;tole ac dia&longs;tole &longs;pirituum ac &longs;anguinis circuitio oritur;
aut plurium circularium motuum commixtione unum tempe
rant motum, ut animalia cum progrediuntur; o&longs;&longs;a &longs;iquidem,
quibus membra &longs;ub&longs;i&longs;tunt, ita à mu&longs;culis commoventur, ut
unum quod que &longs;ui motus centrum con&longs;tituat in eâ finitimi o&longs;&longs;is
parte, cui &longs;ivè
page in&longs;eritur. At motu recto, ut potè brevi&longs;&longs;imo, nihil fertur,
ni&longs;i cui ex naturæ in&longs;tituto cedit quies certo in loco, à quo
ab&longs;tractum fuerit, eóque &longs;ibi redditum &longs;pontè remigrat. Nihil
igitur præter naturam in circuli motu deprehendi pote&longs;t, ex
quo di&longs;par illa intimarum atque extimarum partium velocitas
petenda &longs;it; cum vix alium natura per &longs;e expetat &longs;implicem
motum præter circularem. Cur autem qui &longs;ecundùm rectam
extremæ &longs;emidiametro ad perpendiculum in&longs;i&longs;tentem lineam
fit motus, naturalis cen&longs;eatur? An quia gravia &longs;uis nutibus ad
terræ centrum rectâ feruntur? Semidiametro igitur, ni&longs;i in
verticali plano con&longs;tituatur horizonti parallela, motus qui &longs;e
cundùm lineam circuli Tangentem e&longs;t, præter naturam con
tinget, quippe qui à rectâ, quæ gravia in centrum dirigit, de
flectat: & in circulo horizonti parallelo circumacta &longs;emidiame
ter nullo naturali motu agitabitur; nulla enim recta linea cir
culi Tangens in eo plano e&longs;t, quæ lineæ directionis gravium
congruat: & tamen quemcumque demum &longs;itum circulus eju&longs;
que &longs;emidiameter obtineat, eandem &longs;emper motuum analo
giam &longs;ervant partes pro ratione intervalli à centro, citrà ullam
motuum naturalis, & præter naturam, commi&longs;tionem.
Verùm mirifica &longs;it circuli natura; quid hæc ad explicandam
Mechanicarum motionum cau&longs;am? an ut hanc ignotam fatea
mur, quia admirandam prædicamus? &longs;ed unico argumento,
commenta huju&longs;modi disjiciamus. Si minor potentia majori
e&longs;t hoc virtutis
neque in Circulum referri po&longs;&longs;e: adeóque principium aliud e&longs;&longs;e
magis latè patens, à circulo ab&longs;olutum: Atqui citrà omnem cir
cularem
fe&longs;tum e&longs;t igitur fru&longs;trà ex circulo peti Mechanicarum motio
num principium; &longs;ed illud e&longs;&longs;e, quod à nobis indicatum e&longs;t,
quippe quod, ubicumque reperitur, hoc efficit, ut minor po
tentia majori ponderi motum conciliet, nec is unquam &longs;ine illo
contingit.
A&longs;&longs;umptionis veritas ut innote&longs;cat, ingen&longs;que pondus tardè
movendum à tenui virtute &longs;ine circulari motu propelli po&longs;&longs;e
confirmem, non ego te in &longs;uburbanum campum deducam, ut
tenerrimo germini &longs;uppullulanti incumbentes glebas demùm
loco ce&longs;&longs;i&longs;&longs;e ob&longs;erves, aut marmora Me&longs;&longs;alæ &longs;cindentem capri
ficum obtrudam, turre&longs;que longâ annorum &longs;erie labefactatas
enatis fruticibus atque virgultis; ne mihi fortè herbe&longs;centes
cuneos obtrudas, quos ad vectem, & circulum revocare velis.
Sed age raptandus &longs;it in plano horizontali, aut inclinato, aut
etiam elevandus &longs;it ad perpendiculum cylindrus A. Experire
primùm quanto labore id præ&longs;tes illum trahens illigato fune
in C, & arreptâ extremitate funis B. Tùm in B infixo firmi
ter paxillo ductarius funis alligetur; hic porrò in&longs;eratur annu
lo C optimè ferruminato, & quoad ejus fieri poterit exqui&longs;itè
polito, arreptáque alterâ funis extremitate D iterum trahe cy
lindrum, & quantò minori labore id perficias, tu te ip&longs;e doce
bis. At hîc nulla circuli vides miracula; hîc libra nulla; nullus
hîc vecti locus: motus enim tùm potentiæ trahentis, tùm cy
lindri, rectus e&longs;t. Facilitatis autem di&longs;crimen non ex ullo cir
culari motu, qui nu&longs;quam apparet, &longs;ed ex eo oritur, quòd pri
mùm potentia & onus æqualiter moventur; po&longs;teà verò cylin-
cylindrus venit in B funis ultrà B extenditur juxtà longitudi
nem CB u&longs;que in E; ac propterea motus potentiæ duplus e&longs;t,
&longs;cilicet CE.
Statue item in pariete puncta duo A & B (quo autem majo
re intervallo disjuncta fuerint, res meliùs &longs;uccedet) ibique
clavos rotundos nihil ha
bentes a&longs;peritatis infige.
Tùm pondera duo H &
G æqualia a&longs;&longs;ume, eáque
funiculo nullis nodis a&longs;pe
ro, &longs;ive &longs;erico crudo, &longs;ive
crinibus equinis connexa
impone claviculis A & B,
ut liberè ex iis depen
deant: &longs;uâ autem gravitate
funiculum AB intentum Horizonti parallelum &longs;ervabunt, &
neutro prævalente ob gravitatis æqualitatem prorsùs immota
con&longs;i&longs;tent. Elige jam pondus tertium I, quod alteri datorum
H & G æquale &longs;it, aut etiam &longs;ingulis aliquantò minus; illud
que in E extento funiculo AB adnecte: &longs;tatim pondus I &longs;ecun
dùm rectam EF de&longs;cendens videbis; pondera autem H & G
per rectas HA, & GB a&longs;cendentia, quâ men&longs;urâ funiculi in
flexi partes AF, BF &longs;imul &longs;umptæ excedunt rectam AB. Nul
lus igitur motus circularis hîc e&longs;t; &longs;ed omnes recti ad perpendi
culum, & tamen potentia I minor commovet majus pondus,
quod ex H & G conflatur.
Id autem ideò contingere, quia motus EF de&longs;cendentis I
major e&longs;t motu a&longs;cendentium H & G, hinc manife&longs;tum e&longs;t,
quòd pondus I u&longs;que ad certum terminum de&longs;cendit, ibique
&longs;ub&longs;i&longs;tit: quòd &longs;i illud manu apprehen&longs;um adhuc deor&longs;um
trahens eleves pondera H & G, ubi manum indè ab&longs;traxeris,
pondera H & G prævalent, ac de&longs;cendentia elevant pondus I
ad certum illum terminum, ubi &longs;ponte &longs;ub&longs;titerat: quia nimi
rum ultrà illum terminum non jam major e&longs;t Ratio ponderis I
ad pondera HG, quàm &longs;it Ratio motuum H & G ad motum I.
Hæc autem inferiùs, ubi de librâ & Æquilibrio &longs;ermo erit,
paulò fu&longs;iùs & dilucidiùs explicabuntur; nunc enim &longs;atis e&longs;t
pollere citrà omnem motum circularem.
Ratum itaque e&longs;to ad nullum certum machinæ genus cætera
e&longs;&longs;e revocanda; &longs;ed omnibus commune e&longs;&longs;e principium, ex quo
vires de&longs;umunt; impetûs &longs;cilicet à potentiâ producti proportio
ad ponderis re&longs;i&longs;tentiam (quæ eò minor e&longs;t, quò tardiùs mo
veri debet) ea e&longs;t, quæ motûs facilitatem conciliat; nullus
quippe adeò tenuis impetus reperitur, cui lenti&longs;&longs;imus aliquis
motus non re&longs;pondeat, &longs;i intereà à velociori motu potentia non
prohibeatur. Ubi autem de potentiæ velocitate &longs;ermo e&longs;t, non
ea intelligatur, quæ e&longs;&longs;et, ubi præter &longs;e nihil ip&longs;a moveret, ab
&longs;oluta ab omni re&longs;i&longs;tentiâ; &longs;ed eam velocitatem intellige, quæ
comparatè dicitur, ubi ejus motus cum oneris motu confertur.
Semper tamen impetus, qui in Potentiâ reperitur quatenùs ex
cedit re&longs;i&longs;tentiam ponderis, majorem in eâ intentionem ha
bet, quàm in pondere, quamvis pares entitativè &longs;int impetus
Potentiæ, & oneris. Hæc autem clariùs patebunt lib.4. cap.1.
atque materie.
QUamvis in&longs;tructarum Machinarum vires ad calculos revo
centur in&longs;pectâ earum figurâ, ut Potentiæ atque oneris
motus invicem comparentur; quo tamen loco & &longs;itu Machina
ip&longs;a collocetur, di&longs;piciendum e&longs;t, ut innote&longs;cat, quanta illi vis
inferatur tùm ab oneris gravitate, tùm à potentiæ conatu: ex
hoc &longs;iquidem decernendum erit, quàm &longs;olidam con&longs;trui opor
teat Machinam. Quotus enim qui&longs;que e&longs;t, qui ignoret longè
&longs;olidiorem requiri machinam, &longs;i ex illa dependeat, aut illi in
cumbat onus, quàm &longs;i non machinæ; &longs;ed &longs;ubjecto plano, inni
tatur idem pondus, aut aliunde dependeat? alia &longs;cilicet &longs;unt
gravitatis momenta contrà virtutem &longs;u&longs;tinentem etiam citrà
motum, alia verò momenta, quatenus motui adver&longs;atur.
Machinatore di&longs;ponantur, ut pondus, quàm minimum fieri
po&longs;&longs;it, à machinâ &longs;u&longs;tineatur: hâc enim ratione fiet, ut lon
giùs avertatur periculum luxationis aut fractionis membrorum,
quibus machina di&longs;tinguitur, etiam&longs;i exilior illa fuerit; & ma
chinæ gravitas aliqua &longs;ubtrahetur, dum moles ip&longs;a minuitur,
atque proinde movendi oneris difficultas non augebitur ex ma
chiná; quæ etiam minore impendio parabitur.
Sit exempli gratiâ pondus A, quod &longs;it trochleâ attollendum
in D. Poterit id duplici ratione fieri; primùm raptando illud in
plano Horizontali ita, ut ex B
veniat in C, tùm alligatâ tro
cleâ in I illud attollendo ad
perpendiculum u&longs;que in D:
cum raptatur, totum incumbit
pondus &longs;ubjecto plano; cum at
tollitur, totum ex trochleâ de
pendet. At &longs;i trochleâ utaris,
de cujus firmitate &longs;ubdubites,
& loci di&longs;po&longs;itio ferat, ut po&longs;
&longs;it ex E & H onus &longs;u&longs;pendere,
res faciliùs perficietur. Ponde
ri enim A adnecte funem OE,
ex quo pendere po&longs;&longs;it in E, ac
prætereà tantumdem funis OS liberè vagantis; trochleam au
tem alliga in F: ubi verò ope trochleæ adduxeris pondus ex O
in G, tùm funem OS liberè vagantem eleva, ac benè inten
tum adnecte in H, ut jam pondus ex H dependeat ad perpen
diculum: Ex hoc fiet, ut re&longs;oluto fune OE, liberéque vagan
te, ope trochleæ in F alligatæ adducas pondus ex G in D mul
tò minori labore, quàm &longs;i ex B in C illud raptâ&longs;&longs;es, & ex C
in D &longs;u&longs;tuli&longs;&longs;es. Con&longs;tat autem pondus idem minùs conniti
adversùs lineas FG aut FD, quàm adversùs perpendiculares
HG aut ID, ex iis quæ di&longs;putata &longs;unt lib.
1. cap.
15, ac
propterea etiam minùs dubitari pote&longs;t de trochleæ firmitate.
Hoc autem compendium elevandi pondera perinde, atque
&longs;i per planum inclinatum attollerentur, ea &longs;cilicet &longs;u&longs;pendendo
atque obliquè trahendo, ubi in praxim ritè deduxeris, appa-
neque vincendus &longs;it partium tritus atque conflictus inter pon
dus, ac &longs;ubjectum planum, neque &longs;ternendum &longs;it multo robo
re planum ip&longs;um, quod oneri &longs;u&longs;tinendo non impar &longs;it. At ubi
funem EO, quoad ejus fieri poterit, intenderis, aquá largiter
imbuito; hoc enim fiet, ut &longs;e&longs;e contrahens etiam paulò inten
tior, atque ad de&longs;tinatum opus evadat aptior.
Quæ cum ita &longs;int, alia &longs;e offert methodus elevandi pondera
non levi laboris compendio, &longs;i nimirùm duplex adhibeatur
trochlea, altera quidem in A imminens pon
deri ad perpendiculum, altera verò in B.
Adhibita igitur trochlea B elevabit pondus
ex C in D, ibique totum ex B pendebit:
tùm vici&longs;&longs;im trochleâ A utere, & ex D in E
a&longs;cendet pondus, quod ibi totum ex A pen
debit: iterum igitur adhibe trochleam B, ut
ex E in F a&longs;cendat; atque vici&longs;&longs;im, adhibitâ
trochleâ A a&longs;cendet ex F in G; & &longs;ic de
inceps.
Ubi vides motum ponderis a&longs;cendentis per arcus CDEFG
majorem e&longs;&longs;e quàm &longs;i rectâ ad perpendiculum elevatum fui&longs;&longs;et
ex C in G. Quia verò altitudines perpendiculares &longs;ingulis ar
cubus re&longs;pondentes &longs;ubinde majores fiunt, propterea plus vi
rium à potentia movente adhibendum e&longs;t in progre&longs;&longs;u. Quâ
autem Ratione altitudines illæ perpendiculares cre&longs;cant, faci
lè innote&longs;cet, &longs;i arcuum &longs;ingulorum Sinus ver&longs;os &longs;uis Radiis
re&longs;pondentes ad calculos revocaveris; arcus enim &longs;uperiores &
plurium e&longs;&longs;e graduum, & ex Radio minori, manife&longs;tum e&longs;t:
di&longs;tantia autem parallelarum AC, BD perpendicularium ea
dem &longs;emper e&longs;t; quapropter & æquales lineæ &longs;unt Sinus Recti
arcuum inæqualium in circulis inæqualibus, videlicet arcuum
majorum in circulis minoribus. Quamquam nec omninò ne
ce&longs;&longs;e e&longs;t ità &longs;ingulis tractionibus pondus attollere, ut ad per
pendiculum dependeat, &longs;i maximè trochleæ invicem non mo
dicum di&longs;tarent; &longs;ed &longs;ufficeret alternis operis trochleas agita
re, ut a&longs;cendens pondus modò ad hoc, modò ad illud perpen
diculum accederet, ita tamen ut ultró citróque tran&longs;grediatur
perpendiculum, quod medium cadit inter extremas AC & BD; Cæterùm
&longs;atius e&longs;t A & B parùm di&longs;tare.
Ut autem exemplo aliquo res manife&longs;ta fiat, &longs;tatuamus alti
tudinem AC e&longs;&longs;e pedum 70, di&longs;tantiam verò AB pedum 30,
cui æqualis e&longs;t ea, quæ ex D cadit perpendicularis in AC, &longs;ci
licet DS. Quare in triangulo ASD rectangulo nota e&longs;t Hy
pothenu&longs;a AD, quæ æqualis e&longs;t ip&longs;i AC, & nota e&longs;t Ba&longs;is
SD. Atqui con&longs;tat Perpendiculum AS e&longs;&longs;e medio loco pro
portionale inter &longs;ummam atque differentiam Hypothenu&longs;æ ac
ba&longs;is, &longs;cilicet inter 100 & 40; igitur ducta prima in tertiam,
videlicet ducta &longs;umma in differentiam dabit 4000 Quadratum
Mediæ (hoc e&longs;t perpendiculi AS) cujus Radix ped. 63 1/4 ferè
e&longs;t Perpendiculum AS. Igitur elevatio CS e&longs;t ped. 6 3/4.
Cum itaque BD æqualis &longs;it ip&longs;i AS (jungunt enim paral
lelas æquales AB & SD) iterum in triangulo BVE rectangu
lo nota e&longs;t Hypothenu&longs;a BE ped. 63 1/4, & Ba&longs;is EV e&longs;t ped. 30:
Quare inter &longs;ummam ped. 93 1/4, ac differentiam ped. 33 1/4 media
proportionalis ped. 55. 67″. e&longs;t Perpendiculum BV; atque
adeò elevatio DV e&longs;t ped. 7. 58″. major quàm CS. Et &longs;ic de
reliquis.
At &longs;tatue di&longs;tantiam AB &longs;olùm ped. 20: reperies perpendi
culum AS vix excedere ped. 67; quare elevatio CS erit ped. 3
ferè; ac propterea etiam Perpendiculum BV erit paulò majus
ped. 63. 94″; & elevatio DV ped. 3. 06″; & &longs;ic de cæteris.
Potentiæ verò elevantis motum metitur differentia, quæ
inter lineas BC & BD intercedit: quando autem di&longs;tantia
AB e&longs;t ped. 30, linea BC e&longs;t ped. 76. 15″; at cum e&longs;t ped. 20,
BC e&longs;t ped. 72 4/5. Cum igitur in primo ca&longs;u BD &longs;it ped. 63 1/4,
motus potentiæ e&longs;t ped. (12 9/10); in &longs;ecundo autem ca&longs;u cum BD
&longs;it ped. 67; linea autem BC &longs;it ped. 72 4/5, motus potentiæ e&longs;t
ped. 5 4/5. Quare in primo Ratio motûs Potentiæ ad motum
ponderis e&longs;t (12 9/10) ad 6 3/4, in &longs;ecundo Ratio e&longs;t 5 4/5 ad 3: & factâ
reductione ad alias denominationes, prima Ratio e&longs;t 86 ad 45,
&longs;ecunda Ratio e&longs;t 29 ad 15, quæ &longs;i ad eumdem denominato
rem 45 reducatur, erit 87 ad 45. Con&longs;tat autem majorem e&longs;&longs;e
Rationem 87 ad 45, quàm 86 ad 45. per 8. l. 5. Majorem igi
tur Rationem habet motus Potentiæ ad motum ponderis, quan-
jore; atque adeò major e&longs;t etiam movendi facilitas.
Quòd &longs;i rei hujus minimè dubium experimentum &longs;umere
placeat, ip&longs;i&longs;que oculis rem totam &longs;ubjicere citrà omnem de
ludentis phanta&longs;iæ &longs;u&longs;picio
nem, firmetur in A orbiculus
circà &longs;uum axem ver&longs;atilis, &
ex eo æqualia pondera D & E
funiculo connexa dependeant
ad perpendiculum; quæ prop
ter gravitatis æqualitatem im
mota permanent. Tùm in B
firmetur orbiculus circà &longs;uum
axem pariter ver&longs;atilis, & a&longs;
&longs;umatur pondus C ponderi E
æquale, cui adnectatur funi
culo EBC. Si manu retineas
pondus C, ne gravitet, per
&longs;i&longs;tit pondus E in &longs;uo perpendiculo: jam manu retine
pondus D, ne pror&longs;us moveatur, ac dimitte pondus C, vi
debis hoc quidem de&longs;cendere, pondus verò E a&longs;cendere,
donec ex B dependeat, & in æquilibrio cum pondere C
&longs;ub&longs;i&longs;tat. Iterum retine pondus C, & dimitte pondus D,
pariterque pondus D de&longs;cendens videbis, E verò adhuc
a&longs;cendens; & &longs;ic deinceps u&longs;que eò, dum pondus E uni
cum ambobus D & C æquipolleat, ut &longs;uperiori capite in
dicatum e&longs;t. Id igitur quod à ponderibus D & C præ&longs;tatur,
à quâlibet potentiâ æquali in D & C con&longs;titutâ præ&longs;tari po&longs;&longs;e
manife&longs;tum e&longs;t. Si itaque &longs;implicibus orbiculis fit, ut pondus
æquale po&longs;&longs;it prævalere, multò magis id fiet, &longs;i trochleæ adhi
beantur.
Ex his apparet, quid & in cæteris machinarum generibus,
analogiâ &longs;ervatâ, dicendum &longs;it, ex quarum opportunâ col
locatione facilitas movendi augentur. Si enim, exempli gra
tiâ, cubus A marmoreus elevandus fuerit vecte BC, mul
tò faciliùs id fiet, &longs;i ille &longs;upponatur cubo, quàm &longs;i ex I ad
perpendiculum elevaretur eodem vecte &longs;u&longs;pen&longs;um: ex I &longs;ci
licet totus cubus à vecte &longs;u&longs;tineretur; at &longs;ubjectus vectis
etiam reliquo latere cubus
idem &longs;ubjecto plano incumbat.
Quemadmodum autem non
quemlibet vectem cuilibet
oneri
nes intelligunt; &longs;ed habita ra
tione materiæ, ex quâ con&longs;tat,
congrua &longs;oliditas ei tribuenda
e&longs;t; ita pariter in cæteris omnibus, quæ hùc &longs;pectant (&longs;ive
&longs;int machinarum membra, &longs;ive paxilli &longs;int aut tigilli, quibus
machinæ adnectuntur) materiæ &longs;oliditatem attendendam e&longs;&longs;e
manife&longs;tum e&longs;t, ne frangantur. Et quidem quod ad materiam
attinet, non omnium &longs;olidorum partes pari nexu cohærent,
&longs;ed alia aliis fragiliora &longs;unt: &longs;ic lignum quernum difficiliùs
frangitur, quàm fraxineum aut populeum: neque enim in
omni ligno æque opero&longs;a &longs;imili&longs;que &longs;taminum textura repe
ritur; cum etiam lignum idem quaqua ver&longs;um findi non po&longs;
&longs;it pari facilitate; permagni quippe intere&longs;t, recta ne juxtà
venarum ductum? an obliquè?
&longs;ectio facienda &longs;it.
Id quod
in ip&longs;is quoque lapidibus, atque marmoribus ob&longs;ervare quan
doque nece&longs;&longs;e e&longs;t, ubi non æquè per omnes partes compacta
materia venas habet &longs;ci&longs;&longs;ioni maximè obnoxias. In metallis
pariter eorum natura con&longs;ideranda e&longs;t, molli&longs;ne illa &longs;it, ac
flexibilis? an verò dura?
ut eam, quam &longs;emel induit figu
ram, con&longs;tanter retineat. Ex quo fit, ut pro materiæ di&longs;&longs;i
militudine di&longs;par etiam cra&longs;&longs;ities requiratur: quis enim ne&longs;ciat,
quantum ligneum inter ac ferreum eju&longs;dem molis vectem in
ter&longs;it?
Verùm illud potiùs con&longs;iderandum videtur, quod ad &longs;oli
ditatem ip&longs;am &longs;pectat, etiam&longs;i materies diver&longs;a non &longs;it; pro
variâ enim cra&longs;&longs;itudine mutatur frangendi difficultas; & quia
in mole majori plures in&longs;unt partes divi&longs;ioni re&longs;i&longs;tentes, fran
gendi pariter difficultas augetur pro Ratione multitudinis par
tium, &longs;i cætera paria &longs;int. Dubitare videlicet nemo pote&longs;t à
duplici partium dividendarum numero duplicem oriri re&longs;i&longs;ten
tiam. Si cætera, inquam, &longs;int paria; nam &longs;i filum &longs;ericum ut
rumpatur, requirit vim ut unum, & decem fila &longs;erica paris
decuplam; &longs;i in unum funiculum decem illa fila ritè contor
queantur, multò majorem vim quàm decuplam requiri, ut fu
niculus frangatur, manife&longs;tum e&longs;t: quemadmodum & ligneus
tigillus multo validiùs re&longs;i&longs;tit fractioni, quàm virgarum fa&longs;ci
culus eidem tigillo æqualis; major e&longs;t enim particularum unio,
ubi in unum corpus coale&longs;cant, quàm ubi plura minora corpo
ra con&longs;tituantur.
Hinc &longs;i fuerint duo parallelepipeda quadrata A & B, quorum
latera &longs;int in Ratione quadruplâ, altitudines verò AC, & BD
æquales; con&longs;tat ex 32.
l. 11 ea e&longs;&longs;e inter &longs;e ut ba
&longs;es; ba&longs;es autem &longs;unt qua
drata laterum; igitur pa
rallelepipedum B e&longs;t &longs;ede
cuplum parallelepipedi A.
Finge &longs;exdecim parallele
pipeda ip&longs;i A æqualia in
fa&longs;ciculum colligata, &
&longs;ci&longs;&longs;ionem faciendam jux
ta lineam OS vi oneris in
O po&longs;iti: certum e&longs;t faci
liùs frangi po&longs;&longs;e &longs;exdecim
illa parallelepipeda, quàm
parallelepipedum B illis
omnibus æquale; ut enim &longs;cindatur, curvari oportet vi oneris
incumbentis; illa autem &longs;exdecim faciliùs curvantur quàm
ip&longs;um B. Id quod manife&longs;tum fiat, &longs;i virgam ex &longs;alicto
decerpens, eamque leniter inflectens ob&longs;erves, quâ quidem
parte virga curvata e&longs;t, tenerum corticem in rugas a&longs;&longs;urge
re atque cri&longs;pari, quâ verò parte convexa e&longs;t, corticem
di&longs;trahi atque di&longs;tendi. Ex quo facilè arguimus, quid durio
ribus corporibus contingat, quæ non adeò manife&longs;tè corru
gari po&longs;&longs;unt; flecti &longs;cilicet nequeunt, quin aliqua fiat inte
riorum partium compre&longs;&longs;io, & exteriorum di&longs;tractio. Hinc
in parallelepipedo B, quod flecti intelligitur, ut &longs;cindatur,
partes, quæ circa O, comprimuntur; quæ verò circà S,
di&longs;trahuntur: huic autem motioni repugnant omnes particu-
particulis colligatur. Cum autem &longs;exdecim illa parallelepipe
da minora non &longs;int invicem connexa, quemadmodum particu
læ omnes parallelepipedi B in unam molem coaluerunt, con&longs;tat
pauciores nexus faciliùs, quàm plures, di&longs;&longs;olvi.
Hoc verò ut pleniùs atque apertiùs explicetur, intellige &longs;o
lidum longiu&longs;culum RS in plures tenues laminas plano RI
parallelas divi&longs;um, &longs;ibi
que ita vici&longs;&longs;im con
gruentes, ut earum ex
tremitates con&longs;tituant
planum HI. Omnes
ha&longs;ce laminas &longs;ecun
dùm extremitates ful
cris impo&longs;itas pondus
&longs;uper DC con&longs;titutum
adeò premat, ut cur
vari aliquantulum cogantur. Ob&longs;ervabis illicò extremitates
illas non jam ampliùs in eandem planitiem HI exæquari; &longs;ed
eas quidem laminas, quæ cavitatem &longs;pectant, magis curvari;
minùs verò eas, quæ convexitati re&longs;pondent, ac proptereà ex
timæ laminæ extremitatem ab extremitate intimæ laminæ, quæ
ponderi impo&longs;ito cohæret, magis recedere, quàm interme
diarum extremitates. Con&longs;tat itaque in hoc motu &longs;ingula
rum laminarum particulas, dum curvantur, non iis re&longs;pon
dere adhærentis laminæ particulis, quas priùs contingebant,
cùm omnis curvitatis expertes erant, atque faciliùs potui&longs;&longs;e
&longs;ingulas laminas moveri, quia nullo nexu invicem copulan
tur. Quòd &longs;i ex iis unum &longs;olidum RS planè integrum coa
le&longs;cat, manife&longs;tum e&longs;t planitiem HI permanere, ac propterea,
dum curvatur, nece&longs;&longs;e e&longs;t, ut interiores particulæ invicem
connexæ di&longs;trahantur, cum nequeant aliæ ab aliis &longs;ecedere,
quemadmodum in laminis contingere ob&longs;ervavimus. Hinc
oritur major &longs;olidi, quàm laminarum, re&longs;i&longs;tentia, ne fran
gatur. Non negarim tamen aliquando &longs;atius e&longs;&longs;e duobus me
diocribus tigillis uti, quàm cra&longs;&longs;iore tigno illis æquali; quia
nimirum alterutro labem patiente rima&longs;vè agente, alter faci
liùs integer per&longs;everat; in cra&longs;&longs;iore autem tigno, &longs;i rimam du-
rum aut fibrarum ductum. Cæterum &longs;ublato huju&longs;modi peri
culo, ubi reliqua paria &longs;int, cra&longs;&longs;iora corpora difficiliùs fran
guntur.
Quare &longs;olidorum re&longs;i&longs;tentia, ne frangantur, major e&longs;t
quam pro Ratione &longs;ectionum; hæc &longs;iquidem Ratio &longs;ectionum
&longs;ervari quidem intelligitur, &longs;i limâ aut &longs;errâ &longs;ecari corpora
oporteat; illæ enim tantummodo particulæ re&longs;i&longs;tunt., quæ
&longs;ectionem admittunt; at ubi de fractione agitur, quæ præter
motum particularum, quæ dividuntur, motum etiam aliquem
exigit aliarum, quas comprimi aut di&longs;trahi opus e&longs;t, plus,
minùs, pro Ratione vicinitatis, longè alia e&longs;t Ratio, pro ut
compre&longs;&longs;io illa atque di&longs;tractio particularum faciliùs aut dif
ficiliùs perfici poterit. Hoc autem ex ipsâ figurâ poti&longs;&longs;imùm
pendet: Solidi enim RS &longs;ectio CDE eadem quidem e&longs;t, &longs;i
vè illud circà DE longiorem lineam, &longs;ivè circa CD brevio
rem, curvari debeat, ut frangatur; &longs;ed non eadem e&longs;t in
fractione CD ac in fractione DE frangendi difficultas; nam
cum propiores &longs;int puncto D partes, quæ ad C, quàm quæ
ad E &longs;itæ &longs;unt, con&longs;tat has quidem magis cum circà lineam
CD curvatur &longs;olidum, illas verò, cùm circà lineam DE
curvatur, minùs di&longs;trahi oportere, ut fractio &longs;equatur. Quò
autem magis di&longs;trahi debent particulæ, quæ ex D ver ûs E
recedunt, magis interim comprimi nece&longs;&longs;e e&longs;t eas, quæ ad D
accedunt &longs;ecundùm lineam RO in plano RI. Major igi
tur e&longs;t difficultas, &longs;i circà breviorem lineam CD curve
tur, & fractio &longs;ecundùm longiorem lineam DE &longs;equatur,
quàm &longs;i contrà curvetur circà longiorem DE, & fractio &longs;it
juxtà breviorem CD.
Jam igitur &longs;i duo &longs;olida invicem comparentur, quæ eju&longs;
dem &longs;int materiæ eju&longs;demque longitudinis, & in pari ab ex
tremitatibus di&longs;tantiâ frangi oporteat, &longs;tatuatur in utroque
&longs;olido punctum fractionis, per quod intelligatur planum &longs;e
cans &longs;imiliter inclinatum, facien&longs;que in utroque &longs;olido &longs;uper
ficies, quas vocemus Item planum per quod movetur
Potentia vim frangendi habens, ita productum intelligatur, ut
Ba&longs;ibus prædictis &longs;imili inclinatione occurrens de&longs;cribat &longs;ectio
num lineas, quas vocemus Cra&longs;&longs;ities. Ut &longs;i fuerint duo &longs;oli-
dinis, parieti infixa &longs;ecundùm
æquales partes CI & EH, ut
in punctis I & H fiat fractio,
ex hypothe&longs;i. Si per ea puncta
agantur plana &longs;imiliter inclina
ta, erunt &longs;uperficies IL &
HM, quas vocamus hîc
Jam in extremitatibus D & F
æquè remotis à punctis I & H
&longs;int Potentiæ vim frangendi habentes, & per lineam motûs
huju&longs;modi Potentiarum intelligantur plana cum &longs;imili inclina
tione occurrentia ba&longs;ibus IL & HM, ponamu&longs;que communes
horum planorum &longs;ectiones e&longs;&longs;e lineas parallelas, & æquales li
neis IN & HO; quas &longs;ectiones vocamus
atque pro earum men&longs;urâ u&longs;urpamus lineas IN & HO. Cum
itaque frangendi difficultas oriatur tùm ex numero partium,
quæ &longs;eparandæ &longs;unt, has autem ip&longs;æ Ba&longs;es IL & HM defi
niunt, tùm ex violento motu di&longs;tractionis partium, qui ex ipsâ
&longs;olidorum cra&longs;&longs;itie IN, & HO digno&longs;citur; illud con&longs;equens
e&longs;t, quòd Re&longs;i&longs;tentiæ &longs;olidorum Ratio ea &longs;it, quæ ex Ratione
Ba&longs;ium, & Ratione Cra&longs;&longs;itierum componitur. Hinc e&longs;t quòd
&longs;i Ba&longs;es fuerint &longs;imiles, & quæ e&longs;t Ratio laterum homologo
rum, ea etiam &longs;it Cra&longs;&longs;itierum Ratio, re&longs;i&longs;tentiæ ad fractionem
invicem comparatæ erunt in Ratione triplicatâ laterum homo
logorum; ac propterea cylindrorum re&longs;i&longs;tentia ad fractionem
erit in Ratione triplicatâ Diametrorum, &longs;eu Cra&longs;&longs;itierum.
Hanc, de quâ hactenus nobis &longs;ermo fuit,
tam
plures partes debent præter naturam comprimi, aut di&longs;trahi,
plures &longs;unt re&longs;i&longs;tentiæ; & quò magis hoc motu debent mo
mento eodem præter naturam moveri, eò etiam magis re
&longs;i&longs;tunt: quâ igitur ratione plures &longs;unt re&longs;i&longs;tentes, & quâ Ra
tione magis re&longs;i&longs;tunt, tota re&longs;i&longs;tentiæ ratio componitur; quæ
ex ipsâ corporis &longs;oliditate pendet, nullâ habitâ ratione longi
tudinis ip&longs;ius &longs;olidi: Propterea Nam &longs;i lon
gitudines frangendorum corporum comparemus, quæ &longs;uâ va
rietate mutant frangendi difficultatem, aut facilitatem, re-
pote&longs;t, ut corpus majore re&longs;i&longs;tentiâ ab&longs;olutâ præditum redda
tur magis obnoxium fractioni; longitudo &longs;iquidem auget fran
gendi facilitatem: ideo autem
paratè ad momenta potentiæ &longs;umitur; hæc verò momenta ex
variâ longitudine, &longs;eu di&longs;tantia à puncto fractionis pendere
manife&longs;tum e&longs;t. Sit enim
&longs;olidum AB, quod ita
flectatur, ut fiat fractio
CD: Potentia movens in
B con&longs;tituta dum perficit
&longs;patium BE, di&longs;tractio par
ticularum &longs;olidi fit &longs;olùm
per &longs;patium CD (aut ve
riùs per CHD, nam etiam partes inter C & H di&longs;trahuntur;
Sed hîc claritatis gratiâ &longs;olùm extremæ CD con&longs;iderantur)
quod e&longs;t multo minus &longs;patio BE &longs;ecundùm Rationem HD ad
HE. At &longs;i &longs;olidum frangendum &longs;it AF, aut &longs;i &longs;it totum AB,
tamen Potentia movens &longs;it &longs;olùm applicata in F, Potentia perfi
ciens &longs;patium FG (quod e&longs;t minus quàm BE in Ratione HF
ad HB) major e&longs;&longs;e debet quàm Potentia in B &longs;ecundùm Ratio
nem Reciprocam motuum BE & FG, ut &longs;equatur idem motus
di&longs;tractionis partium CD; nam ex 8. l. 5. minor e&longs;t Ratio FG
ad CD, quàm &longs;it Ratio BE ad eandem CD. Con&longs;tat igitur
à longitudine augeri facilitatem frangendi, ac proinde Re
&longs;i&longs;tentiam hanc Re&longs;pectivam e&longs;&longs;e &longs;ecundùm Reciprocam Ra
tionem longitudinum.
Ex quo obiter apparet, cur &longs;olida Horizonti perpendicularia
magis re&longs;i&longs;tant fractioni, &longs;i potentiæ motus, &longs;eu conatus, &longs;it ad
perpendiculum Horizonti: quia videlicet in huju&longs;modi motu
ad perpendiculum æqualiter moveri oportet Potentiam cum
&longs;olidi particulis, quæ di&longs;trahi aut comprimi debent: ut autem
Potentia &longs;uperet vim re&longs;titivam, aut major e&longs;&longs;e debet Ratio
motûs potentiæ ad motum corporis re&longs;i&longs;tentis, quàm &longs;it Ratio
virium re&longs;i&longs;tendi ad virtutem movendi, aut virtus movendi ab
&longs;olutè major e&longs;&longs;e debet vi re&longs;i&longs;tendi: Cum itaque in motu per
pendiculari intercedere non po&longs;&longs;it motuum inæqualitas, ne
ce&longs;&longs;e e&longs;t virtutem movendi vehementer augeri, ut &longs;uperet vim,
hantur, aut comprimantur.
Hinc ex ha&longs;tâ ad perpendiculum &longs;u&longs;pensâ pendebit ingens
&longs;axum, & tigillum perpendiculariter terræ in&longs;i&longs;tentem pre
met moles, penè dixerim, immen&longs;a, citrà ha&longs;tæ aut ti
gilli fractionem: quia omnes ha&longs;tæ atque tigilli partes &
æqualiter cum onere &longs;u&longs;pen&longs;o aut incumbente moveri de
berent, & omnes æqualiter re&longs;i&longs;tunt di&longs;tractioni aut com
pre&longs;&longs;ioni: At &longs;i ad horizontem inclinata aut parallela fue
rint huju&longs;modi &longs;olida (ha&longs;ta videlicet atque tigillus) non
e&longs;t æqualis omnium partium di&longs;tractio aut compre&longs;&longs;io, mi
nùs enim di&longs;trahuntur, quæ puncto H proximæ &longs;unt, quam
quæ ad D accedunt (concipe H in media cra&longs;&longs;itie) con
trà verò illæ magis, hæ minùs comprimuntur; quemad
modum neque motui di&longs;tractionis aut compre&longs;&longs;ionis e&longs;&longs;et
æqualis motus oneris deorsùm urgentis in ha&longs;tæ, vel tigil
li non perpendicularium extremitate con&longs;tituti, &longs;ed multò
major e&longs;&longs;et hîc oneris motus. Quoniam verò rerum natu
ra magis repugnat corporum penetrationi, ad quam quodam
modo accedere videtur compre&longs;&longs;io, quàm corporum unito
rum divi&longs;ioni, ubi vacui metus ab&longs;it; hinc e&longs;t majorem
molem faciliùs &longs;u&longs;tineri à fulcro ad perpendiculum &longs;ubjecto,
quàm &longs;u&longs;pendi ex &longs;olido perpendiculari citrà fractionis pe
riculum. Quamvis negandum non &longs;it ad huju&longs;modi facili
tatem, quam experimur in &longs;u&longs;tinendo potiùs, quàm in re
tinendo onere, conferre plurimum, quòd tellus, cui ful
crum infigitur, demùm non &longs;ub&longs;idit; at laqueare &longs;eu for
nix ex quo &longs;olidum pendet onere prægravatum, tantam
gravitatem non ita facilè ferre pote&longs;t. Quare ad tollenda
in &longs;uperiores ædificiorum partes ingentia &longs;axa multo cau
tiùs atque tutiùs ij operantur, qui longam trabem, aut plu
ra tigna ritè connexa, qua&longs;i navis malum rudentibus u&longs;
quequaque firmatum, ne à perpendiculo deflectat, &longs;ta
tuunt, cui &longs;uperiorem trochleam adnectant; quàm qui tra
bem Horizonti parallelam parieti infigunt ad idem munus
præ&longs;tandum; hæc &longs;iquidem horizonti parallela magis fractio
ni obnoxia e&longs;t, quàm perpendicularis; præterquam quod
parietem aliquatenus labefactare pote&longs;t, cum habeat ratio-
ni&longs;i huic periculo ex arte obviam eatur.
Comparatis itaque invicem &longs;olidorum frangendorum lon
gitudinibus, hoc e&longs;t intervallis inter fractionum puncta &
locum, ubi potentia vim frangendi habens con&longs;tituta intel
ligitur, quò major e&longs;t longitudo, eò minor e&longs;t re&longs;i&longs;tentia
&longs;olidi, ne frangatur. Qua propter ubi duo data &longs;olida con
ferantur, quæcumque demùm illa &longs;int, non &longs;olùm eorum
Re&longs;i&longs;tentia Ab&longs;oluta, quæ ex Rationibus Ba&longs;ium, & Cra&longs;
&longs;itierum componitur, attendenda e&longs;t, &longs;ed etiam Re&longs;i&longs;tentia
Re&longs;pectiva, quæ ex longitudinibus pendet: atque adeò
adæquata Ratio re&longs;i&longs;tentiæ, ne frangantur, ea e&longs;t, quæ
componitur ex Rationibus Ba&longs;ium & Cra&longs;&longs;itierum atque ex
Ratione longitudinum Reciprocè &longs;umptarum: cùm enim
longitudini majori re&longs;pondeat minor re&longs;i&longs;tentia, manife&longs;tum
e&longs;t longitudinum Rationem e&longs;&longs;e Reciprocè &longs;umendam, ut
re&longs;i&longs;tentiæ, quæ ex illis oritur, Ratio habeatur. Hinc e&longs;t
fieri aliquando po&longs;&longs;e, ut &longs;olidum cra&longs;&longs;ius minùs re&longs;i&longs;tat
fractioni, quàm &longs;ubtilius, &longs;i hoc breve &longs;it, illud verò valdè
longum, &longs;i videlicet longitudo cra&longs;&longs;ioris ad longitudinem
&longs;ubtilioris Rationem habeat majorem, quàm &longs;it ea, quæ ex
Rationibus Ba&longs;ium, & Cra&longs;&longs;itierum componitur. Sic &longs;i duo
fuerint cylindri, & alter triplo cra&longs;&longs;ior fuerit reliquo, &longs;ed
etiam trigecuplo longior fuerit illo, minùs etiam fractioni
re&longs;i&longs;tet; quia re&longs;i&longs;tentia ab&longs;oluta majoris cylindri ad mino
rem e&longs;t ut 27 ad 1, &longs;ed re&longs;i&longs;tentia Re&longs;pectiva eju&longs;dem ma
joris ad minoris re&longs;i&longs;tentiam pariter re&longs;pectivam e&longs;t ut 1 ad
30: Ratio ergo ex his Rationibus 27 ad 1, & 1 ad 30
Compo&longs;ita, e&longs;t Ratio 27 ad 30, hoc e&longs;t 9 ad 10, ac propterea
major cylindrus re&longs;i&longs;tit fractioni ut 9, minor verò fractioni
re&longs;i&longs;tit ut 10.
De&longs;ine jam mirari, &longs;i quando paxillum maximis viribus
re&longs;i&longs;tere videris; quia nimirùm potentia, quæ motum co
natur, proximè applicata e&longs;t parieti aut plano, cui paxil
lus infigitur: quòd &longs;i remotior illa fuerit, etiam minùs hic
re&longs;i&longs;tet. Sic defixo in terram paxillo AB, cui funis AC al
ligatur, experientia docet paxillum eò re&longs;i&longs;tere validiùs, quò
propiùs ad A alligatur funis, debiliùs autem re&longs;i&longs;tere, quò
in A nimirùm motus
potentiæ trahentis vix
excederet motum pa
xilli, qui ibi flectere
tur ex hypothe&longs;i; at
fune in B po&longs;ito, po
tentia ibi con&longs;tituta,
& per funem applica
ta multò velociùs mo
veretur, quàm paxilli
partes propè A, quæ
ibi flecterentur.
Quòd &longs;i loci conditio, aut ip&longs;a oneris movendi con&longs;titutio
id exigat, ut funis propè B alligetur, & de paxilli AB firmi
tate dubitetur, paxillum alterum DE paulò remotiorem com
modo loco depange ita, ut funis primùm in D firmetur, de
inde circa B convolutus extendatur, pro ut operis faciendi ra
tio fieret.
Eâdem ratione &longs;i tigillus, ex quo onus dependere debet, pa
rieti &longs;it infixus, & &longs;it GH, fractioni magis erit obnoxius, quò
propiùs accedet pondus ad H:
propterea aut ei &longs;ubjicitur brevior
tigillus IR omninò contiguus,
aut &longs;upponitur fulcrum OS in
clinatum; quod fractionem eò va
lidiùs impediet, quò minùs di&longs;ta
bunt H & S, & quò acutior fue
rit angulus, quem fulcrum SO
cum pariete con&longs;tituit, &longs;eu, quod
eôdem recidit, quò magis ad
recti anguli quantitatem acce
det angulus GSO. Quæ omnia
ita ex dictis aperta &longs;unt, ut ulte
riori explicatione non egeant.
Sed & illud hîc, ubi de Re&longs;i&longs;tentiâ Re&longs;pectivâ &longs;ermo e&longs;t,
adjiciendum videtur, quòd ex &longs;olâ majori longitudine hæc non
minuitur, ni&longs;i cùm longitudo &longs;olidi ad perpendiculum in&longs;i&longs;tit
&longs;ubjecto plano; & tantùm Potentia oblique atque in tran&longs;
ver&longs;um trahens applicata extremitati longioris &longs;olidi plus ha
bet momenti, quàm applicata extremitate brevioris, quin
velociùs, & faciliùs movetur &longs;ecundùm Rationem longitu
dinum illarum. At quando &longs;olida &longs;unt horizonti parallela,
aut ad illum ita inclinata, ut centrum gravitatis partis illius,
quæ erumpit ex corpore, cui &longs;olidum infigitur, non immi
neat ba&longs;i &longs;u&longs;tentationis, non &longs;ola longitudo attendenda e&longs;t,
&longs;ed & ip&longs;a gravitas, quæ etiam nullo addito extrin&longs;eco mo
tore &longs;ua habet momenta, quibus deor&longs;um connititur. Ex
quo fit pro majori gravitate etiam frangendi facilitatem au
geri, ip&longs;a nimirum gravitas e&longs;t potentia conjuncta, quæ au
getur pro ratione materiæ; materia autem augetur pro ra
tione longitudinis (cætera &longs;iquidem paria e&longs;&longs;e hîc claritatis
gratiâ, ponamus) ac propterea longius pri&longs;ma comparatum
cum breviori pri&longs;mate, eo quòd majorem habeat gravita
tem, minùs re&longs;i&longs;tit fractioni &longs;ecundùm Reciprocam Ratio
nem longitudinum. Atqui Ratio motûs huju&longs;modi Potentiæ
conjunctæ e&longs;t &longs;ecundùm Rationem longitudinum, & ex
dictis Ratio Re&longs;i&longs;tentiæ in ordine ad huju&longs;modi motum e&longs;t
permutatim ac Reciprocè &longs;ecundùm eandem longitudinum
Rationem: igitur Ratio duplicatur, & re&longs;i&longs;tentia longioris
ad re&longs;i&longs;tentiam brevioris e&longs;t &longs;ecundùm &longs;ubduplicatam Ratio
nem longitudinum reciprocè &longs;umptarum. Id quod etiam
hinc con&longs;tat, quia cùm &longs;ingula illius longitudinis puncta
&longs;uam habeant gravitatem, &longs;ua omnibus in&longs;unt momenta pro
Ratione di&longs;tantiæ à puncto quod e&longs;t veluti centrum motûs;
ergo aggregata momentorum &longs;unt ut &longs;ectores ab illis longi
tudinibus tanquam à Radiis de&longs;cripti: &longs;unt autem &longs;imiles
&longs;ectores in duplicatâ Ratione Radiorum. Quare &longs;i longitudi
nes &longs;int ut 3 ad 2, Re&longs;i&longs;tentia re&longs;pectiva longioris ad re&longs;i&longs;ten
tiam brevioris e&longs;t ut 4 ad 9. Tota igitur &longs;olidorum re&longs;i&longs;ten
tia, ne frangantur, componitur ex Rationibus Ba&longs;ium, &
Cra&longs;&longs;itierum, & ex &longs;ubduplicatâ Ratione longitudinum per
mutatim ac reciprocè &longs;umptarum.
Ex his itaque, quæ de &longs;olidorum re&longs;i&longs;tentiâ, ne frangan
tur, hactenùs di&longs;putata &longs;unt, conjecturam facilè accipiet
quæque machinarum membra, quóve loco collocanda &longs;int,
ut & materia & forma re&longs;pondeant fini, in quem machinæ
de&longs;tinantur: neque enim &longs;atis e&longs;t concinno, & eleganti dia
grammate machinam oculis repræ&longs;enta&longs;&longs;e, eju&longs;que vires ad
calculos revocâ&longs;&longs;e, quantum quidem ex machinæ figurâ col
ligitur, &longs;i demùm, in&longs;tituto motu machina pondere prægra
vata luxetur.
Illud tamen præterea Machinator animadvertat, oportet,
quod &longs;pectat ad momenta virium, quas potentia movens
exercet; neque enim &longs;ola ponderis gravitas machinam, aut
corpus, cui machina alligatur, aut innititur, urget aut pre
mit, &longs;ed & ip&longs;a potentia, dum adversùs ip&longs;um pondus co
natur machinam movens, aliquando auget gravitatem ex
oppo&longs;itâ parte, adeò ut & huic & ponderi re&longs;i&longs;tere debeat
machina, aut id, quod machinam retinet. Si enim fuerit
vectis AB in
nixus &longs;uper ba
culum CD, ex
B pendeat glo
bus plumbeus
E, & extremi
tas A quie&longs;cat
aliquo corpore
retinente, ut &longs;i
fuerit parieti in
fixa; &longs;olo globo E gravitante minus periculum &longs;ube&longs;t fractio
nis tùm vectis, tùm baculi CD &longs;u&longs;tentantis, quàm &longs;i in A
&longs;it potentia F; cujus conatus deor&longs;um oppo&longs;itus conatui de
or&longs;um ponderis E faciliùs curvitatem, aut etiam demùm
fractionem vectis efficere pote&longs;t in I, ut patet; immò & ba
culus CD &longs;u&longs;tentans vectem, non &longs;olùm momenta ponderis E,
&longs;ed & momenta Potentiæ F, quæ in I uniuntur, in &longs;e recipit;
atque adeò utri&longs;que ferendis par e&longs;&longs;e debet.
Simile quiddam ob&longs;ervare e&longs;t, &longs;i ex orbiculo O, in clavo
M &longs;u&longs;pen&longs;o, circà &longs;uum axem ver&longs;atili, dependeat pondus S,
& Potentia in R deor&longs;um conata cogat pondus S a&longs;cendere:
certum e&longs;t enim ab axe orbiculi, & à clavo M &longs;u&longs;tineri non
&longs;olùm pondus S, &longs;ed & Poten
tiam, quæ e&longs;t in R. Contrà ve
rò &longs;i orbiculus V &longs;it adnexus pon
deri T, funis autem orbiculo in
&longs;ertus alligetur clavo in N, & po
tentia P &longs;ur&longs;um trahat, con&longs;tat ab
axe quidem orbiculi &longs;u&longs;tineri &longs;o
lum pondus T; à clavo verò N
non totum pondus T &longs;u&longs;tineri,
&longs;ed ejus &longs;emi&longs;&longs;em, nam etiam Po
tentia P &longs;u&longs;tinet pondus. Validior
igitur e&longs;&longs;e debet clavus M quàm
clavus N, hic enim ponderis &longs;e
mi&longs;&longs;em fert, ille verò plus quàm
duplum. Potentia enim R major
e&longs;t pondere S.
Quòd &longs;i tàm pondera S & T,
quàm clavi M & N, atque Po
tentiæ R & P non in plano Ver
ticali, &longs;ed in Horizontali con&longs;tituantur, certum e&longs;t pondera
S & T non &longs;u&longs;pen&longs;a &longs;ed jacentia, nihil adversùs clavos M &
N; aut adversùs &longs;uorum orbiculorum O & V axes conari, im
mò neque adversùs Potentias R & P; quandoquidem toto ni&longs;u
plano &longs;ubjecto incumbunt, nullámque exercent Activam Re
&longs;i&longs;tentiam; &longs;ed Formalem tantummodo, quâ repugnent Po
tentiis moventibus: quæ quidem re&longs;i&longs;tentia, tùm ex ip â pon
derum gravitate, tùm ex attritu &longs;ubjecti plani componitur.
Clavorum igitur M & N ea &longs;it, oportet, &longs;oliditas atque firmi
tas, quæ potentiarum R & P conatibus re&longs;pondeat; ne forte
clavi ip&longs;i frangantur faciliùs, aut revellantur, quàm pondera
&longs;uo loco dimoveantur. Sed hæc innui&longs;&longs;e &longs;at fuerit, ut &longs;ingula
diligenter à machinatore circum&longs;picienda e&longs;&longs;e intelligatur; ne
que tamen in his ad nau&longs;eam diutiùs immorandum.
an componere.
EX iis, quæ de Machinarum viribus di&longs;putata &longs;unt &longs;atis
liquet nullum dari finitum Pondus quod data Potentia mo
vere non po&longs;&longs;it &longs;i congruens machina adhibeatur: cum etenim
data &longs;it Ratio Ponderis ad Potentiam, eo artificio Machina
di&longs;ponatur, ut Ratione illâ datâ fiat major Ratio motûs Potentiæ
ad motum Ponderis; & Pondus cedet Potentiæ moventi. Sic
vici&longs;&longs;im &longs;i oblata fuerit machina, examinandus primùm e&longs;t lo
cus, ubi Potentia applicanda e&longs;t, ubi Pondus collocandum;
tùm utriu&longs;que motûs rationes ineundæ: & pronunciabis majo
rem requiri rationem Potentiæ ad Pondus, quàm &longs;it Ratio mo
tûs Ponderis ad motum Potentiæ. Sit enim ex.
gr. motuum hu
ju&longs;modi Ratio, quæ e&longs;t 3 ad 8; Potentia vim movendi habens
ut 3 non movebit Pondus, cujus vis re&longs;i&longs;tendi, & momentum,
&longs;it ut 8; &longs;ed opus e&longs;t, ut illa major &longs;it quàm 3. At neque Po
tentiam augere potes, ut oportet, neque Ponderi quicquam de
trahere: vide igitur utrum fieri po&longs;&longs;it, ut mutetur in machinâ
motuum Ratio, aut Potentiæ motum augendo, aut ponderis
motum minuendo.
Hinc manife&longs;tum e&longs;t machinam majorem non plus afferre
facilitatis præ minore, &longs;i illæ quidem omninò &longs;imiles fuerint
(modò utraque &longs;atis &longs;olida &longs;it, ne fractioni &longs;it obnoxia) mo
tuum enim Ratio eadem e&longs;t in utráque. Sic Vectis 100 pal
morum &longs;i ita ab hypomochlio di&longs;tinguatur in partes ut hinc
palmos 20, hinc 80 relinquat, non majorem movendi faci
litatem præbebit, quàm vectis palmorum quinque ita divi
&longs;us ab hypomochlio, ut hinc palmus unus, hinc verò quatuor
relinquantur. Ut igitur longior ille Vectis utilior accidat, &longs;i
hypomochlium quidem transferri queat, remove illud à Po
tentiâ, & admove Ponderi, motuumque Ratio augebitur; pa
tet &longs;cilicet majorem e&longs;&longs;e Rationem 85 ad 15, quam 80 ad 20:
Quod &longs;i verò hypomochlium ita fixum &longs;it ac vecti adnexum,
palmi 80 ut priùs, hinc autem &longs;int palmi (14 2/17), & eadem erít
Ratio, quæ e&longs;t 85 ad 15. Quare breviore vecte plus ponderis
movebis, quàm longiore; vis enim, quæ longiore illo 100 pal
morum movebat pondus librarum 100, breviore hoc palmo
rum (94 2/17) movebit libras 141 2/3: Quia quamvis in utroque Vecte
hypomochlium habente po&longs;t palmum octuage&longs;imum, Potentia
eodem &longs;emper motu moveatur, non tamen idem e&longs;t ponderis
motus, qui in minore vecte minor e&longs;t, in majore major, ac
proinde motûs Potentiæ ad motum Ponderis Ratio major e&longs;t in
minore, minor in majore vecte. Quod &longs;i demùm nec hypo
mochlium transferre, nec vecte mutilato uti liceat, licebit &longs;a
nè fu&longs;tem, vel quid &longs;imile, firmiter ad alligatum Vecti adjun
gere, potentiamque ab hypomochlio longiùs removere: opor
teret autem additamentum huju&longs;modi e&longs;&longs;e palmorum 33 1/3; nam
ut 15 ad 85, ita 20 ad 113 1/3; adeóque totus vectis e&longs;&longs;et pal
morum 133 1/3.
Porrò hîc ob&longs;erva, quantò facilius &longs;it ponderis motum mi
nuere, quàm potentiæ motum augere: in allato &longs;iquidem
exemplo, manente eodem potentiæ motu, minuitur ponderis
motus decurtato vecte ac diminuto palmis (5 15/17); manente au
rem eodem ponderis motu augetur Potentiæ motus acuto vecte
palmis 33 1/3: Quia nimirum in Ratione majoris Inæqualitatis &longs;i
Con&longs;equens terminus minor minuatur, aut Antecedens termi
nus major augeatur, fit adhuc major Inæqualitas; ut autem
eadem Ratio &longs;ervetur aucto Antecedente ac diminuto Con&longs;e
quente, manife&longs;tum e&longs;t, quæ pars Con&longs;equentis integri e&longs;t
con&longs;equens diminutus, eam debere e&longs;&longs;e partem Antecedentis
aucti Antecedentem datum: atqui Antecedens datus e&longs;t major
dato Con&longs;equente; igitur plus addendum e&longs;t Antecedenti,
quàm dematur Con&longs;equenti. Sic data &longs;it Ratio 8 ad 6: Con
&longs;equens bifariam &longs;ecetur, eju&longs;que &longs;emi&longs;&longs;is fiat novus Con&longs;e
quens; erit Ratio 8 ad 3 majoris adhuc inæqualitatis; hæc enim
e&longs;t dupla &longs;uperbipartiens tertias, illa verò erat &longs;olùm &longs;e&longs;qui
tertia. Ut igitur retento priori Con&longs;equente 6 fit eadem Ratio
dupla &longs;uperbipartiens tertias, &longs;icut Con&longs;equens fuit bifariam
divi&longs;us, ita datus Antecedens 8 e&longs;t duplicandus, ut &longs;it Ratio
quàm &longs;it &longs;emi&longs;&longs;is Con&longs;equentis minoris qui demitur. In re au
tem no&longs;trâ &longs;emper Ratio motûs Potentiæ per machinam vali
dioris factæ ad motum dati ponderis e&longs;t Ratio Majoris inæqua
litatis: Quapropter &longs;atius e&longs;t Ponderis motum minuere, quam
potentiæ motum auctâ machinâ augere.
Hæc quidem, quæ in vecte propo&longs;ita facilè ac in promptu
e&longs;t per&longs;picere, in cæteris pariter mechanicis Facultatibus, ut
in Trochleis, Cochleâ, & reliquis intelligenda &longs;unt, ut ex iis,
quæ inferiùs dicentur, &longs;uo loco manife&longs;tum fiet. Sed quoniam
ad ponderis motum extenuandum certos quo&longs;dam fines ip&longs;a
machinarum materia præ&longs;cribit; neque enim quemadmodum
quantitatem omnem, & corporum molem in &longs;ubtiliores, ac
&longs;ubindè &longs;ubtiliores partes mente concidimus, ita etiam id re
ipsâ perficere atque in praxim deducere po&longs;&longs;umus: propterea
ut plurimum cogimur Potentiæ velociorem motum conciliare,
ut majorem obtineat Rationem ad motum Ponderis. Quis ete
nim non inca&longs;&longs;um uti po&longs;&longs;it Vecte, cujus hypomochlium à
pondere &longs;atis gravi non ampliùs di&longs;tet, quàm per digiti &longs;emi&longs;
&longs;em? aut Cochleam adhibere, cujus &longs;piras intervallum capilla
ceum &longs;ecernat?
Verùm cum id duplici methodo præ&longs;tare po&longs;&longs;imus, videlicet
aut Machinam ip&longs;am, &longs;pecie non mutatâ, augentes, aut illam
ex pluribus membris componentes, &longs;ive eju&longs;dem generis &longs;int,
&longs;ive diver&longs;i; operæ pretium fuerit perpendere, maju&longs;-ne in
augmento? an verò in compo&longs;itione?
compendium inveniatur.
dem Facultatis &longs;pecies immutata permanet, factâ &longs;olum partis
alicujus acce&longs;&longs;ione; ut &longs;i, quia Vectis ju&longs;to brevior e&longs;t, Poten
tiæ ab hypomochlio di&longs;tantiam longiorem facias; cum Tro
chleæ adhibeantur oneri movendo impares, amplificatis locu
lamentis orbiculorum numerum augeas; quia Cochlea ob &longs;pi
rarum raritatem minùs valida e&longs;t quàm oporteat, lineam ip&longs;am
ita inclines, ut &longs;pi&longs;&longs;ioribus &longs;piris circumducatur. At verò
po&longs;ita
adjiciuntur, aut generis eju&longs;dem, ut cum Vectis Vecti, Co
chleæ Cochlea, Trochleis Throchleæ adjunguntur; aut diver
&longs;i generis, ut cum facultates ip&longs;æ permi&longs;centur, vecti trochleas,
gendo. Prioris Compo&longs;itionis intrà idem genus &longs;pecimen ali
quod exhibui in
&longs;uis locis de eâ redibit &longs;ermo: Po&longs;terioris autem Compo&longs;itionis
diver&longs;arum Facultatum, ubi de &longs;ingulis di&longs;purabimus, exem
pla aliqua &longs;ubjiciemus, ut di&longs;cat Tyro Machinarum vires ritè
ad calculos revocare, &longs;olertiamque machinandi acquirat.
Quamvis autem quæ&longs;tio hæc multò dilucidiùs explicaretur,
&longs;i unamquamque Facultatem &longs;ingillatim attingeremus, quàm
&longs;i unâ comprehen&longs;ione omnia complectamur; hîc tamen
doctrinæ ratio exigit, ut dimi&longs;&longs;is rivulis fontem ip&longs;um aperia
mus, ex quo in Machinam Compo&longs;itam vis major, quàm in
Amplificatam, majore compendio derivatur. Et quidem cum
res tota ex potentiæ atque Ponderis motuum Ratione pendeat,
quamdiu in &longs;implici aliquâ facultate con&longs;i&longs;timus, motus Po
tentiæ ad motum Ponderis &longs;implicem habet Rationem; &longs;i verò
Facultas una cum aliâ quâpiam facultate conjungitur, atque
connectitur, jam Potentiæ motus ad motum ponderis eam ha
bet Rationem, quæ ex &longs;ingularum facultatum rationibus com
ponitur. Voco autem
inter ip&longs;os Potentiæ ac Ponderis motus intercederet, &longs;i facul
tas illa &longs;olitaria adhiberetur; Atqui Ratio hæc motuum in &longs;in
gulis Facultatibus modum recipit ex Facultatis ip&longs;ius partibus,
quarum altera ad Potentiam, àd Pondus altera &longs;pectare vide
tur; ut per &longs;ingulas Facultates eunti con&longs;tabit. In Vecte enim
Ponderis ab hypomochlio di&longs;tantia pertinet ad Pondus, Poten
tiæ autem di&longs;tantia ab eodem hypomochlio penes potentiam
e&longs;t: In Trochleis ip&longs;arum Trochlearum di&longs;tantia Pondus re&longs;pi
cit; funis autem explicatio Potentiam: In Axe in Peritrochio
cra&longs;&longs;ities Axis Ponderi, Peritrochij amplitudo Potentiæ tribui
tur: In Cuneo longitudo ad Potentiam &longs;pectat, cra&longs;&longs;ities ad
Pondus: In Cochleâ demùm &longs;piræ circumductæ perimeter ad
Potentiam attinet, extremitatum &longs;piralis lineæ intervallum, ad
Pondus. Manife&longs;tum e&longs;t igitur, ubi &longs;implex motuum Ratio in
&longs;ingulis Facultatibus augenda fuerit, manente eâ parte, quæ
ad Pondus &longs;pectat, nece&longs;&longs;ariò ita augendam e&longs;&longs;e partem reli
quam, quæ Potentiæ tribuitur, ut majori illi motuum Rationi
re&longs;pondeat. Sic dato Vecte palmorum &longs;ex, quo potentia mo-
deris ab hypomochlio di&longs;tantia, & motuum Ratio e&longs;&longs;e debeat
vigecupla, &longs;atis con&longs;tat totum vectem requiri palmorum 21, ut
unus Ponderi cedat, Potentiæ autem viginti.
At verò &longs;i motuum Ratio ex Rationibus componenda &longs;it, &longs;a
tisfuerit datæ Facultati minorem Rationem continenti, quàm
oporteat, Facultatem aliam adjicere, cujus Ratio cum priori
Ratione compo&longs;ita quæ&longs;itam Rationem con&longs;tituat. Sic dato
Vecti quintuplam rationem continenti adjunge aliam quamli
bet facultatem quadruplæ Rationis; ex quadruplâ enim Ratio
ne & quintuplâ componitur Ratio vigecupla quæ&longs;ita. Ita au
tem &longs;ecunda hæc Facultas priori Facultati adnectenda e&longs;t, ut
quemadmodum duorum Magnetum oppo&longs;iti poli junguntur,
Au&longs;tralis videlicet unius Aquilonari alterius, &longs;ic duarum Fa
cultatum oppo&longs;itæ partes connectantur, ut &longs;cilicet quo loco ad
priorem Facultatem applicanda e&longs;&longs;et Potentia, eidem admo
veatur locus Ponderi in &longs;ecundâ Facultate de&longs;tinatus: proinde
&longs;iquidem &longs;e res habebit, atque &longs;i pondus diminutum pro Ra
tione prioris facultatis, videlicet &longs;ub quintuplum, in &longs;ecun
dam hanc Facultatem transferretur, in quâ ejus motus ad mo
tum Potentiæ Rationem haberet &longs;ubquadruplam: re enim ve
râ duabus hi&longs;ce Facultatibus junctis, Potentiæ motus vigecu
plus e&longs;t ad motum Ponderis; nam Pondus in vectis extremita
te alterâ con&longs;titutum quintuplo tardiùs movetur, quàm reli
qua vectis extremitas; hæc autem po&longs;teriori Facultati loco
Ponderis adjuncta quadruplo tardiùs movetur quàm Poten
tia; igitur Ponderis motus vigecuplo tardior e&longs;t motu Po
tentiæ.
Statuamus exempli gratiâ &longs;ecundam hanc Facultatem Vecti
adjunctam e&longs;&longs;e pariter Vectem eju&longs;dem generis quinque pal
morum ita ab hypomochlio di&longs;tinctum in partes, ut hæ in qua
druplâ &longs;int Ratione: Ecce quanto compendio rem a&longs;&longs;equamur;
id enim quod &longs;implici Vecte palmorum 21 præ&longs;tandum e&longs;&longs;et,
compo&longs;itis vectibus duobus altero palmorum &longs;ex, altero palm.
quinque perficimus, &longs;ervatâ &longs;emper eâdem Ponderis ab hypo
mochlio di&longs;tantiâ, nimirum palmi unius. Hæc tamen de duo
bus hi&longs;ce vectibus dicta ita intelliges velim, ut ad motum &longs;im
pliciter pertineant; non verò ad motûs quantitatem; &longs;atis enim
&longs;ecundo hoc vecte, ad quam promoveretur Vecte palmorum 21:
Verùm hîc &longs;ola movendi facilitas con&longs;ideratur. Quòd &longs;i non
alterum Vectem adhibeas; &longs;ed aliud facultatis genus, ut Tro
chleas binis orbiculis in&longs;tructas, & Vecti in loco Potentiæ ad
nexas, multò adhuc faciliùs movebitur Pondus, cujus motus
erit &longs;ubvigecuplus motûs Potentiæ funem Trochlearum tra
hentis, & tantus erit Ponderis motus, quantus e&longs;&longs;et, &longs;i extre
mitati Vectis palmorum &longs;ex apponeretur Potentia quadrupla
datæ Potentiæ. Idem planè de cæteris dicendum Faculta
tibus.
Hinc manife&longs;tum e&longs;t compo&longs;itis tribus, quatuorve, aut plu
ribus Facultatibus, Rationem Compo&longs;itam motus potentiæ ad
motum Ponderis fieri multò majorem; cui &longs;i æqualem Ratio
nem habere velimus unicâ atque &longs;implici Facultate, hujus
magnitudinem aliquando enormem fieri nece&longs;&longs;e e&longs;&longs;et; ut &longs;uis
locis infrà declarabitur.
In eo igitur elucebit Machinatoris indu&longs;tria, &longs;i Facultates
ip&longs;as aptè congruenterque di&longs;ponat, atque permi&longs;ceat, &longs;pecta
tâ materiæ &longs;oliditate, &longs;patij amplitudine, Ponderis po&longs;itione,
Potentiæ virtute, temporis ad movendum conce&longs;&longs;i opportuni
tate: hæc enim omnia attenti&longs;&longs;imè perpendenda &longs;unt; ne, dum
nimis &longs;ollicitè laborem imminuere &longs;tudet, motum plus æquo
imminuens, tardioremque efficiens temporis jacturam faciat,
aut totum &longs;patium machina implens in eas angu&longs;tias Potentiam
moventem conjiciat, ut motum expeditè perficere nequeat.
ONera &longs;i ex alio in alium locum deportanda fuerint, gemi
no labore opus e&longs;t, conatu videlicet, quo &longs;u&longs;tineantur,
& impetu, quo transferantur: proptereà &longs;atius e&longs;t ita res di&longs;po
nere, ut vires omnes ad transferendum exerceantur, citrà co
natum &longs;u&longs;tinendi; ut eâ ratione vel gravius onus vel idem mul-
pariter atque transferre cogeretur. Quoniam verò (cum one
ra &longs;ubjecto plano impo&longs;ita illud premant, atque tùm onerum
tùm &longs;ubjecti plani facies, quæ &longs;e invicem contingunt, non ita
læves &longs;int, ut partes omnes in rectum directæ nihil habeant
a&longs;peritatis; quin immò ut plurimum, & &longs;alebris impedita via
&longs;it, & movendi corporis partes aliæ præ aliis extent atque emi
neant) ex mutuo prominentium particularum tritu atque con
flictu difficultas ad movendum oriretur; idcircò optimo con&longs;i
lio factum e&longs;t, ut oneribus ip&longs;is &longs;ubjiciantur Cylindri aut Rotæ,
quæ dum in gyrum aguntur, conflictum illum partium tollunt,
qui vitari non po&longs;&longs;et, &longs;i onera &longs;uper plano raptarentur. Hinc Ci
&longs;ia, Sarraca, Vehes, Carri & genus omne plau&longs;trorum. Id quod
etiam homines ip&longs;i, ut terre&longs;tre iter commodiùs habeant, &
minori jumentorum labore illud perficiant, quàm &longs;i iis in&longs;i
dentes veherentur, &longs;uos in u&longs;us retulerunt: Hinc Belgæ &longs;ua
e&longs;&longs;eda, Galli petorita & rhedas, Hi&longs;pani pilenta, Itali carpen
ta; & pro &longs;uâ qui&longs;que voluntate diver&longs;a vehiculorum genera
excogitârunt, quæ &longs;ubjectis rotis aguntur: dum enim Rota
convertitur, eju&longs;que curvaturæ partes aliis atque &longs;ubinde aliis
&longs;ubjectæ planitiei partibus aptantur, adeóque currus promove
tur, &longs;olus rotæ modiolus axis ambitum axungiâ lubricum terit;
ex quo tritu aut nulla aut levis mora motui infertur.
Illud autem e&longs;t omnibus explorati&longs;&longs;imum, & quotidiano ex
perimento confirmatum, quo majoribus rotis in&longs;tructi currus
(ni&longs;i di&longs;crimen aliquod in cæteris intercedat) multò faciliùs
trahuntur, pa&longs;&longs;imque ob&longs;ervatur Romæ in vulgaribus illis vehi
culis (ab antiquis Ci&longs;iis aut parum aut nihil di&longs;tant) quæ cum
ex celeberrimi Architecti Bonarotæ præ&longs;cripto duas ingentes
rotas habeant, tantis ponderibus onu&longs;ta cernuntur, ut miracu
lo proximum videatur ab unico equo tam ingentia onera trahi
po&longs;&longs;e: id quod alibi neutiquam fieri pote&longs;t, ubi minoribus Rotis
vehicula huju&longs;modi in&longs;tructa longè minoribus oneribus defe
rendis paria &longs;unt, &longs;i unicus equus adhibeatur.
Hujus rei cau&longs;am indaganti acquie&longs;cendum non e&longs;t iis, qui
illam ex rationibus Vectis petendam e&longs;&longs;e exi&longs;timant, perinde
atque &longs;i rotæ majoris &longs;emidiameter e&longs;&longs;et longior Vectis, mino
ris verò brevior; ac proptereà majore rotâ faciliùs moveretur
ciliùs pondera moventur, quàm breviore. Hoc, inquam, 1/4
veritate abe&longs;&longs;e palam fiet, &longs;i animadvertamus potentiam tra
hentem medio temone applicatam e&longs;&longs;e axi, cui pariter axi in
nititur onus; atque adeò tùm onus tùm Potentiam concipi
qua&longs;i in Rotæ centro, cujus &longs;emidiametri altera extremitas hy
pomochlij punctum de&longs;ignaret. Atqui Vectis, in quo Potentia
& onus ab hypomochlio eandem aut æqualem di&longs;tantiam ha
bent, parùm aut nihil habet utilitatis: immò in Vecte, quâ
vectis e&longs;t, tria puncta diver&longs;a tribuenda &longs;unt Potentiæ, oneri,
& Hypomochlio, ut infrà, ubi de Vecte di&longs;putabitur: in Rotâ
autem duo tantummodo puncta con&longs;iderantur, &longs;cilicet cen
trum & &longs;emidiametri extremitas. Igitur in Rotâ ratio Vectis
non invenitur, ideóque neque major Rota accipienda e&longs;t qua
&longs;i longior Vectis. Aliundè itaque petendam e&longs;&longs;e cau&longs;am, cur
majores rotæ præ minoribus motum juvent, manife&longs;tum e&longs;t.
Et primùm quidem, quod ad moram illam attinet, quæ ex
modioli Rotæ atque axis tritu oritur, eam minorem e&longs;&longs;e in ma
joribus Rotis, &longs;atis con&longs;tàt, &longs;i attendamus axis cra&longs;&longs;itiem, non
Rotæ magnitudini re&longs;pondere, &longs;ed oneris gravitati, quam opus
e&longs;t &longs;u&longs;tinere; quapropter axi &longs;atis valido pro ratione ponderis
&longs;u&longs;tinendi parùm refert, utrùm Rota, cujus radij bipalmares
&longs;int, an verò tripalmares, infigatur: manente igitur codem axe
aut major, aut minor Rota vehiculo &longs;ubjici pote&longs;t. Sed quo
niam Rota major, cujus diameter &longs;e&longs;quialtera e&longs;t minoris, dum
conver&longs;ionem unam perficit, &longs;patium quoque &longs;e&longs;quialterum
decurrit, eumdem tamen axem, quem minor Rota, terit, hinc
fit, per 8. lib.
5. eumdem axis ambitum ad majoris Rotæ peri
metrum (hoc e&longs;t ad ejus motum) minorem habere rationem
quàm ad perimetrum minoris Rotæ (hoc e&longs;t ad minorem mo
tum) atque adeò tritus ille modioli, & axis minùs impedit ma
jorem motum quàm minorem.
Deinde, ut cap.16. lib.1. &longs;ubindicatum e&longs;t &longs;uperiùs, majo
res rotæ efficiunt, ut axis magis à terrâ di&longs;tet; ac proinde te
mo, cui alligatus e&longs;t equus, vel &longs;ubjecto plano parallelus e&longs;t,
vel minimùm à paralleli&longs;mo recedit: ex quo fit tractionem aut
parallelam e&longs;&longs;e, aut &longs;altem minùs obliquam, quam &longs;i Rota mi
nor e&longs;&longs;et, & axis depre&longs;&longs;ior: quò autem minor e&longs;t tractionis
citato explicatum e&longs;t.
Ad hæc viarum a&longs;peritatem impedimento e&longs;&longs;e nemo ne&longs;cit;
offendicula autem, in quæ vehiculorum Rotæ incurrunt, ma
gis ob&longs;i&longs;tere minori Rotæ, quàm majori, facilè o&longs;tenditur; hîc
enim pariter (id quod de magnitudinibus demon&longs;trat Eucli
des lib.
5. prop.
8.) idem majorem habet Rationem ad minus,
quàm ad majus. Nam &longs;i
Rotæ minoris &longs;emidiame
ter CB fuerit, majoris au
tem CD, & in planis pa
rallelis BA, DE volvantur,
ut impedimentum &longs;imile &longs;i
militerque po&longs;itum inve
nient, multò majus e&longs;&longs;e
oportet illud, quod majori
Rotæ objicitur, quàm quod
minori. Sit enim minoris
offendiculum GI; ducatur
ex centro per I recta, quæ
&longs;it CIE &longs;ecans majoris Rotæ peripheriam in H: erit igitur ar
cus IB &longs;imilis arcui HD, & ille quidem minor, hic verò ma
jor, ut manife&longs;tum e&longs;t. Ducatur in planum perpendicularis
HF, & hoc erit impedimentum majoris Rotæ &longs;imile impedi
mento minoris IG, nam &longs;imilem arcum à conver&longs;ione circà
centrum cum plani contactu impedit; nece&longs;&longs;e quippe e&longs;t Ro
tam majorem converti circà punctum H, &longs;icut & minorem cir
cà punctum I, ut tran&longs;grediantur ob&longs;i&longs;tens offendiculum.
Porrò lineam HF majorem e&longs;&longs;e quàm IG &longs;ic o&longs;tenditur.
Quo
niam AB & ED parallelæ &longs;unt, triangula CBA, & CDE
&longs;imilia &longs;unt: ergo per 4. lib.6. ut CB ad CD, hoc e&longs;t ut CI
ad CH, ita CA ad CE; & permutando ut CI ad CA, ita
CH ad CE; & dividendo ut CI ad IA, ita CH ad HE: at
CI minor e&longs;t quàm CH; igitur per 14. lib.5. etiam IA minor
e&longs;t quàm HE. Item quia AB & ED ex hypothe&longs;i parallelæ
&longs;unt, recta IE in illas incidens facit angulos IAG & HEF
æquales per 29. lib.
1. &longs;unt autem triangula IGA & HFE
rectangula ad G & F ex con&longs;tructione; &longs;unt igitur &longs;imilia, & IA ad IG, ita HE ad HF: quare cum ex
dictis IA minor &longs;it quàm HE, erit per 14.lib.5. etiam IG mi
nor quàm HF.
Cum itaque HF major &longs;it quàm IG (a&longs;&longs;umptâ DM æqua
li ip&longs;i IG, & ductâ perpendiculari MS, donec occurrat peri
phæriæ in S) inter Tangentem ED & arcum circuli &longs;tatuatur
perpendicularis SL æqualis ip&longs;i IG; & ex centro C ducatur
per S recta CO. In triangulo igitur CEO angulus internus
E, per 16. lib.
1; minor e&longs;t externo SOL; igitur etiam angu
lus SOL major e&longs;t quàm IAG: adde utrique angulum
rectum, ergo duo SLO, SOL &longs;imul majores &longs;unt duobus
IGA, IAG &longs;imul; ac propterea etiam externus LSC major
e&longs;t externo GIC per 32.lib.1. Quapropter &longs;emidiameter CS
obliquior incidit in offendiculum SL, quàm &longs;emidiameter CI
incidat in æquale offendiculum IG: minùs igitur impeditur
Rotæ majoris conver&longs;io, quàm minoris, quippe cui minus di
rectè opponatur æquale offendiculum.
Præterea cum trahendi difficultas hinc oriatur, quòd Rota
incurrens in ob&longs;tantem lapidem, aut quid &longs;imile, jam non cir
cà &longs;uum centrum convoluta aptatur &longs;ubjecto plano, &longs;ed, dum
Rota adhæret atque in&longs;i&longs;tit offendiculo; nece&longs;&longs;e e&longs;t plau&longs;trum
cum impo&longs;ito onere elevari pro objecti impedimenti altitudi
ne; faciliùs ab eâdem Potentia elevatur plau&longs;trum onu&longs;tum, &longs;i
major fuerit Rota, quàm &longs;i minor, quia videlicet motus Poten
tiæ ad eandem elevationem majorem habet Rationem in Ro
tâ majore quàm in minore, cum illâ enim plus movetur,
quàm cum i&longs;tâ. Sit majoris Rotæ impedimentum LS pla
nè æquale impedimento GI minoris; producatur perpendicu
laris LS in T, & perpendicularis GI in V: tùm intervallo SC
de&longs;cribatur arcus CT, & intervallo IC de&longs;cribatur arcus CV.
Certum e&longs;t in motu Rotæ majoris propter obicem LS manente
puncto S transferri centrum C in T, ita ut ST &longs;it Rotæ &longs;emi
diameter æqualis &longs;emidiametro CD, & &longs;imiliter in motu Rotæ
minoris propter offendiculum GI manente puncto I transferri
centrum C in V, ita ut IV æqualis &longs;it &longs;emidiametro CB. Quo
niam verò CD, VG, TL ad angulos rectos &longs;ubjecto plano in
&longs;i&longs;tunt, & parallelæ &longs;unt, anguli alterni VIC, ICB æquales
&longs;unt per 29. lib.1, eorumque men&longs;uræ, arcus videlicet VC &
TSC, SCD, eorumque men&longs;uræ arcus TC & SD, &longs;unt
æquales. Atqui arcus SD major e&longs;t quàm IB; igitur & arcus
TC major e&longs;t quàm VC; hi autem arcus TC & VC re&longs;pon
dent motui Potentiæ trahentis: longiore igitur ac majore mo
tu Potentiæ fit eadem elevatio, ac proinde faciliùs in Rotâ ma
jore quàm in minore. Porrò arcum SD majorem e&longs;&longs;e arcu IB,
magi&longs;que di&longs;tare punctum S à puncto D, quàm punctum I à
puncto B, illicò manife&longs;tum fiet, &longs;i duos circulos datis duobus,
æquales de&longs;crip&longs;eris &longs;e intùs contingentes, & ad contactüs
punctum lineam Tangentem duxeris, quocumque enim po&longs;ito
minoris circuli offendiculo inter Tangentem, & circulum mi
norem interjecto, illud idem offendiculum longiùs à con
tactûs puncto removendum videbis, ut inter Tangentem
eandem, & circulum majorem interjici po&longs;&longs;it: Id quod adeò
manife&longs;tum e&longs;t, ut non &longs;it in eo explicando diutiùs immo
randum.
Quòd &longs;i ad calculos rem hanc curiosiùs revocare libeat, &longs;ic
ex gr. Rotæ minoris &longs;emidiameter CA pedum duorum, &longs;cilicet
digitorum 32, offendiculi verò DE
altitudo digitorum 4. Cum igitur
FD & CA parallelæ &longs;int, &longs;icut &
FC ac DA per 34. lib.
1. FD &
CA æquales &longs;unt, remanetque EF
digit.28, & e&longs;t Sinus anguli FCE,
quo cognito innote&longs;cit complemen
tum, arcus &longs;cilicet quæ&longs;itus EA.
Fiat itaque ut CE ad EF, hoc e&longs;t
ut 32 ad 28, &longs;eu ut 8 ad 7, ita
100000. Radius ad 87500 Sinum
arcûs gr.61. 2′ 42″; erit enim quæ&longs;i
tus arcus EA gr. 28. 57′ 18″. Jam verò po&longs;itâ &longs;emidiametro
CA digitorum 32, fiat ut 113 ad 355, ita data &longs;emidiameter
digit. 32 ad &longs;emiperipheriam circuli digitorum ferè 100 1/2, &longs;ci
licet 100. 53″: ergo arcus EA e&longs;t proximè digitorum 16.
At Rotæ majoris &longs;emidiameter BA &longs;it &longs;e&longs;quialtera (quic
quid &longs;it quòd figura &longs;olùm exprimat &longs;e&longs;quiquartam) pedum
&longs;cilicet trium, hoc e&longs;t digitorum 48, & offendiculum GH 4. Quare HI e&longs;t digit.
44 Sinus anguli IBH,
ex quo innote&longs;cet arcus complementi HA. Fiat ut BH 48 ad
HI 44, &longs;eu ut 12 ad 11, ita Radius 100000 ad 91666 Sinum
arcûs gr. 66. 26′. 33″; & e&longs;t quæ&longs;itus arcus HA gr.23.33′.27″.
Jam &longs;it ut 113 ad 355, ita &longs;emidiameter 48 ad &longs;emiperiphe
riam digitorum 150 4/5 ferè: igitur arcus HA e&longs;t proximè di
gitorum 20. Cum itaque dum onus elevatur ut 4, Potentia
in minore Rotâ moveatur ut 16, in majore autem ut 20
(ut paulò &longs;uperiùs o&longs;ten&longs;um e&longs;t motum centri æqualem e&longs;&longs;e
arcubus EA, & HA) facilitas movendi, quæ hinc oritur, erit
ut 5 ad 4.
Ex his manife&longs;tum e&longs;t, in vehiculis, quæ quatuor rotis
in&longs;truuntur, quarum binæ, priores minores &longs;unt, po&longs;teriores
verò majores, faciliùs &longs;uperari impedimenta à po&longs;terioribus
rotis quàm à prioribus, ac propterea minori labore currum ab
equis trahi, quàm &longs;i po&longs;teriores prioribus e&longs;&longs;ent æquales. Id
quod opportunè factum e&longs;t, quia ut plurimum (quemadmo
dum in antiquioribus Rhedis viatoriis cernere e&longs;t) in po&longs;te,
riorem potiùs, quàm in anteriorem currus partem, onus reji
citur, atque adeò po&longs;terior axis magis premitur: quæren
dum igitur fuit aliquod laboris compendium. Quamquam
non negarim alio pror&longs;us con&longs;ilio primùm excogitatam hanc
Rotarum inæqualitatem; ut nimirum onus con&longs;titutum qua&longs;i
in plano trahentem versùs inclinato, faciliùs quoque illum
ex impre&longs;&longs;o anterioris tractionis impetu &longs;equeretur, &longs;i in pla
nitie quidem tractio fieret; ubi verò &longs;uperandus e&longs;&longs;et clivus,
ut minùs adversùs trahentem repugnaret onus &longs;e ip&longs;um in
proclive urgendo; nam &longs;i Rotæ æquales e&longs;&longs;ent, longè faciliùs
vehiculum in po&longs;teriora relaberetur, pro ip&longs;ius clivi inclina
tione, cui parallelum e&longs;&longs;et planum oneri &longs;ubjectum in&longs;i&longs;tens
axibus æqualium Rotarum: at Rotis inæqualibus po&longs;itis, &
po&longs;terioribus quidem majoribus, planum, cui onus incumbe
re intelligitur à po&longs;teriori axe ad anteriorem deductum minùs
inclinatur, quàm collis proclivitas ferat; ac propterea trahen
tibus equis minùs repugnat. Licèt autem non &longs;emper a&longs;cen
dendum &longs;it in colles & clivos, quorum a&longs;cen&longs;us manife&longs;tè ar
duus e&longs;t atque difficilis, rarò tamen, aut ferè nunquam, adeò
æquata e&longs;t viarum planities, quin leviter &longs;altem inflexæ modò
de&longs;cen&longs;uum vici&longs;&longs;itudine non modicè utilis e&longs;t illa Rotarum
inæqualitas.
Hinc manualia illa curricula (&longs;eu ru&longs;ticæ vehes) quæ binis
brachiis in&longs;tructa unicam habent in anteriore parte rotam &
&longs;ublevatis brachiis conver&longs;a Rotâ promoventur, faciliùs
con&longs;trui po&longs;&longs;ent, &longs;i propè vectorem duæ e&longs;&longs;ent Rotæ majores
illâ anteriore Rotâ, ita ut harum diameter triplex e&longs;&longs;et diame
tri illius: hunc enim unicus homo multò majus pondus trans
ferre pote&longs;t vel impellendo, cùm in planitie e&longs;t, aut clivum
a&longs;cendit, vel trahendo, cùm ex declivi de&longs;cendit; levatur &longs;i
quidem labore &longs;u&longs;tinendi, & omnes vires exercet impellendo
aut trahendo; & illa Rotarum inæqualitas in causâ e&longs;t, cur fa
ciliùs impellatur pondus versùs illam partem, in quam incli
natur.
Et quoniam in Rotarum inæqualium mentionem incidi, il
lud hîc pariter ob&longs;ervandum videtur, commodiùs currum mo
veri, cùm anteriores Rotæ à po&longs;terioribus aliquantulùm di&longs;tant,
quàm cùm valdè vicinæ &longs;unt (ubi tamen reliqua omnia paria
fuerint, neque aliud præter Rotarum di&longs;tantiam, intercedat
di&longs;crimen) &longs;i in planitie quidem, & viâ minimum flexuosâ de
ducendus &longs;it. Quia nimirum quo propiores fuerint axes, pla
num, cui onus incumbit, magis inclinatur, ac propterea an
teriores Rotas premens adversùs &longs;ubjectam tellurem minus
obliquè conatur, ideóque pondus illam validiùs urgens majo
rem creat movendi difficultatem: contrà verò &longs;i axes invicem
paulò remotiores fuerint, minùs inclinato plano, minor e&longs;t
priorum rotarum pre&longs;&longs;us in &longs;ubjectam tellurem. Sic &longs;i Rotæ
fuerinc A & B, pla
num, cui onus in&longs;i
det, e&longs;t AB, at &longs;i Ro
tæ fuerint A & C,
planum e&longs;t AC, quod
utique minùs incli
natum e&longs;t, magi&longs;que
accedit ad paralleli&longs;mum cum Horizonte DE, atque adeò
Rota B magis terram premit, quàm Rota C. Si enim in utro
que plano pondus fuerit &longs;imiliter po&longs;itum (puta circà me-
det ad angulos magis inæquales in planum AB magis inclina
tum, quàm in AC minùs inclinatum, atque momentum gra
vitatis ponderis magis accedet ad B quàm ad C, ut infrà &longs;uo
loco explicabitur, & &longs;ubindicatum e&longs;t &longs;uperiùs lib.1. cap.
14.
§. Hinc Hamburgen&longs;ia plau&longs;tra, quibus
merces Hamburgo Norimbergam devehuntur, longiora &longs;unt,
quia nec altiores clivi in itinere frequentes occurrunt, nec
angu&longs;tæ &longs;unt viarum flexiones, ex quibus oriatur aut a&longs;cen
dendi, aut plau&longs;trum inflectendi difficultas. Quare illis &
majora onera imponi po&longs;&longs;unt, & &longs;ex equi non bini & bini, &longs;ed
&longs;inguli recto ordine adjunguntur; quo fit ut non in diver&longs;a
trahentes, omninò &longs;imili impetu currum deducant. Quòd &longs;i
viæ plus haberent difficultatis tùm ex clivis, tùm ex flexioni
bus, non expediret tàm longa plau&longs;tra con&longs;truere, nec equos
tam longâ &longs;erie di&longs;ponere, ut cuique rem vel leviter con&longs;ide
ranti &longs;tatim patebit.
motum præ&longs;tent.
ADeò ingentia aliquando pondera transferenda proponun
tur, ut ea carris imponere tran&longs;vehenda aut nimis opero
&longs;um &longs;it, aut periculo non vacet, ne rotarum axes pondere præ
gravati diffringantur, aut propter &longs;oli mollitudinem rotæ de
vorentur: propterea rationem aliquam inire oportet, quâ voti
compotes &longs;imus, citrà huju&longs;modi pericula. Et quidem &longs;i cor
pus teres &longs;it, nec viarum &longs;alebræ, aut angu&longs;tiæ impedimento
&longs;int, ip&longs;um ver&longs;ari in gyrum poterit &longs;imili artificio, quo ad
deportandos Ephe&longs;um ex lapicidinis &longs;capos columnarum cen
tum viginti &longs;eptem altitudine pedum &longs;exaginta u&longs;us e&longs;t Cte&longs;i
phon Gno&longs;&longs;ius (&longs;ic eum vocat Plinius lib.
7. cap.
37. cum Vi
truvio lib.10. cap. 6, quem tamen idem Plinius lib.36. cap.14.
templo con&longs;truendo præfectus, & quidem felici eventu: ca
pitibus enim &longs;caporum, ubi axis extremitates de&longs;inebant, &longs;ub
&longs;cudis in modum in&longs;eruit, atque implumbavit ferreos axes:
tùm de materiâ trientali &longs;capos (hoc e&longs;t ligneos tigillos cra&longs;&longs;i
tudinis unciarum quatuor pedis, &longs;eu pollicum quatuor) duos
longiores juxtà columnæ longitudinem, duo&longs;que breviores
tran&longs;ver&longs;arios ita compegit, ut parallelogrammum con&longs;tituen
tes columnam po&longs;&longs;ent complecti; medii&longs;que tran&longs;ver&longs;ariis
ferreas armillas in&longs;eruit, quibus axes ferrei infigebantur,
adeò ut liberè ver&longs;ari po&longs;&longs;ent, cum boves traherent; quem
admodum & in gyrum volvuntur cylindri marmorei aut la
pidei, quorum u&longs;us e&longs;t in exæquandis ambulationibus. E&longs;t
autem maximè veri&longs;imile, & probabile, ita firmiter
ligneum illud parallelogrammum fui&longs;&longs;e compactum, ut non
&longs;olùm extremis tran&longs;ver&longs;ariorum capitibus anterioribus alli
gari po&longs;&longs;ent boves; &longs;ed etiam per totam anterioris &longs;capi lon
gitudinem di&longs;tribui, ut faciliùs columna transferretur.
Pro&longs;perum exitum con&longs;ecuta &longs;caporum vectura animum
adjecit Methageni Cte&longs;iphontis filio, ut paternam in
du&longs;triam æmularetur in Epi&longs;tyliis vehendis: cum enim ho
rum figura non ea e&longs;&longs;et, quæ perinde atque cylindrica vol
vi po&longs;&longs;et, duabus rotis pedum circiter duodenûm &longs;ingula
epi&longs;tylia firmiter inclu&longs;it; rotarumque centris ferreos axes
infixit, qui in armillis &longs;imilem haberent ver&longs;ationem, ac
dictum e&longs;t in &longs;caporum vecturâ. Cum enim boves ligneo
parallelogrammo alligati traherent, Rotæ volvebantur, at
que cum illis pariter epi&longs;tylia Rotis cohærentia in gyrum
ver&longs;abantur; quippe quæ in &longs;ubjectum &longs;olum non incurre
bant, cum &longs;olæ Rotæ terram attingerent. Hâc methodo
corporibus, quæ non &longs;unt ad volubilitatem rotundata, faci
lem conyer&longs;ionem conciliare po&longs;&longs;umus; ex Rotis nimirum &
pondere moles una compingitur, cujus extremitatibus cylin
dricis tota innititur, nihilque refert, cujus demum figuræ &longs;it
pars media, &longs;cilicet pondus, modò hæc à &longs;olo aliquantulum
di&longs;tans motum non impediat. Quâ autem ratione aut Rotæ
con&longs;truantur, aut illis onus includatur, artificis &longs;eu architecti
&longs;olertiæ relinquitur.
Methagenis artificium imitatus Paconius, te&longs;te Vitruvio
lib.
10. cap.
6. lapideam ba&longs;im longam pedes duodecim, la
tam pedes octo, & altam pedes &longs;ex Apollinis colo&longs;&longs;o re&longs;ti
tuendam, duabus Rotis pedum circiter quindecim, &longs;imili
ter inclu&longs;it: &longs;ed aliâ ratione ac Methagenes deducere &longs;tatuit.
A Rotâ ad Rotam circâ lapidem fu&longs;os &longs;extantales, hoc e&longs;t
cra&longs;&longs;itudinis pollicum duorum, ad circinum compegit ita, ut
fu&longs;us à fu&longs;o non di&longs;taret pedem unum. Tùm circà fu&longs;os fu
nem involvit, qui bobus trahentibus explicabatur, & con
vertebantur Rotæ. Verùm quia funis circumvoluti &longs;piræ ad
unam, aut ad alteram partem &longs;pectabant, non poterat viâ
rectâ ad lineam deduci moles illa; &longs;ed modò in hanc, mo
dò in illam partem deflectebat, ut opus e&longs;&longs;et retroducere,
adeò ut ducendo & reducendo pecuniam contriverit, & ope
ram lu&longs;erit Paconius. Potui&longs;&longs;et tamen huic malo occurrere,
nec &longs;ui inventi laude fraudari, &longs;i circà fu&longs;os non unicum,
&longs;ed duplicem funem ita involvi&longs;&longs;et, ut funium &longs;piris vel ab
extremitatibus fu&longs;orum, vel à medio, incipientibus, funis
uterque paribus &longs;emper intervallis à &longs;ibi proximâ Rotâ di&longs;ta
rent; &longs;ic enim factum fui&longs;&longs;et, ut boves æqualiter utrumque
funem trahentes, æqualiterque evolventes, molem illam rectâ
viâ deducerent.
Quamquam autem &longs;uâ laude non careant huju&longs;modi arti
ficum inventa, expediti&longs;&longs;imè tamen, & citrà impendium, one
ra ingentia traducuntur &longs;ubjectis cylindris, qui pondere pre&longs;&longs;i,
cùm illud trahitur, convertuntur. Palangas peculiari voca
bulo Veterès dixere fre&longs;tes teretes, qui navibus &longs;ubjiciuntur,
cùm attrahuntur ad pelagus, vel cùm ad littora &longs;ubducuntur;
ut apud Nonium Marcellum legi&longs;&longs;e me memini. Neque aliud
quidpiam cen&longs;endus e&longs;t Cæ&longs;ar intellexi&longs;&longs;e, ubi lib.
3. Belli
Civil. &longs;cribit
lis
&longs;olitam. loco
interiorem partem tran&longs;duxit. Sunt autem &longs;cytalæ ut apud Sui
dam, rotunda & polita ligna: aliquid tamen peculiare. ad
dit Ari&longs;toteles in Mechan. quæ&longs;t. 11. quærens,
talas faciliùs portantur onera quàm &longs;uper currus, cum tamen ij
magnas habeant rotas, illæ verò pu&longs;illasScytalis nimirum pu-
non eas quidem circà axem,
&longs;ed cum axe ip&longs;o, cui adnectun
tur, ver&longs;atiles; cuju&longs;modi e&longs;
&longs;ent in hoc &longs;chemate rotulæ A
& B cum &longs;uo axe connexæ.
Porrò duplicem huju&longs;modi &longs;cytalarum u&longs;um con&longs;idero: &longs;i
enim onus impo&longs;itum incumbat Rotulis ip&longs;is, vel quia plana
&longs;it ejus &longs;uperficies, vel quia tabulato fuerit &longs;uperpo&longs;itum,
perinde res &longs;e habet, atque &longs;i cylindrus e&longs;&longs;et, cujus diameter
idem e&longs;&longs;et cum rotularum diametro: neque tunc admodum
refert, cuju&longs;nam figuræ &longs;it axis, quem onus non tangit, &longs;i
ve rotundus ille &longs;it, &longs;ive angulatus. At &longs;i onus ip&longs;i axi in
cumbat, promineantque hinc & hinc rotulæ, omninò ne
ce&longs;&longs;e e&longs;t axem rotundum e&longs;&longs;e, ut fieri po&longs;&longs;it rotularum con
ver&longs;io, atque ita longum, ut inter rotulas onus laxè interci
piatur; maximè quippe cavendum e&longs;t, ne rotulæ onus con
tingant, alioquin ex mutuo conflictu mora non mediocris
motui crearetur. Ideò autem excogitatæ videntur huju&longs;mo
di &longs;cytalæ, ut minimâ &longs;ui parte &longs;ecundùm extremitates tan
gerent &longs;ubjectum planum, atque adeò in pauciora incurre
rent offendicula, quàm cylindri totâ &longs;ua longitudine incum
bentes plano. Sed illæ ab u&longs;u artificum jam diù intermi&longs;&longs;æ
locum &longs;implicibus cylindris conce&longs;&longs;ere, quippe qui ob con
tinentem &longs;ibique &longs;emper &longs;imilem figuram &longs;olidiores &longs;unt, &
periculo carent, cui obnoxiæ &longs;unt &longs;cytalæ, ne videlicet Ro
tulæ illæ labem aliquam faciant cum rotunditatis, atque adeò
etiam motûs, detrimento. Illud verò commodum, quod ex
offendiculorum evitatione oriebatur, obtinemus pariter, &longs;i
duplicem planorum tigillorum &longs;eriem &longs;ub&longs;ternamus capitibus
cylindrorum; hinc enim fit, ut viarum &longs;alebræ evitentur, &
Cylindri modicâ &longs;ui parte contingant &longs;ubjectos tigillos, qui
viam planam & æquabilem con&longs;tituentes moram nullam mo
tui injiciunt.
Sed & in hoc cylindrorum u&longs;u communiter cen&longs;etur ali
quid ine&longs;&longs;e facilitatis majoris ad onera deducenda, quàm &longs;i
illa currui imponerentur; tùm quia currui &longs;ua ine&longs;t gravitas,
quæ unâ cum impo&longs;itâ &longs;arcinâ majus onus con&longs;tituit, ac
impendit laborem; at &longs;ubjectis oneri cylindris, horum gra
vitas nihil officit trahenti: Tùm quia currûs Rotæ, cum &longs;int
circà &longs;uum axem, cui infiguntur, mobiles, aut hûc & illuc
nutant, &longs;i laxa &longs;int capita, nec clavo exqui&longs;itè coërceantur,
aut &longs;i arctiùs axi cohæreant, axem quem complectuntur, &
clavum quo coërcentur, validiùs terunt; & ex utroque hoc
capite movendi difficultas oritur, cùm aliquid impre&longs;&longs;i im
petûs aut in illâ incon&longs;tantiâ, aut in hoc conflictu contera
tur: nihil autem huju&longs;modi cylindris contingit. Tùm etiam
quia Rotæ modiolus ab axe premitur, & deor&longs;um pondere
urgente, & antror&longs;um impetu ad anteriora trahente; ex quo
quantum difficultatis in movendo oriatur, hinc manife&longs;tum
e&longs;t, quod ni&longs;i axungiâ aut amurcâ illinantur curruum axes,
ægrè convertuntur rotæ, & den&longs;o &longs;tridore, quantus &longs;it par
tium tritus atque conflictus, te&longs;tatum faciunt. At Cylindri
quantumvis ab onere premantur, nullo pingui liquore obli
nendi &longs;unt, ut lubrici fiant; nulla enim impo&longs;iti oneris a&longs;pe
ritas cylindrorum conver&longs;ionem impedire pote&longs;t. Nam &longs;i fue
rit ingens lapis AB cylin
dris &longs;ubjectis impo&longs;itus, &
cylindri punctum C cen
gruat puncto A lapidis, dia
metri CD altera extremitas
D tangit &longs;ubjectum planum;
cum verò &longs;axum ex B ver
sùs A propellitur, &longs;eu tra
hitur ex A, ita cylindrus
convertitur, ut DF ar
cus &longs;en&longs;im ad &longs;ubjectum
planum, contrà verò arcus CE ad impo&longs;itum &longs;axum accom.
modetur, citrà omnem &longs;axi & cylindri affrictum.
Hinc tamen aliquid etiam incommodi cylindris adhæret, &longs;i
cum plau&longs;trorum rotis conferantur; hæ &longs;cilicet motum con
tinuant, cum &longs;ine fine volvantur, quippe quæ axi infixæ, im
po&longs;ito oneri pariter, ut ita loquar, cohærent; illos verò, ni
mirum cylindros, onus dum promovetur, po&longs;t &longs;e relinquit; ac
proinde aut cylindrorum copia non exigua &longs;uppetere debet,
relinquuntur, &longs;ubinde transferendi &longs;unt, ut iterùm oneri
&longs;ubjiciantur. Verùm hæc alterna cylindrorum tran&longs;latio non
adeò gravis e&longs;t; quin plus habeat adjumenti, quàm incom
modi; cum enim plurimùm referat, utrùm qui &longs;ubjicitur cy
lindrus, reliquis po&longs;terioribus cylindris parallelus, an obli
quus &longs;tatuatur, ut onus ad lineam viâ rectâ deducatur, aut
motus &longs;ui ve&longs;tigium inflectat; facillimum e&longs;t opportunâ cylin
dri tran&longs;lati collocatione parallelâ, aut obliqua, de&longs;tinatum
oneris motum admini&longs;trare.
Illud autem non immeritò hîc examinandum occurrit, utrùm
majores cylindri minoribus potiores cen&longs;endi &longs;int, & an præ&longs;tet
&longs;ubjicere oneri cylindrum GI majorem, an verò minorem
GH. Et quidem &longs;i figuræ dumtaxat magnitudo atque parvi
tas &longs;pectetur, hoc unum di&longs;crimen invenio, quòd ad certam
motûs men&longs;uram perficiendam crebriùs volvi oportet cylin
drum minorem, quàm majorem; onus verò à &longs;ubjecto plano
di&longs;tare majoris diametri GI intervallo potiùs, quàm minoris
GH, non video, quid conferat ad motûs facilitatem; tantum
enim promovetur onus, quantus e&longs;t peripheriæ arcus, cui illud
in motu aptatur, eíque æqualis e&longs;t arcus oppo&longs;itus, qui plano
pariter in motu congruit: ac propterea parum refert, utrùm
eadem arcus men&longs;ura &longs;it majoris circuli pars minor, an minoris
circuli pars major.
Verùm &longs;i qua inter motum occurrant offendicula, hæc
minùs officere majori cylindro, quàm minori, dicendum e&longs;t,
quemadmodum & de rotis majoribus dictum e&longs;t &longs;uperiori ca
pite; &longs;iquidem majoris cylindri diameter obliquior incidit in
idem offendiculum, quod minùs directè opponitur motui, &
longiore motu Potentiæ fit eadem ponderis elevatio, ut ibi ex
plicatum e&longs;t.
Aliud e&longs;t præterea, nec &longs;anè nullius momenti, quod majo
ri cylindro incitatiorem dat volubilitatem; quòd videlicet
(quemadmodum & globo majori contingit) major cylindrus,
quamvis Geometricam Rotunditatem non a&longs;&longs;equatur, tamen
propiùs accedit ad figuram exqui&longs;itè Rotundam, quàm mi
nor: &longs;i enim à circulo Geometricè perfecto æqualiter recedant
utriu&longs;que cylindri majoris ac minoris ba&longs;es, non tamen æqua-
lus, in minori minor, atque adeò ille magis, quàm hic, ad
rotunditatem accedit. In majori autem circulo angulum, qui
peripheriam complectitur, majorem e&longs;&longs;e palam e&longs;t, quia idem
exce&longs;&longs;us majori Radio additus con&longs;tituit &longs;ecantem anguli mi
noris, quàm &longs;i minori Radio addatur; ac propterea angulus
Complementi major e&longs;t in majori, quàm in minori. Id quod,
per &longs;e quidem &longs;atis clarum, dilucidiùs explicabitur, &longs;i ex mi
nore circulo extet particula, cu
jus altitudo &longs;it ON, ex majore
autem circulo æqualis altitudo
emineat IM. Ductis Tangen
tibus & Radiis, certum e&longs;t Se
cantis exce&longs;&longs;um ON &longs;upra Ra
dium LO minorem, habere
majorem Rationem ad &longs;uum
Radium, quàm habeat æqualis
exce&longs;&longs;us IM ad &longs;uum Radium
LI majorem ex 8.lib.5. E&longs;t igi
tur MLP angulus minor angulo NLS, & Complementum
LMP majus e&longs;t Complemento LNS quare totus angulus
VMP major e&longs;t toto angulo TNS, ac proinde magis ad ro
tunditatem accedit.
CIrculi motus, ob id ip&longs;um quia circulus e&longs;t, circa &longs;uum
centrum perficitut eâ ratione, ut &longs;uperiores partes pro
grediantur, inferiores retrocedant, anteriores de&longs;cendant,
po&longs;teriores a&longs;cendant, &longs;ervatâ &longs;emper pari oppo&longs;itorum pro
gre&longs;sûs atque regre&longs;sûs, de&longs;censûs atque a&longs;censûs men&longs;urâ;
pro ut unicuique rem vel leviter con&longs;ideranti patet. Quare
dum in gyrum circulus agitur, centrum quidem manet, reli
quæ verò partes ita &longs;ingulæ ex alio in alium locum &longs;ibi invi-
volvitur, omninò non mutet. Quemadmodum ob&longs;ervare e&longs;t
in Solis orbitâ, quam Eclipticam vocant; hæc enim diurnâ
conver&longs;ione circa Mundi axem Solem &longs;ecum rapiens à &longs;uo lo
co non recedit, Sole ab ortu in Occa&longs;um commigrante: id
multò magis in &longs;ingulorum circulorum circà &longs;ua centra revo
lutione manife&longs;tum apparet. Quod &longs;i circulus aut horizonti
parallelus, aut illi ad perpendiculum in&longs;i&longs;tens, raptetur; mo
tus ille nihil habet circulari affine, cum circà centrum non
perficiatur, &longs;ed &longs;ingula circuli puncta &longs;olo motu recto unâ cum
centro moveantur.
Sin autem axis circulo ver&longs;atili infixus trahatur, jam circu
lus & cum-axe pariter movetur, & circa axem volvitur: atque
adeò &longs;ingularum circuli partium motus is e&longs;t, qui ex recto cen
tri, & circulari ip&longs;ius orbitæ componitur. Hinc &longs;emicirculi
&longs;uperioris partes cum progrediantur versùs cumdem locum, ad
quem centrum tendit, &longs;uum motum motui centri addunt:
Contrà verò inferioris &longs;emicirculi partes retrocedentes &longs;uum
motum à centri motu detrahunt. Rotæ igitur puncta omnia,
dum currus trahitur, &longs;i non &longs;ummatim tota revolutio, &longs;ed par
ticulatim, accipiatur, non æquali velocitate moventur. Sit
explicandi gratiâ,
circulus BD AE,
cujus centrum C
moveatur ver&longs;us F,
& &longs;it tangens GA,
cui in motu appli
catur ip&longs;ius circu
li orbita; in quâ
accipiatur &longs;extans
hinc & hinc AD,
& AE. Igitur in
Conver&longs;ione, dum
Centrum C trahitur ad F, punctum D venit in G, & arcus
DA æqualis e&longs;t rectæ GA, cui in motu &longs;ubinde per partes
congruit: atque adeò, quarum partium &longs;emidiameter CA
e&longs;t 21, earum arcus AD, & recta AG e&longs;t 22, & motus cen
tri illi æqualis CF e&longs;t pariter 22. Quoniam verò in motu or-
dit juxta men&longs;uram &longs;inûs SE (qui ad Radium CA 21 e&longs;t ut 18)
hinc e&longs;t po&longs;t conver&longs;ionem, in qua D e&longs;t in G, punctum A
ita a&longs;cendi&longs;&longs;e, ut &longs;it in lineâ HE parallelâ Tangenti GA, &longs;ed
motui centri tantum detraxerit, quantus e&longs;t &longs;inus SE. Quia
igitur Radius CD ubi congruit punctis FG, &longs;ecat in H
rectam HE, &longs;umatur HI æqualis &longs;inui SE, & puncti A totus
progre&longs;&longs;us remanet SI partium 4, quarum SH, &longs;eu CF e&longs;t 22.
Quare A e&longs;t in I, quando D e&longs;t in G.
Contrà verò in &longs;uperiore &longs;emicirculo &longs;umatur item ex B
hinc, & hic &longs;extans BK & BL; atque in conver&longs;ione ubi cen
trum C venerit in F, & punctum orbitæ D in G, erit K in O,
& diameter DK &longs;ecabit parallelam KN in M. Igitur punctum
B ita de&longs;cendit ad parallelam NK, ut motui centri CF, hoc
e&longs;t BO &longs;eu RM, addiderit &longs;uum progre&longs;&longs;um juxta men&longs;uram
RL Sinum Sextantis BL, hoc e&longs;t 18. Venit igitur B in N;
atque additis RM 22, & MN 18, totus progre&longs;&longs;us puncti B
e&longs;t RN 40. Comparatis itaque invicem curvis lineis AI &
BN, manife&longs;tum e&longs;t puncta B & A non æque velociter mo
veri, cum eodem temporis &longs;patio inæqualia loci &longs;patia per
currant.
Eadem erit methodus, &longs;i reliquorum orbitæ punctorum ve
locitates aut tarditates con&longs;iderandæ &longs;int: &longs;i tamen adverteris
non eandem e&longs;&longs;e omnium circuli Quadrantum rationem in de
terminandâ men&longs;ura motûs addendi, aut demendi motui cen
tri. Nam in anteriori Quadrante &longs;uperioris &longs;emicirculi, & in
po&longs;teriori Quadrante inferioris &longs;emicirculi, men&longs;ura progre&longs;
sûs addendi in illo, & regre&longs;&longs;us demendi in i&longs;to, attendenda
e&longs;t ex Sinu Recto arcûs, qui de&longs;cribitur in motu circa cen
trum à puncto, cujus velocitas inquiritur, aut tarditas: Et
quidem integer Sinus Rectus accipitur, &longs;i punctum à &longs;ummo
vertice de&longs;cendens, vel ab infimo contactûs puncto a&longs;cendens
movetur, ut ex B vel ex A: &longs;in autem punctum con&longs;ideretur,
quod intrà eo&longs;dem Quadrantes di&longs;tet ab extremitatibus diame
tri &longs;ubjecto plano in&longs;i&longs;tentis, puta L aut E, quæ moventur in
V, aut in P, progre&longs;sûs aut regre&longs;sûs men&longs;ura de&longs;umitur ex dif
ferentiâ Sinuum Rectorum, qui re&longs;pondent arcubus BL & BV,
aut arcubus AE & AP. In po&longs;teriori verò Quadrante &longs;upe-
culi, progre&longs;&longs;us addendus, aut regre&longs;&longs;us demendus, motui
centri, men&longs;uram de&longs;umit ex Sinubus Ver&longs;is, aut ex eorum
differentiâ, pro ut puncti motus a&longs;cendens aut de&longs;cendens in
cipit ab extremitate Quadrantis, aut à loco medio, ut facilè
cuique con&longs;tat: neque enim &longs;chema multiplici linearum de&longs;
criptione ad confu&longs;ionem implere operæ pretium e&longs;t.
Cum itaque in oppo&longs;itis Quadrantibus &longs;imilem men&longs;uram
recipiant incrementa atque decrementa &longs;ive à &longs;inubus Rectis,
&longs;ive à Ver&longs;is, addenda aut demenda motui centri, mani
fe&longs;tum e&longs;t punctum quodlibet in integrâ conver&longs;ione demùm
progre&longs;&longs;um fui&longs;&longs;e pari men&longs;urâ cum motu centri. Si enim Al
gebricè &longs;tatuatur motus Centri Z, incrementum in &longs;uperiore
&longs;emicirculo addendum +A, decrementum in inferiore &longs;emicir
culo tollendum — A; manife&longs;tum e&longs;t totum motum, qui com
ponitur, Z +A — A non e&longs;&longs;e ni&longs;i Z.
His ita con&longs;titutis, quæ ita clara &longs;unt, ut nihil habere vi
deantur dubitationis, nec in controver&longs;iam vocari queant, jam
eximendus e&longs;t &longs;crupulus, quem philo&longs;ophantibus injecit Ari
&longs;toteles Mechanic. quæ&longs;t. 24. de circulorum concentricorum
motu, quando alter ad alterius motum promoto communi cen
tro movetur. Sit
enim major circu
lus, cujus Radius
CB, minor autem,
cujus Radius CS;
quos tangant pa
rallelæ BF & ST,
quibus item recta
per centrum ducta
parallela &longs;it CO,
quam videlicet per
currit centrum,
dum trahitur. Ne
gari non pote&longs;t in
hâc circulorum tractione & conver&longs;ione peripherias tùm ma
joris, tùm minoris Circuli &longs;uis Tangentibus ita coaptari, ut
factâ Quadrantis BD conver&longs;ione, fiat pariter Quadrantis SI
& centrum C in O, atque adeò Radius CD matato &longs;itu factus
&longs;it OF. Major igitur Quadrans percurrit &longs;patium BF, & mi
nor &longs;patium ST. At quia æquales rectæ OF & CB perpen
diculares &longs;unt ad eandem rectam BF, ctiam &longs;unt parallelæ,
jungúntque parallelas ST & BF, quæ propterea etiam &longs;unt
æquales, ex 34. lib.1. Igitur arcus SI minor arcu BD, coap
tatur &longs;patio æquali ip&longs;i arcui Quadrantis BD, cui &longs;upponitur
æqualis recta BF. Quarum itaque partium 7 e&longs;t Radius CB,
earum e&longs;t Quadrans BD, hoc e&longs;t recta BF 11, e&longs;tque pariter
ST 11. At quarum partium 7 e&longs;t Radius CB, earum &longs;it Ra
dius CS 4; igitur Quadrans SI e&longs;t 6 3/7 multo minor quàm
recta ST, cui ip&longs;e Quadrans SI in motu congruit.
Id enim verò tantum præ &longs;e fert difficultatis, ut mirum &longs;it,
quot Ixiones rota hæc torqueat, & quàm varias in partes &longs;e alij
aliter ver&longs;ent; quorum &longs;ententias &longs;i examinare liberet, in lon
gum nimis &longs;ermonem me vocaret i&longs;ta di&longs;putatio, nec &longs;atis &longs;ci
rem, utrùm plus aliquid lucis propo&longs;itæ quæ&longs;tioni affunderetur.
Quid igitur probabilius dicendum videatur, paucis expono.
Priùs tamen ob&longs;erva in dictâ Quadrantis revolutione, quan
do Centrum C venerit in O, & D in F, & in I in T, tunc
punctum B e&longs;&longs;e in E (e&longs;t enim OE æqualis Radio CB) atque
punctum S in V (e&longs;t &longs;cilicet OV æqualis Radio CS) ita
ut B a&longs;cendat per curvam BE, punctum autem S a&longs;cendat
per curvam SV, & &longs;imiliter punctum D de&longs;cendat per cur
vam DF, punctum verò I de&longs;cendat per curvam IT. Ex quo
patet punctum S minoris circuli plus promoveri, quàm
punctum B majoris circuli; hujus enim progre&longs;&longs;us e&longs;t CE, il
lius autem e&longs;t CV: & pari ratione con&longs;tat magis ad anterio
ra promoveri punctum I minoris circuli, cujus progre&longs;sûs men
&longs;ura e&longs;t IO, quàm punctum D majoris circuli, cujus progre&longs;
&longs;us e&longs;t DO.
Et hæc quidem, quando centri motus legem accipit à pe
ripheriâ majoris circuli; ad cujus motum minor circulus con
centricus movetur; eo quod major circulus in&longs;i&longs;tit &longs;ubjecto pla
no, cui orbita &longs;ubinde coaptatur rectam lineam &longs;ibi æqualem
de&longs;ignans ex hypothe&longs;i, dumque movetur, &longs;ecum rapit interio
rem circulum.
Quod &longs;i minor circulus in&longs;i&longs;tat &longs;ubjecto &longs;ibi plano, legem
que det motui centri; quia minor peripheria de&longs;ignat rectam
&longs;ibi æqualem, res contrario modo procedit, quia dum ad mi
noris circuli motum circulus major movetur, hujus orbita de
&longs;ignat in plano &longs;ubjecto lineam minori peripheriæ æqualem.
Hinc &longs;i arcus SI de&longs;ignat rectam SG &longs;ibi æqualem, ubi I ve
nerit in G, etiam D erit in H, atque totus Quadrans BD de
&longs;ignabit &longs;olùm rectam BH æqualem rectæ SG. Erit igitur
recta SG æqualis Quadranti SI 6 2/7; cui pariter æqualis e&longs;t
BH: Ex quo fit punctum B, quia di&longs;tat à centro C partibus 7,
non &longs;olùm non procedere in revolutione Quadrantis; &longs;ed re
trocedere per 5/7 interea, dum commune centrum C promove
tur per 6 2/7.
Non ab&longs;imili ratione punctorum B, & S jam in E & V
tran&longs;latorum motus per con&longs;equentes circuli Quadrantes, do
nec integra revolutio perficiatur, con&longs;iderandus e&longs;t: & quæ
de uno puncto cuju&longs;que circuli deprehenduntur, de &longs;ingulis
eju&longs;dem orbitæ punctis dicta faciliùs intelliguntur, quàm ut
uberiori explicatione opus &longs;it.
Ex his apertè liquet eam lineam rectam in &longs;ubjecto plano de
&longs;ignari à peripheriâ tùm majoris, tùm minoris circuli, quæ
æqualis &longs;it motui centri, prout ille legem accipit à majore aut
à minore orbitâ, ad cujus motum altera movetur; ac proinde
modò longiori, modò breviori lineæ rectæ in motu coaptantur
ambæ peripheriæ; ut enim rectè loquitur Ari&longs;toteles loc. cit.
que moverit alter, tantum alter movebitur.
Cur igitur parem lineam rectam de&longs;ignat in plano utraque
orbita major & minor? con&longs;tat ex dictis: quia nimirum cu
ju&longs;libet circuli quodlibet punctum dum trahitur &longs;imul, & vol
vitur, promovetur non ni&longs;i pro ratione motûs centri: &longs;ed con
centricorum circulorum unum & idem e&longs;t centrum; ergo uni
cus e&longs;t centri motus, & &longs;ecundùm unam eandemque men&longs;u
ram motûs centri, omnia puncta tùm majoris, tùm minoris or
bitæ, demum ab&longs;olutâ conver&longs;ione, promota &longs;unt; &longs;ingulorum
enim incrementa, dum &longs;uperiorem &longs;emiperipheriam motu
de&longs;cribunt, ab oppo&longs;itis decrementis eli&longs;a in inferioris &longs;emipe-Nil
itaque mirum, &longs;i tres lineæ, quarum primam centrum percur
rit, &longs;ecundam orbita minor de&longs;ignat, tertiam orbita major, pla
nè æquales &longs;unt; pendent enim ab unico & communi motu
centri, cui nihil additur, aut demitur ex integrâ conver&longs;ione
circa centrum, &longs;ivè illa latiùs excurrat in majore circulo, &longs;ivè
arctiùs in minore coërceatur.
At, inquis, difficile e&longs;t cogitatione a&longs;&longs;equi, & oratione ex
plicare, quî fieri po&longs;&longs;it, ut peripheriâ utráque &longs;ubjectum &longs;ibi
planum &longs;emper tangente, nullóque puncto manente &longs;ine mo
tu, ita ut plana &longs;ubjecta ab aliis &longs;ubinde atque aliis punctis tan
gantur, pauciora puncta minoris peripheriæ totidem punctis
rectæ lineæ coaptentur, ac plura puncta majoris peripheriæ.
Sunt qui difficultatem hanc declinant ad&longs;truentes infinita
puncta tùm in circulorum peripheriis, tùm in lineis rectis, ne
ganté&longs;que inter infinitas multitudines, quæ invicem compa
rentur, affirmari po&longs;&longs;e totidem in unâ infinitâ multitudine, ac
in aliâ pariter infinitâ unitates reperiri, nulla enim e&longs;t infiniti
ad infinitum Ratio, ac proinde nulla fieri pote&longs;t, perinde ac in
multitudinibus finitis, comparatio minoris, aut majoris, aut
propriè, &, ut aiunt, po&longs;itivè æqualis. Hæc tamen (quamvis
quod ad infinita Ratione carentia &longs;pectat, à me ultrò admit
tantur, Rationem &longs;cilicet habere dicuntur inter &longs;e magnitudi
nes, idem & de multitudinibus dicendum, quæ po&longs;&longs;unt mul
tiplicatæ &longs;e mutuò &longs;uperare, ut definit Euclides lib.5. ubi au
tem nullus e&longs;t terminus, ut in infinito, nullus pariter exce&longs;&longs;us
intercedere pote&longs;t quavis factâ multiplicatione) non facient
&longs;atis comparanti omnia puncta unius lineæ cum omnibus
punctis alterius lineæ, non quâ infinitæ punctorum multitudi
nes &longs;unt, &longs;ed quâ finitæ magnitudines ex punctis illis quan
tumvis infinitis con&longs;tituuntur: finitas autem magnitudines
comparari invicem po&longs;&longs;e, ac Rationem inter&longs;e habere nemo
negaverit. Supere&longs;t igitur explicandum, quomodo peripheria
minor coaptetur lineæ rectæ æquali illi eidem, cui commen&longs;u
ratur peripheria major.
Propterea, duce Galilæo Dialog.1. de motu, ob&longs;ervant &longs;imi
lium polygonorum concentricorum motum ac conver&longs;ionem,
in quâ polygonum, ex quo centri motus legem accipit, &longs;ingu-
integrâ conver&longs;ione linea recta &longs;ubjecti plani &longs;it æqualis peri
metro polygoni: at non item partes omnes lineæ, cui alterum
polygonum in motu coaptatur, &longs;i unica comprehen&longs;ione &longs;u
mantur, lineam æqualem polygoni majoris perimetro con&longs;ti
tuunt. Res, clarita
tis gratia, explicetur
in Hexagonis, quo
rum commune cen
trum &longs;it A, & latera
BC, DE incumbant
parallelis lineis BH,
DK. Det primùm le
gem motui centri po
lygonum exterius, & majus, fiatque conver&longs;io circa punctum
C, demùm latus CF congruet rectæ CH, & centrum A per
arcum AF erit tran&longs;latum in F; latus verò minoris polygoni
EG congruet parti IK, intactam relinquens partem EI, ita
tamen; ut tota EK æqualis &longs;it ip&longs;i CH. Id quod e&longs;t mani
fe&longs;tum, quia factâ tran&longs;latione centri in F, &longs;emidiameter, quæ
ex F pertingit ad H, e&longs;t parallela ip&longs;i AC, cum ad &longs;imiles an
gulos incidat in &longs;ubjectam lineam; &longs;unt autem parallelæ etiam
AF, DK, & BH; igitur tres lineæ AF, EK, CH &longs;unt æqua
les, ex 34. lib.1. Atqui quod uni lateri contingit, etiam reli
quis lateribus commune e&longs;t; igitur factá integrâ conver&longs;ione
Hexagonum majus de&longs;ignabit lineam &longs;extuplicem ip&longs;ius CH
æqualem toti perimetro, & Hexagonum minus percurret li
neam &longs;imiliter ip&longs;ius EK &longs;extuplicem, quæ æqualis e&longs;t perime
tro majoris Hexagoni, &longs;umendo tàm partes lineæ DK, quas
intactas relinquit, quàm quæ tangunrur. Cæterùm &longs;i eæ &longs;o
lùm, quæ ab Hexagono minore tanguntur, accipiantur, patet
illas &longs;imul &longs;umptas non e&longs;&longs;e majores perimetro eju&longs;dem mino
ris Hexagoni.
Deinde polygonum interius & minus det legem motui cen
tri, & conver&longs;io fiat circa punctum E, po&longs;tquam latus EG
congruit lineæ EI, & centrum e&longs;t in G (in hoc enim exem
plo ad vitandam in Schemate confu&longs;ionem literarum a&longs;&longs;ump-
licet minotis &longs;ubdupla &longs;unt laterum majoris) cum interim
punctum C retroce&longs;&longs;erit in L, & demum latus CF congruat
lineæ LM. Igitur majus polygonum &longs;olùm de&longs;ignat in motu,
quo progreditur, lineam CM æqualem lateri minoris polygoni
EI; & factâ integrâ conver&longs;ione, de&longs;ignata erit linea &longs;extuplex
ip&longs;ius CM & ip&longs;ius EI; atque adeò utrumque polygonum
æqualem lineam progrediendo de&longs;ignat.
Hæc quæ de Hexagonis concentricis exempli gratiâ dicta
&longs;unt, de omnibus &longs;imilibus atque concentricis polygonis dicta
intelliguntur, quotcumque &longs;int laterum. Jam verò Authores
illi concipiunt circulos tanquam polygona infinitorum late
rum: & quemadmodum minus polygonum totidem &longs;patia &longs;ub
jectæ lineæ intacta relinquit, totidemque tangit, quot habet
latera; ita pariter in circuli minoris conver&longs;ione, infinita &longs;pa
tia vacua non-quanta (ne &longs;cilicet &longs;i quanta e&longs;&longs;ent, opus e&longs;&longs;et
lineâ infinitâ) intermi&longs;ta &longs;patiis, quæ tanguntur, ad&longs;truunt,
adeò ut demùm ex omnibus &longs;patiis tactis &longs;imul & intactis coa
le&longs;cat linea æqualis ei, quæ tangitur à majore peripheriâ ma
joris circuli.
Mihi tamen arridere non pote&longs;t illa loquendi formula, quæ
circulum polygonum infinitorum (& quidem infinitorum &longs;im
pliciter) laterum dicit. Polygonum enim utique regulare cir
culus e&longs;&longs;et; polygonum autem e&longs;&longs;e non pote&longs;t illud, quod angu
lis caret; neque anguli e&longs;&longs;e po&longs;&longs;unt, ubi non e&longs;t lineæ ad li
neam inclinatio; in peripheriâ verò circuli linea nulla e&longs;&longs;e po
te&longs;t, e&longs;&longs;ent &longs;iquidem infinitæ lineæ æquales invicem, quæ uti
que con&longs;tituerent exten&longs;ionem &longs;impliciter infinitam. Quod &longs;i
infinita dixeris puncta; non e&longs;t puncti ad punctum inclinatio,
quæ po&longs;&longs;it angulum con&longs;tituere, ac proinde circulus non e&longs;t po
lygonum infinitorum laterum, ni&longs;i vocabulis ad opinandi li
centiam immoderatè abutamur. Adde quod omnia diametri
puncta ad omnia puncta peripheriæ e&longs;&longs;ent in Ratione, quam
Archimedes
tionem 7 ad 22, & Rationem 71 ad 223: non igitur infinita e&longs;&longs;e
po&longs;&longs;unt aut diametri, aut peripheriæ, aut utriu&longs;que puncta; ab
infinitis enim Rationem omnem ablegant iidem Authores. Si
quomodo ad circulos concentricos traducantur ea, quæ de po
lygonorum concentricorum conver&longs;ione con&longs;iderata &longs;unt.
Quòd &longs;i circulum ita in polygonum convertamus, ut nec
illi fixum definitumque laterum numerum tribuamus, nec &longs;im
pliciter infinitum; &longs;ed liceat minora &longs;emper atque minora late
ra concipere, ut laterum ip&longs;orum numerus &longs;emper augeatur,
ita ut non &longs;impliciter infinitus, &longs;ed indefinitus dicatur, non
abnuo: propo&longs;ita enim difficultas &longs;atis commodè hâc ratione
explicabitur. Verùm in hac laterum extenuatione, &longs;i ad mini
mam exten&longs;ionem deveniamus, quæ à puncto phy&longs;icè non dif
ferat; non infinitus e&longs;t huju&longs;modi punctorum numerus, &longs;ed
certus e&longs;t atque definitus: Necip&longs;is punctis, &longs;eu minimis Phy
&longs;icis &longs;ua figura detrahenda e&longs;t, in majori enim peripheriâ mi
nùs curvantur interiùs, minú&longs;que convexa &longs;unt exteriùs, pro
piú&longs;que ad lineam rectam accedunt; in minori autem orbitâ
puncta hæc circularia curvantur magis, magi&longs;que convexa &longs;unt
exteriùs, & à rectitudine magis deflectentia ita ab&longs;unt à &longs;ub
jectâ rectâ lineâ, ut, dum conver&longs;io fit circuli, & trahitur, de&longs;
cribant in motu lineam curvam magis ob&longs;ecundantem motui
centri, quàm quæ de&longs;cribitur à punctis &longs;imiliter po&longs;itis in ma
jore peripheriâ.
Cærerùm cavendum e&longs;t maximè ab eo, quod quia &longs;ube&longs;t
æquivocationi, difficultatem in hâc quæ&longs;tione auget; illud au
tem e&longs;t, quod punctum peripheriæ cum puncto lineæ Tangen
tis perperam comparatur, qua&longs;i in contactu coæquarentur; id
quod à veritate longè abe&longs;t; &longs;e enim contingunt circulus & li
nea incommen&longs;urabiliter, &longs;i contactus præcisè &longs;pectetur: at &longs;i
contactus & motus componantur, jam quædam exten&longs;io conci
pitur, quæ aliquâ ratione comparari pote&longs;t cum &longs;patio lineæ,
quæ tangitur, quatenùs huic aut illi parti lineæ in motu coapta
tur circulus, aut ejus pars. Quare circuli minoris, qui ad ma
joris circuli motum movetur, &longs;ingula puncta non aptè compa
rantur cum &longs;ingulis &longs;ubjectæ rectæ lineæ punctis, qua&longs;i circuli
punctum, quod e&longs;t tertium à contactu, antequam incipiat mo
tus, in conver&longs;ione tangat tertium rectæ lineæ punctum; &longs;ed
tanget forta&longs;&longs;e quintum aut &longs;extum pro ratione magnitudinis
obiervare e&longs;t; quò enim majus e&longs;t interius polygonum, eò
etiam minora &longs;unt intervalla, quæ intacta relinquuntur. Ex
quamvis in circuli contactu intervalla huju&longs;modi intacta non
admittantur, non e&longs;t tamen abs re puncto circuli, quod volui
tur &longs;imul & trahitur cum ip&longs;o circulo, vim tribuere tangendi
plus quàm unum &longs;ubjectæ rectæ lineæ punctum, quemadmo
dum majoris peripheriæ punctum in motu contingit ex punctis
&longs;ubjectæ lineæ rectæ non communicantibus minus quàm unum,
&longs;i ad interioris circuli motum circulus exterior moveatur: nam
ad majoris, & exterioris motum minor, & interior promovetur;
ad minoris verò & interioris motum major & exterior circulus
retroagitur. Quapropter &longs;i interior circulus in primo ca&longs;u ve
lociùs, & exterior in &longs;ecundo ca&longs;u tardiùs movetur comparatè
ad &longs;patium collocatum cum eorum peripheriis, nil mirum in
motu perfici ab illius puncto Phy&longs;ico plus &longs;patij, quàm ferat
ejus magnitudo, ab hujus autem puncto Phy&longs;ico minus &longs;patij:
in continuâ enim quantitate partes minores &longs;ubinde ac minores
vera, ut opinor, Philo&longs;ophia admittit. Sed quia hæc e&longs;&longs;et in
finita, concertationumque plena di&longs;putatio, &longs;atis ea &longs;int, quæ
diximus, & ad utiliora gradum faciamus.
EXPLICATIS &longs;uperiore Libro Cau&longs;is motûs Ma
chinalis, ordinis ratio po&longs;tularet, ut ad ip&longs;as Ma
chinas, &longs;eu, ut ab Antiquioribus apud Pappum
lib.8. Collect. Mathem. prop.10. vocantur, Facul
tates, ad quas Machinamenta ab artificibus exco
gitata reducuntur, aut ex quibus hæc componuntur, exami
nandas & explicandas progrederemur: Et fortè alicui videatur
ab in&longs;tituto no&longs;tro alienum libram hîc con&longs;iderare, quippe quæ
non ad motum oneribus conciliandum inventa e&longs;t, ideóque
nec inter Facultates enumeratur, &longs;ed u&longs;um omnem habet in
motu prohibendo, ubi factum fuerit ponderibus æquilibrium.
Nec eo quidem con&longs;ilio libræ momenta hic expendo, ut indè
Vectis rationes explicentur (quemadmodum non paucis placet)
non enim Vectis vires ad libræ Rationes revocandas exi&longs;timo,
cum &longs;ua cuique Facultati cau&longs;a in&longs;it, communis illa quidem,
&longs;ed quæ perinde in Vecte reperitur, atque &longs;i nulla pror&longs;us
exi&longs;teret libra. Verùm eatenus libram Mechanicæ contem
plationi in&longs;erendam cen&longs;eo, quatenus non minoris artis e&longs;t ea,
quæ in motum prona &longs;unt, cohibere & &longs;i&longs;tere, quàm onera
quie&longs;centia per vim &longs;uo loco dimovere: Cum maximè ad libram
pertineat Statera, in qua modicum pondus multò majori pon
deri æquipollet, æquatis in di&longs;pari gravitate gravitationum Præterquam quod
explicato æquilibrio, faciliùs declaratur in motu Machinali,
quid præ&longs;tet major illa Ratio momentorum agendi ad momen
ta re&longs;i&longs;tendi, quàm &longs;it reciproca Ratio gravitatum, &longs;eu vi
rium oppo&longs;itarum, ab&longs;olutè &longs;umptarum extrà machinam; ex
qua majore Ratione momentorum, etiam Potentiæ moventis
virtus innote&longs;cit. Nihil autem officit libræ dignitati, quod
Cain authorem agno&longs;cere videatur, qui, ut Jo&longs;ephus lib.
1.
excogitatis men&longs;uris & ponderibus immutavit, pri&longs;linamque &longs;inceri
tatem & genero&longs;itatem ignaram talium artium, in novam quan
dam vir&longs;utiam depravavit. Quid enim &longs;i quis præclaro artifi
cio ex naturæ the&longs;auris deprompto abutatur? Dolos & fallacias,
aut errores, quibus in&longs;ici pote&longs;t libræ u&longs;us, ideò retegemus:
ut nimirum quod Ju&longs;titiæ commutativæ &longs;ymbolum datur, om
ni inju&longs;titiæ &longs;u&longs;picione vacet. Cæterùm quæ nobis ine&longs;t arbi
trij libertas, poti&longs;&longs;ima naturæ rationis compotis prærogativa,
libræ, aut &longs;tateræ jure merito comparatur, quâ iniqui abuten
tes dicuntur P&longs;alm. 61.
S. Ba&longs;ilius hom. in P&longs;alm. 61. ait
quædam e&longs;t à Conditore omnium apparata, per quam rerum naturam
po&longs;&longs;is probè digno&longs;cere.
quæ &longs;ufficiens di&longs;crimen boni, ac mali demon&longs;trat. Corporea enim
pondera in libræ lancibus probamus; quæ verò ad in&longs;tituendam vi
tam eligenda veniunt, per liberum arbitrium di&longs;cernimus: quod &
&longs;tateram nominavit, quòd momentum æquale ad utrumlibet po&longs;&longs;it
capere.
EO con&longs;ilio in&longs;tituta e&longs;t libra, ut certis, ac notis ponderi
bus, ignotæ gravitatis quantitas indagetur, quæ demùm
innote&longs;cit, cum æquatis hinc & hinc ponderum libræ adnexo
rum momentis, neutro prævalente, libra con&longs;i&longs;tit. In hoc
mùm
ditur in C, quod,
bræ dicitur, non quia &longs;it ne
ce&longs;&longs;ariò Centrum gravitatis li
bræ, &longs;ed quia e&longs;t Centrum,
circa quod agitur, &longs;eu ver&longs;a
tur jugum, infixo nimirum in C axiculo, qui &
Græcis apud Ari&longs;torelem in quæ&longs;t. Mechan.
Partes autem jugi videlicet CA, & CB.
etiam ab aliquibus Ex medio jugi ad per
pendiculum a&longs;&longs;urgit lingula CD, quæ in&longs;eritur an&longs;æ EF com
plectenti capita axiculi, adeò ut &longs;u&longs;pensâ ex F an â, quæ ho
rizonti ad perpendiculum immineat, tùm demùm intelligatur
factum æquilibrium, cum lingula an&longs;æ congruit, & jugum
con&longs;i&longs;tit horizonti parallelum. Utrùm autem
&longs;it ip&longs;a lingula, an verò an&longs;a, non conveniunt Authores: li
tem Grammaticis dirimendam relinquo.
Extremis brachiorum punctis A & B adnectitur utrumque
pondus, tam notum, quod e&longs;t alterius men&longs;ura, quàm igno
tum; cujus gravitas examinatur. Nihil autem refert, an pon
dera uncinis adnexa dependeant, an verò lancibus indè pen
dentibus imponantur; id quod vulgare e&longs;t magi&longs;que u&longs;itatum,
& libræ fecit nomen Illud enim præcipuum e&longs;t, ac
maximè attendendum, quòd omnia hinc & hinc æqualia &longs;int,
nimirum pondus unius lancis cum funiculis &longs;eu catenulis æqua
le &longs;it ponderi alterius lancis cum &longs;uis appendiculis (pondus, in
quam, ponderi æquale &longs;it; nil enim intere&longs;t æquales ne? an
inæquales fuerint utriu&longs;que lancis funiculi &longs;ecundùm longitu
dinem, modò in æquali di&longs;tantiâ à centro adnectantur) & bra
chium alterum majus non &longs;it reliquo brachio non &longs;olùm quoad
gravitatem, quæ materiæ jugi ine&longs;t, &longs;ed poti&longs;&longs;imùm quoad
ip&longs;orum brachiorum longitudinem.
Porrò hæc brachiorum longitudo non e&longs;t de&longs;umenda, ut ita
loquar, materialiter, à centro jugi ad extremitatem, ubi mate
ria de&longs;init, ex quâ con&longs;tat, &longs;ivè ferrum &longs;it, &longs;ivè lignum, &longs;ivè
aliud quidpiam: &longs;ed brachiorum longitudinem definiunt
di&longs;tantiam à centro omnino æqualem e&longs;&longs;e oportet. Huju&longs;modi
autem puncta non alia &longs;unt, quàm puncta contactûs jugi & an
nulorum &longs;eu uncinorum illi infixorum, quibus deinde lances
aut pondera adnectuntur. Hoc illud e&longs;t, in quo maxima arti
ficis indu&longs;tria, atque diligentia collocanda e&longs;t, ut exacti&longs;&longs;imam
brachiorum æqualitatem a&longs;&longs;equatur.
Data itaque hac, quam diximus, brachiorum æqualitate, &longs;i
æqualia pondera hinc & hinc addantur, manife&longs;tum e&longs;t jugum
libræ ex aginâ &longs;u&longs;pen&longs;um ad neutram partem inclinari, &longs;ed ma
nere horizonti parallelum; fieri namque non pote&longs;t, ut extremi
tas altera de&longs;cendat, quin oppo&longs;ita extremitas cum adnexo pon
dere a&longs;cendat, & quidem æquali motu propter brachiorum
æqualitatem. Finge enim pondus B de&longs;cendere in F, utique
pondus A a&longs;cendet in E, at
que de&longs;cribent arcus BF &
AE æquales, quippe qui
æqualibus angulis ad verti
cem in C &longs;ubtenduntur, &
ab æqualibus radiis CB, CA
de&longs;cribuntur. At æqualis e&longs;t in B vis de&longs;cendendi atque in A
repugnantia ad a&longs;cendendum; illa igitur præpollere non pote&longs;t.
Siquidem vis de&longs;cendendi componitur ex ponderis gravitate,
& non impeditâ motûs naturalis velocitate; repugnantia verò
ad a&longs;cendendum componitur & ex ponderis contranitentis gra
vitate, & ex velocitate motûs præter naturam: &longs;unt autem gra
vitates ex hypothe&longs;i æquales, motus etiam per arcus BF & AE
e&longs;&longs;ent æquales; ac proinde vis tendendi deor&longs;um inveniens
æqualem oppo&longs;itam repugnantiam ad motum &longs;ur&longs;um nequit illi
imprimere impetum, quo per vim moveatur: ut enim &longs;equa
tur motus, aut gravitates di&longs;pares e&longs;&longs;e oportet, aut motuum Po
tentiæ moventis & Ponderis moti velocitates inæquales, ut ma
jor &longs;it Ratio huju&longs;modi velocitatum, quàm &longs;it reciproca Ratio
gravitatum: alioquin nulla e&longs;&longs;et virium movendi & re&longs;i&longs;tentiæ
inæqualitas, ubi omnia e&longs;fent æqualia. Cum itaque in librâ &longs;ic
con&longs;titutâ intercedat omnimoda æqualitas & brachiorum, qui
bus definitur motus, & gravitatum, quæ &longs;ibi invicem æquali
ter ob&longs;i&longs;tunt, ac proinde eadem &longs;it reciproca Ratio gravitatum
ce&longs;&longs;e e&longs;t; & in alteram partem &longs;i inclinerur, manife&longs;tum e&longs;t in
illâ lance plus ponderis fui&longs;&longs;e impo&longs;itum, quàm in reliquâ.
Ut autem quàm exacti&longs;&longs;imè ponderum ignota gravitas exa
minari queat, opus e&longs;t ut axiculus jugo infixus (&longs;altem in &longs;upe
riore parte, cui &longs;capus incumbit) exqui&longs;itè cylindricam figu
ram obtineat; hinc enim fiet, ut cum rotundo foramine &longs;capi
contactus fiat in lineâ, quamcumque tandem po&longs;itionem ha
beat ip&longs;e &longs;capus: nam quemadmodum ex prop.
13. lib.
3. duo
circuli &longs;e intùs contingentes tangunt in puncto, ita duæ &longs;uper
ficies cylindricæ, cava altera, altera convexa, &longs;e tangunt in li
neà. Id &longs;i fiat facilè ab æquilibrio deflectet &longs;capus, &longs;i vel modi
ca intercedat ponderum inæqualitas. At &longs;i angulatus fuerit axi
culus, vel &longs;uperior foraminis pars rotunditatem non fuerit a&longs;&longs;e
cuta, jam non in unâ lineâ, &longs;ed in pluribus contactus fieret, at
que adeò iners e&longs;&longs;et ad motum &longs;eapus, etiam&longs;i non omninò
æqualia e&longs;&longs;ent pondera lancibus impo&longs;ita.
Quare artifices illos non probo, qui axem ita ef&longs;ormant, ut
&longs;uperior pars in aciem de&longs;inat, illud &longs;ibi per&longs;uadentes, quod
minore partium conflictu &longs;e tangentes axis & &longs;capus faciliorem
relinquant in alterutram partem motum libræ. Id quod ut ve
rum &longs;it, non tamen vacat periculo, ne, dum axis capita in&longs;e
runtur an&longs;æ, acies illa planè &longs;ursùm non dirigatur, &longs;ed modi
cum in alterutram partem vergat: quæ declinatio &longs;i contingat,
foramen autem exactè rotundum fuerit, miraculo proximum
cen&longs;e, &longs;i libra vacua æquilibrium con&longs;tituat, ita ut lingula ritè
collocata congruat an&longs;æ; acies &longs;i quidem illa dividit inæquali
ter &longs;capi longitudinem, & brachium alterum altero longius e&longs;t,
atque præponderat. Hoc vitium ubi libra contraxerit, inepti
artifices nihil &longs;u&longs;picati ab axe malè conformato, aut perperam
di&longs;po&longs;ito, ortum duxi&longs;&longs;e, vel brachium extenuant, vel lancem
immutant, donec æquilibrium inveniant. Verùm libram hu
ju&longs;modi dolo&longs;am e&longs;&longs;e inferiùs con&longs;tabit propter brachiorum in
æqualitatem: quæ quidem levem infert ponderum differen
tiam in rebus exigui momenti contemnendam; &longs;ed in iis, quæ
exqui&longs;itam ponderis men&longs;uram exigunt, non leve damnum
hinc pote&longs;t emergere.
Quod &longs;i axis non &longs;it an&longs;æ, &longs;ed &longs;capo, firmiter infixus, volua-
magis arridet) jam non &longs;uperior; &longs;ed inferior axiculi pars at
tendenda e&longs;t; quippe quæ inferiorem foraminum an&longs;æ partem
contingit; & eadem, quæ de &longs;uperiore parte dicebantur, ob
&longs;ervanda &longs;unt. Illud tamen præterea in an&longs;æ foraminibus ob
&longs;ervandum venit, quod eorum infima pars ita &longs;it con&longs;tituta, ut
axis illis incumbens parallelus &longs;it horizonti, quando an&longs;a &longs;u&longs;
penditur, ut liberè pendeat, vel ita collocatur, ut ad perpen
diculum horizonti immineat: alioquin axe inclinato, jugum
urgeret aiteram an&longs;æ partem, ab alterâ recederet; ex quo jugi
cuman, conflictu aliqua motui difficultas crearetur.
Jam verò quod ad pondera attinet, &longs;upervacaneum e&longs;t mo
nere non omnia pondera omnibus libris convenire: quamvis
enim libra, quâ libra e&longs;t, nuliam pror&longs;us re&longs;puat ponderum gra
vitatem, &longs;ed omnem quorumcumque ponderum æqualitatem
apta &longs;it indicare &longs;uo æquilibrio; quia tamen ex materiâ con&longs;tat,
quæ definitam habet &longs;oliditatem atque partium firmitatem (ut
nihil dicam de certis atque definitis viribus retinentis an&longs;am,
& cum ansâ libram, ac utrumque pondus) fieri pote&longs;t, ut adeò
gravia lancibus imponantur onera, quæ brachiorum rectitudi
nem inflectant, & eorum æqualitatem corrumpant: Quare te
nuioribus libris parva pondera examinantur, cra&longs;&longs;ioribus ma
jora. Illud potiùs cavendum e&longs;t, ne pondera, quibus tanquam
men&longs;urâ utimur, fallacia &longs;int, quia fal&longs;a, aut excedendo legi
timam gravitatis quantitatem, aut ab illâ deficiendo.
Quamvis autem tot pondera minimæ men&longs;uræ adhibere po&longs;
&longs;emus, quot numerare oporteret ad explorandam propo&longs;itæ
gravitatis ignotæ quantitatem, hoc tamen valde incommodum
e&longs;&longs;et: quid enim, &longs;i lanius carnem in macello vendens grana
numerare cogeretur, quæ æquilibrium cum carne con&longs;tituunt? &longs;ed & inutilis e&longs;&longs;et labor, nam multa &longs;unt, quorum quantitas
non e&longs;t ad vivum re&longs;ecanda, & minuti&longs;&longs;imæ particulæ fru&longs;tra
inve&longs;tigantur. Subtilitas hæc relinquatur gemmariis, aurifici
bus, auríque monetalis cu&longs;oribus, quibus damnum e&longs;&longs;et minu
tias contemnere. Quamquam nec i&longs;tis author fuerim, ut &longs;in
gularibus granis uterentur, &longs;ed potiùs ponderibus, quæ pluri
bus granis æquivalerent; &longs;i enim &longs;ingula grana à legitimo pon
dere dificiunt per cente&longs;imam grani partem, quæ facilè &longs;ensûs
gri grani deficientis; & in uncia libræ Romanæ ponderalis ad
monetam pertinentis cum grana 576 contineantur, in uncia
auri error e&longs;&longs;et granorum ferè &longs;ex deficientium, & in integrâ
librâ, quæ e&longs;t granorum 6912, e&longs;&longs;et error granorum 69; qui
tamen error vix contingat, &longs;i a&longs;&longs;umatur integra uncia, aut li
bra: illud &longs;i quidem, quod &longs;olitarium præ &longs;ua tenuitate in con
&longs;pectum non cadit, cum pluribus &longs;imilibus conjunctum evadit
demum notabile atque con&longs;picuum. Quare ad paranda pon
dera huju&longs;modi &longs;ubtiliora, a&longs;&longs;ume laminam metallicam ponde
re unius libræ, &longs;ed æquabiliter exten&longs;am, eju&longs;que duodecimam
partem accipe; hæc erit Uncia, quam &longs;epones. Alterius Unciæ
octavam partem a&longs;&longs;umens habebis Draclimam. Drachmæ pars
tertia dabit &longs;crupulum. Scrupuli &longs;emi&longs;&longs;is e&longs;t obolus.
Oboli
triens e&longs;t &longs;iliqua. Demùm &longs;iliquæ quadrans e&longs;t Granum.
Ex
hac minutâ divi&longs;ione &longs;atis con&longs;tat, quàm obnoxiæ errori &longs;int
minores particulæ præ majoribus; idemque error, qui in unciâ
fingularis e&longs;&longs;et, & ut nullus con&longs;ideraretur, toties repetitus,
quot grana in unciâ continentur, jam non e&longs;&longs;et contemnen
dus. Id autem dictum intelligatur etiam in majoribus ponde
ribus, ubi unciæ non reputantur, &longs;atius e&longs;&longs;e majora pondera
habere, quàm minimam men&longs;uram &longs;æpiùs multiplicatam a&longs;
&longs;umere.
Sed quoniam adhuc incommodum accideret tot habere
men&longs;uras, quæ juxta &longs;eriem naturalem numerorum cre&longs;cerent,
ut propo&longs;itæ paucitatis examinandæ quantitas indagetur, ob
&longs;ervatum e&longs;t non leve compendium, quod offert progre&longs;&longs;io
Geometrica ab unitate incipiens, & in Ratione dupla aut tri
plâ progrediens. Nam maximum terminum progre&longs;&longs;ionis du
plæ &longs;ibimet ip&longs;i additum &longs;i mulctaveris unitate, & in progre&longs;
&longs;ione triplâ maximo termino unitate mulctato &longs;i re&longs;idui &longs;emi&longs;
&longs;em addideris, numerum habebis gravitatum omnium, quæ
paucis illis ponderibus examinari po&longs;&longs;unt. Sic dentur octo pon
dera in Ratione duplâ incipiendo ab uncia 1; octavum e&longs;t
unc. 128: hunc numerum duplica, & à 256 aufer unitatem,
reliquus numerus 255 indicat octo illis ponderibus po&longs;&longs;e in li
brâ examinari omnes gravitates ab uncia 1 ad uncias 255. Si
mili modo in Ratione triplâ dentur quatuor pondera 1. 3. 9. 27.
tus numero 27 dat 40: cujus igitur gravitatis e&longs;t primum pon
dus ut 1, tot gravitates u&longs;que ad 40 examinari po&longs;&longs;unt illis &longs;olis
quatuor ponderibus. Præ&longs;tat autem uti ponderibus in Ratio
ne duplâ, quia licèt plura pondera requirantur, omnia tamen
&longs;eor&longs;im in propriâ libræ lance collocantur: at &longs;i Ratio ponde
rum &longs;it tripla, aliquâ commutatione uti nece&longs;&longs;e e&longs;t, ut in ad
jecta Tabella ob&longs;ervabis, quæ u&longs;que ad numerum 40. exten
ditur: Ubi etiam vides in Ratione triplâ &longs;ufficere quatuor pon
dera 1. 3.9. 27, at in duplâ exigi &longs;ex videlicet 1. 2. 4. 8. 16. 32.
TABELLE WAR HIER
TABELLE WAR HIER
At contingere pote&longs;t paratis hi&longs;ce ponderibus in Ratione
duplâ aut triplâ aliquid abundare, & maximum terminum cæ
teris additum excedere quæ&longs;itum numerum, (ut hic, &longs;i opus
e&longs;&longs;et provenire &longs;olum ad 40, maximus terminus 32 e&longs;t abun
dans) proptereà retentâ cæterorum &longs;ummâ adde aliud pondus,
ut quæ&longs;itum numerum compleat, & e&longs;t illud, quo opus e&longs;t;
&longs;ic 1. 2. 4. 8. 16. conficiunt &longs;ummam 31; aufer 31 ex 40, re&longs;i
duum e&longs;t 9; &longs;it igitur &longs;extum pondus 9, & &longs;atis erit u&longs;que ad
40; quia cum habeantur reliquis ponderibus omnes numeri
infra 31, jam ex 23 & 9 fit 32, ex 24 & 9 fit 33, & &longs;ic de re
liquis deinceps. Idem dic de aliâ qualibet &longs;ummâ majore
quàm ferant data pondera, minore tamen quàm opus &longs;it, &longs;i
adhuc unum pondus in eâdem progre&longs;&longs;ione adderetur; &longs;ufficit
enim re&longs;iduum. Exemplum habes in &longs;uperiore Tabella pon
derum in Ratione triplâ, ubi quatuor conficiunt 40, &longs;ed &longs;i ad
deretur quintum in eadem Ratione 81, e&longs;&longs;et nimis magnum,
52, & quia inter 40 & 52 differentia e&longs;t 12, quintum pondus
ut 12 &longs;ufficiet. Hinc quia ad libram requiruntur &longs;olum 24 &longs;e
munciæ, ad unciam 24 &longs;crupuli, ad &longs;crupulum 24 grana, &longs;i
pondera &longs;int in Ratione triplâ, &longs;ufficiunt tria ponderâ 1. 3.9.
quæ conficiunt 13, & quartum pondus &longs;it 11, ut compleatur
&longs;umma 24: & in Ratione duplâ &longs;ufficiunt quatuor pondera
1. 2. 4. 8. quæ conficiunt 15, & quintum pondus 9 complens
&longs;ummam 24. illud e&longs;t, quod requiritur, ut ex adjectis Tabel
lis liquet.
TABELLE WAR HIER
TABELLE WAR HIER
Unum hîc, ubi de Ponderibus &longs;ermo e&longs;t, obiter moneo, libræ
nomen apud Romanos æquivocum fui&longs;&longs;e, alia enim erat libra
Ponderalis aridorum, alia Men&longs;uralis liquidorum (& poti&longs;&longs;i
mum olei, quod cornu librali metiebantur) quam inci&longs;is & in
&longs;culptis lineis in uncias 12 partiebantur, quemadmodum & li
bra pondo in uncias pariter 12 di&longs;tinguebatur: &longs;ed inter utram
que libram, &longs;i materia ip&longs;a ad pondus revocabatur, non exi
guum erat di&longs;crimen; ut enim ex proprio experimento te&longs;ta-per genera.
Libra men&longs;ura &longs;olùm uncias decem continebat, quarum li
bra pondo erat duodecim: quapropter uncia men&longs;uralis ad un
ciam ponderalem erat ut 5 ad 6 &longs;pectatâ gravitate & quantita
te materiæ.
USus libræ brachiorum inæqualium minùs nece&longs;&longs;arius e&longs;t,
ac propterea neque communis aut vulgaris, ni&longs;i quatenus
ad &longs;tateram traductus e&longs;t: illam tamen hîc con&longs;iderare erit
operæ pretium, ut æquilibrij rationes magis innote&longs;cant. Sit
libra AB, cujus centro C
dividatur jugum in brachia
inæqualia CA & CB.
Certum e&longs;t, etiam &longs;i nul
lum addatur pondus, ju
gum ex centro C &longs;u&longs;pen
fum retinere non po&longs;&longs;e po
&longs;itionem AB horizonti pa
rallelam; quia licet punctum C &longs;it centrum motûs libræ, non
e&longs;t tamen centrum gravitatis illius; hoc enim e&longs;t in puncto ju
gum (quod hîc æquabiliter ductum ponitur) bifariam dividen
te, videlicet in I, quod æquales gravitates IA & IB cir
cum&longs;tant. Verùm interim ex hypothe&longs;i fingamus lineam AB
omni gravitate carentem; & in ip&longs;is libræ extremitatibus &longs;ta
tuamus pondera eam inter &longs;e reciprocè Rationem habentia,
quæ e&longs;t Ratio brachiorum, & ut CA ad CB, ita &longs;it pondus B
ad pondus A. Pondera hæc, quæ in lancibus libræ vulgaris
æqualium brachiorum magnam momentorum inæqualitatem
haberent, quia inæqualiter gravia, hîc æquilibrium con&longs;ti
tuunt, quamvis inæquales &longs;int eorum gravitates ab&longs;olutæ, quia
libræ brachia reciprocè: &longs;ecundùm eandem Rationem in
æqualia: quatenus enim alligantur pondera hæc extremita-
pote&longs;t in alterutram partem inclinari, cum neutrum pon
dus po&longs;&longs;it ab altero a&longs;&longs;umere vim, qua &longs;ursùm moveatur,
majorem oppo&longs;itâ virtute innatá de&longs;cendendi, qua repu
gnat, ne elevetur. Sit CA ad CB ut 1 ad 4, & vici&longs;&longs;im pon
dus B ut 1 ad pondus A ut 4. Si gravitates dumtaxat con
&longs;iderentur, virtus ponderis A e&longs;t ut 4, virtus verò ponderis B
ut 1: &longs;ed quia à centro motûs C retinentur, nec liberè rectâ viâ
moveri po&longs;&longs;unt, impedimentum recipiunt pro brachiorum lon
gitudine, minû&longs;que impeditur de&longs;cen&longs;us aut a&longs;cen&longs;us rectus
ponderis, quod longiori brachio adjacet, magis, quod brevio
ri. Illud igitur pondus, quod majori brachio adnectitur, &longs;i
de&longs;cendat, magis de&longs;cendit, &longs;i a&longs;cendat, magis a&longs;cendit; quod
verò breviori, &longs;i a&longs;cendat, minùs a&longs;cendit, & &longs;i de&longs;cendat,
minùs de&longs;cendit: atque adeò &longs;i B de&longs;cenderet in E, men&longs;ura
de&longs;censùs e&longs;&longs;et perpendicularis EG, a&longs;&longs;en&longs;um autem ponderis
A in D metiretur perpendicularis DF: idem dic &longs;i A de&longs;cen
deret, & B a&longs;cenderet. Porrò DF & EG &longs;unt in Ratione
brachiorum CA & CB ut patet, quia triangula rectangula
CFD, & CGE, præter rectos angulos ad F & G æquales, ha
bent etiam æquales ad C angulos ad verticem, & per 32. lib.
1.
&longs;unt æquiangula; igitur per 4 lib.
6. ut CD ad CE, ita DF
ad EG; at CD æqualis e&longs;t ip&longs;i CA, & CE ip&longs;i CB (e&longs;t enim
eadem linea, quæ mutatâ po&longs;itione AB venit in DE) igitur
ut CA ad CB ita DF ad EG. Quare ratione po&longs;itionis pon
dus B vim habet de&longs;cendendi, & re&longs;i&longs;tit a&longs;cen&longs;ui, ut 4, pon
dus autem A vim habet de&longs;cendendi, ac proinde etiam re
&longs;i&longs;tendi, ne a&longs;cendat, &longs;olùm ut 1.
Cum itaque momentum de&longs;cendendi (idem e&longs;to judicium
de momento repugnantiæ, ne a&longs;cendat) componatur tùm ex
gravitate ponderis, tùm ex propen&longs;ione ad motum, hoc e&longs;t ex
motûs, qui con&longs;equi po&longs;&longs;et, velocitate, manife&longs;tum e&longs;t gravi
tatem ut 4, cujus motus e&longs;&longs;et ut 1, nec po&longs;&longs;e vincere gravitatem
ut 1, cujus motus e&longs;&longs;et ut 4, nec vici&longs;&longs;im po&longs;&longs;e ab illâ vinci;
e&longs;t &longs;iquidem inter gravitatem quadruplum &longs;emel, & gravita
tem &longs;ubquadruplam quater Ratio æqualitatis; victoria autem
obtineri non pote&longs;t, ni&longs;i intercedat virium inæqualitas. Si
enim pondera e&longs;&longs;ent æqualia, ponderis A re&longs;i&longs;tentia ratione
tis, ergo re&longs;i&longs;tentia e&longs;t æqualis: item &longs;i longitudines e&longs;&longs;ent
æquales, re&longs;i&longs;tentia ponderis B e&longs;&longs;et &longs;ubquadrupla ratione
gravitatis, &longs;ed quadruplicatur ratione di&longs;tantiæ CB; ergo in B
e&longs;t æqualis.
Neutrum igitur pondus pote&longs;t oppo&longs;ito ponderi impetum
imprimere, quo elevetur; quia nimirum unaquæque gravitas
majorem impetum alteri communicare non pote&longs;t, quàm po&longs;
&longs;it ip&longs;a concipere, ac propterea impetus gravitatis B, quæ e&longs;t
ut CA, potens conari deor&longs;um ut GE, &longs;i imprimeretur gravi
tati A, quæ e&longs;t ut CB, deberet illam elevare ut FD: Atqui
gravitas ip&longs;ius A, quæ e&longs;t ut CB, conatur deorsùm ut FD, &
ejus impetus &longs;i gravitati B, quæ e&longs;t ut CA, imprimeretur, il
lam elevare deberet ut GE: igitur in unaquâque gravitate
æqualis e&longs;&longs;et eju&longs;dem conatus deorsùm & vis illata nitens &longs;ur
sùm, nec plus præ&longs;tare po&longs;&longs;et impetus impre&longs;&longs;us, quàm innatus.
Utraque igitur con&longs;i&longs;tere debet, & neutra impetum acquirit,
aut ab alterâ impetum accipit, quia fru&longs;tra e&longs;&longs;et impetus acqui
&longs;itus aut impre&longs;&longs;us, quem nullus con&longs;equi pote&longs;t motus. Quare
cum eadem &longs;it gravitatum Ratio ut CA ad CB, atque motuum
reciprocè ut FD ad GE, ex 16 lib.
6. rectangulum &longs;ub extre
mis CA, hoc e&longs;t pondere B, ut 1, & motu GE, ut 4, æquale
e&longs;t rectangulo &longs;ub mediis CB, hoc e&longs;t pondere A ut 4, & mo
tu FD ut 1: &longs;unt igitur æqualia momenta, quæ componuntur
ex gravitate ut 1 & motu ut 4, atque ex gravitate ut 4 &
motu ut 1.
Ex his aperti&longs;&longs;imè liquet, cur &longs;uperiori capite tantopere in
culcata &longs;it brachiorum æqualitas in libræ jugo, ut ex æquili
brio innote&longs;cat propo&longs;iti ponderis ignota gravitas; hæc enim
æqualis cen&longs;etur notæ gravitati, ubi cùm oblato pondere illa
æquâ lance libratur: quia &longs;cilicet, &longs;i inæqualia e&longs;&longs;ent brachia,
inæquales e&longs;&longs;ent propen&longs;iones ad motum, &longs;eu motuum veloci
tates, quæ ad componendam momentorum Rationem concur
runt; adeóque fieri non po&longs;&longs;et, ut æquales e&longs;&longs;ent gravitates in
lancibus; nam minor gravitas ex brachio longiore plus habet
momenti, quàm ex breviore, pro ratione inæqualitatis brachio
rum. Verum e&longs;t libram huju&longs;modi brachiorum inæqualium
vacuam po&longs;&longs;e priùs ad æquilibritatem reduci, deinde, illâ &longs;ic
pro Ratione inæqualium brachiorum, & ex æquilibrio argui
ponderum illorum Rationem, non tamen æqualitatem: &longs;edar
tificium hoc, quod peritioribus nihil officeret, an&longs;am non mo
dicam furacibus, & dolo&longs;is mercatoribus præberet decipiendi
imperitos; quamvis enim libræ huju&longs;modi æquilibri impo&longs;itis,
hinc & hinc ponderibus adhuc fieret æquilibrium, &longs;ignum
quidem e&longs;&longs;et æqualibus momentis addita e&longs;&longs;e æqualia momen
ta gravitatis, non tamen verùm e&longs;&longs;et additas e&longs;&longs;e æquales gra
vitates, ut rudioribus forta&longs;&longs;e videretur. Hinc e&longs;t libram bra
chiorum inæqualium in u&longs;u non e&longs;&longs;e, ne locus pateat dolis.
Dixi autem expre&longs;sè priùs &longs;tatuendam e&longs;&longs;e libræ vacuz
æquilibritatem, deinde &longs;umenda pondera reciprocè pro Ratio
ne longitudinis brachiorum: ni&longs;i etenim priùs æquilibritas illa
&longs;tatueretur, &longs;i pondera impo&longs;ita e&longs;&longs;ent reciprocè in Ratione
longitudinis brachiorum, &longs;emper pondus minus additum bra
chio longiori præponderaret, quia etiam ip&longs;a brachij longioris
gravitas &longs;ua habet momenta, & quidem non modica, majora
momentis brachij brevioris, quæ omninò computanda &longs;unt:
nam &longs;i ponderum in ea Ratione reciprocè po&longs;itorum momenta
&longs;int æqualia, illi&longs;que adjiciantur inæqualia gravitatis bra
chiorum momenta, manife&longs;tum e&longs;t momentorum &longs;ummam, cui
plus additur, majorem e&longs;&longs;e reliquâ, cui additur minus.
Sed quænam &longs;unt, & quanta utriu&longs;que brachij momenta?
Ut hæc inve&longs;tigemus, & certâ ratione definiamus, ponamus
jugum ip&longs;um &longs;ecundùm &longs;uas omnes partes uniu&longs;modi, & gravi
tatem æquabiliter fu&longs;am per totam illius longitudinem. Sit igi
tur datum pri&longs;ma AB, quod
in quinque partes æquales
dividatur, &longs;ingulas pondoli
bram unam; & per &longs;ingula
gravitatis centra ducatur
recta
dùm rectam HI, à qua pars
una C ab&longs;cinditur à reliquis, totius pri&longs;matis &longs;u&longs;pen&longs;io, ita ut
centrum motûs &longs;it in S. Proculdubio unaquæque pars à cæteris
&longs;ejuncta &longs;i appenderetur &longs;ecundùm longitudinem jugi
quod infigeretur per centra gravitatum
centro motûs. Quid autem refert (quod quidem attinet ad
hanc momentorum Rationem) &longs;i in unum continuum corpus
unitæ illæ partes coagmententur, an verò divi&longs;æ &longs;olo contactu
&longs;ibi invicem adhæreant? eadem quippe e&longs;t gravitas &longs;ingulis in
&longs;ita, eadem &longs;ingularum à centro di&longs;tantia. Cum itaque centra
gravitatum
C & D æquiponderant: at di&longs;tantia
ergo momentum partis E triplum e&longs;t momenti partis C; &longs;imi
lique ratione pars F habet momentum quintuplum, & pars G
&longs;eptuplum. Igitur componendo, momentum totius aggregati
quatuor partium D, E, F, G, e&longs;t &longs;edecuplum momenti partis
C; neque enim &longs;ingulæ partes ex hoc quod cum cæteris pen
deant, illi&longs;que cohæreant, &longs;uum amittunt momentum. Hinc
fit momenta brachiorum e&longs;&longs;e inter &longs;e ut Quadrata longitudi
num eorumdem brachiorum: &longs;iquidem o&longs;tenditur &longs;ingularum
partium momentum cre&longs;cere &longs;ecundùm Rationem numero
rum imparium, prout &longs;ecundùm eandem Rationem cre&longs;cunt
di&longs;tantiæ centrorum gravitatis illarum. Sic brachiorum
longitudines &longs;i e&longs;&longs;ent in Ratione 2 ad 7, illorum momenta
ratione &longs;uæ gravitatis innatæ & ratione po&longs;itionis e&longs;&longs;ent ut 4
ad 49.
Hæc Ratio momentorum in Ratione Quadratorum longi
tudinis, &longs;i res attentè perpendatur, omnibus e&longs;t manife&longs;ta:
Nam &longs;ingulorum brachiorum gravitates juxta hypothe&longs;im
æquabiliter fu&longs;æ per totum libræ jugum Rationem inter &longs;e
habent, quam illorum longitudinis propen&longs;iones ad motum,
&longs;eu, quod eòdem recidit, di&longs;tantiæ à centro motûs eandem
pariter Rationem habent, quam brachiorum longitudines:
Quoniam igitur (ut &longs;æpiùs dictum e&longs;t, &longs;æpiú&longs;que iterùm
inculcandum) momenta componuntur ex gravitatibus ratio
ne materiæ, & ex propen&longs;ionibus ad motum ratione &longs;itûs &longs;eu
po&longs;itionis, componuntur duæ Rationes longitudinum; atque
adeó momentum unius brachij ad momentum alterius bra
chij e&longs;t in duplicata Ratione &longs;uarum longitudinum, hoc
e&longs;t, ut ip&longs;arum longitudinum Quadrata. Id quod adhuc ul
teriùs &longs;ic explicari po&longs;&longs;e videtur. Sit libræ jugum M. N, &
motûs centrum O: intelligatur moveri, ut obtineat po&longs;itio-Momentum brachij minoris OM referre videtur
&longs;ector MOP, momentum verò brachij majoris ON referre
videtur &longs;ector NOR; &longs;ingularum
quippe partium motus ab arcu
de&longs;criptus illarum momentum ob
oculos ponit, & totius brachij mo
mentum illius motus, &longs;cilicet &longs;ector
in motu de&longs;criptus. At ob æquali
tatem angulorum ad verticem in
O, &longs;ectores MOP, NOR &longs;unt &longs;i
miles, &, quia uterque &longs;ector e&longs;t
&longs;imilis pars &longs;ui circuli, eam inter &longs;e habent &longs;ectores Rationem,
quæ e&longs;t circulorum, per 15.lib.5. circuli autem &longs;unt in dupli
catâ Ratione diametrorum, ex 2.lib.12. &longs;eu Radiorum OM
& ON; igitur & &longs;ectores &longs;unt in duplicatâ Ratione OM ad
ON, hoc e&longs;t quadrati OM ad quadratum ON.
At quæris.
In propo&longs;ito pri&longs;mate AB, momentum brachij
SA ad momentum brachij SB e&longs;t ut 1 ad 16: An, ut ha
beatur æquilibrium in S, addendum erit in A pondus libra
rum 15? quandoquidem pars C e&longs;t libræ unius, reliquum au
tem brachium lib.
4, & longitudo SB e&longs;t quadrupla longitu
dinis SA.
Hoc &longs;anè non e&longs;t iis, quæ dicta &longs;unt, con&longs;equens, necex illis
efficitur: aliud quippe e&longs;t momenta brachiorum e&longs;&longs;e ut 1 ad 16,
aliud verò perinde &longs;e habere, atque &longs;i ex brachiorum gravita
te carentium extremitatibus penderent libræ 1 & 16, ut ad
æquilibrium con&longs;tituendum opus &longs;it breviori brachio addere
libras 15. Primum illud verum e&longs;t, etiam &longs;i extremitatibus ad
necti intelligamus hinc quidem libræ &longs;emi&longs;&longs;em; hinc verò li
bras octo, mane &longs;cilicet eadem Ratio 1 ad 16. Alterum à for
mà veritatis prorsùs alienum videtur, nam licet libræ 4 in ex
tremitate B po&longs;itæ æquivaleant libræ unciæ &longs;imul cum pondere
lib.15. in extremitate A; non e&longs;t tamen eadem ratio librarum 4
&longs;ecundùm longitudinem brachij SB di&longs;tributarum; quo enim
propiores &longs;unt partes centro motûs, eò minus habent mo
menti: non igitur libræ 4 &longs;ic di&longs;tributæ æquivalent libris
16, nec addendum erit pondus librarum 15 in oppo&longs;itâ extre
mitate ad æquilibrium con&longs;tituendum, quandoquidem nec ip&longs;a
beret &longs;i totâ ex A penderet.
Equidem ex his, quæ paulò ante dicebam de &longs;ectoribus re
ferentibus momenta brachiorum, aliquando eò deveni, ut &longs;u&longs;
picarer totam gravitatem brachij ON (idem dic de reliquo
OM) intelligendam e&longs;&longs;e ibi exercere totum momentum, ubi
e&longs;t qua&longs;i centrum omnium &longs;uorum momentorum, hoc e&longs;t, ubi
momenta bifariam dividuntur. Si autem &longs;ector NOR refert
totum momentum brachij ON; non e&longs;t intelligendum cen
trum hoc momentorum e&longs;&longs;e punctum L, ubi e&longs;t &longs;emi&longs;&longs;is bra
chij ON; quia Sector LOQ ad Sectorem NOR e&longs;t in Ra
tione Quadrati OL ad Quadratum ON, quod e&longs;t illius qua
druplum. Quod &longs;i inter OL & ON &longs;umatur media propor
tionalis OV, jam &longs;ector VOT e&longs;t ad Sectorem NOR in du
plicatâ Ratione Radiorum OV, & ON, hoc e&longs;t ut OL ad
ON, hoc e&longs;t ut 1 ad 2; ac propterea Sector VOT æqualis e&longs;t
Trapezio NVTR; proinde in V videbantur divi&longs;a æqualiter
momenta, Hinc arguebam vel totam brachij gravitatem cen
&longs;endam e&longs;&longs;e &longs;ua exercere momenta in puncto di&longs;tantiæ à centro
motûs mediæ proportionalis inter &longs;emi&longs;&longs;em brachij & totam
brachij longitudinem, vel in extremitate brachij cen&longs;en
dam e&longs;&longs;e pendere gravitatem, quæ medio loco proportiona
lis &longs;it inter totam brachij eju&longs;dem gravitatem & ejus &longs;e
mi&longs;&longs;em.
Verùm, ut quod res e&longs;t &longs;incerè eloquar, quamvis in Secto
ribus illis, quos paulò ante commemorabam, imaginem
quandam momentorum gravitatis &longs;ecundùm brachiorum
longitudinem di&longs;tributæ agno&longs;cerem, non tamen in re
Phy&longs;icâ &longs;atis fidebam Geometricæ illi commentationi: quip
pe qui ob&longs;ervabam à Sectoribus quidem poni ob oculos Ra
tionem momentorum &longs;ingulorum brachiorum ex motu, qui
idem e&longs;t, &longs;ivè multa, &longs;ivè modica &longs;it gravitas, &longs;ivè in uno,
&longs;ivè in alio puncto con&longs;tituta intelligatur, non tamen defi
niri ip&longs;ius gravitatis momenta. Quare &longs;atius duxi ad experi
menta potiùs confugere, ut hinc lux aliqua &longs;uboriretur, qua
gravitatis quæ&longs;ita momenta innote&longs;cerent.
Primùm igitur a&longs;&longs;umptus e&longs;t ligneus cylindrus, cujus dia
meter CE unc. 1. 06″ pedis Romani antiqui, & addito in A
factum e&longs;t in B puncto. Fuit autem longitudo BA unciarum
pedis Romani 7 2/5 BC ve
rò unc.(42 17/50). Re&longs;ecto de
mùm &longs;ubtili&longs;&longs;imè cylindro,
repertum e&longs;t pondus AB
unciarum 2 1/8, pondus an
tem BC unc. 13 1/2. Hisob
&longs;ervatis cum nullus dubitarem, quin momenta brachiorum
e&longs;&longs;ent ut quadrata longitudinum, ip&longs;as longitudines AB
unc. 7 2/5, & BC unc.(42 17/50) ad unicam
delicet (370/50) & (2117/50): & a&longs;&longs;umptis numeratorum Quadratis 136900
atque 4481689 hanc po&longs;ui Rationem momentorum. Tùm &longs;ic
ratiocinatus &longs;um Algebricè; ut 136900 ad 4481689, ita mo
mentum BA 1 ℞ ad 32.73″ ℞ momentum BC. Cum igitur
æqualitas e&longs;&longs;et inter momentum brachij BC, & momentum
brachij BA plus ip&longs;o pondere D; hæc enim con&longs;tituebant
æquilibrium, æquatio Algebricè e&longs;t inter momentum BC
32. 73″ ℞ & BA + D, hoc e&longs;t 1 ℞ + unc. 40 1/2: & per An
tithe&longs;im demptâ utrinque 1 ℞, æquatio e&longs;t inter 37. 73″ ℞ &
unc. 40 1/2. Factâ itaque numeri ab&longs;oluti 40 1/2 divi&longs;ione per nu
merum Radicum prodit pretium 1 ℞ pondo unc.1.27″, quod e&longs;t
momentum brachij BA; ac proinde momentum brachij BC:
e&longs;t pondo unc.41. 57″. Quare perinde e&longs;t atque &longs;i gravitas
unc. 1. 27″ poneretur in extremitate Alineæ Mathematicæ, ac
in extremitate C poneretur gravitas unc. 41. 57″.
At in A fuit
additum pondus unc. 40 1/2: ergo momentum brachij BC æqui
valet ponderi D, & præterea unc.1.07″, qui e&longs;t &longs;emi&longs;&longs;is gravitatis
brachij AB ob&longs;ervatæ unc. 2 1/8, hoc e&longs;t in cente&longs;imis paulò ul
tra 2. 12″. Si verò momentis brachij BA pondo unc.
1.27″ ad
datur gravitas D pondo unc. 40. 50″, fit aggregatum 41.77″,
quod excedit inventum momentum brachij BC unc.41.57″.
exce&longs;&longs;u (20/100) unciæ: quæ di&longs;crepantia facillimè potuit oriri ex
aliquâ exili, ac minime notabili differentiâ vel in dimetiendis
brachiorum longitudinibus, vel in ponderandis eorum gravi
tatibus; cum maximè re&longs;egmina illa, & &longs;cobs, non computa
rentur in gravitate. Quod &longs;i fiat ut longitudo BC 2117 ad
dus in B unc. 7. 30″, con&longs;tat e&longs;&longs;e ferè &longs;emi&longs;&longs;em gravitatis
unc. 13 1/2: &longs;ed e&longs;t exce&longs;&longs;us &longs;emunciæ ob minùs accuratam ob
&longs;ervationem.
Qua propter aliud experimentum quàm accurati&longs;&longs;imè in&longs;ti
tui ligneo parallelepipedo, cujus longitudo palmorum Roma
norum 7. unc.6. 566‴, ejus verò pondus lib.
1. unc.1 1/4. Alte
ri extremitati additus e&longs;t
plumbeus cylindrus ad per
pendiculum pendens, cujus
pondus unc. 20. Impo&longs;itum
e&longs;t parallelepipedum rotun
do claviculo ferreo, qui horizonti parallelus erat, & factum
e&longs;t æquilibrium in puncto, ubi tota longitudo in duas partes
dividebatur, quarum minor ponderi adhærens fuit men&longs;urâ
unc. 18 1/6, partes verò major fuit men&longs;urâ palm.
6. unc.2/5. Cum
itaque longitudo CB ob&longs;ervata fuerit unciarum men&longs;uralium
72. 40″, & AC unciarum men&longs;uralium 18. 16″, in eadem
pariter Ratione ponuntur brachiorum gravitates ab&longs;olutæ.
Quare CB pondo unc.
1059, AC verò pondo unc.
2. 66″.
Igitur ut longitudinis BC quadratum 52417600 ad longitudi
nis AC quadratum 3297856, ita momentum BC 1 ℞ ad
(3297856/52417600) ℞ momentum brachij AC: cui additur cylindrus D
unc.20: E&longs;t ergo æquatio inter AC + D, hoc e&longs;t (3297856/52417600) ℞ +
unc. 20.00″ & 1 ℞; & factâ Antithe&longs;i e&longs;t æquatio inter
unc. 20.00″ & (49119744/52417600) ℞: demum in&longs;titutâ divi&longs;ione con&longs;urgit
pretium 1 ℞, hoc e&longs;t momentum BC, unc. 21. 342‴ & paulo
amplius: atque momentum brachij AC e&longs;t pondo unc.1.343‴,
cui additâ gravitate cylindri fit &longs;umma unc. 21. 343‴ planè
æqualis momento brachij BC.
Et ut hanc operandi methodum confirmarem, iterum in&longs;ti
tui argumentationem a&longs;&longs;umendo quadrata gravitatum utriu&longs;
que brachij, &longs;unt enim ex hypothe&longs;i gravitates in Ratione lon
gitudinum. Cum igitur &longs;it CB pondo unc.
10. 50″; & AC
pondo unc. 2. 66.″ fiat ut quadratum CB 1121481 ad quadra
tum AC 70756, ita ip&longs;ius CB momentum 1 ℞ ad (70756/1121481) ℞ Quoniam verò AC + D hoc e&longs;t (70756/1121481) ℞
20.00″ æquatur momento BC hoc e&longs;t 1 ℞, factâ per
Antithe&longs;in communi &longs;ubtractione (70756/1121481) ℞, remanet æquatio
inter pondus unc. 20.00″ & (1050725/1121481) ℞, & factâ divi&longs;ione emer
git pretium 1 ℞, hoc e&longs;t momentum BC pondo unc. 21. 347‴.
atque adeò momentum ip&longs;ius AC e&longs;t pondo unc.
1. 347″; cui
&longs;i addatur cylindri D gravitas unc. 20, totum momentum in A
e&longs;t unc. 21. 347‴, omnino æquale momento ip&longs;ius B: id quod
ab initio vix &longs;perare audebam, cum hæc operatio à &longs;uperiore
differat &longs;olùm per (1/1000). Hîc pariter brachij AC gravitas ab&longs;o
luta pondo unc. 2. 66″. habet momentum unc.
1. 347‴, cum
ejus &longs;emi&longs;&longs;is &longs;it unc. 1. 330‴, quæ e&longs;t minima atque prorsùs
contemnenda differentia: quî enim fieri potuit, ut, quantali
bet adhiberetur diligentia in metiendo, & ponderando, ne
pilum quidem à verò aberrarem? aut quis omninò certus &longs;it
omnes parallelepipedi partes æquali prorsùs fui&longs;&longs;e præditas gra
vitate, itaut quæ pars ad arboris radicem vergebat, non fuerit
paulò den&longs;ior, aut interiùs nodulum aliquem latentem habue
rit, quo factum fuerit, ut vera gravitas in&longs;tituto calculo non
exacti&longs;&longs;imè re&longs;ponderet? &longs;imili ratione &longs;emi&longs;&longs;is gravitatis bra
chij BC intelligitur in extremitate B: nam fiat ut longitudo
BC 72. 40″ ad longitudinem AC 18.16″, ita reciprocè pon
dus in A unc. 21. 347‴ ad pondus in B unc.
5. 354‴: erat au
tem brachij BC gravitas ab&longs;oluta unc. 10. 59″ cujus, &longs;emi&longs;&longs;is
5. 295‴. differt ab invento pondere &longs;olùm per (50/1000) unciæ, hoc
e&longs;t ferè &longs;e&longs;qui&longs;crupulum, &longs;eu grana 34.
Ex his quidem &longs;atis apparebat brachij gravitatem in libræ
jugo intelligendam e&longs;&longs;e, qua&longs;i ejus &longs;emi&longs;&longs;is in ipsâ extremitate
con&longs;titueretur, &longs;eu, quod idem e&longs;t, tota gravitas brachij ad
mediam longitudinem applicaretur (eadem &longs;iquidem e&longs;&longs;e mo
menta totius gravitatis in dimidiatâ di&longs;tantiâ, ac dimidiæ gra
vitatis in totâ di&longs;tantiâ, ex &longs;æpiùs dictis e&longs;t manife&longs;tum) mihi
tamen &longs;atisfactum non exi&longs;timabam, ni&longs;i ulteriore experimento
veritatis ve&longs;tigia per&longs;equerer. Quare eundem plumbeum cy
lindrum, cujus longitudo erat palmi 1. unc. 1. (9/10), ita in extre
mitate A collocavi, ut &longs;uper AI jaceret, & factum e&longs;t æquili
brium in E, eratque EA longitudo unc. (22 4/10). Tùm divi&longs;o bi
fariam in O &longs;patio AI, quod cylindrus jacens occupabat, ex
brium exacti&longs;&longs;imè in E, &longs;icut priùs, cum jacebat &longs;uper AI.
Deinde cylindrum eumdem iterum parallelepipedo impo&longs;ui ja
centem, &longs;ed ea ratione illum ultrò citróque promovebam, ut
omnino propè fulcrum con&longs;i&longs;teret, donec demùm factum e&longs;t
æquilibrium in H, & fuit HA palm.2. unc.(10 7/10): Factâ verò
&longs;u&longs;pen&longs;ione cylindri ex L, ita ut HL e&longs;&longs;et dimidiata cylin
dri jacentis longitudo, æquilibrium pariter in H factum e&longs;t.
Relictâ igitur illâ &longs;ectorum analogiâ, deprehendi per illas
quidem ob oculos poni motum, non verò momentum, &longs;eu pro
pen&longs;ionem ad motum, quæ ex di&longs;tantiâ à centro motûs in ipsâ
longitudine definienda e&longs;t: & quod ad gravitatem attinet, nul
lus mihi relictus e&longs;t dubitandi locus ita computandam e&longs;&longs;e to
tius brachij gravitatem per ip&longs;um æquabiliter diffu&longs;am, qua&longs;i
tota in dimidiatâ di&longs;tantiâ à centro motûs collocaretur: quam
vis enim particularum gravium, quæ ultrâ &longs;emi&longs;&longs;em longitudi
nis magis à centro removentur, momentum cre&longs;cat pro Ratio
ne di&longs;tantiæ, reliquarum tamen numero æqualium citrà longi
tudinis &longs;emi&longs;&longs;em centro propiorum momentum &longs;imiliter pro
Ratione minoris di&longs;tantiæ minuitur; ac proptereà tantùm i&longs;ta
momenta &longs;imul &longs;umpta decre&longs;cunt, quantum illa &longs;imul &longs;umpta
augentur. Ex quo oritur quædam qua&longs;i æqualitas, perinde at
que &longs;i momenta omnia majora & minora in illam particulam
confluerent, quæ media e&longs;t Arithmeticè inter extrema (mo
menta &longs;i quidem ratione di&longs;tantiæ Arithmeticè cre&longs;cunt, prout
Arithmeticè ip&longs;a di&longs;tantia cre&longs;cit) hæc autem e&longs;t in &longs;emi&longs;&longs;e
longitudinis brachij. Ex quo iterum confirmatur momenta
brachiorum e&longs;&longs;e ut quadrata longitudinum; &longs;unt enim in du
plicatâ Ratione illarum; &longs;emi&longs;&longs;es quippè &longs;unt in Ratione inte
grarum longitudinum, gravitates &longs;unt in Ratione earumdem
longitudinum, ergo Ratio compo&longs;ita e&longs;t duplicata eju&longs;dem Ra
tionis longitudinum.
Hinc datâ jugi æquabilis, & uniformis gravitate ab&longs;olutâ,
& datâ Ratione longitudinum brachiorum inæqualium libræ,
dividatur data gravitas &longs;ecundùm datam Rationem brachio
rum: tùm fiat ut longitudo minor ad longitudinem majorem,
ita dimidia gravitas majoris brachij ad aliud, ex quo quarto ter-
&longs;iduum indicabit pondus addendum extremitati brachij mino
ris, ut fiat æquilibrium cum &longs;olâ gravitate brachij longioris.
Vel potiùs fiat ut quadratum longitudinis brachij minoris ad
differentiam inter quadrata brachiorum, ita &longs;emi&longs;&longs;is gravitatis
brachij minoris ad pondus ip&longs;i addendum.
QUamvis libro primo plura de Gravitatis centro, prout hu
jus operis in&longs;tituto congruebat, di&longs;putata &longs;int, eorum ta
men plenior explicatio ex his, quæ duobus præcedentibus ca
pitibus dicta &longs;unt, petenda e&longs;t, &longs;i quidem Phy&longs;icam æquilibrij
cau&longs;am no&longs;&longs;e velimus. Neque enim Gravitatis centrum illud
e&longs;t, quod æquales gravitates, &longs;ed quod æquales gravitationes,
aut æqualia gravitatis momenta, hoc e&longs;t æquales ad de&longs;cen
dendum propen&longs;iones ac vires circum&longs;tant. Nam gravitas câ
Ratione per univer&longs;um corpus grave di&longs;tribuitur, quâ Ratio
ne materia ip&longs;a, cui illa ine&longs;t, diffu&longs;a intelligitur; quæ &longs;i uniu&longs;
modi &longs;it & homogenea, ibi centrum habet, ubi e&longs;t molis ip&longs;ius
centrum; ubi &longs;iquidem bifariam moles & materia, ibi pariter
gravitas illi in&longs;ita bifariam dividitur. Quoniam verò fieri po
te&longs;t, ac &longs;æpiùs contingit, materiam quidem corporis & molem
invariatam permanere, figuram autem mutari; ex quo nunc in
hanc, nunc in illam partem migrat gravitatis centrum, quia
alia atque alia fiunt gravitatis momenta pro variâ corporis &longs;e
cundùm &longs;uas partes po&longs;itiones; proptereà huju&longs;modi momento;
rum æqualitas ex libræ Rationibus de&longs;umenda e&longs;t, &longs;ivè æqua
lium, &longs;ivè inæqualium brachiorum libra intelligatur, prout va
ria corporis gravis &longs;u&longs;pen&longs;io aut &longs;u&longs;tentatio contingit.
Sed quia in communi u&longs;u non adeò frequens e&longs;t illa &longs;u&longs;pen
&longs;io, qua corpus pendeat qua&longs;i ex puncto lineæ directionis tran
&longs;euntis per centrum gravitatis, & ad univer&longs;i centrum de
ductæ, aut illa &longs;u&longs;tentatio, qua corpus grave acuti&longs;&longs;imo apici
ita plerumque &longs;u&longs;penditur, aut &longs;u&longs;tinetur corpus, ut ductâ per
Gravitatis centrum lineâ, aut ex hujus extremitatibus tan
quam polis illud &longs;u&longs;pendatur, aut &longs;ubjecto fulcro lineæ huic
parallelo illud &longs;u&longs;tineatur; ideò huju&longs;modi lineam per centrum
gravitatis ductam liceat appellare
diameter qua&longs;i in librâ locum Axis &longs;eu Aginæ obtinet, corporis
verò partes hinc & hinc po&longs;itæ rationem habent brachiorum
libræ, atque pro di&longs;tantiarum &longs;eu longitudinum Ratione &longs;ua
habent momenta. Sit propo&longs;itum Trapezium, cujus gravita
tis centrum C puncto re&longs;pondeat, &
&longs;u&longs;tineatur &longs;ecundùm rectam lineam
ACN (&longs;imilis e&longs;&longs;et philo&longs;ophandi
ratio, &longs;i a&longs;&longs;umeretur recta RCS) quæ
proptereà
dicitur, quia &longs;icut circuli diameter
per centrum ducta illum in &longs;emicircu
los æquales di&longs;tinguit, ita hæc per
gravitatis centrum tran&longs;iens dividit
Trapezium in momenta æqualia, itaut in neutram partem in
clinetur, juxta dicta de centro Gravitatis. Sed cur fiat æquili
brium intelliges ex Rationibus libræ Brachiorum inæqualium:
ducatur enim ad rectam AN per C perpendicularis DCE, &
fiunt brachia CD, CE inæqualia; &longs;unt igitur momenta CE
longioris majora momentis CD brevioris. Ductis verò ip&longs;i
DE parallelis BF & ML, &longs;ecatur diameter gravitatis AN in
punctis H & I: quare inæqualia &longs;unt brachia HB longius, &
HF brevius, & vici&longs;&longs;im IM e&longs;t brevius, & IL longius: Ex quo
fit momenta in L & E majora e&longs;&longs;e momentis in M & D, at mo
mentum in F minus e&longs;&longs;e momento in B; atque adeò compo
nendo majora cum minoribus ex eâdem parte, fieri compo&longs;i
tum momentum unius partis æquale toti momento oppo&longs;itæ
partis. Vel &longs;i non placeat particulatim Trapezium di&longs;tinguere
qua&longs;i in tot libras, quot ductæ intelliguntur parallelæ, dic to
tius gravitatis ADN &longs;emi&longs;&longs;em intelligi in D, & totius gravi
tatis AEN &longs;emi&longs;&longs;em intelligi in E; & quamvis pars ADN ab
&longs;olutè & &longs;eor&longs;im accepta major &longs;it & gravior parte AEN ab&longs;o
lutè &longs;umptâ, quia tamen &longs;unt reciprocè in Ratione di&longs;tantia-
minùs gravis ex po&longs;itione majorem habet propen&longs;ionem ad mo
tum, qui e&longs;&longs;et velocior; partis verò gravioris minor e&longs;t propen
&longs;io ad motum, qui e&longs;&longs;et tardior; atque adeò hæc minùs re&longs;i&longs;tit
ratione motûs, magis autem ratione gravitatis; at illa ex adver
&longs;o magis re&longs;i&longs;tit ratione motûs, &longs;ed minùs ratione gravitatis,
&longs;ervatâ reciprocè eâdem Ratione inter gravitates & motus. Nil
igitur mirum &longs;i æquatis hinc & hinc viribus agendi, & re&longs;i&longs;ten
di &longs;equatur con&longs;i&longs;tentia.
Hinc manife&longs;tum e&longs;t, cur mutatâ figurâ centrum gravitatis
ad eam partem transferatur, quæ longiùs à &longs;u&longs;tentationis vel
&longs;u&longs;pen&longs;ionis loco recedit; quia nimirum cre&longs;cunt ex illâ parte
comparatè ad oppo&longs;itam momenta ratione di&longs;tantiæ majoris, ac
proinde, ut fiat momentorum æqualitas, centrum ad illam par
tem &longs;ecedit. Sic ce&longs;pitantes à naturâ docentur in partem op
po&longs;itam illi, in quam inclinantur, brachium illicò extendere,
ut brachij gravitas longiùs à corpore tran&longs;lata plus habeat mo
menti, quàm cùm reliquo corpori adhæret, atque hinc &longs;equa
tur centri gravitatis in illam partem tran&longs;latio. Veritas hæc &longs;a
tis nota e&longs;t ip&longs;is funambulis, cùm corpus univer&longs;um &longs;uper ex
tento fune librant; neque enim temerè crura & brachia exten
dunt aut contrahunt, &longs;ed certâ lege, ut centrum momento
rum gravitatis totius corporis hac vel illâ ratione di&longs;po&longs;iti im
mineat, & incumbat funi. Sic plumbeæ virgæ rectæ ex medio
&longs;u&longs;pen&longs;æ, & in æquilibrio manentis, &longs;i brachium alterum in
flexeris, fieri non pote&longs;t, ut reliquum brachium rectum &longs;ervet
po&longs;itionem horizonti parallelam, &longs;ed deor&longs;um inclinabitur, qun
cum longius &longs;it brachio inflexo, majora habet momenta ac
prævalet. Quod &longs;i ob inæqualem virgæ cra&longs;&longs;itiem non planè
ad mediam illius longitudinem facta &longs;it &longs;u&longs;pen&longs;io, &longs;ed æquili
britas contingat in puncto, quod propius e&longs;t cra&longs;&longs;iori extremi
tati virgæ, factâ alterutrius brachij inflexione tollitur æquili
brium, quia non jam ampliùs eadem e&longs;t reciprocè Ratio longi
tudinum, quæ & gravitatum.
Ex his pariter con&longs;equens e&longs;t aliquando minimam virtutem
&longs;atis e&longs;&longs;e ad dimovenda ab æquilibrio ingentia corpora, &longs;i ita
&longs;u&longs;tineantur, ut fulcrum vel in puncto, vel in lineâ contingant:
quoniam &longs;i corpus grave in&longs;i&longs;tat apici coni, aut pyramidis, aut
corpus &longs;ive planâ, &longs;ive &longs;phæricè cavâ, &longs;ive &longs;phæricam æmulan
te &longs;uperficie, contactus in puncto efficitur, ac propterea qua
cunque in extremitate corporis addatur vis movendi, æquili
brium tollitur, & quidem eò faciliùs, quo magis à puncto con
tactûs extremitas illa removetur; in illâ quippe di&longs;tantiâ vis mo
vendi apta velociorem motum efficere, quàm &longs;i propior e&longs;&longs;et,
plus habet momenti: Id quod adhuc faciliùs accidit, &longs;i ab ex
tremitate, ubi vis movendi applicatur, ductâ per contingentis
fulcri punctum rectâ lineâ ad oppo&longs;itam extremitatem, inæqua
liter divi&longs;a &longs;it in puncto contactûs, & vis ip&longs;a movendi in magis
di&longs;tante extremitate con&longs;tituta fuerit; tunc enim non &longs;ua tan
tùm momenta addit, &longs;ed illa multiplicat pro Ratione exce&longs;sûs
&longs;uæ di&longs;tantiæ; quemadmodum de inæqualibus libræ brachiis
dictum e&longs;t. Sin autem fulcrum &longs;u&longs;tinens, quod horizonti paral
lelum ponitur, &longs;it acies pri&longs;matis, aut latus pyramidis jacentis,
aut portio cylindrica &longs;eu conica jacens; tunc in lineâ fit con
tactus, &longs;i vel plana &longs;it, vel circulariter concava corporis in
&longs;i&longs;tentis &longs;uperficies: &longs;ed &longs;i vis movendi, quantacumque &longs;it, ad
datur &longs;ecundùm rectam lineam, quæ efficit Gravitatis diame
trum, puta in A vel N, non mutat æquilibritatem, &longs;i fulcrum
congruit toti diametro AN: &longs;i verò fulcrum brevius e&longs;t quàm
AN, & ex. gr. congruit &longs;olùm ip&longs;i AI, jam centrum motûs e&longs;t
I, & oportet vim movendi tantam e&longs;&longs;e in N, ut aggregatum ex
parte MLN ac virtute additâ in N habeat ad partem MAL re
liquam majorem Rationem, quàm &longs;it Ratio di&longs;tantiæ IA ad
di&longs;tantiam IN. Quare in huju&longs;modi contactu lineari vis mo
vendi, æquilibrium facilè tollens, e&longs;&longs;e debet ad latus diametri
gravitatis, & pro ratione di&longs;tantiæ majus erit momentum; ma
ximum autem erit momentum in E di&longs;tantiâ maximâ.
Non igitur facilè inter fabulas rejicienda &longs;unt, quæ Atlas
Sinicus pag.32. de Montibus circa urbem Peking loquens ait,
&longs;ummitate ingens e&longs;t lapis, qui minimo contactu movetur ac titubat:
fieri &longs;iquidem potuit, ut lapis ille in infimâ parte excavatus in
nitatur &longs;ubjecto &longs;axo, à quo vel in puncto, vel in lineâ tanga
tur, &longs;icuti dictum e&longs;t; & cum &longs;it perfectè libratus, modico im
pul&longs;u tangentis, quâ &longs;altem parte ad illum patet acce&longs;&longs;us, po-
obeundo lapidem quâcumque in parte tangatur, &longs;equitur illius
trepidatio, &longs;ignum e&longs;t contactum &longs;ubjecti fulcri e&longs;&longs;e in puncto.
Simili ratione explicanda &longs;unt, quæ idem Atlas Sinicus in XI
Provincia Fokien habet pag. 125, ubi ait,
Changcheu Orientalem partem mons e&longs;t Cio dictus, in quo lapida,
e&longs;&longs;e &longs;cribunt altum perticas quinque, cra&longs;&longs;um decem & octo, qui quo
ties tempe&longs;tas imminet, titubat omninò, ac movetur:
pis in perfecto æquilibrio con&longs;titutus &longs;uprà fulcrum, à quo in
puncto, vel in lineâ tangatur, & forta&longs;&longs;e etiam ab eodem fulcro
di&longs;tinctus in longitudines inæquales, violento impul&longs;u hali
tuum aut infernè &longs;ubeuntium, aut ex &longs;uperiore nubium parte
obliquè reflexorum, facilè moveri pote&longs;t ac titubare, &longs;i extre
mitas à fulcro remotior impellatur.
Et quoniam de Sinen&longs;ibus mentio incidit, non injucundum
fuerit hîc aliud addere pertinens ad eorum indu&longs;triam in &longs;er
vando æquilibrio. Idem Atlas Sinicus, cum &longs;ermo e&longs;t de Pro
vincia Peking, ubi &longs;olum e&longs;&longs;e areno&longs;um atque plani&longs;&longs;imum
te&longs;tatur, hæc habet pag.28.
non infrequens, nec incommodus e&longs;t. Plau&longs;trum adhibent cum unâ
rotâ ita con&longs;titutum, ut uni illius medium occupandi, & qua&longs;i equo
in&longs;idendi &longs;it locus, aliis duobus ab utroque latere ad&longs;identibus; auri
ga plau&longs;trum retro ligneis vectibus urget ac promovet non &longs;ecurè mi
nùs, quàm velociter. Si rem conjecturis indagare liceat, ego ro
tam concipio ita inclu&longs;am ligneo loculamento majoris &longs;egmen
ti circuli figuram habente, ut huic in&longs;itus &longs;it rotæ axis, ad dex
tram autem & ad lævam extantia tabulata tantæ latitudinis,
ut quis modò propè rotam, modò longiùs ad&longs;idere queat ad
æquilibrium con&longs;tituendum inter duos viatores inæqualiter
graves: Aurigæ locus e&longs;t in &longs;uprema parte loculamenti, cui
qua&longs;i equitans in&longs;idet, bino&longs;que contos, &longs;eu vectes concinnè
locatos, ut manubrium ante &longs;e habeat, extremitas altera (for
ta&longs;sè in acumen de&longs;inens, ut leviter &longs;olo infigatur) po&longs;t &longs;e ter
ram re&longs;piciat, utrâque manu apprehendens &longs;olum obliquè pre
mit, & currum in anteriora velociter promovet. Id quod nemi
ni difficile videatur, qui &longs;æpiùs ob&longs;ervaverit à puero fabri
lignarij aut ferrarij rotam curulem identidem impul&longs;am per
urbis vias velociter deduci; quæ dum impre&longs;&longs;o impetu veloci-
nutabunda inclinetur, ob velocem conver&longs;ionem immunis e&longs;t à
ca&longs;u: quemadmodum etiam &longs;tanneum aut argenteum orbem
apici cultri impo&longs;itum, &longs;i in gyrum velociter agatur, à ca&longs;u im
munem videmus, etiam&longs;i punctum &longs;u&longs;tentationis non exacti&longs;&longs;i
mè centro re&longs;pondeat. Sic aliquis &longs;uppo&longs;itam &longs;phærulam altero
pede, etiam &longs;ummis digitis premens, celeriter in gyrum totum
corpus contorquet, qui non ita facilè citrà cadendi periculum
eidem &longs;phærulæ in&longs;i&longs;tens quietus con&longs;i&longs;teret; ipsâ nimirum
conver&longs;ionis celeritate gravitatis propen&longs;ionem eludente. Non
ab&longs;imili igitur ratione in huju&longs;modi rotæ Sinici plau&longs;tri conver
&longs;ione veloci deteritur, quicquid in alterutram partem inclinatio
nis oriretur vel ex modicâ viæ inæqualitate, vel ex æquilibrio
non adeò exactè &longs;ervato, ut etiam con&longs;i&longs;tente plau&longs;tro in&longs;iden
tes viatores con&longs;i&longs;terent æqualiter librati ab&longs;que alicujus artifi
cij &longs;ub&longs;idio: Quod artificium in promptu e&longs;&longs;e non dubito; ne
que enim Sinen&longs;es ita &longs;ibi præfidentes exi&longs;timo, ut aliquâ ratio
ne &longs;ibi non præcaveant à periculo casûs, &longs;i fortè rotun in obicem
incurrente plau&longs;trum &longs;eu loculamentum in anteriorem, aut in
po&longs;teriorem partem improvisâ inclinatione convertatur. Sed
&longs;ingula per&longs;equi nec otium e&longs;t, nec operæ pretium: quapropter
generatim dicendum corporis æquilibrium ibi fieri, ubi in duas
partes ita di&longs;tinguitur, ut illarum gravitates &longs;int reciprocè in
Ratione longitudinum &longs;eu di&longs;tantiarum à puncto &longs;u&longs;pen&longs;ionis
&longs;eu &longs;u&longs;tentationis, quemadmodum in librâ dictum e&longs;t. Quare &longs;i
tota moles propo&longs;ita eâdem gravitatis &longs;pecie prædita fuerit, nec
facile &longs;it in illâ centrum gravitatis invenire, quia nimis irregu
laris e&longs;t, di&longs;tingue illam in duas partes, & &longs;ingularum inventa
centra gravitatis junge rectâ lineâ, quæ qua&longs;i libræ jugum divi
datur in reciprocâ Ratione illarum partium; e&longs;t enim punctum
illud, in quod cadit divi&longs;io, punctum æquilibrij, & centrum gra
vitatis totius. Sic Trapezij, NPMQ in
venies punctum æquilibrij, &longs;i duorum
triangulorum NQM, NPM, in quæ di
viditur, &longs;ingularia centra gravitatis inve
nias O & B: hæc jungantur rectâ OB;
tum fiat ut triangulum NQM ad trian
gulum NPM, ita reciprocè BD ad DO,
quæ&longs;itum. At &longs;i Trapezio addatur triangulum NLP eju&longs;dem
&longs;pecificæ gravitatis, emergit Pentagonum irregulare LPMQN:
inveniatur additi trianguli centrum &longs;ingulare gravitatis A, &
jungatur recta AD; tùm fiat ut Trapezium ad triangulum ad
ditum, ita reciprocè AS ad SD, & e&longs;t punctum S centrum
commune gravitatis totius Pentagoni, in quo fit æquilibrium;
perinde enim e&longs;t ac &longs;i in jugo libræ AD inæqualiter di&longs;tributæ
appenderetur ex A quidem triangulum NLP; ex D verò Tra
pezium NQMP, quæ in illis di&longs;tantiis à centro motûs æqualia
haberent momenta.
Quòd &longs;i tota moles propo&longs;ita con&longs;tet partibus non eju&longs;dem
&longs;pecificæ gravitatis, non jam &longs;atis e&longs;t inveni&longs;&longs;e &longs;ingularia cen
tra, ut ducatur jugum libræ illa connectens, & notam e&longs;&longs;e Ra
tionem molis ad molem; &longs;ed prætereà opus e&longs;t notam habere
Rationem gravitatis &longs;pecificæ ad gravitatem &longs;pecificam; quiz
Ratio gravitatum ab&longs;olutarum componitur ex Rationibus
quantitatum, & gravitatum &longs;ecundùm &longs;peciem. Quamobrem
&longs;i additum triangulum habeat &longs;pecificam gravitatem majorem
gravitate &longs;pecificâ Trapezij, quia hoc ligneum e&longs;t, illud fer
reum, non cadet in S punctum æquilibrij, &longs;ed accedet ad
punctum A, quia factâ huju&longs;modi Rationum compo&longs;itione,
minor e&longs;t inæqualitas gravitatum ab&longs;olutarum; &longs;i enim Trape
zium excedit mole Triangulum, cedit illi &longs;pecificâ gravitate.
Ponamus namque Rationem molis Trapezij ad molem Trian
guli e&longs;&longs;e ut & ad 2; &longs;pecificæ verò gravitatis Rationem ut 5 ad
42, gravitas ab&longs;oluta Trapezij lignei e&longs;t ut 35, gravitas Trian
guli ferrei ut 84: &longs;unt igitur gravitates in Ratione 5 ad 12: di
vidatur itaque jugum AD in I reciprocè, ut &longs;it AI 5, ID 12,
& erit I centrum gravitatis compo&longs;itæ, ac punctum æquilibrij,
quia ab illo inæquales gravitates habent &longs;uas di&longs;tantias in Ra
tione reciprocâ ip&longs;arum gravitatum. Eadem e&longs;t in corporibus
omnibus Ratio, & methodus deprehendendi punctum æqui
librij, &longs;eu centrum gravitatis, per quod deinde duci pote&longs;t dia
meter gravitatis, ut fiat opportuna &longs;u&longs;pen&longs;io.
Quia tamen aliquando evenit &longs;u&longs;pen&longs;um corpus aut &longs;u&longs;ten
tatum, dum po&longs;itionem horizonti parallelam &longs;ervare contendit,
aliquod incommodum &longs;ubire in motu corporis, cui innititur;
perpetuò &longs;ervet. Rem exemplo declaro.
In pyxide nauticâ in
&longs;i&longs;tit cu&longs;pidi acus magnetica æqualibus momentis librata, ut
horizonti parallela jaceat, quamcumque in partem dirigatur.
Si alicui navis plano pyxis ip&longs;a adhæreret ita, ut infimâ &longs;ui par
te illi congrueret, quamcumque in partem navis inclinaretur,
ip&longs;um pariter pyxidis fundum inclinari manife&longs;tum e&longs;t, & alte
ri acûs magneticæ po&longs;itionem horizonti parallelam &longs;ervantis
extremitati occurrens illius motum impediret, aut &longs;altem retar
daret. Ut igitur &longs;emper pyxis tùm acui magneticæ, tùm hori
zonti parallela con&longs;i&longs;tat, &longs;u&longs;pendenda fuit, non quidem funi
culo, ne incertis motibus jactaretur, &longs;ed duobus polis, &longs;uper
quibus opportunè ver&longs;aretur æqualiter librata. Verùm duobus
hi&longs;ce polis non tollitur omne incommodum; &longs;i etenim poli
re&longs;piciant navis latera, elevatâ aut depre&longs;sâ prorâ juvant, &longs;ed
navi in dextrum aut in &longs;ini&longs;trum latus inclinatâ, alter deprime
retur, alter elevaretur, ni&longs;i & ip&longs;i infigerentur circulo &longs;uper
alios polos proram & puppim re&longs;picientes ver&longs;atili. Sit pyxis
ip&longs;a ABCD, in qua venti de&longs;
cripti &longs;int, & in centro O acus
magnetica volubilis in&longs;i&longs;tat: py
xidem circulus EIFH com
plectatur, cui poli D & B facilè
ver&longs;atiles infigantur, ut inclinatâ
navi in A vel in C pyxis horizon
ti parallela maneat; & ut eumdem
paralle i&longs;mum &longs;ervet, etiam &longs;i na
vis in B aut D inclinetur, circu
lus ille EIFH duos pariter polos
facilè ver&longs;atiles habeat in E & F
externæ pyxidi immobili infixos:
hac enim ratione fiet, ut in quacumque navis inclinatione
pyxis nautica à &longs;uo paralleli&longs;mo & æquilibrio non recedat.
Hoc eodem artificio con&longs;truitur luceina ferreo aut æneo
globo inclu&longs;a multipliciter perforato, ut fumo exitus pateat,
quæ citrà effu&longs;ionem olci in &longs;olo rotata non extinguitur; e&longs;t &longs;i
quidem va&longs;culum plumbeum, ut &longs;ua gravitate &longs;ecuriùs deor
&longs;um vergat, polis ver&longs;atilibus &longs;u&longs;pen&longs;um in circulo, qui pariter
demum &longs;caphio, &longs;eu inferiori hemi&longs;phærio globi, cui includi
tur, eâ di&longs;po&longs;itione, ut quemadmodum pyxidis nauticæ hic
de&longs;criptæ ambitus in quatuor partes di&longs;tinguitur à polis, ita lu
cernæ hujus ambitus in octo partes à polis di&longs;tribuatur, atque
proinde facilior &longs;it globi in omnem partem volutatio citrà peri
culum inclinationis va&longs;culi oleum cum ellychnio continentis.
Nec pluribus opus e&longs;t hîc explicare, quàm proclive &longs;it arti
ficium hoc ad plura traducere, quorum u&longs;us e&longs;t in plano hori
zontali, ne libellâ &longs;emper & normâ indigeamus, ut illa ritè
collocentur: ut &longs;i horologium horizontale &longs;tatuendum &longs;it quo
cumque in plano, &longs;it illud pyxidi inclu&longs;um cum circulo, quem
admodum de pyxide nauticâ dictum e&longs;t: &longs;i lectulum viatorium
in rhedâ &longs;ternere oporteat, in quo citrà jactationem, etiam viâ
&longs;alebrosâ, quie&longs;cere liceat, ferreo parallelogrammo complecte
re lectulum ex polis &longs;u&longs;pen&longs;um circâ medium eo loco, ut cor
pus in lectulo jacens &longs;it horizonti parallelum, ip&longs;um verò paral
lelogrammum polis rhedæ infixis & ver&longs;atilibus ad caput & ad
pedes &longs;u&longs;pendatur: & alia huju&longs;modi, quæ facilè pro rerum
opportunitate excogitari po&longs;&longs;unt.
Verùm quàm facilè e&longs;t &longs;uper polos in æquilibrio con&longs;tituere
corpora gravitatis centrum habentia vel in ipsâ &longs;u&longs;tentationis
lineâ, vel infrà illam, tam multis difficultatibus implicitum
opus e&longs;t in æquilibrio &longs;tatuere corpus, cujus gravitatis cen
trum in parte &longs;uperiori reperitur, & quidem maximè &longs;i mul
tùm inde removeatur; tunc enim &longs;u&longs;&longs;icit vel minima inclinatio,
ut totum corpus revolvatur, cum ex alterâ parte &longs;int plura gra
vitatis momenta, quàm in oppo&longs;itâ.
Nam &longs;i corpus BC, cujus centrum gravitatis &longs;it A, &longs;u&longs;pen
datur &longs;uper polis in I, quando axi &longs;u&longs;tentanti ad perpendiculum
re&longs;pondet centrum gravitatis A, ma
net æquilibrium, &longs;ed factâ corporis
inclinatione, ut A recedat à perpen
diculo, jam versùs C plures &longs;unt
partes gravitatis de&longs;cendentes, quàm
versùs B &longs;int partes a&longs;cendentes, &
illæ velociùs moventur deor&longs;um,
quàm hæ &longs;ur&longs;um; quapropter illæ
revolvitur. Id quod multò magis contingit in Acrobarycis, quæ
nimirum gravitatem in &longs;ummitate habent, ut &longs;i corpori BC in
&longs;uperiori parte adnexa e&longs;&longs;et pyramis D; cum enim totius com
po&longs;itæ molis ex &longs;olido BC, & pyramide D, centrum commu
ne gravitatis non e&longs;&longs;et in A, &longs;ed adhuc &longs;uperius procul à polo
I, qui e&longs;t centrum motûs, factâ levi inclinatione multo plus
gravitatis e&longs;&longs;et ex parte C, quàm ex oppo&longs;ità B, ut con&longs;tat:
nam quò altius & remotius e&longs;t centrum gravitatis, eò faciliùs
linea directionis cadit extra punctum vel lineam &longs;u&longs;tentationis,
facta pari inclinatione.
Liceat autem hîc obiter, qua&longs;i cerollarij loco, attingere
æquilibria corporum humido in&longs;identium, & Acrobary corum
fluitantium, in quibus pariter Rationes libræ agno&longs;centur, &longs;i
rectè perpendatur, ubi fiat &longs;u&longs;tentatio. In omni igitur corpo
re fluitante duplex pars con&longs;ideranda e&longs;t, & quæ intrá humi
dum mergitur, & quæ in aëre extat: illa quidem utpote &longs;ecun
dùm &longs;peciem minùs gravis, quàm humor, levitat, hæc verò
aëre gravior gravitat: Quare & illa &longs;uum habet centrum levi
tatis, & hæc centrum gravitatis; nec po&longs;&longs;et corpus datam po&longs;i
tionem &longs;ervare, ni&longs;i in eâdem lineâ perpendiculari ad univer&longs;i
centrum tendente e&longs;&longs;et utrumque centrum & levitatis & gra
vitatis; cumque par &longs;it virtus a&longs;cendendi virtuti de&longs;cendendi,
neutrâ prævalente, & &longs;ibi vici&longs;&longs;im utrâque ob&longs;i&longs;tente, con&longs;i&longs;tit
corpus. Quòd &longs;i non in eodem perpendiculo &longs;it utrumque
centrum, utrumque &longs;uâ viâ pergere pote&longs;t, illud a&longs;cendendo,
hoc de&longs;cendendo. Sic baculum rectum in aquam immittens,
manúque retinens, ne in alterutram partem inclinetur, mergi
quidem illum videbis pro Ratione &longs;pecificæ &longs;uæ gravitatis, quæ
minor e&longs;t &longs;pecificâ gravitate aquæ, &longs;ed erectus non manebit,
ni&longs;i quandiù retinueris; nam ubi illum dimi&longs;eris, &longs;tatim cen
trum gravitatis de&longs;cendet, & levitatis centrum a&longs;cendet, quia
vel exiguus aquæ motus partem immer&longs;am inclinans &longs;atis e&longs;t,
ut centra illa non eidem perpendiculo re&longs;pondeant; ac prop
terea demùm baculus jacens innatabit.
Quie&longs;cente igitur corpore in humoris &longs;uperficie, mani
fe&longs;tum e&longs;t centrum gravitatis partis extantis in eodem perpen
diculo e&longs;&longs;e cum centro levitatis partis demer&longs;æ. Quare &longs;i
pars demer&longs;a & levitans &longs;it EC, pars verò extans in aëre &
gravitans &longs;it AF, centrum gravi
tatis e&longs;t G, centrum levitatis e&longs;t
H, quæ &longs;ibi directè adver&longs;antia
in oppo&longs;itas partes conantur
æqualibus viribus, atque prop
terea nullus &longs;equitur motus.
Quòd &longs;i aut H recederet versùs
D, aut G versùs B, & hoc po&longs;&longs;et
de&longs;cendere, & illud a&longs;cendere
neutro contranitente.
Jam verò quie&longs;centi pri&longs;mati imponatur aliquod pon
dus, certum e&longs;t partem in aëre extantem, conflatam ex
parte pri&longs;matis & ex addito pondere, graviorem e&longs;&longs;e, ac
proinde prævalere viribus partis in aquâ levitantis, illam
que deprimere, quoadu&longs;que fiat æqualitas inter levitatem
& gravitatem. Sed multùm intere&longs;t, utrùm additi pon
deris centrum gravitatis in eodem perpendiculo &longs;it cum cen
tro gravitatis G, ut rectâ deprimatur pri&longs;ma infrà &longs;uperfi
ciem aquæ; an verò &longs;it extrà illud perpendiculum; id
quod &longs;i accidat, commune centrum gravitatis transfertur ver
&longs;us A, aut B. Sit ex.
gr. ad partes A propè S; cumque non
immineat puncto H centro levitatis, de&longs;cendit pri&longs;ma ad partes
A, & oppo&longs;ita pars a&longs;cendit, ita ut E deprimatur infrà &longs;uperfi
ciem aquæ, F veró emergat. Sed dum ad partes CF pri&longs;ma
emergit ex aquâ, ad partes autem DE deprimitur, centrum levi
tatis non manet in H, &longs;ed ad majorem partem depre&longs;&longs;am &longs;ecedit,
donec fiat V, atque in eodem
tatis S; & tunc quie&longs;cit pri&longs;ma, nec amplius demergitur in E,
aut emergit ex F. Su&longs;tinetur itaque centrum gravitatis S à cen
tro levitatis V, & vici&longs;&longs;im centrum levitatis V retinetur à cen
tro gravitatis S; & fit tùm inter gravitates, tùm inter levitates
æquilibrium, quia gravitas in A major minùs di&longs;tat à puncto,
vel potiusà lineâ &longs;u&longs;tentationis factâ à plano tran&longs;eunte per V,
& gravitas in B minor magis di&longs;tat; ideóque neutra prævalet:
& &longs;imiliter ievitas in DE major minùs di&longs;tat à lineâ detentio
nis facta à plano tran&longs;eunte per S, ac levitas minor in C magis
re&longs;t minori virtuti repugnanti ad de&longs;cendendum velociùs.
Quemadmodum verò &longs;i tantum ponderis adderetur in A, ut
centrum commune gravitatis non po&longs;&longs;et imminere centro levi
tatis partis demer&longs;æ, nemo non intelligit futuram omnimodam
depre&longs;&longs;ionem partis A infrà &longs;uperficiem aquæ, & omnimodam
emer&longs;ionem oppo&longs;itæ partis C; ita in Acrobarycis fluitantibus
manife&longs;tum e&longs;t, quò altiùs attollitur gravitas, eò faciliùs factâ
inclinatione transferri commune centrum gravitatis ultrà per
pendiculum, in quo e&longs;t centrum levitatis partis demer&longs;æ. Sic
&longs;i ju&longs;to longior &longs;it in navi malus, factâ ex fluctibus inclinatione
in latus, aut &longs;altem impul&longs;u venti &longs;uprema carba&longs;a implentis,
facilis erit navis &longs;ubmer&longs;io, quia plus momentorum gravitatis
e&longs;t ex alterâ parte, quàm ex oppo&longs;itâ, tran&longs;lato in navis latus,
aut ultra illud, centro gravitatis totius partis extantis in aëre.
Sed de his, Deo dante, pleniùs in Hydro&longs;taticis di&longs;&longs;erendum
erit, ubi o&longs;tendetur ad navium &longs;tabilitatem nece&longs;&longs;ariam e&longs;&longs;e
eam centrorum di&longs;po&longs;itionem, ut centrum gravitatis totius na
vis cum omnibus impo&longs;itis &longs;it infrà centrum levitatis partis de
mer&longs;æ in eodem perpendiculo, in quo pariter erit centrum gra
vitatis partis extantis.
redeat.
NEmini dubium e&longs;&longs;e pote&longs;t æquilibrium tolli ob momento
rum gravitatis inæqualitatem, vel quia in una libræ æqui
libris lance additum e&longs;t pondus, vel quia altera jugi extremi
tas, alicujus elevantis aut deprimentis vi, recedit à po&longs;itione
horizonti parallelâ. Illud in quæ&longs;tionem revocati pote&longs;t, an
&longs;ublato ponderis exce&longs;&longs;u, aut ce&longs;&longs;ante impul&longs;u extrin&longs;eco, li
bra redeat ad æquilibrium, & po&longs;itionem horizonti parallelam
&longs;ibi ip&longs;a re&longs;tituat. Certè Keplerus in A&longs;tronomiâ Opticâ cap.1.
ad horizontis æquilibrium redituram,
&longs;ed rerum naturæ, &longs;ed utilitati generis humani bellum indicere. At
ex adver&longs;o Authores ferè omnes, qui de his accuratiùs &longs;crip&longs;e
runt, triplicem libræ &longs;peciem di&longs;tinguentes unam tantummo
do agno&longs;cunt, quæ &longs;e re&longs;tituat horizonti parallelam. Hoc &longs;i
quidem tanquam certum a&longs;&longs;umunt, corpus quodcumque gra
ve, quod &longs;u&longs;pen&longs;um, aut &longs;u&longs;tentatum liberè in aëre pendeat,
in cò tantum &longs;itu quie&longs;cere, in quo gravitatis centrum cum &longs;u&longs;
pen&longs;ionis aut &longs;u&longs;tentationis puncto in eâdem directionis lineâ
reperiatur; de&longs;cendit enim quantum pote&longs;t, neque ei opponi
tur punctum &longs;u&longs;pen&longs;ionis aut &longs;u&longs;tentationis, ni&longs;i in eodem per
pendiculo ad univer&longs;i centrum ducto utrumque &longs;it. Cumitaque
libra &longs;it corpus grave &longs;u&longs;pen&longs;um, & &longs;uum habeat centrum gra
vitatis, tunc demùm quie&longs;cet, ubi eam po&longs;itionem obtinuerit,
in quâ &longs;u&longs;pen&longs;ionis punctum, & gravitatis centrum in eâdem
&longs;int directionis lineâ. Punctum verò &longs;upen&longs;ionis libræ non il
lud hîc intelligitur, ex quo pendet an&longs;a, cui libra in&longs;eritur, &longs;ed
ip&longs;a Agina, &longs;eu &longs;partum, ut Ari&longs;totelico vocabulo utar, e&longs;t &longs;u&longs;
pen&longs;ionis punctum; ex illo enim proximè libra &longs;u&longs;penditur.
Hinc oritur triplex libræ &longs;pecies, quia tripliciter componi
po&longs;&longs;unt centrum motûs, & centrum gravitatis; primò &longs;cilicet
po&longs;&longs;unt in uno eodemque puncto convenire, deinde centrum
motûs pote&longs;t e&longs;&longs;e &longs;uperius, demum inferius centro gravitatis.
Et quidem &longs;i unum idemque punctum &longs;it motûs & gravita
tis centrum A, & æqualibus brachiis AB, AC æqualia &longs;int
adnexa pondera B & C, uti
que æquilibrium horizonta
le manet, propter momento
rum æqualitatem tùm ratio
ne gravitatum æqualium,
tùm ratione æqualium pro
pen&longs;ionum ad motum. Si
igitur applicatâ manu in B
deprimatur libra, ut &longs;it DE;
amotâ manu, cur redeat libra ad priorem po&longs;itionem BC?
adhuc enim momenta utrinque &longs;unt æqualia, & tantumdem
a&longs;cendere deberet D, quantum de&longs;cenderet E: par igitur e&longs;t
que prævalente fiet in eo &longs;itu DE con&longs;i&longs;tentia.
Attamen huic argumentationi, quamvis legitimæ, non ac
quie&longs;cunt nonnulli, qui libram huju&longs;modi in quácumque po&longs;i
tione quie&longs;centem &longs;e vi&longs;uros de&longs;perant, quia nunquam vide
runt: quare potiùs cau&longs;am inquirunt, cur ad æquilibrium re
deat libra æqualium brachiorum, quamvis ex medio jugo &longs;u&longs;
pendatur. Exi&longs;timant aliqui po&longs;&longs;e vim argumenti eludi, &longs;i con
cedant quidem in uno eodemque puncto convenire centrum
motûs & centrum gravitatis jugi, non tamen libræ: nam &longs;i
præter jugum a&longs;&longs;umantur etiam uncini aut lances, quibus ad
nectuntur aut imponuntur pondera, multò magis &longs;i eadem pon
dera a&longs;&longs;umantur, centrum gravitatis huju&longs;ce molis compo&longs;itæ
reperiri a&longs;&longs;erunt infrà ip&longs;um jugum, ac propterea nullam e&longs;&longs;e
huju&longs;modi primam &longs;peciem libræ.
Sit libræ jugum AB; centrum motûs & gravitatis jugi &longs;it C:
pendeant lances D & E, &longs;ingularúmque cum &longs;uis appendiculis
gravitas &longs;it æqualis gra
vitati jugi, ut facere con
&longs;ueverunt accuratiores
monetarij. Lancium igi
tur &longs;imul &longs;umptarum
commune gravitatis cen
trum e&longs;t in F: jungantur
centra gravitatum C &
F; & erit demum totius
libræ vacuæ DABE
commune gravitatis cen
trum in G. Quod &longs;i lan
cibus D & E imponan
tur æqualia pondera,
commune centrum gravitatis erit inter G & F, atque quò gra
viora erunt pondera, eò propiùs accedet ad F. E&longs;t igitur ma
nife&longs;tum centra motûs & gravitatis totius libræ non in eodem
puncto convenire, &longs;ed gravitatis centrum e&longs;&longs;e infrà centrum
motûs, &longs;eu &longs;partum C.
Verum effugium hoc nullum e&longs;&longs;e cen&longs;eo: inclinetur enim
libra, & acquirat po&longs;itionem HI, jam HM & IN lineæ di-
& &longs;unt parallelæ, quia ambæ perpendiculares ad horizontem;
ac propterea ex 33. lib.1. æquales &longs;unt ac parallelæ HI & MN.
Cumque CF linea directionis centri gravitatis jugi &longs;it ii&longs;dem
HM & IN parallela, & exeat ex C medio rectæ HI, cadet
pariter in medium rectæ MN ex 34 lib.1. & idem punctum F
e&longs;t commune centrum gravitatum M & N; atque proinde li
bræ MHIN commune centrum gravitatis erit in eadem rectâ
lineá CF. Si itaque quie&longs;cit corpus grave &longs;u&longs;pen&longs;um, quando
in eâdem directionis linea e&longs;t punctum &longs;u&longs;pen&longs;ionis, & gravi
tati, centrum, etiam in po&longs;itione HI deberet libra quie&longs;cere,
e&longs;to in C non conveniant contra motûs & gravitatis totius
libræ.
Nicolaus Tartalea lib.
8. quæ&longs;ito 32. ideo libram ad paralle
li&longs;mum horizontis redire exi&longs;timat, quia in inclinatione jugi
putat majora e&longs;&longs;e momenta brachij elevati, quàm depre&longs;&longs;i.
Id quod hâc methodo conatur o&longs;tendere.
Si ex C æqualiter
di&longs;tent pondera æqualia A & B,
fuerintque ab æquilibrio remota,
de&longs;cribunt circulum, in quo
&longs;umptis partibus æqualibus, dum
A de&longs;cendit ex F in A, vis de
&longs;cendendi e&longs;t NO, at ex A in G
vis de&longs;cendendi e&longs;t OP major,
quàm NO, ut con&longs;tat ex doctri
nâ Sinuum. Similiter vis de&longs;cen
dendi ip&longs;ius B ex I in B e&longs;t KL
major, quàm LM vis de&longs;cenden
di ex B in H. E&longs;t autem KL ip&longs;i OP, & LM ip&longs;i ON
æqualis; igitur OP e&longs;t etiam major, quàm LM. Cum itaque
in &longs;itu ACB pondus B gravitet &longs;olùm ut LM, & pondus A
gravitet ut OP, major e&longs;t potentia ip&longs;ius A, quàm ip&longs;ius B:
igitur ad æquilibrium de&longs;cendere oportet pondus A.
Sed peccat hæc Tartaleæ argumentatio, quia in pondere B
non e&longs;t con&longs;ideranda vis de&longs;cendendi in H, &longs;ed repugnantia
ad a&longs;cendendum in I, &longs;ecundùm quam ob&longs;i&longs;tit oppo&longs;ito pon
deri A; hujus autem re&longs;i&longs;tentiæ men&longs;ura e&longs;t LK æqualis ip&longs;i
OP potentiæ &longs;eu propen&longs;ioni ip&longs;ius A ad de&longs;cendendum:
quamdiu hæc æqualitas permanet.
Joannes Keplerus A&longs;tronomiæ Opticæ loco citato, cur libræ
brachia revolvantur ad æquilibrium, infert ex eo, quòd altero
brachiorum prægravato additione ponderis, ita jugum libræ
con&longs;i&longs;tit, ut quod e&longs;t gravius non planè imum locum petat,
& quod e&longs;t levius, non planè in apicem attollatur. Cujus rei
cau&longs;am inquirens &longs;tatuit libræ jugum
CD bifariam in A divi&longs;um; & centro
A de&longs;cripto circulo ducit perpendicu
lum BAF: ex quo manife&longs;tum e&longs;t
neutrum pondus po&longs;&longs;e deprimi infra F,
aut attolli &longs;upra B. Sed quia pondus D
ponitur gravius, quàm pondus C, &
utrumque naturâ &longs;uâ ad imum tendit,
contenduntque invicem, partiuntur
inter &longs;e de&longs;cen&longs;um BF in proportione,
quâ ip&longs;a &longs;unt: adeò ut BH de&longs;cen&longs;us
ponderis C &longs;it ad BG de&longs;cen&longs;um ponderis D, ut pondus C ad
pondus D. E&longs;t autem FG linea æqualis lineæ BH, quia ex
æqualibus AB & AF auferuntur æqualia latera AH & AG,
cum enim triangula CHA, DGA rectangula &longs;int, & angu
los ad verticem A æquales habeant, & latera AC, AD æqua
lia; etiam per 26. lib.1. latus AH e&longs;t æquale lateri AG. Igitur
ut pondus C ad pondus D, ita FG ad GB.
Ducatur ex F ad AD perpendicularis FK: &longs;imiliter triangula
AGD, AKF rectangula, &
cum latere AF æquali lateri AD, per eandem 26.lib.1.
tera
Igitur propter
linea æqualis lineæ KC. Quare ut
ad GF, hoc e&longs;t ita KC ad KD: ac propterea factâ jugi &longs;u&longs;pen
&longs;ione in K pondera C & D inæqualia &longs;ecundùm Rationem bra
chiorum reciprocè po&longs;ita æquiponderabunt & con&longs;i&longs;tent. Cum
igitur in hac eâdem Ratione &longs;it de&longs;cen&longs;us BH & BG, ut e&longs;t
pondus C ad pondus D, fiet con&longs;i&longs;tentia in &longs;itu CAD.
per &longs;ub&longs;umptionem patet,
doctrinam conatus &longs;um paulo clariùs exponere,
in circulo fieri de&longs;cen&longs;us par e&longs;t.
Meam hebetudinem di&longs;&longs;imulare non po&longs;&longs;um, qui huju&longs;ce
Keplerianæ argumentationis vim &longs;atis a&longs;&longs;equi non valeo: quid
enim, &longs;i fieret æquilibrium horizontale ponderum, facta in K
&longs;u&longs;pen&longs;ione? an propterea con&longs;equens e&longs;t fieri æquilibrium
etiam in &longs;itu CAD, ni&longs;i aliunde probetur? &longs;ed quod ad rem
no&longs;tram attinet, pondera alligata, & adnexa libræ non ita con
&longs;ideranda &longs;unt, ut ambo de&longs;cendant, &longs;i comparatè &longs;umantur,
&longs;ed alterius propen&longs;io ad motum deor&longs;um comparanda e&longs;t cum
alterius repugnantiâ ad motum &longs;ur&longs;um, & vici&longs;&longs;im hujus pro
pen&longs;io ad de&longs;cendendum cum illius re&longs;i&longs;tentiâ, ne a&longs;cendat.
Quapropter &longs;i ex D pondere majore auferatur exce&longs;&longs;us &longs;upra
pondus C, & fiant æqualia pondera, non po&longs;&longs;unt ad æquili
brium horizontale redire, ni&longs;i C de&longs;cendat, D verò a&longs;cendat:
Cum autem hujus a&longs;cen&longs;us GA &longs;it æqualis de&longs;cen&longs;ui HA, nul
la e&longs;t ratio, cur propen&longs;io ponderis C vincere debeat æqualem
ponderis D re&longs;i&longs;tentiam.
Deinde quid intelligendum e&longs;t, cum dicitur ip&longs;ius C de&longs;cen
&longs;us e&longs;&longs;e BH, ip&longs;ius verò D de&longs;cen&longs;us e&longs;&longs;e BG? ex B enim non
utrumque de&longs;cendit, &longs;ed alterutrum: & &longs;i pondus D de&longs;cendi&longs;
&longs;et ex B, ex adver&longs;o pondus C a&longs;cendi&longs;&longs;et ex F; cúmque illius
de&longs;cen&longs;us e&longs;&longs;et BG, hujus a&longs;cen&longs;us e&longs;&longs;et FH; &longs;unt autem BG
& FH æquales. Quòd &longs;i non motus præcedens, &longs;ed &longs;ola pro
pen&longs;io ad de&longs;cendendum & repugnantia ad a&longs;cendendum con
&longs;ideretur pro ratione po&longs;itionis, pondus D habet men&longs;uram
propen&longs;ionis ad de&longs;cendendum, non motum (qui forta&longs;&longs;e tran
&longs;iit) ex B in D, &longs;ed quem in eo &longs;itu po&longs;&longs;et perficere ex D in F:
atque adeò ip&longs;ius D de&longs;cen&longs;us e&longs;t GF, eju&longs;que re&longs;i&longs;tentia, ne
a&longs;cendat u&longs;que ad &longs;ummum e&longs;t GB, & vici&longs;&longs;im ponderis C pro
pen&longs;io ad de&longs;cendendum non e&longs;t ex B in C, &longs;ed ex C in F, &longs;i
u&longs;que ad imum de&longs;cendat, habens men&longs;uram HF, ejus verò
repugnantiam ad a&longs;cendendum metitur HB. E&longs;t igitur mani
fe&longs;tum uniu&longs;cuju&longs;que ponderis propen&longs;ionem habere oppo&longs;i
tam re&longs;i&longs;tentiam æqualem (e&longs;t enim propen&longs;io GF æqualis re
&longs;i&longs;tentiæ HB, & propen&longs;ioni HF æquali e&longs;t re&longs;i&longs;tentia GB)
ac proinde nullum &longs;equi po&longs;&longs;e motum ponderum æqualium à
centro A æqualiter di&longs;tantium. At, inquis, quid cau&longs;æ e&longs;t,
habemus? &longs;ed omnis libra ea e&longs;t, ut vel ad æquilibrium redeat,
vel omninò quantum pote&longs;t de&longs;cendat, qua parte habet bra
chium inclinatum Re&longs;pon&longs;io in promptu e&longs;t; quia &longs;cilicet dif
ficillimum e&longs;t duo illa puncta exqui&longs;itè convenire, hoc e&longs;t cen
trum motus & centrum gravitatis, nimirùm punctum illud,
quod brachiorum longitudinem di&longs;criminat. Quod &longs;i vel mi
nimum duo illa centra di&longs;crepent, natura omnes &longs;ui juris api
ces exacti&longs;&longs;imè per&longs;equitur, & e&longs;t &longs;partum non in medio, &longs;ed
aut in &longs;uperiore, aut in inferiore parte jugi (&longs;i quidem brachia
&longs;int æqualia; nam &longs;i ad latus e&longs;&longs;et in eadem recta linea, libra e&longs;
&longs;et inæqualium brachiorum, & tunc non adnexorum ponderum
æqualitas e&longs;&longs;et con&longs;ideranda, &longs;ed eorum Ratio, &longs;umpta recipro
cè brachiorum Ratione) ex quo &longs;equitur aut reditus ad æquili
brium, aut ulterior de&longs;cen&longs;us brachij inclinati.
Hinc e&longs;t de illâ duplici tantummodo libræ &longs;pecie locutum
fui&longs;&longs;e Ari&longs;totelem in Mechan.
detur &longs;peculantis intellectûs terminis coërceri, nunquam in
praxim ni&longs;i fortuito deducenda. Non enim &longs;atis e&longs;t accurati&longs;
&longs;imè inquirere centrum gravitatis jugi, ut illud &longs;it pariter cen
trum motûs, &longs;ed nece&longs;&longs;e e&longs;t punctum hoc in eádem rectá lineâ
e&longs;&longs;e, quæ jungit puncta contactuum jugi & annulorum, ex
quibus lances dependent: nam ni&longs;i hoc contingat, centrum il
lud gravitatis a&longs;&longs;umptum non e&longs;t punctum, à quo brachiorum
longitudines di&longs;criminantur, ut inferiùs con&longs;tabit dilucidiùs
ex iis, quæ de librâ curvâ dicentur.
Quærendum e&longs;t itaque, cur libra aginam habens in &longs;upe
riore loco, &longs;i ab æquilibrio horizontali dimoveatur, ad illud re
deat. Et ne locus æquivocationi pateat, dum ad hoc de
mon&longs;trandum a&longs;&longs;umuntur puncta notabili intervallo inter &longs;e
di&longs;tantia (ne videlicet linearum brevitas confu&longs;ionem aut ob
&longs;curitatem pariat) ob&longs;erva lingulæ nomine non eam &longs;olùm par
tem intelligi, quæ &longs;upra libræ jugum intrà an&longs;am excurrens
extat; &longs;ed lingulæ, &longs;eu, ut aliis placet, trutinæ pars e&longs;t etiam
linea, quæ in ip&longs;a jugi cra&longs;&longs;itie de&longs;cripta intelligitur perpendi
cularis ad lineam longitudinis brachiorum, & tran&longs;iens per
centrum motûs. Quare hujus lineæ pars intercepta inter cen
trum motûs, & lineam longitudinis brachiorum, &longs;ivè exigua
rationem attinet) nihil intere&longs;t, nam eadem planè &longs;emper e&longs;t
ratio, atque demon&longs;tratio. Sit libra æqualium brachiorum
AB, cujus puncto medio C in
&longs;i&longs;tat perpendicularis CD, & &longs;it
in ipsâ jugi cra&longs;&longs;itie centrum mo
tûs punctum D, impo&longs;iti&longs;que
æqualibus ponderibus in A & B,
maneat in æquilibrio horizonta
li AB. Deprimatur extremitas A,
ut veniat in E, reliqua extremitas
B a&longs;cendit in F, & C venit in G.
Non pote&longs;t igitur manere libra in po&longs;itione EF &longs;ublato de
primente in E, &longs;ed manentibus æqualibus ponderibus redit ad
æquilibrium, séque re&longs;tituit in AB; tùm quia centrum gravi
tatis non e&longs;t in lineâ directionis tran&longs;eunte per D punctum
&longs;u&longs;pen&longs;ionis, tùm poti&longs;&longs;imum quia momenta ip&longs;ius F majora
&longs;unt momentis ip&longs;ius E ratione po&longs;itionis & propen&longs;ionis ad
motum; pote&longs;t enim F de&longs;cendere juxta men&longs;uram FH, dum
E a&longs;cendit juxta men&longs;uram EI; e&longs;t autem major Ratio motûs
FH ad motum EI, quam &longs;it Ratio ponderum, quæ e&longs;t Ratio
æqualitatis, nimirum ut FG ad GE. Nam per 8 lib.5. FO ad
GE majorem habet Rationem quàm FG ad GE, & FO ad
OE majorem habet Rationem quàm FO ad GE; ergo multo
major e&longs;t Ratio FO ad OE, quàm FG ad GE. At &longs;imilia
&longs;unt triangula FHO, EIO, quia æquiangula (nam propter
paralleli&longs;mum linearum directionis FH & IE, alterni E & F,
& alterni I & H, qui etiam recti ponuntur, & qui ad verticem
O, æquales &longs;unt) igitur per 4.lib.
6. ut FO ad OE, ita FH
ad EI. E&longs;t igitur major Ratio de&longs;censûs FH ad a&longs;cen&longs;um EI,
quàm &longs;it Ratio ponderum, quæ e&longs;t ut FG ad GE.
Hinc patet clara &longs;olutio quæ&longs;tionis à Keplero propo&longs;itæ:
quia &longs;i pondus E majus &longs;it pondere F, illud non ad imum lo
cum de&longs;cendet, &longs;ed ibi libra obliquè &longs;ub&longs;i&longs;tet, ubi pondera
erunt in Ratione reciprocâ motuum; quando &longs;cilicet ratione
po&longs;itionis ita propen&longs;io ad de&longs;cendendum ponderis F erit ad
re&longs;i&longs;tentiam ponderis E, ne a&longs;cendat, ut e&longs;t vici&longs;&longs;im pondus E
ad pondus I: & tunc perpendicularis linea directionis ex D
&longs;itæ libræ & ponderum. Cujus rei argumentum e&longs;t mani
fe&longs;tum, quod libra quie&longs;cens in po&longs;itione EF &longs;i moveatur ab
aliquo deprimente ulteriùs aut elevante, &longs;ibi relicta non minùs
redit ad eumdem &longs;itum obliquum, quam redeat ad æquilibrium
horizontale, &longs;i pondera &longs;int æqualia. Quæ omnia ex dictis pla
na &longs;unt & aperta; &longs;ed an hoc idem rite probaverit Keplerus,
viderint alij.
Eadem philo&longs;ophandi ratio erit in librâ brachiorum inæqua
lium LM, in qua &longs;int pondera L & M (computatis ip&longs;orum
brachiorum gravitatibus juxta
momenta, quæ habent in illâ eâ
dem longitudine, ut dictum cap.2.
hujus libri) reciprocè in Ratione
brachiorum NM & NL. Depri
matur L in P, & elevabitur M in
Q, & N in V.
Dico libram &longs;ummoto deprimen
te, ad æquilibrium LM redituram.
Ducantur perpendiculares PT & QR, productâ LM horizon
tali, &longs;i opus fuerit. Triangula SQR, SPT &longs;unt &longs;imilia; igitur
per 4 lib.6. ut QS ad SP, ita ponderis Q propen&longs;io ad de&longs;cen
dendum QR, ad ponderis P re&longs;i&longs;tentiam, ne a&longs;cendat, PT.
E&longs;t autem major Ratio QR ad PT, quàm &longs;it ponderis P ad
pondus Majorem autem e&longs;&longs;e
Rationem &longs;ic o&longs;tenditur. Pondus P ad pondus Q e&longs;t ut NM
ad NL ex hypothe&longs;i, hoc e&longs;t ut QV ad VP: &longs;ed per 8. lib.
5.
major e&longs;t Ratio QS ad VP, quàm QV ad VP, & major Ra
tio QS ad SP, quàm QS ad VP: igitur major e&longs;t Ratio QS
ad SP, quàm QV ad VP, hoc e&longs;t quàm pondus P ad pon
dus
ad PT; igitur major e&longs;t Ratio de&longs;censûs QR ad a&longs;cen&longs;um PT,
quàm &longs;it Ratio ponderis P ad pondus Q: Ergo vis de&longs;cendendi
major e&longs;t; quàm oppo&longs;ita re&longs;i&longs;tentia, ac proptereà re&longs;tituet &longs;e
libra in æquilibrio horizontali.
Ex his manife&longs;tum e&longs;t rem contrario modo &longs;e habere, quan
do &longs;partum e&longs;t in cra&longs;&longs;itie jugi ira collocatum, ut &longs;it infra li
neam, quæ con&longs;tituit longitudinem brachiorum; tunc enim al-
priorem paralleli&longs;mum cum horizonte, ut potiùs, nullo ulteriùs
deprimente, brachium inclinatum de&longs;cendat omninò, donec
impediatur ab ansá, in quam incurrit alterum brachium eleva
tum: quod &longs;i &longs;uperiori aut inferiori brachio nullum occurreret
impedimentum, ita fieret totius libræ conver&longs;io & revolutio,
ut &longs;partum e&longs;&longs;et in loco &longs;uperiore, & tunc demùm in æquili
brio horizontali jugum quie&longs;ceret. Quæ omnia licet per&longs;picua
&longs;int, &longs;i &longs;uperiores duæ figuræ invertantur, clarioris tamen ex
plicationis gratiâ, &longs;it iterum jugum AB
æqualiter divi&longs;um in C, & in perpen
diculari CD &longs;it axis, & centrum mo
tûs inferiùs in D: po&longs;itis æqualibus
ponderibus A & B &longs;it æquilibrium ho
rizontale: & quoniam æqualia &longs;unt
pondera, atque æquales ad motum pro
pen&longs;iones, centrumque gravitatis e&longs;t
in eâdem perpendiculari lineâ di
rectionis cum puncto &longs;u&longs;tentationis D, manent in æquilibrio.
Deprimatur A in E, elevatur pariter B in F, & C deprimitur
in G. Dico libram, &longs;i &longs;ibi ip&longs;a dimittatur, non redituram ad po
&longs;itionem AB &longs;upra punctum D; &longs;ed pondus E ulteriùs de&longs;cen
&longs;urum. Ductis enim perpendicularibus EI & FH, propen&longs;io
ponderis F ad motum deor&longs;um, ut &longs;e re&longs;tituat in priore æqui
librio, e&longs;t FH, re&longs;i&longs;tentia ponderis E ad motum &longs;ur&longs;um e&longs;t
EI. E&longs;t autem major Ratio re&longs;i&longs;tentiæ EI ad propen&longs;ionem
deor&longs;um FH, quàm &longs;it Ratio ponderis F ad pondus E, aut vi
ci&longs;&longs;im; hæc enim æqualia &longs;unt ex hypothe&longs;i, & e&longs;t eorum Ra
tio ut AC ad CB, hoc e&longs;t ut EG ad GF: Non igitur pote&longs;t à
pondere F, cujus momenta minora &longs;unt elevari pondus E, cu
jus momenta &longs;unt majora ex di&longs;po&longs;itione ad motum. Con&longs;tat
verò major Ratio re&longs;i&longs;tentiæ EI ad propen&longs;ionem FH, quàm
ponderis F ad pondus E, quia in triangulis OIE, & OHF &longs;i
milibus eâdem e&longs;t Ratio EI ad FH, quæ e&longs;t EO ad OF; &longs;ed
ex 8 lib.5. EO ad OF majorem habet Rationem quam EG ad
GF: igitur major e&longs;t Ratio EI ad FH, quam EG ad GF, hoc
e&longs;t ponderis ad pondus. De&longs;cendet itaque E, & nullo occur
rente obice ea fiet totius libræ revolutio circà centrum D, ut
punctum &longs;u&longs;tentationis, fiat punctum &longs;u&longs;pen&longs;ionis libræ. Ea
dem dicta intelligantur de librâ brachiorum inæqualium, quæ
&longs;upervacaneum e&longs;t iterum inculcare.
Oblata itaque librâ facilè digno&longs;ces, cujus &longs;peciei illa &longs;it,
quamvis ob punctorum propinquitatem, &longs;cilicet centri mo
tûs, & puncti brachiorum longitudinem di&longs;criminantis, non
valeat oculus dijudicare: impo&longs;itis enim æqualibus ponderi
bus, ut habeat æquilibrium horizontale, aliquantulum depri
me alterutrum brachiorum, & &longs;ublato deprimente, &longs;i quidem
man&longs;erit obliqua (id quod rari&longs;&longs;imè continget) pronunciabis
centrum motûs convenire cum puncto brachiorum longitudi
nem di&longs;criminante: &longs;in autem ad æquilibrium redierit, cen
trum motûs erit in &longs;uperiore loco; &longs;i ulteriùs de&longs;cenderit, cen
trum motûs erit infra lineam longitudinis brachiorum. Vel
etiam facto æquilibrio horizontali, adde pondus alteri lanci;
&longs;i de&longs;cendat ita, ut jugum oblique con&longs;i&longs;tat aut magis aut mi
nùs, prout major aut minor factus e&longs;t exce&longs;&longs;us ponderis, pro
nunciabis centrum motûs e&longs;&longs;e in &longs;uperiore loco: at &longs;i factâ
ponderum inæqualitate lanx gravior u&longs;que ad imum deprima
tur, quantùm pote&longs;t, indicabit centrum motûs e&longs;&longs;e in inferio
re loco, aut convenire cum puncto brachia di&longs;criminante: &longs;ed
hoc ultimum temerè non affirmabis, ni&longs;i re&longs;titutâ ponderum
æqualitate, &longs;equatur quies in quacumque po&longs;itione, aut con
versâ deor&longs;um ansâ non contingat obliqua jugi con&longs;i&longs;tentia:
&longs;i enim factâ an&longs;æ &longs;u&longs;pen&longs;ione centrum illud fui&longs;&longs;et in inferio
re loco, factâ conver&longs;ione e&longs;&longs;et in &longs;uperiore loco, & continge
ret æquilibrium in po&longs;itione obliquâ.
QUamvis ad ponderum examen in&longs;tituendum rarò contin
gere po&longs;&longs;it, ut librâ Curvâ uti cogamur, quia tamen in
machinamentis aliquibus ita aut loci angu&longs;tiæ, aut opportuna
ra, ut & libræ Rationes &longs;erventur, & tamen jugi rectitudo nul
la appareat; non erit hî inutile libram curvam examinare, ut,
&longs;i quando eâ uti contigerit, innote&longs;cat, quænam &longs;int brachio
rum, & motuum Rationes. Libram autem curvam voco, quæ
a communi formâ deflectens latera habet non in directum po&longs;i
ta, &longs;ed in angulum concurrentia, aut in arcum &longs;inuata, quo
rum extremitates &longs;ivè &longs;ur&longs;um, &longs;ivè deor&longs;um re&longs;piciunt: factâ
enim &longs;u&longs;pen&longs;ione &longs;ive ubi angulum latera con&longs;tituunt, &longs;ivè in
aliquo arcûs puncto, ea fieri pote&longs;t hinc & hinc ponderum ad
ditio, quam horizontale æquilibrium con&longs;equatur. Sed quia
imperitis fucum facere po&longs;&longs;et apparens hæc laterum longitudo,
caveant, ne ex illis jugum libræ deductum intelligant: contin
gere &longs;cilicet pote&longs;t, ut planè varia &longs;it huju&longs;modi libræ forma,
& magnitudo, idem tamen &longs;it &longs;emper libræ jugum, in quo
brachia de&longs;umenda &longs;unt.
Sint enim in angulum compacta duo latera recta AB &
AC; non e&longs;t tota jugi magnitudo computanda ex horum late
rum longitudinibus; &longs;ed ex ipsâ extre
mitatum B & C di&longs;tantiâ BC; quæ &longs;em
per eadem e&longs;t, &longs;ivè &longs;it arcus BEFC,
&longs;ivè alia &longs;int latera DB & DC, aut
GB & GC, atque &longs;u&longs;pen&longs;io fiat &longs;ivè
in A, &longs;ivè in D, &longs;ivè in G, &longs;ivè in quo
cumque alio puncto, quod &longs;it intra &longs;pa
tium à lineis AB, AC, BC comprehen&longs;um. E&longs;t igitur idem
jugum BC, quia in B & C adnexa intelliguntur pondera, eo
rúmque di&longs;tantia, prout libræ adnectuntur, ca e&longs;t, quæ jugi
longitudinem determinat. Verùm an libra æqualium &longs;it po
tiùs, quàm inæqualium brachiorum, definiendum e&longs;t ex
puncto &longs;u&longs;pen&longs;ionis, à quo ad extremitates B & C deducen
dæ &longs;unt rectæ lineæ; quæ &longs;i æquales fuerint, libra e&longs;t æqualium
brachiorum; &longs;in autem inæquales, inæqualium. Hinc &longs;i late
ra AB & AC jungantur tran&longs;ver&longs;ario HI, in eoque &longs;umatur
punctum &longs;u&longs;pen&longs;ionis D, nil refert æqualia-ne, an inæqualia
&longs;int latera AB & AC? &longs;ed attendenda e&longs;t æqualitas aut in
æqualitas linearum ex D ductarum ad extremitates B & C.
Neque me arguas, quòd dixerim jugum e&longs;&longs;e BC, & attenden-
nis ductarum, puta DB & DC; brachia &longs;iquidem in ip&longs;o jugo
con&longs;ideranda &longs;unt; illæ
mune præter puncta extrema B & C. Quamvis enim lineæ hu
ju&longs;modi brachia libræ non &longs;int, &longs;i res proprie con&longs;ideretur, in&longs;e
runt tamen æqualitatem aut inæqualitatem brachiorum, qua
tenus ex puncto &longs;u&longs;pen&longs;ionis D ducta intelligitur ad BC jugum
perpendicularis DM, quæ jugum dividit in partes BM & CM
æquales aut inæquales. Nam quia triangula BMD & CMD
&longs;unt rectangula, quadrato BD, ex 47. lib.1. æqualia &longs;unt duo
quadrata DM & MB, & quadrato DC æqualia &longs;unt duo qua
drata DM & MC. Si igitur lineæ DB & DC æquales &longs;unt,
carum pariter quadrata &longs;unt æqualia; ex quibus dempto com
muni quadrato DM, remanent quadrata BM & CM æqualia,
ac proinde lineæ MB & MC æquales. Si verò lineæ BD &
CD &longs;unt inæquales, quadrata carum &longs;unt inæqualia; ex qui
bus dempto communi quadrato DM, re&longs;idua &longs;unt quadrata
BM & CM inæqualia, eorumque latera (&longs;cilicet lineæ MB &
MC) inæqualia erunt pronuncianda.
Brachia itaque hujus libræ curvæ propriè &longs;umpta non illa
&longs;unt, quæ apparent, & quia ex illis libræ curvæ moles con&longs;tat,
vulgariter hoc vocabulo donantur; &longs;ed &longs;unt &longs;egmenta lineæ
jungentis extremitates, quibus pondera adnectuntur; in quæ
&longs;egmenta dividitur à perpendiculo, quod ad illam ducitur ex
puncto, quod e&longs;t motûs centrum. Cum igitur punctum hoc,
quod tanquam centrum legem dat motui, &longs;it extrà lineam ex
tremitates illas jungentem, aut in &longs;uperiore, aut in inferiore
loco crit; ac proptereà altera erit ex duabus illis &longs;peciebus li
bræ, de quibus capite &longs;uperiore &longs;ermo fuit, habentibus &longs;par
tum aut &longs;uprà, aut infrà; & huic curvæ ea omnia convenient,
quæ ibi dicta &longs;unt, ut fiat æquilibrium horizontale, aut obli
quum. Si enim &longs;it libræ &longs;ca
pus rectus AB bifariam divi
&longs;us, centrum motûs habens
in C & pondera adnexa in D
& E æqualia, habet æquilibrium horizontale, ad quod redit, &longs;i
ab illo dimoveatur; & &longs;i pondera D & E &longs;int inæqualia, ha
bet æquilibrium obliquum pro Ratione di&longs;criminis ponderum,
puncta contactuum, quibus pondera adnectuntur. Facta au
tem figuræ conver&longs;ione, ut C &longs;it in inferiore loco, & linea DE
in &longs;uperiore, in &longs;olo æquilibrio horizontali manet, à quo &longs;i re
moveatur, ad illud non redit, neque ullum habet æquilibrium
in po&longs;itione obliquâ, ut dictum e&longs;t. Jam ex jugo AB omnia
&longs;uperflua re&longs;ecentur, & remaneant virgulæ CD & CE con
nexæ in C centro motûs: manife&longs;tum e&longs;t non e&longs;&longs;e immutata
ponderum momenta, & eundem e&longs;&longs;e motum libræ curvæ DCE
ac rectæ AB; &longs;ivè C intelligatur in parte &longs;uperiori, &longs;ivè in in
feriori. Quare & de hac curvâ, quod ad æquilibrium &longs;pectat,
eadem dicenda &longs;unt, quæ de librâ &longs;partum &longs;uperiùs aut inferiùs
habente &longs;unt dicta.
Et quidem &longs;i latera illa, quibus libra curva con&longs;tat, &longs;ecun
dùm longitudinem æqualia &longs;int, & paris gravitatis, additis
hinc & hinc æqualibus ponderibus fiet æquilibrium horizonta
le; quia vera linea jugi in &longs;egmenta æqualia dividitur, &longs;unt au
tem omnes Rationes Æqualitatis, omninò &longs;imiles. At &longs;i late
ra illa &longs;int inæqualia, non erunt addenda reciprocè pondera
(etiam computatâ ip&longs;orum laterum gravitate) in Ratione illa
rum longitudinum; &longs;ed in Ratione &longs;egmentorum jugi, ut fiat
æquilibrium: quia ex laterum illorum inæqualitate &longs;tatim qui
dem infertur etiam veram lineam jugi dividi in &longs;egmenta in
æqualia; &longs;ed non illico con&longs;equens e&longs;t &longs;imilem e&longs;&longs;e Rationem
Inæqualitatis: Immò &longs;i inæqualia &longs;int illa latera, fieri omnino
non pote&longs;t, ut &longs;egmenta, quæ fiunt à perpendiculari cadente
in ba&longs;im, videlicet in lineam jugi, &longs;int in eâdem Ratione; alio
quin &longs;i ba&longs;is &longs;egmenta e&longs;&longs;ent in Ratione laterum adjacentium,
angulus, ex quo perpendicularis demittitur, e&longs;&longs;et bifariam
&longs;ectus, per 3 lib.6. atque adeò duo triangula haberent duos an
gulos duobus angulis æquales, nimirum rectum & acutum, at
que latus haberent commune; ergo per 26.lib.1. & reliqua late
ra e&longs;&longs;ent æqualia, contra hy
pothe&longs;im. Sit enim libra cur
va laterum inæqualium BAC,
linea recta BC e&longs;t vera linea
jugi, in quam cadens perpen
diculum AD definit brachio-Non e&longs;t autem DB ad DC
ut BA ad AC, alioquin angulus BAC e&longs;&longs;et bifariam &longs;ectus,
& duo triangula DAB, DAC haberent præter rectos ad D,
ctiam acutos ad A æquales, atque latus AD commune, ac
proinde e&longs;&longs;ent etiam latera BA & AC æqualia contra hypo
the&longs;im.
Sunt igitur anguli ad A inæquales, & minor e&longs;t, qui adja
cet minori lateri AC, quàm qui adjacet majori lateri AB: quia
in triangulo BAC major e&longs;t angulus C oppo&longs;itus majori lateri
BA, quàm angulus B oppo&longs;itus minori lateri AC, ex 18.lib.1.
igitur in triangulis BDA, CDA rectangulis ad D, comple
mentum CAD minus e&longs;t complemento BAD. Qua propter
&longs;i angulus BAC &longs;it bifariam dividendus, recta AE auferet ali
quid ex majore angulo BAD, & con&longs;tituens angulum BAE
cadet in ba&longs;im inter B & D. E&longs;t itaque, per 3.lib.6. ut BA ad
AC, ita BE ad EC: &longs;ed minor e&longs;t Ratio BE ad EC quàm BD
ad EC, & multo minor quàm BD ad DC. per 8.lib.5. igitur
minor e&longs;t Ratio BA ad AC, quàm &longs;it Ratio brachij BD ad
brachium DC. Si igitur pondera in C & B e&longs;&longs;ent reciprocè ut
BA ad AC, haberent minorem Rationem, quàm BD ad DC,
ac propterea non e&longs;&longs;ent apta ad con&longs;tituendum æquilibrium
horizontale. Retento igitur pondere B, augendum e&longs;&longs;et pon
dus C, vel retento pondere C, minuendum e&longs;&longs;et pondus B, ut
e&longs;&longs;ent in reciprocâ Ratione brachiorum BD & DC.
Hinc etiam con&longs;tat retentis eodem latere AB eadémque li
neâ horizontali BC cum eodem angulo B, &longs;i velis uti minori
pondere, quod cum pondere B faciat æquilibrium, addendum
e&longs;&longs;e in A latus majus latere AC, puta latus AF, itaut tota BF
&longs;it jugi longitudo, & brachia &longs;int BD & DF. Manife&longs;tum e&longs;t
autem ex 8.lib.5. majorem Rationem e&longs;&longs;e eju&longs;dem BD ad DC
minorem, quàm ad DF majorem; ad pondera debent e&longs;&longs;e in F
& B ut BD ad DF; igitur minus pondus in F æquivalet eidem
ponderi B, cui in C æquivalet pondus majus. Porrò nemini
dubium e&longs;&longs;e pote&longs;t, an latus AF majus &longs;it latere AC, quippe
quod in triangulo CAF opponitur angulo obtu&longs;o ACF, per
19.lib.1.
Sed &longs;i res fuerit in praxim deducenda, indicare oportet, quâ
methodo utendum &longs;it, ut quæ&longs;itam ponderum Rationem, hoc
concipitur ad laterum extremitates jungedas deductum. Hæc
autem e&longs;&longs;e poterit praxis. Laterum AB & AC longitudine
metire, tùm ex B ad C extentum funiculum ad &longs;imilem men
&longs;uram revoca. His paratis certum e&longs;t hanc jugi longitudinem
communiter majorem e&longs;&longs;e longitudine &longs;ingulorum laterum,
&longs;emper tamen &longs;altem alterius, tanto exce&longs;&longs;u, ut po&longs;&longs;it ab ea au
ferri pars, de quâ mox dicetur; debet &longs;cilicet excedere me
diam proportionalem inter aggregatum laterum, & eorum dif
ferentiam. Cum enim linea jugi à perpendiculo cadente ex
angulo verticali dividenda &longs;it, utrumque latus cum jugo facit
angulos acutos; alioquin &longs;i alteruter angulorum rectus e&longs;&longs;et,
aut linea jugi non e&longs;&longs;et parallela horizonti, aut latus e&longs;&longs;et idem
perpendiculum; & &longs;i obtu&longs;us e&longs;&longs;et, perpendiculum caderet ex
tra lineam extremitates jungentem. Debet igitur tanta e&longs;&longs;e
jugi longitudo, ut differentia partium, in quas dividitur ad
differentiam laterum &longs;it ut &longs;umma laterum ad totum jugum.
Quare fiat ut jugi longitudo funiculo deprehen&longs;a ad laterum
&longs;ummam, ita laterum differentia ad partem auferendam ex
longitudine jugi; cujus re&longs;iduum bifariam divi&longs;um dabit mi
noris brachij longitudinem. Hujus operationis ratio manife&longs;ta
e&longs;t ex corollario primo prop.
36.lib.3, & ex 3. eju&longs;dem lib.3. Sit
exempli gratia latus AB partium 20, latus AC partium 9,
di&longs;tantia BC partium 23. Fiat ut 23 ad 29 &longs;ummam laterum,
ita laterum differentia 11 ad (13 20/23) partem auferendam ex jugi
longitudine 23: Re&longs;iduum partium (9 3/23) bifariam dividatur, &
ejus &longs;emi&longs;&longs;is (4 13/23) e&longs;t longitudo brachij minoris DC; quod reli
quum e&longs;t jugi partium (18 10/23) dat longitudinem alterius brachij
majoris BD. E&longs;t igitur brachiorum (atque adeò etiam ponde
rum reciprocè) Ratio ut 424 ad 105.
Quod &longs;i his cognitis inve&longs;tigare oporteat, quanta &longs;it hujus
lineæ horizontalis BC di&longs;tantia à puncto &longs;u&longs;pen&longs;ionis A, ni
mirum quanta &longs;it perpendicularis AD, &longs;tatim ex 47. lib.1. in
note&longs;cet, &longs;i ex quadrato lateris AC 81 auferas brachij DC
quadratum (20 445/529); nam re&longs;iduum (60 84/529) e&longs;t quadratum perpen
diculi AD, quod proinde e&longs;t partium (7 17/23) proximè.
At &longs;i pro ratione tui in&longs;tituti nimia &longs;it hujus perpendiculi
nus di&longs;tare à puncto &longs;u&longs;pen&longs;ionis A, jam con&longs;tat latera AB
& AC explicanda in majorem angulum; quapropter etiam
major erit jugi longitudo, ex 24.lib.1. Sit ergo definita per
pendiculi AD altitudo partium 4: hujus quadratum 16 aufer
ex 81 quadrato lateris AC, & re&longs;iduum 65 e&longs;t quadratum bra
chij minoris DC, quod idcircò e&longs;t partium (8 1/16) &longs;erè. Simili
ter ip&longs;ius AD quadratum 16 aufer ex 400 quadrato lateris AB,
& re&longs;iduum 384 e&longs;t quadratum brachij majoris BD, quod e&longs;t
partium (19 23/39) proximè; & totum jugum BC e&longs;t partium (27 25/39).
Quare brachi BD ad brachium DC Ratio e&longs;&longs;et ut 764 ad 314,
quæ reciprocè e&longs;&longs;et & ponderum.
Ex quibus per&longs;picuum e&longs;t, po&longs;itis ii&longs;dem libræ curvæ late
ribus, di&longs;parem e&longs;&longs;e ponderum Rationem: in priore enim po&longs;i
tione Ratio e&longs;t 424 ad 105, hoc e&longs;t proxime ut 4 ad 1. in po&longs;te
riore po&longs;itione, ubi in majorem angulum latera explicantur,
Ratio e&longs;t 764 ad 314, hoc e&longs;t ut 2. 43 ad 1; quæ minor e&longs;t
Ratio, quàm prior ut 4 ad 1. Si autem latera eadem e&longs;&longs;ent in
directum con&longs;tituta, e&longs;&longs;et ponderum Ratio ut 20 ad 9, hoc e&longs;t
ut 2. 22′ ad 1; quæ e&longs;t minima Ratio omnium, quæ intercede
re po&longs;&longs;unt inter pondera æquilibrium horizontale con&longs;tituen
tia ex illorum laterum extremitatibus: quæ extremitates quo
minus di&longs;tabunt, inflexis &longs;ubinde lateribus, eo majus pondus
requiretur in extremitate lateris brevioris, ut æquè ponderet
cum uno eodemque pondere collocato in extremitate lateris
longioris.
Porrò ubi de ponderum Ratione &longs;ermo e&longs;t, cave ne ip&longs;orum
laterum inæqualium libræ curvæ gravitatem contemnas; &longs;i
enim æqualia illa e&longs;&longs;ent, æqualia quoque e&longs;&longs;ent eorum mo
menta tùm ratione gravitatis, tum ratione po&longs;itionis, nam per
pendiculum caderet in medium jugum, & latera e&longs;&longs;ent &longs;imi
liter inclinata, ac proinde &longs;ola ponderum æqualitas &longs;pectaretur:
at laterum huju&longs;modi inæqualium momenta &longs;unt ex utroque
capite inæqualia, videlicet & ratione gravitatis in&longs;itæ, quæ ex
hypothe&longs;i &longs;ingulis lateribus ine&longs;t pro Ratione molis inæqualis,
& ratione po&longs;itionis, quæ valde diver&longs;a e&longs;t, cùm non &longs;int late
ra illa &longs;imili angulo ad perpendiculum inclinata; &longs;ed magis in-
gulum: pro variâ autem inclinatione ip&longs;am eju&longs;dem lateris gra
vitatem varia obtinere momenta manife&longs;tum videtur. Pona
mus laminam metallicam AB clavo
infixam in A, circa quem qua&longs;i cen
trum de&longs;cribat &longs;emicirculum BDC.
Si obtineat perpendicularem po&longs;itio
nem AB, tota gravitas innititur clavo
A &longs;u&longs;tinenti, & nullam vim habet de
&longs;cendendi; &longs;imiliter in perpendiculari
po&longs;itione AC tota gravitas retinetur à
clavo A, nec pote&longs;t de&longs;cendere. At &longs;i
po&longs;itionem habeat AD horizonti pa
rallelam, omnino nec &longs;u&longs;tinetur, nec
retinetur à clavo, &longs;ed toto conatu &longs;uas
de&longs;cendendi vires exerit. In locis igi
tur intermediis partim &longs;u&longs;tinetur aut
retinetur à clavo A, partim conatum
deor&longs;um exercet: &longs;ic ex B veniens in E &longs;u&longs;tinetur juxta men
&longs;uram FE, & deor&longs;um tendit juxta men&longs;uram GE; at ex B ve
niens in H &longs;u&longs;tinetur juxta men&longs;uram IH, & deor&longs;um tendit
juxta men&longs;uram KH. Simili modo contingit in quadrante in
feriore; nam in po&longs;itione AL retinetur juxta men&longs;uram IL,
nec de&longs;cen&longs;um pote&longs;t habere ni&longs;i ut LM; atque in O impedi
mentum à retinente e&longs;t ut FO, conatum deor&longs;um metitur ON.
Quia &longs;cilicet &longs;i ab aliquo &longs;u&longs;tineatur in L, perinde &longs;e habet ac
&longs;i e&longs;&longs;et in plano habente inclinationis angulum CAL; in quo
plano gravitatio e&longs;t ad gravitationem in perpendiculo ut Ra
dius ad &longs;ecantem, &longs;eu ut Sinus Complementi ad Radium, hoc
e&longs;t ut IL ad AL: ac propterea vires clavi retinentis in eâ in
clinatione ad vires retinentis in perpendiculo debent e&longs;&longs;e ut IL
ad AC, hoc e&longs;t ad AL: At gravitatio, quâ urgetur planum
inclinatum, e&longs;t ut PC Sinus Ver&longs;us anguli inclinationis, qui
planè æqualis e&longs;t ip&longs;i LM. Cùm autem hîc nullum habeatur
&longs;ubjectum planum, quod prematur à gravitante laminâ metal
licâ, exerit hunc conatum deor&longs;um adversùs aliud oppo&longs;itum
pondus, quod elevare conatur, vel cui conanti re&longs;i&longs;tit, ne ab
eo elevetur. Si igitur in lineá AC perpendiculari lamina AC
nitentis, & in AL impeditur, ac retinetur &longs;ecundum men&longs;u
ram IL, fiat ut AC ad IL, ita tota gravitas laminæ ad aliud,
& prodibit quantitas gravitationis contra retinentem, re&longs;i
duumque LM erit illa gravitatio, quæ con&longs;ideranda e&longs;t in eâ
po&longs;itione inclinata AL.
Sed quoniam AL à centro motûs A di&longs;tantiam habet AI,
comparanda erit hæc di&longs;tantia cum di&longs;tantia oppo&longs;iti lateris li
bræ, ut habeantur momenta invicem comparata. Ob&longs;ervan
dum tamen e&longs;t non rem perinde &longs;e habere, ac &longs;i tota gravita
tio laminæ inclinatæ AL po&longs;ita e&longs;&longs;et in L, atque adeò in di&longs;tan
tià AI; &longs;ed quia di&longs;tribuitur &longs;ecundùm totam ip&longs;am longitudi
nem AL, & partes remotiores plus habent momenti, quàm
propiores centro, juxtà Rationem di&longs;tantiarum, proptereà vel
tota gravitas lateris AL, quæ e&longs;t LM, intelligenda e&longs;t in me
dia di&longs;tantiâ inter A & I, vel &longs;emi&longs;&longs;is gravitationis AL, hoc e&longs;t
&longs;emi&longs;&longs;is ip&longs;ius LM, intelligendus e&longs;t in I, quemadmodum hu
jus libri 3. cap.
2. dictum e&longs;t totam gravitatem AD intelligen
dam in mediâ di&longs;tantiâ inter A & D, aut ejus &longs;emi&longs;&longs;em in ex
tremitate D. Quamvis autem ex inclinatione CAL oriatur
di&longs;tantia AI, hæc tamen venire pariter in computationem
debet, quia comparari debent hæc momenta cum momentis
di&longs;tantiæ oppo&longs;itæ, quæ momenta orta ex Ratione di&longs;tantiarum
eadem &longs;unt, &longs;ive AL &longs;it lamina, &longs;ive trabs; quamquam valde
di&longs;pares &longs;int gravitates, quæ a&longs;&longs;umendæ &longs;unt ex eâdem inclina
tione; ac propterea & LM indicans gravitationem comparatè
ad totam gravitatem ab&longs;olutam, & AI definiens momentum
ex di&longs;tantiâ, con&longs;iderari debent. Hoc pacto habetur totum
momentum lateris AL; &longs;imiliterque habebitur momentum la
teris oppo&longs;iti. Ex quo patet laterum inclinatorum in librâ cur
vâ momenta componi & ex Ratione di&longs;tantiarum, & ex Ratio
ne momenti, quod habent &longs;ingula latera ex inclinatione ad
perpendiculum.
At &longs;ubdubitas, utrùm i&longs;ta, quæ hîc dicuntur, cum iis aptè
cohæreant, quæ lib.1. cap.15. dicta &longs;unt, ubi ponderis in L
con&longs;tituti vires ad de&longs;cendendum definiri diximus à Sinu an
guli declinationis à perpendiculo CAL, qui æqualis e&longs;t ip&longs;i
AI: hîc verò laminæ AL gravitationem con&longs;tituimus ex
neâ IL.
Quapropter ob&longs;erva non eandém e&longs;&longs;e rationem gravitationis
lateris AL libræ, atque ponderis adnexi in extremitate L; hu
jus enim momenta perinde computantur, ac &longs;i e&longs;&longs;et in I; quia
&longs;cilicet AI æqualis e&longs;t brachio libræ PL, & planum inclina
tum, in quo pondus L con&longs;titutum intelligitur, non e&longs;t AL,
&longs;ed Tangens in L ad angulos rectos, ut loco citato explicatum
e&longs;t. At libræ latus AL &longs;uam habens gravitatem aliter &longs;e habet:
nam quemadmodum &longs;i inniteretur clavo in A, non tamen illi
infigeretur, atque ab aliquo &longs;u&longs;tineretur in puncto L, certum
e&longs;t planum inclinatum, in quo moveretur, e&longs;&longs;e AL, contra
quæ momenta de&longs;cendendi in plano inclinato reluctatur clavus
in A po&longs;itus, & retinens; ita &longs;ublato &longs;u&longs;tinente in L, & po&longs;ito
contranitente reliquo latere libræ, non tollitur munus clavi A
retinentis, &longs;ed &longs;ub&longs;tituitur latus illud oppo&longs;itum loco &longs;u&longs;tinen
tis in L: igitur contra illud latus hoc latus AL exercet eadem
momenta gravitationis, quæ exerceret adversùs &longs;u&longs;tinentem
in L, hoc e&longs;t in planum inclinatum; quæ momenta ea &longs;unt,
quæ remanent demptis IL momentis gravitationis in plano in
clinato, nimirum re&longs;iduum LM. Quia verò qui &longs;u&longs;tineret la
tus AL in L, non e&longs;&longs;et unicum &longs;u&longs;tinens, &longs;ed planum inclina
tum e&longs;t AL, & ita latus retinetur in clavo A, ut etiam ab eo
aliquatenus &longs;u&longs;tineatur, atque adeò lamina inclinata &longs;u&longs;tinea
tur à duobus in A & L, retineaturque &longs;olùm ab A; propterea
non totum momentum LM, &longs;ed ejus &longs;emi&longs;&longs;em accipiendum
diximus, ut habeantur momenta, quibus contranititur oppo
&longs;itum latus, &longs;i addantur momenta, quæ oriuntur ex di&longs;tantiâ à
centro motûs, ut dictum e&longs;t.
Hæc autem ut exemplo clariora fiant, &longs;int eadem, quæ priùs
in præcedente figurâ po&longs;ita &longs;unt, latera libræ curvæ BAC, lon
gius BA partium 20, brevius CA partium 9, & quidem in eâ
po&longs;itione, ut perpendiculum AD cadens in jugum &longs;it partium
(7 17/23), & brachium jugi DC adjacens minori lateri &longs;it partium
(4 13/23), reliquum verò jugi brachium DB partium (18 10/23). Primùm
quære momenta laterum ex eorum inclinatione: Cumque per
pendiculum AD &longs;it æquale Sinui Complementi anguli incli-
dem anguli inclinationis, &longs;cilicet differentia inter AD & AC,
quæ e&longs;t partium (1 6/23): & &longs;imili methodo Sinus Ver&longs;us anguli in
clinationis DAB e&longs;t partium (12 6/23). Ratio igitur gravitationis
lateris AB ad gravitationem lateris AC ex inclinatione e&longs;t ut
282 ad 29; Ratio momentorum ex di&longs;tantiâ à centro, ut &longs;upra
diximus, e&longs;t ut 424 ad 105. Compo&longs;itis igitur duabus hi&longs;ce
Rationibus, e&longs;t totius momenti lateris AB ad totum momen
tum lateris AC Ratio ut 119568 ad 3045, hoc e&longs;t in minimis
terminis ut 39. 267″ ad 1. Sit igitur gravitas ab&longs;oluta lateris
AB unciarum 20; gravitatio re&longs;pondens &longs;emi&longs;&longs;i Sinus Ver&longs;i an
guli inclinationis e&longs;t unciarum (6 3/23). Item gravitas ab&longs;oluta la
teris AC &longs;it unc. 9: gravitatio re&longs;pondens &longs;emi&longs;&longs;i Sinus Ver&longs;i
anguli inclinationis e&longs;t unc. (29/46). Hæc gravitatio (29/46) ducatur in
di&longs;tantiam à perpendiculo partium (4 13/23), & e&longs;t momentum
2.878‴. Similiter gravitatio unc. (6 3/23) ducatur in di&longs;tantiam à
perpendiculo partium (18 10/23), & e&longs;t momentum 113.013‴. Di
vi&longs;o itaque majore numero 113013 per minorem 2878, in mi
nimis terminis Ratio e&longs;t ut 39.268″ ad 1: quæ minimùm differt
à priore illa Ratione propter neglectas fractiunculas in divi
&longs;ionibus.
Nunc inquiramus, quantum ponderis addendum &longs;it lateri
minori, ut fiat æquilibrium cum &longs;olâ majoris lateris gravitate.
Statuatur pondus addendum Algebricè 1 ℞, cujus di&longs;tantia à
perpendiculo cum &longs;it partium (4 13/23), ponderis additi momentum
e&longs;t (105/23) ℞ addendum momento lateris minoris invento. Quare
2.878‴ + (105/23) ℞ æquantur momento 113.013‴ lateris majoris:
& utrinque demptis 2.878‴, remanet æquatio inter (105/23) ℞ &
110.135‴. Demum in&longs;titutâ divi&longs;ione prodit
ponderis addendi, unciarum 24 1/8. Huic itaque ponderi additâ
gravitatione lateris minoris AC unc. (29/46) hoc e&longs;t in mille&longs;imis
630‴, erit in C totum pondus unc. 24.755‴; & in B intelli
gitur gravitas unc. (6 3/23), hoc e&longs;t in mille&longs;imis unc. 6.130‴ ferè.
Vides igitur hæc pondera e&longs;&longs;e reciprocè po&longs;ita in Ratione
di&longs;tantiarum DB & DC: & quamvis demum in his Ratio
nibus non &longs;ibi exacti&longs;&longs;imè re&longs;pondeant numeri, &longs;atis pa-
culis.
Cæterùm hæc tam minutè per&longs;equi in librâ curvâ, cujus
latera non adeò notabili gravitate &longs;unt prædita, labor quidem
videtur inutilis: &longs;ed quoniam huju&longs;modi libræ præcipuus u&longs;us
e&longs;&longs;e pote&longs;t in machinationibus, ubi latera libræ &longs;unt tigilli cra&longs;
&longs;iores non mediocris gravitatis, operæ pretium fuit indicare,
quâ methodo ip&longs;orum laterum gravitates & momenta compu
tari oporteat, ut non ca&longs;u, &longs;ed ex certâ ratione pondera collo
centur, & æquipondia &longs;tatuantur.
IN&longs;trumenti cuju&longs;que bonitas æ&longs;timatur ex fine, ad quem fuit
in&longs;titutum, prout ad illum a&longs;&longs;equendum aptum fuerit, aut
ineptum, eóque melius cen&longs;etur in&longs;trumentum, quò certiùs
per illud propo&longs;itus finis obtinetur; quemadmodum per &longs;ingu
la eunti facilè con&longs;tabit. Ut igitur exacti&longs;&longs;imum libræ genus
innote&longs;cat, &longs;atis patet inquirendum e&longs;&longs;e, quænam libra facilli
mè ab æquilibrio recedat; quo rece&longs;&longs;u indicans vel minimam
ponderum inæqualitatem, etiam &longs;uo æquilibrio exqui&longs;itam
ponderum æqualitatem o&longs;tendit; id quod per libram ve&longs;tiga
mus. Hîc autem de librâ æqualium brachiorum &longs;ermo e&longs;t, quâ
communiter uti &longs;olemus: quamquam aliqua etiam ad libram
inæqualium brachiorum proportione traduci queant. Ex du
plici capite libram, quà libra e&longs;t, ponderum gravitates præ aliis
libris exqui&longs;itè examinare contingit, videlicet aut ex brachio
rum longitudine, aut ex &longs;parti, &longs;eu centri motûs, po&longs;itione;
reliqua enim impedimenta, aut adjumenta materiam potiùs &longs;e
quuntur, quàm libræ formam.
Et quidem quod ad brachiorum longitudinem &longs;pectat, adeò
certum Ari&longs;toteli videtur majoribus libris, majori &longs;cilicet bra
chiorum longitudine præditis, accuratiùs examinari ponde
rum æqualitatem, ut in Mechanicis quæ&longs;tionibus hoc primum
&longs;am autem ex eo de&longs;umendam putat, quòd &longs;partum &longs;it cen
trum, brachia verò qua&longs;i lineæ à centro exeuntes; & quia Ra
dij longiores ab eodem centro cum brevioribus exeuntes &longs;i pa
riter moveantur, majorem arcum de&longs;cribunt, propterea etiam
citius moveri nece&longs;&longs;e e&longs;t extremitatem libræ, quò plus à &longs;parto
di&longs;ce&longs;&longs;erit. Hinc e&longs;t in minore librâ po&longs;&longs;e aliquando ex tenui
inæqualitate ponderum fieri motum non con&longs;picuum, atque
adeò illam occultè di&longs;cedere ab æquilibrio; id quod in majore
librâ contingere non pote&longs;t, quia longioris brachij extremitas
notabili motu inclinatur. Sit enim li
bra longior AB, cujus &longs;partum &longs;it C;
moveatur, & de&longs;cribat arcus BG, &
AF, qui &longs;unt multò magis con&longs;picui
& majores, quàm qui à librâ minore
DE habente idem motûs centrum C,
de&longs;cribantur arcus EI & DH. Con
&longs;tat igitur motum puncti E pror&longs;us fugere omnem oculorum
aciem, &longs;i motus extremitatis B vix &longs;it con&longs;picuus. Ex quo il
lud etiam con&longs;equens e&longs;t, quod major libra clariùs indicat
æquilibrium.
Verùm &longs;i hæc ita accipiantur, prout communi huic inter
pretationi &longs;ube&longs;t Ari&longs;toteles, vix aliquid habent momenti:
quis enim pondera vix inæqualia bilance &longs;ubtiliter examinans
jugi extremitates re&longs;picit, ut videat, an lineæ horizonti paral
lelæ congruat jugum? & non potiùs lingulam CO con&longs;iderat,
an cum ansâ perpendiculari illa conveniat? Quod &longs;i lingula at
tendatur, idem e&longs;t ejus motus &longs;ive longior &longs;it libra AB, &longs;ive
brevior DE; factâ enim inclinatione aut majore motu BG, aut
minore motu EI, eadem e&longs;t lingulæ po&longs;itio CS. Hoc tantùm
habent emolumenti brachia longiora, quod faciliùs dividuntur
bifariam æqualiter quàm breviora: & &longs;i minimum aliquod di&longs;
crimen intercedat, hoc minorem habet Rationem ad bra
chium longiùs, quàm ad brevius. Quare aliâ ratione acci
pienda e&longs;t libra: nam &longs;i in uno eodemque puncto C conveniant
&longs;partum & jugi divi&longs;io, aut &longs;partum &longs;it inferius, &longs;ive longiora,
&longs;ive breviora &longs;int brachia, ponderum inæqualitas illicò inno
te&longs;cit, quia extremitas præponderans, ad imum locum, quan-Locutus igitur videtur Ari&longs;toteles de
librâ &longs;partum habente in &longs;uperiore jugi loco extrà lineam, quæ
jugi longitudinem definit.
Sit iterum libra longior AB, & brevior DE, utraque bifa
riam divi&longs;a in C; & &longs;it linea lingulæ perpendicularis CK, in
quâ &longs;umatur &longs;partum, &longs;eu motús
centrum O, & re&longs;iduum OK &longs;it
lingula, ex cujus declinatione à
perpendiculo an&longs;æ, digno&longs;citur
&longs;ublatum æquilibrium. Sit pondus
A ad pondus B ut 5 ad 3: centrum
gravitatis jugi & ponderum commune non pote&longs;t e&longs;&longs;e C, quod
brachia CA & CB æqualia con&longs;tituit; &longs;ed erit ut pondus A ad
pondus B, ita reciprocè longitudo BG ad longitudinem GA,
eritque punctum G centrum gravitatis, nec libra con&longs;i&longs;tet, ni
&longs;i recta GOH fiat perpendicularis horizonti: lingula igitur
OK declinabit à perpendiculo an&longs;æ juxta angulum HOK.
Eadem pondera transferantur in minorem libram DE; & &longs;i
fiat ut pondus D 5 ad pondus E 3, ita EF ad FD, erit F cen
trum gravitatis libræ DE & ponderum: quare libra non con
&longs;i&longs;tet, ni&longs;i recta FOI &longs;it horizonti perpendicularis, & tunc à
perpendiculo declinabit lingula OK juxta angulum IOK.
Quoniam verò e&longs;t ut 4 ad 1, ita AC ad CG, ita DC ad CF,
& AC major e&longs;t quàm DC, erit etiam ex 14 lib.5. GC major
quàm FC; igitur angulus COF minor e&longs;t angulo COG, pars
minor toto; ac proinde ad verticem angulus KOI minor e&longs;t
angulo KOH. Po&longs;itis igitur ponderibus ii&longs;dem in libræ lon
gioris AB extremitatibus, declinabit lingula à perpendiculo,
cum eo con&longs;tituens angulum majorem, quàm &longs;it angulus ab
eadem lingulâ con&longs;titutus cum perpendiculo, quando ponde
ra illa inæqualia adnectuntur libræ breviori DE. Hinc e&longs;t
quòd &longs;i inæqualitas ponderum exigua &longs;it, centrum gravitatis
in utrâque librâ non multùm recedat à puncto C, parùm in ma
jore, minimùm in minore, ac proinde lingulæ deflexio forta&longs;&longs;e
inob&longs;ervabilis erit in minore librâ, quæ in majore evadet nota
bilis atque con&longs;picua. Hinc etiam patet, cur extremitas A
de&longs;cendens magis moveatur, quàm extremitas D minoris li
bræ; quia &longs;cilicet angulus OGA, per 16. lib.1. major e&longs;t quàm
linea AG cum illà faciens obtu&longs;iorem angulum, magis depri
metur infrà lineam AB horizontalem.
Sed jam inquirendum e&longs;t, utrùm expediat centrum motûs
magis di&longs;tare à lineâ jugi, an verò illi propiùs admoveri, ut
clariùs innote&longs;cat rece&longs;&longs;us jugi ab æquilibrio horizontali: illa
quippe &longs;parti po&longs;itio eligenda e&longs;t, quæ etiam minimum mo
tum indicet notabili lingulæ declinatione. Dico itaque &longs;par
tum lineæ jugi proximum utilius e&longs;&longs;e, quàm remotum. Sit
enim libra AB bifariam in C di
vi&longs;a, & ex hoc puncto exeat per
pendicularis CI; in quâ pro cen
tro motûs eligatur punctum S;
ponantur verò pondera A & B ita
e&longs;&longs;e inæqualia, ut centrum gravi
tatis commune &longs;it D. Igitur DSR
e&longs;t linea, quæ facta perpendicularis con&longs;tituit cum lingulâ SI
angulum ISR. Deinde reliquis omnibus manentibus, &longs;it cen
trum motûs O remotius à lineâ jugi, & linea DOV facta per
pendicularis declinabit à lingulâ OI juxta angulum IOV,
quem con&longs;tat e&longs;&longs;e minorem angulo ISR; nam angulus DSC
externus major e&longs;t interno DOS, per 16. lib.
1. e&longs;t autem huic
ad verticem IOV, & illi ad verticem ISR; igitur ISR angu
lus e&longs;t major angulo IOV.
Quòd &longs;i centrum motûs adhuc propiùs admoveatur medio
jugi puncto C, adhuc majorem angulum con&longs;tituet cum lin
gulâ, ac proptereà adhuc multò notabilior erit deflexio lingu
læ à perpendiculo, etiam &longs;i exiguus &longs;it motus ex eo, quod cen
trum gravitatis D proximè accedat ad punctum C: e&longs;t &longs;iqui
dem extrà controver&longs;iam, quò minor e&longs;t ponderum inæquali
tas, eò etiam minorem e&longs;&longs;e puncti D à puncto C di&longs;tantiam.
Ex quo manife&longs;tum evadit exiguam ponderum differentiam
non digno&longs;ci, &longs;i &longs;partum notabili intervallo rece&longs;&longs;erit à lineâ ju
gi; hæc enim &longs;parti di&longs;tantia habet rationem Radij, di&longs;tantia
centri gravitatis à medio jugi locum obtinet Tangentis; igitur
&longs;i fiat major &longs;parti di&longs;tantia, eadem Tangens ad majorem Ra
dium minorem Rationem habebit, atque adeò &longs;ubtendet mul
tò acutiorem angulum, qui proptereà minùs ob&longs;ervari poterit.
currant minima &longs;parti à jugo di&longs;tantia, & ob longitudinem ma
jorem brachiorum libræ major centri gravitatis di&longs;tantia à me
dio jugi puncto, patet multò faciliùs digno&longs;ci inæqualia e&longs;&longs;e
pondera, quia majore angulo linea deflectit à perpendiculo; &
po&longs;ito minimo Radio Tangens major angulo majori opponitur.
Hæc quidem de libra &longs;partum habente &longs;uprà lineam jugi
dicta accommodari po&longs;&longs;unt libræ &longs;partum habenti infrà jugi li
neam, &longs;i eadem &longs;chemata inver&longs;o &longs;itu po&longs;ita intelligantur: quò
enim maiore angulo deflectit à perpendiculo linea jungens gra
vitatis centrum, & centrum motus, eò faciliùs brachium, in
quo e&longs;t gravitatis centrum, inclinatur. Verùm &longs;i duplex hæc
libræ &longs;pecies, quæ &longs;uprà, & quæ infra jugi lineam &longs;partum ha
bet, invicem comparetur, &longs;atis apertum e&longs;t multò faciliùs à
po&longs;teriore h`c &longs;pecie indicari ponderum inæqualitatem; quia
videlicet &longs;i centrum gravitatis in alterutram partem vel mini
mùm recedat à medio jugi, non ampliùs imminet &longs;parto in eo
dem perpendiculo, neque pote&longs;t &longs;u&longs;tineri, &longs;ed illicò, quantùm
pote&longs;t ad imum locum de&longs;cendit. At in priore illa &longs;pecie libræ
&longs;partum in &longs;uperiore loco habentis, recedente in alterutram
partem centro gravitatis, de&longs;cendit illud quidem; &longs;ed non ni&longs;i
pro ratione exce&longs;sûs ponderis; qui de&longs;cen&longs;us inob&longs;ervabilis erit,
&longs;i exigua &longs;it ponderum differentia. Hinc non &longs;emel animadver
ti accurati&longs;&longs;imas bilances, quibus aurearum monetarum ponde
ra examinantur, eas e&longs;&longs;e, quæ &longs;partum in inferiore loco habent;
lanx enim, quæ pondere prægravatur, ad imum, quantùm po
te&longs;t de&longs;cendit: factâ autem libræ conver&longs;ione ita, ut an&longs;a infe
riùs &longs;u&longs;tentata libram &longs;u&longs;tineat, ii&longs;demque ponderibus impo&longs;i
tis, lanx prægravata non de&longs;cendit ad imum locum; &longs;ed manet
libra in obliquâ po&longs;itione, quæ ponderum inæqualitati congruè
re&longs;pondet; &, &longs;i ea &longs;it ponderum inæqualitas, quæ omnem ob
&longs;ervantis &longs;ubtilitatem effugiat, videtur libra in æquilibrio hori
zontali po&longs;ita, cum tamen in priore &longs;itu, antequam libra inver
teretur, non po&longs;&longs;et in ullo æquilibrio con&longs;i&longs;tere.
Non ita tamen hæc dicta intelligi velim, ut nulla &longs;it habenda
ratio materiæ, ex qua libra con&longs;tat; hæc &longs;iquidem tantæ gravi
tatis e&longs;&longs;e pote&longs;t, ut axem vehementiùs premens motum aliqua
tenus impediat, ac propterea levis illa virtus effectiva motus,
retur, ex hâc pre&longs;&longs;ione, & prominularum particularum &longs;e vi
ci&longs;&longs;im contingentium conflictu elidatur, atque jugi æquili
brium horizontale permaneat. Gravitatem autem motui im
pedimento e&longs;&longs;e ex eo con&longs;tat,
e&longs;t, movetur libra, quàm cum pondus habet,
Ari&longs;toteles 9. 10. Mechan. Cui tamen in a&longs;&longs;ignandâ hujus
difficultatis causa non aquie&longs;co, licet ultrò concedam
trarium ei, ad quod vergit onus, movere difficile e&longs;&longs;e
vacuæ lances minùs graves &longs;unt, impo&longs;ito autem pondere fiunt
graviores, & proptereà lanx elevanda facta gravior difficiliùs
movetur contra in&longs;itam gravitati propen&longs;ionem, etiam vici&longs;&longs;im
lanx deprimenda facta gravior ex adnexo pondere faciliùs ob
&longs;ecundat naturali gravium propen&longs;ioni, atque adeò augere de
beret movendi facilitatem, vel &longs;altem hanc imminui non per
mitteret. Non aliunde igitur ortum ducere videtur huju&longs;mo
di difficultas movendi libram onu&longs;tam, quàm ex majore pre
mentis gravitatis conatu: pre&longs;&longs;ione autem motum impediri quis
neget, &longs;i &longs;uper planam &longs;uperficiem continuo lævore lubricam
ducat regulam metallicam exqui&longs;ite politam, quam nunc te
nui, nunc validiori conatu premat? utique percipiet pro vario
prementis conatu aliam atque aliam e&longs;&longs;e trahendæ regulæ me
tallicæ difficultatem.
Adde graviori libræ cra&longs;&longs;iorem axem, ut ei proportione
re&longs;pondeat, nece&longs;&longs;ariò adjungi; hic autem &longs;i non &longs;it exqui&longs;itè
cylindricus, quâ parte fit contactus, &longs;ed aliquatenùs angulatus
duobus in locis contingat, &longs;atis manife&longs;tè apparet magis impe
diri motum libræ, quàm &longs;i axis tenuior e&longs;&longs;et, atque &longs;ubtilior;
licet enim hic pariter &longs;imilique ratione angulatus e&longs;&longs;et, quia
tamen anguli minùs di&longs;tarent invicem, quàm in axe cra&longs;&longs;iore,
minùs etiam libræ conver&longs;ionem impedirent. Idem accidit, &longs;i
axis quidem cylindricus, foramen autem, cui axis in&longs;eritur,
non exqui&longs;itè rotundum &longs;ed angulatum fuerit. Cur autem libræ
conver&longs;io impediatur, &longs;i fiat contactus in duobus punctis, pa
làm e&longs;t; quia nimirum quamdiu centrum gravitatis compo&longs;itæ
interjicitur inter duos illos contactus (vel &longs;altem linea directio
nis per illud centrum ducta tran&longs;it per intervallum illud duo
rum contactuum) non pote&longs;t fieri libræ in alterutram partem
ri lanci addi nece&longs;&longs;e e&longs;t, ut centrum gravitatis omninò cadat
extrà illud &longs;patium, quod à contactibus comprehenditur.
Hinc patet, cur libræ cra&longs;&longs;iores, & majores ingentibus &longs;ar
cinis onu&longs;tæ inertes fiant ad motum, etiam &longs;i adnexis ponderi
bus in&longs;it aliquot unciarum, aliquando forta&longs;&longs;e etiam librarum,
di&longs;paritas. Contrà verò aurificibus, & gemmariis, quibus mi
nutias contemnere damno e&longs;&longs;et, valdè exiguæ libræ in u&longs;u
&longs;unt; quippè quæ &longs;ubtili&longs;&longs;imo axe contentæ &longs;unt, & levi jugo
con&longs;tant, cujus gravitati æqualis e&longs;t &longs;ingularum lancium gra
vitas: quare cum nec vehemens pre&longs;&longs;io contingat, nec axis
adeò tenuis facilè angulos admittat, exilioribus huju&longs;modi li
bris etiam minima ponderum inæqualitas exploratur, &longs;i cæte
róqui fuerint ritè con&longs;tructæ.
At quærat hîc qui&longs;piam.
Proponitur libra, quæ vacua æqui
librium o&longs;tendit, nec ita gravis e&longs;t, ut de validiore axis pre&longs;
&longs;ione dubitetur: ut inquiratur, quàm facilè mobilis illa &longs;it, alte
ri lanci &longs;ingula &longs;ubinde grana delicatè imponuntur, quot &longs;atis
&longs;int ad primò tollendum æquilibrium, tùm aliâ librâ tenuiori
examinatum granorum omnium pondus (rejecto ultimo grano,
cujus additione primò facta e&longs;t libræ inclinatio) deprehendi
tur unciæ unius, exempli gratiâ. Quæritur, an, &longs;i eidem lanci
imponantur merces, & oppo&longs;itæ lanci legitima pondera, &longs;it
&longs;emper numeranda uncia una amplius, ut verum mercis pon
dus habeatur; quandoquidem deprehen&longs;um e&longs;t non mutari
æquilibrium, ni&longs;i uncia addatur.
Ut quæ&longs;tioni &longs;atisfaciam, tanquam certum &longs;tatuamus hanc
libræ inertiam non oriri ex multâ jugi & lancium gravitate
axem premente; &longs;i enim ex huju&longs;modi pre&longs;&longs;ione oriretur, ad
ditis hinc & hinc ponderibus multò major fieret pre&longs;&longs;io, ex
quâ movendi difficultas major crearetur; & &longs;i minorem pre&longs;&longs;io
nem vix unius unciæ exce&longs;&longs;us vincit, utique majorem pre&longs;&longs;io
nem non ni&longs;i plurium unciarum exce&longs;&longs;us vincere poterit. De
finire autem huju&longs;modi pre&longs;&longs;ionum vires motum libræ retar
dantes, meæ tenuitatis non e&longs;t; quippè qui nec divinare au
deo, nec certam rationem pre&longs;&longs;iones illas dimetiendi invenio.
Illud igitur reliquum e&longs;t, &longs;eclusâ pre&longs;&longs;ione, quòd axis con
tactus non omninò in unico puncto, &longs;ed in pluribus fiat, ac
primò di&longs;po&longs;ita &longs;it libra ad recedendum ab æquilibrio. Hoc au
tem indicat, libræ pror&longs;us vacuæ centrum gravitatis e&longs;&longs;e inter
extrema puncta contactûs axis; &longs;ed additâ unciâ compo&longs;itæ gra
vitatis centrum convenire cum extremo puncto contactûs
axis.
Quærendum e&longs;t igitur, quo intervallo extremum hoc
punctum, quod etiam e&longs;t gravitatis centrum, di&longs;tet à medio
jugi puncto. Id quod ut innote&longs;cat, ob&longs;ervetur jugi & lan
cium gravitas; tùm in extremitatibus jugi intelligatur &longs;emi&longs;&longs;is
&longs;ingulorum brachiorum, & addatur &longs;ingularum lancium gra
vitas: &longs;int autem hinc & hinc ex. gr. unciæ duodecim tota gra
vitas: alteri addatur uncia, & erunt hinc quidem unciæ 12;
hinc verò unciæ 13. Quare jugum reciprocè di&longs;tinguatur in
duas partes, quarum altera &longs;it 13, altera 12: igitur punctum
hoc divi&longs;ionis jugi di&longs;tat à medio jugi puncto parte unâ quin
quage&longs;imá totius longitudinis eju&longs;dem jugi: hæc &longs;iquidem lon
gitudo di&longs;tincta intelligitur in partes 25 æquales; punctum
medium ab extremitate di&longs;tat partibus 12 1/2, centrum gravita
tis compo&longs;itæ di&longs;tat partibus 12; igitur punctorum i&longs;torum in
tervallum e&longs;t (1/50).
Jam imponatur alteri lanci merx, quæ cum pondere le
gitimo lib.
2. faciat æquilibrium: aio non po&longs;&longs;e pronuncia
ri mercem e&longs;&longs;e unc. 25: nam &longs;i ponatur merx unc.
25: ad
ditâ gravitate lancis & brachij unc. 12 ex hypothe&longs;i, hinc
quidem e&longs;&longs;ent unciæ 37, hinc verò unciæ 36; igitur divi
&longs;o jugo in partes 73, centrum gravitatis di&longs;taret à medio jugi
puncto parte (1/146). At punctum extremum contactûs axis & jugi
di&longs;tat parte (1/50), igitur multo majus pondus &longs;upra unciam adden
dum e&longs;t merci, ut æquilibrium exqui&longs;itè faciat cum pondere
legitimo lib.
2. Nimirum in&longs;tituenda e&longs;t analogia ut 12 ad 13,
ita unciæ 36 ad uncias 39; dempto igitur pondere lancis & bra
chij libræ, quantitas mercis e&longs;t unc. 27. Ex quo liquet, quò
majora pondera lancibus imponuntur, eò majorem e&longs;&longs;e diffe
rentiam à pondere legitimo. Hinc ulteriùs patet huju&longs;modi
librâ &longs;atius e&longs;&longs;e multam mercem &longs;imul ponderare, quàm per
partes: pone enim e&longs;&longs;e uncias 12 legitimi ponderis, cum quo
ad 13, ita unc. 24 ad 26, & demptis unciis 12 ad brachium &
lancem &longs;pectantibus, remanent mercis unciæ 14: quare bis
facta ponderatione erit differentia unc. 4; unica autem ponde
ratio dabat tantum uncias 3: quia videlicet &longs;ingulis vicibus ad
ditui id, quod re&longs;pondet gravitati lancis oppo&longs;itæ; atque adeò
differentia &longs;æpiùs repetita major e&longs;t, quàm &longs;implex: &longs;ic qua
tuor libris ponderis legitimi re&longs;ponderent in altera lance mer
cis lib.4.unc. 5; quòd &longs;i quatuor vicibus operando &longs;ingulas libras
expendi&longs;&longs;es, differentia demùm e&longs;&longs;et unciarum 8.
Unum ad huc &longs;upere&longs;&longs;e videtur hîc ob&longs;ervandum, quoniam
longioribus brachiis exqui&longs;itiùs indicari æquilibrium diximus:
cavendum &longs;cilicet, ne in aliud incommodum incidamus, quo
illud idem pereat, quod per&longs;equimur. Si enim longiora fiant
brachia, additur gravitas, quæ magis axem premens motui ali
quam difficultatem creat: quod &longs;i retentâ eâdem brachiorum
gravitate illorum cra&longs;&longs;ities extenuetur, & in longitudinem ex
tendantur, vide ne nimis exilia evadant ita, ut flexioni obnoxia
&longs;int, vel &longs;uâ ip&longs;orum, vel expendendorum ponderum gravita
te. Præterquam quod longiora brachia plus habere videntur
momenti ad premendum axem, etiam &longs;i par &longs;it longiorum at
que breviorum libræ brachiorum gravitas ab&longs;oluta; cujus &longs;e
mi&longs;&longs;is in extremitate brachij longioris plùs habet momenti ad
de&longs;cendendum, quàm in extremitate brevioris. Et &longs;i longior
ha&longs;ta ex medio &longs;u&longs;pen&longs;a faciliùs &longs;ponte &longs;uâ flectitur circa me
dium (id quod breviori non accidit) indicio e&longs;t obicem reti
nentem magis premi; idem igitur & axi libræ contingere po
te&longs;t, cujus pre&longs;&longs;io major e&longs;&longs;e videtur ex longioribus brachiis,
etiam&longs;i in cæteris nullum intercedat di&longs;crimen. Sic Ari&longs;tote
les quærit quæ&longs;t. 27. Mechan.
pondus, difficiliùs &longs;uper humeros ge&longs;&longs;atur, etiam &longs;i medium qui&longs;piam
illud ferat, quam &longs;i brevius &longs;it?
buit validiori vibrationi extremitatum magis di&longs;tantium ab hu
mero &longs;u&longs;tinente: &longs;ed hoc non ni&longs;i in motu contingit, & cùm
flexile e&longs;t pondus, cuju&longs;modi e&longs;&longs;et longior ha&longs;ta aut bractea
ferrea mediocris cra&longs;&longs;itiei. Certè longiori columnæ marmoreæ
jacenti, cujus medio recens fulcrum &longs;ubjectum fuit, jam pu
tre&longs;centibus extremis fulcris, &longs;ua longitudo obfuit, ut frange-
&longs;ecundum &longs;peciem marmore non ita facilè accidi&longs;&longs;et: non ni&longs;i
quia gravitas magis à fulcro di&longs;tans plùs habet momenti, etiam
&longs;i non contingat vibratio corporis, quemadmodum in motu.
Illud po&longs;tremò non omittendum, quod ad lingulam perti
net, hanc enim longiu&longs;culam e&longs;&longs;e præ&longs;tat, quàm brevem, ut
vel levi inclinatione libræ, apex lingulæ magis con&longs;picuo mo
tu extra an&longs;am ad latus &longs;ecedat, & &longs;ublatum æquilibrium indi
cet. Dum tamen lingulæ longitudinem affectas, cavendum,
ne illa momentum addat &longs;uâ gravitate brachio, quod inclina
tur; quamvis enim hoc nihil referat, ubi &longs;ublatum horizontale
æquilibrium indicatur; in librâ tamen, quæ in æquilibrio obli
quo pote&longs;t con&longs;i&longs;tere, videretur indicare majorem ponderum
inæqualitatem, quàm revera &longs;it. Cæterùm communiter libræ
hoc periculo vacant; &longs;ola enim ponderum æqualitas horizonta
li æquilibrio inquiritur, non ponderum Ratio obliquo æquili
brio inve&longs;tiganda proponitur: quare communiter nil de lingu
læ gravitate timendum e&longs;t, quod nos &longs;olicitos habeat.
Quare præter exqui&longs;itam brachiorum æqualitatem, & accu
ratam lingulæ cum ip&longs;o jugo po&longs;itionem ad angulos rectos, ad
libram exacti&longs;&longs;imam con&longs;tituendam concurrunt brachiorum &
lingulæ longitudo, jugi & lancium modica gravitas, axis &longs;ub
tilitas, &longs;parti & jugi quàm maxima propinquitas, & ip&longs;ius
&longs;parti infrà jugi lineam po&longs;itio. Quæ tamen omnia cum rectâ
ratione &longs;unt admini&longs;tranda, ut ponderibus examinandis pro
portione re&longs;pondeant libræ partes; majoribus enim &longs;arcinis va
lidior axis, & cra&longs;&longs;iora libræ brachia conveniunt; & &longs;ic de
reliquis.
LIbram dolo&longs;am voco, quæ &longs;olitariè accepta &longs;inè ponderi
bus ju&longs;ta apparet, & æquilibrium o&longs;tentat, re tamen verâ
inju&longs;ta e&longs;t, quia adnexis ponderibus &longs;uo æquilibrio non tribuit
to æquilibrio. Quare nullus mihi &longs;ermo de iniquorum vendi
torum &longs;ycophantiis, quibus, ju&longs;tam licèt libram adhibentes,
rudem ac &longs;implicem emptorem circumveniunt, aut imprimen
do impetum &longs;ur&longs;um brachio, cui legitimum pondus adnectitur,
ut merx præponderare videatur, aut ponderibus iniquis & ju&longs;to
minoribus utendo, aut &longs;ubjectam men&longs;am, cui lanx mercis in
cumbit, materiâ aliquatenus tenaci illinendo, ut &longs;ublatâ in
aërem librâ priùs attollatur lanx ponderis quàm mercis, quæ
omninò præponderans apparet, &longs;i libra &longs;partum habeat infra
jugum, aut &longs;imiles impo&longs;turas excogitando: &longs;ed de illis tantum
deceptionibus agendum, quæ ex ip&longs;ius libræ con&longs;tructione,
aut po&longs;itione ortum habere po&longs;&longs;unt.
Et primò quidem &longs;e offert dolus, cujus meminit Ari&longs;toteles
quæ&longs;t.1.Mechan. familiaris eo tempore vendentibus purpuram,
& ea, quorum modica quantitas pretium exigebat non contem
nendum: hi enim librâ utebantur, quæ brachiis non omninò
paribus con&longs;tabat, ita tamen, ut hæc inæqualitas non &longs;e oculis
&longs;tatim proderet. Ut autem lateret dolus, &longs;capum &longs;eu jugum
libræ ex ligno con&longs;truebant, cujus partes omnes non eandem
&longs;pecificam gravitatem obtinerent, quamvis nulla &longs;ecundùm
molem diver&longs;itas intuenti occurreret: quia enim nodi, & partes
radici propiores, ut potè magis den&longs;æ, graviores &longs;unt, quàm
reliquæ partes à radice remotiores & nodis carentes, partem il
lam graviorem breviori brachio tribuebant, vel &longs;i materia pla
nè uniu&longs;modi e&longs;&longs;et, & æquabili gravitate prædita, breviori
brachio aliquid plumbi infundebant, ut materiæ gravitate mo
mentum, quod ratione po&longs;itionis deerat, &longs;upplente, appareret
æquilibrium lancium in vacuâ librâ. Sed ubi demum merx
lanci longioris brachij imponebatur, hæc erat ju&longs;to minor,
quamvis cum oppo&longs;ito pondere e&longs;&longs;et æquilibris; non enim erat
illi æqualis, &longs;ed in Ratione reciprocâ longitudinis brachij mi
noris ad longitudinem majoris. Hûc &longs;pectat inæqualitas bra
chiorum orta ex eo, quòd jugi ferrei pars altera ex validiore, &
diuturniore percu&longs;&longs;ione mallei facta den&longs;ior, etiam gravior e&longs;t;
nam puncto longitudinem jugi bifariam dividenti non re&longs;pon
det centrum gravitatis; &longs;ed recedit à medio versùs extremita
tem den&longs;iorem, atque graviorem; ac proptereà, ut æquili-
nem jugi. Similiter &longs;i jugi quidem materia æquabiliter &longs;it gra
vis, &longs;ed brachiorum inæqualitatem &longs;uppleat lancium gravitas
reciprocè inæqualis; æquilibris erit libra vacua; &longs;ed damno
emptoris merx longiori brachij adnectitur. Quare ut pateat
dolus, facto æquilibrio inter mercem ac pondus, &longs;tatim com
muta lances, & pondus majus ex longiore brachio multò plus
habebit momenti, quàm merx ex brachio breviore: idcircò,
&longs;i ex pondere dematur, quantùm &longs;atis &longs;it ad æquilibrium cum
merce iterum &longs;tatuendum, plus mercis habebit emptor, quàm
pro oppo&longs;iti ponderis men&longs;urâ.
Secundò &longs;it jugi materia planè æquabilis, & ab axe jugum
dividatur omnino bifariam: &longs;ed puncta contactuum annulo
rum, ex quibus pendent lances, non æqualiter di&longs;tent à me
dio: etiam&longs;i lancis propioris gravitas &longs;uppleat momentum, quod
dee&longs;t ratione &longs;itûs, & æquilibris appareat libra vacua, non ta
men æqualia pondera lancibus impo&longs;ita con&longs;tituent æquili
brium, &longs;ed illud gravius apparebit, quod ex di&longs;tantia majore
appendetur: & &longs;i pondera æquilibrium faciant, inæqualia
erunt reciprocè juxtà Rationem inæqualitatis di&longs;tantiarum à
medio. Similiter igitur facto ponderum æquilibrio, lances
commuta, & quidem &longs;i po&longs;t commutationem iterum æquili
brium fiat, ju&longs;ta e&longs;t libra, &longs;ecùs verò &longs;i alterum gravius appa
reat, quod priùs æquale videbatur.
At quæris, quá methodo po&longs;&longs;is deprehendere, quanta &longs;it bra
chiorum inæqualitas, quando quidem non habetur æquili
brium po&longs;t factam lancium commutationem, & planè ignora
tur, quanta &longs;it mercis gravitas. Ut quæ&longs;tioni &longs;atisfaciam, acci
pio legitima pondera, & primùm facto æquilibrio ob&longs;ervo legi
timi ponderis quantitatem: Commuto deinde lances, & cum
non fiat æquilibrium cum eâdem merce, tantum accipio legi
timi ponderis, quantum requiritur ad æquilibrium. Demum
inter hæc duo pondera legitima invenio terminum medio loco
proportionalem, & hoc e&longs;t mercis pondus, quod collatum cum
alterutro ex legitimis ponderibus dat reciprocè longitudinis
brachiorum Rationem. Hanc methodum e&longs;&longs;e certam patet,
quia cum bis fiat æquilibrium, bis inter pondera e&longs;t eadem Ra
tio reciproca brachiorum. Sint brachia, quæ brevitatis gratia
mum in S ad mercem in R: & factâ commutatione ponitur
merx in S, & iterum fit ut R ad S, ita reciprocè merx eadem
in S ad &longs;ecundum pondus legitimum in R: igitur, per 11.lib.5.
ut primum pondus ad mercem, ita merx ad &longs;ecundum pondus:
&longs;unt autem nota duo pondera legitima; igitur & innote&longs;cit mer
cis gravitas: quæ &longs;i comparetur ut con&longs;equens terminus cum
primo pondere, aut ut Antecedens cum &longs;ecundo pondere, ha
bebitur Ratio R ad S. Sit itaque ex.
gr. in primo æquilibrio
primum pondus legitimum unc. 72, in &longs;ecundo æquilibrio &longs;e
cundum pondus legitimum &longs;it unc. (69 18/100). E&longs;t ergo merx me
dio loco proportionalis unc. (70 576/1000); ac propterea R ad S e&longs;t
ut 72 ad (70 1576/1000), aut ut (70 576/1000) ad (69 18/100), hoc e&longs;t ut 4500 ad
4411. Sit demum totius jugi longitudo di&longs;tincta in partes 200:
addantur termini Rationis inventæ, & fiat ut 8911 ad 4411
ita 200 ad 99, & hæc e&longs;t longitudo brachij brevioris, erit au
tem longioris brachij longitudo partium 101: di&longs;tat ergo &longs;par
tum à puncto medio per unam ducente&longs;imam partem totius ju
gi. Quòd &longs;i res &longs;ubtili&longs;&longs;imè ad calculos revocanda e&longs;&longs;et, hujus
ducente&longs;imæ partis gravitas, quæ e&longs;t &longs;emi&longs;&longs;is gravitatis diffe
rentiæ brachiorum e&longs;&longs;et computanda, atque &longs;ubducenda, vel
addenda, ut mercis pondus exqui&longs;itè innote&longs;cat.
Tertiò.
Accidere pote&longs;t lingulam ex medio libræ &longs;capo a&longs;
&longs;urgere ad angulos rectos, lineamque lingulæ tran&longs;euntem per
centrum motûs ita occurrere lineæ jungenti puncta, ex quibus
lances pendent, ut eam bifariam æqualiter dividat, in eam ta
men ad angulos inæquales cadat. Aio nec brachia e&longs;&longs;e verè
æqualia, nec lingulam, quamvis an&longs;æ congruens videatur, in
dicare æquilibrium horizontale, e&longs;&longs;e veram lingulam, etiam&longs;i
pondera in eo æquilibrio con&longs;i&longs;tentia &longs;int æqualia, & non in
Ratione brachiorum.
Sit &longs;capus libræ AB, ex quo perpendicularis a&longs;&longs;urgat lingula
CD, & ex D per O centrum mo
tûs ducta recta linea occurrat li
neæ SV tangenti extrema puncta,
ex quibus lances pendent, eam
que bifariam dividat in I: &longs;ed quo
niam punctum S e&longs;t paulò altiùs
Dico lineam SV e&longs;&longs;e quidem jugum, &longs;ed brachia non e&longs;&longs;e æqua
lia, non enim &longs;unt IS & IV: quandoquidem ductis rectis OS
& OV, e&longs;t libra curva SOV latera habens inæqualia, SO
minus, & VO majus. Nam in triangulis SIO, VIO latus
IS ex hypothe&longs;i e&longs;t æquale lateri IV, latus IO commune e&longs;t,
angulus SIO e&longs;t ex hypothe&longs;i minor, quàm angulus VIO;
ergo per 24.lib.1. ba&longs;is SO minor e&longs;t ba&longs;i VO. Igitur ex O
perpendicularis linea cadens in jugum SV dividit illud in bra
chia inæqualia, & perpendiculum ex O cadit inter S & I, pu
ta in H, quia ex hypothe&longs;i angulus SIO e&longs;t acutus. Vera
igitur lingula non e&longs;t ID, &longs;ed linea, quæ ad angulos rectos
in&longs;i&longs;tens jugo SV ex H per O ducitur. Quare &longs;i CD con
gruit an&longs;æ perpendicularis horizonti, jugum SV non e&longs;t ho
rizonti parallelum, non e&longs;t igitur æquilibrium horizontale, &longs;ed
obliquum: quia tamen e&longs;t I centrum commune gravitatis pon
derum æqualium in S & V, ac per illud tran&longs;it perpendicu
lum ex O cadens in horizontem, proptereà po&longs;&longs;unt e&longs;&longs;e ponde
ra æqualia, & æquilibrium o&longs;tendere, quod modicá obliquita
te inclinatum mentiatur æquilibrium horizontale. At &longs;i alia
fieret hypothe&longs;is, &longs;cilicet lineam jugi SV non dividi æqualiter,
pondera non e&longs;&longs;ent æqualia, &longs;ed e&longs;&longs;ent reciprocè in Ratione
motuum, quos perficere po&longs;&longs;ent extremitates S & V, juxta &longs;u
periùs dicta cap.
4. hujus lib.
3.
Vitium igitur hujus libræ non in eo con&longs;i&longs;tit, quòd ponde
ra non &longs;int æqualia, &longs;ed quòd indicet æquilibrium horizontale,
cum &longs;it obliquum, & pondera æqualia nunquam po&longs;&longs;int ad
æquilibrium horizontale devenire; ut enim hoc fieret, ponde
ra e&longs;&longs;e oporteret inæqualia reciprocè in Ratione brachiorum
SH & HV. Quòd &longs;i contingat punctum O centrum motûs,
e&longs;&longs;e idem cum puncto I, pondera æqualia verè habebunt æqui
librium horizontale; &longs;ed lingula CD declinabit ab ansâ, qua&longs;i
æquilibrium non e&longs;&longs;et. Libræ huju&longs;modi vitium deprehendi
non pote&longs;t ponderum commutatione in lancibus; quia cùm
æqualia ex hypothe&longs;i &longs;int pondera, eadem utrobique habent
momenta, &longs;ervant quippè eamdem di&longs;tantiam, & æqualiter
&longs;unt ad motum di&longs;po&longs;ita. Rarò tamen continget jugum SV
planè æqualiter dividi à lineâ lingulæ ad angulos obliquos in-
proptereà facta ponderum in lancibus commutatione prodet &longs;e
momentorum inæqualitas.
Quartò.
Libra, quam diuti&longs;&longs;imè ju&longs;tam expertus es, pote&longs;t
momento à &longs;ua ju&longs;titiâ deficere, &longs;i vel modicum inflectatur al
terutrum brachiorum, vel &longs;i utrumque non æqualiter flectatur;
hinc enim oritur brachiorum inæqualitas; quam deprehendes
commutatis ponderibus in utrâque lance; quæ &longs;cilicet æquili
brium con&longs;tituebant propter reciprocam Rationem brachio
rum, quibus adnectebantur, non ampliùs eandem &longs;ervant in
aliâ po&longs;itione Rationem.
Quintò.
Axis, qui duobus in punctis contingat (&longs;cio con
tactum fieri in linea; &longs;ed puncta a&longs;&longs;umo in ip&longs;is lineis, per quæ
tran&longs;it planum perpendiculare ad horizontem, in quo e&longs;t linea
jugi) vel quia ip&longs;e e&longs;t angulatus, vel quia foramen, cui in&longs;eri
tur, non exqui&longs;itè rotundum, quâ &longs;altem parte fit contactus,
libram con&longs;tituit dolo&longs;am: quia videlicet duo illa puncta axis
perinde &longs;e habent, ac &longs;i duo e&longs;&longs;ent centra motûs. Manife&longs;tum
e&longs;t autem eandem jugi lineam non po&longs;&longs;e in duobus punctis
æqualiter dividi. Tripliciter pote&longs;t hoc fieri.
Primò unum ex
his punctis pote&longs;t exactè re&longs;pondere medio jugi; &longs;ecundò po
te&longs;t utrumque hoc punctum æqualiter à medio jugi di&longs;tare;
Tertiò po&longs;&longs;unt ab eodem medio hinc & hinc inæqualiter
di&longs;tare.
Sit linea jugi AB, cujus medium C: puncta contactuum
axis, ex quibus ad jugum ducitur perpendicularis, ea &longs;int pri
mò, ut re&longs;pondeant in jugo punctis
C & D. Si lanci in B imponatur le
gitimum pondus, tùm in A ponatur
merx u&longs;que ad æquilibrium, à quo
proximè recederet, &longs;i aliquid am
plius mercis adderetur, fiet æqualitas, quia ex C puncto æqua
liter ab extremitatibus di&longs;tante fit &longs;u&longs;pen&longs;io libræ. At &longs;i po&longs;itâ
primùm merce in A, deinde legitima pondera addantur in B,
utique plura pondera, quàm par &longs;it, addentur: quia videlicet
non inclinabitur libra infrà B, ni&longs;i ponderum ad mercem Ra
tio excedat Rationem reciprocam brachiorum AD ad DB; e&longs;t
enim D qua&longs;i centrum motûs.
Deinde puncta illa contactuum axis po&longs;&longs;unt re&longs;pondere jugi
punctis E & D æqualiter à medio C di&longs;tantibus: & tunc, ut
tollatur æquilibrium, nece&longs;&longs;e e&longs;t tantum ponderis uni lanci ad
dere, ut pondera &longs;int in majori Ratione, quàm &longs;it Ratio reci
proca brachiorum; erit &longs;i quidem extremitas A proxime di&longs;po
&longs;ita, ut facto additamento gravitatis inclinetur, &longs;i fuerit ut BE
ad EA, ita pondus in A ad pondus in B; & vici&longs;&longs;im extremitas
B erit proximè di&longs;po&longs;ita, ut auctà gravitate inclinetur, &longs;i ut AD
ad DB ita pondus in B ad pondus in A. Quia autem ex hypo
the&longs;i DC & EC æquales &longs;unt, etiam re&longs;idua EA & DB æqua
lia &longs;unt, item AD & BE: quapropter ut AD ad DB, ita BE
ad EA; ex quo con&longs;equens e&longs;t ex &longs;olâ lancium commutatione
(&longs;i centrum motûs modò &longs;it D, modò &longs;it E) non po&longs;&longs;e digno&longs;ci
hoc libræ vitium, &longs;icut digno&longs;ceretur in primo ca&longs;u, &longs;i ut AD
ad DB, ita pondus in B ad pondus in A; factâ enim lancium
commutatione, pondus ex B in A tran&longs;latum præponderaret
ex centro motûs C, cum tamen in priori po&longs;itione circa cen
trum motûs D non tolleret æquilibrium.
Similiter in tertio ca&longs;u, quando puncta contactuum axis e&longs;
&longs;ent F & D à medio C inæqualiter di&longs;tantia, & ut AF ad FB,
ita pondus in B ad pondus in A daret æquilibrium; factá pon
derum in lancibus commutatione non maneret æquilibrium,
quia pondus tran&longs;latum in B ad pondus tran&longs;latum in A po&longs;t
hanc commutationem adhuc e&longs;&longs;et ut BF ad FA; &longs;ed ad æqui
librium circa D centrum motûs deberet e&longs;&longs;e ut AD ad DB,
e&longs;t autem BF prima major, quàm AD tertia, & FA &longs;ecunda
minor e&longs;t, quàm DB quarta; igitur e&longs;t major Ratio BF ad FA,
quàm AD ad DB: igitur pondus, quod priùs erat in B, tran&longs;la
tum in A impar e&longs;t ad æquilibrium con&longs;tituendum.
Ad digno&longs;cendum, an libra hoc vitio laboret, uti poteris hac
methodo. Lancibus impone pondera, ut fiat æquilibrium: tùm
lances commuta; & &longs;iquidem iterum fiat æquilibrium, adde
alteri lanci aliquid ponderis, à quo &longs;i libra inclinetur, aufer ad
ditum pondus, & oppo&longs;itæ lanci impone; quæ &longs;i per&longs;i&longs;tat non
inclinata, adde adhuc pondus, quantum ferre pote&longs;t citrà in
clinationem: iterum commutatis lancibus, nullo pacto manere
æquilibrium videbis, & indicio erit contactum axis fieri in
duobus punctis, quorum alterum re&longs;pondet medio jugi &longs;iqui-
e&longs;t primus ca&longs;us. Quòd &longs;i facto æquilibrio, alterutri lancium
addas pondus, & æquilibrium maneat, adde quantum &longs;atis e&longs;t,
ut libra &longs;it proximè inclinanda in eam partem, &longs;i adhuc pondus
adderetur, tùm oppo&longs;itæ lanci &longs;imiliter additum pondus &longs;i non
tollat æquilibrium, indicat inter puncta contactuum axis e&longs;&longs;e
medium punctum C, quod bifariam dividit jugum: & videbis
po&longs;&longs;e &longs;ine &longs;ine alternis additamentis augeri pondera &longs;ingularum
lancium, quia commune centrum gravitatis modò migrat ad
unum punctum contactûs, modò ad aliud extremum. Sed ad
interno&longs;cendum, utrùm puncta hæc æqualiter, an inæqualiter
à puncto C medio di&longs;tent, ob&longs;erva additamenta illa, æqualia ne
&longs;int? an inæqualia?
Nam ut centrum gravitatis migret ex D in
E, & iterum ex E in D, æqualia addenda &longs;unt primùm in B,
deinde in A, pondera. At ut migret gravitatis centrum ex D
in F, plus addendum e&longs;t ponderis in A, quàm addatur in B, ut
migret ex F in D; quia &longs;cilicet B magis di&longs;tat à D centro mo
tús, quàm A di&longs;tet ab F centro motûs: igitur plus ponderis ad
dendum e&longs;t in A, ut habeat momentum æquale momento pon
deris additi in B. Hoc vitium minoribus libris, quarum exilis
e&longs;t axis, non facilè inerit; majores libræ, quæ cra&longs;&longs;iori axe in
digent, illi obnoxiæ e&longs;&longs;e po&longs;&longs;unt, ni&longs;i artificis indu&longs;tria in eo ex
poliendo &longs;olicita fuerit. Sed quid &longs;i axis, quâ parte contingit,
in angulum &longs;implicem de&longs;inat, non tamen in eum cadat per
pendicularis linea lingulæ, quæ jugum bifariam dividit? Jam
con&longs;tat à centro motûs dividi jugum in brachia inæqualia, ac
proptereà æquilibrium horizontale e&longs;&longs;e non po&longs;&longs;e, inter pon
dera verè æqualia.
Sextò.
Si libra exacti&longs;&longs;imè habens brachia æqualia, & lin
gulam perpendicularem, & lances æquales, & funiculorum
pondera æqualia, habeat tamen funiculum alterum altero lon
giorem, incumbátque plano horizontali, impo&longs;itis æqualibus
ponderibus non apparebit æquilibrium, &longs;i centrum motûs fue
rit in medio jugi puncto, vel infrà illud; &longs;ed ad illam partem
inclinabitur, quæ breviorem funiculum habuerit. Hoc ideò
accidit, quia libram attollens extendit breviorem funiculum
longiori adhuc langue&longs;cente, ac proinde pondus huic lanci im
po&longs;itum non re&longs;i&longs;tit &longs;ur&longs;um trahenti, ni&longs;i cum funiculus i&longs;te
re&longs;i&longs;tentiâ, dum ex alterâ, quæ funiculum habet breviorem,
invenit re&longs;i&longs;tentiam; atque alterâ extremitate manente, alterâ
a&longs;cendente, jugum inclinatur, extento demùm utroque funi
culo lanx utraque attollitur. Sed quia ex hypothe&longs;i omnia &longs;unt
æqualia, vel remanet jugum in eâdem po&longs;itione inclinatum,
&longs;i punctum libræ brachia di&longs;terminans congruat centro motûs,
vel pars inclinata ulteriùs de&longs;cendit, &longs;i &longs;partum &longs;it inferiùs po
&longs;itum.
Hinc pondera apparent inæqualia, quamvis verè æqualia
&longs;int; & non rarò accidit monetas aliquas aureas tanquam le
ves rejici, quamvis reverâ &longs;int ju&longs;ti & legitimi ponderis; quia
lancis, cui imponuntur, funiculus longior e&longs;t, & libra ad hanc
partem, in quâ e&longs;t pondus, inclinatur; ideóque tribuitur mo
netæ levitas, quia libra vacua in aëre &longs;u&longs;pen&longs;a ju&longs;ti&longs;&longs;ima appa
ret. Vici&longs;&longs;im igitur pote&longs;t fieri, ut moneta levis appareat præ
ponderans, in librâ &longs;partum inferiùs habentè, &longs;i moneta levis
fuerit impo&longs;ita lanci, cujus funiculus brevior e&longs;t; factâ &longs;cilicet
jam jugi ad hanc partem inclinatione, cum po&longs;tea lanx utra
que à plano &longs;eparatur, legitimum pondus, quod gravius qui
dem e&longs;t, non pote&longs;t de&longs;cendere, ni&longs;i attollat oppo&longs;itam lan
cem, cujus a&longs;cendentis motus major e&longs;&longs;e deberet motu legitimi
ponderis de&longs;cendentis; ac proptereà ni&longs;i &longs;it major Ratio pon
deris ad monetam, quàm motûs monetæ a&longs;cendentis ad motum
ponderis de&longs;cendentis, moneta videbitur præponderans: &
tanti&longs;per latebit dolus, dum facta fuerit in lancibus ponderis,
& monetæ commutatio: apparebit &longs;iquidem levius id, quod
in lance pendet ex funiculo longiore. Quòd &longs;i libra huju&longs;modi
funiculis inæqualibus in&longs;tructa &longs;partum haberet in loco &longs;upe
riore, initio quidem impo&longs;ita æqualia pondera apparerent in
æqualia, quia non viderentur æquilibria, &longs;ed demùm &longs;e libra in
æquilibrio con&longs;titueret, &longs;i verè omnia æqualia &longs;int, ut fert hy
pothe&longs;is. At &longs;i, ut non paucis venditoribus vulgare e&longs;t, ita li
bra &longs;it con&longs;tituta, ut lanx altera, cui legitimum pondus impo
nitur juxtà quæ&longs;itam mercis quantitatem, &longs;ubjecto piano in
&longs;i&longs;tat, altera merci de&longs;tinata in aëre pendeat, lingulâ an&longs;æ
congruente, quæ æquilibrium o&longs;tendit; &longs;it verò funiculus lan
cis plano incumbentis forta&longs;sè non &longs;atis extentus (quia ita con-
trahat) merx videbitur præponderans, etiam&longs;i non &longs;it major
legitimo pondere; quia deor&longs;um &longs;uá gravitate connitens, dum
pondus ex alterâ parte re&longs;i&longs;tit, inclinat lingulam, & oppo&longs;itæ
lancis funiculum extendit.
Septimò.
Ex ip&longs;o plano, cui libra incumbit, antequam at
tollatur, oriri pote&longs;t fallacia æqualibus ponderibus inæqualita
tem tribuens, etiam&longs;i nullum libræ in&longs;it vitium aut ratione in
æqualitatis brachiorum, aut ratione lingulæ perperam inclina
tæ ad jugum, aut ratione axis angulati, aut ratione funiculo
rum inæqualium. Nam &longs;i planum ab horizonte deflectat, & ad
illum inclinetur; cùm ad perpendiculum an&longs;a attollitur, funi
culi pariter horizonti perpendiculares intelliguntur, & quia
æquales &longs;unt, jugum libræ e&longs;t parallelum plano, ac proptereà
perpendiculum an&longs;æ ad angulos inæquales incidit tùm in ju
gum libræ, tùm in planum inclinatum; lingula igitur, quæ ju
go in&longs;i&longs;tit ad angulos rectos, declinat ab ansâ, & &longs;ublatâ in
aërem librâ, inclinatur lingula ad depre&longs;&longs;iorem plani partem,
manetque inclinata, quamvis pondera æqualia &longs;int, &longs;i centrum
motûs & punctum brachia di&longs;terminans in codem puncto con
veniant; &longs;i verò &longs;partum inferius &longs;it, adhuc magis inclinatur,
videturque lanx illa omninò præponderans: at &longs;i &longs;partum in &longs;u
periore loco fuerit, libra primùm inclinata, demùm in aëre &longs;u&longs;
pen&longs;a ad æquilibrium horizontale veniet.
Octavò.
Si contingat ita pondus in lance collocari, ut ip&longs;ius
ponderis &longs;ingulare centrum gravitatis non omninò in eodem
perpendiculo &longs;it cum puncto jugi, ex quo lanx illa dependet,
æquilibrium non indicabit æqualitatem ponderum in utráque
lance po&longs;itorum: Nam &longs;i linea directionis per huju&longs;modi cen
trum gravitatis tran&longs;iens incurrat in jugi punctum, quod &longs;it
centro motûs vicinius, quàm punctum extremum brachij, op
po&longs;itæ lancis pondus erit minus; &longs;in autem occurrat lineæ jugi
(quæ producta intelligitur) remotiùs à centro motûs, oppo&longs;itæ
lancis pondus erit majus; quia &longs;cilicet hæc centri gravitatis
ponderis collocatio perinde &longs;e habet, atque &longs;i brachium illud
aut imminutum &longs;it, aut auctum: quapropter etiam pondera
æquilibria &longs;unt in Ratione reciprocâ brachiorum, ut ex &longs;æpius
dictis liquet. Hinc &longs;i pondus præter opinionem gravius aut le-
aliter in lance di&longs;pone, ut centro gravitatis ponderis facilè im
mineat punctum jugi, ex quo lanx illa &longs;u&longs;penditur; & tunc
certior fies, an verè gravitas illa ponderi in&longs;it, an verò irrep
&longs;erit fallacia ex ineptâ ip&longs;ius ponderis po&longs;itione priori. Hoc
tamen intellige, quando ex huju&longs;modi po&longs;itione &longs;equeretur in
æqualis velocitas motuum oppo&longs;itorum ponderum.
HActenùs de librâ &longs;ermo fuit, in quâ, cum brachia æqua
lia &longs;int, legitimum pondus e&longs;t æquale gravitati rei, cujus
quantitatem ex gravitate inve&longs;tigamus: & quidem quando exi
gua, vel etiam mediocria &longs;unt pondera, res commodè huju&longs;
modi bilance perficitur; at ubi ingentium &longs;arcinarum quanti
tas examinanda e&longs;t, prorsùs incommodum e&longs;&longs;et opportunas bi
lances aut habere, aut adhibere: quot enim & quanta pondera
parare oporteret, ut centenas aliquot fæni libras, &longs;eu mercato
rios fa&longs;ces, &longs;eu &longs;accos farinæ plenos expenderemus? & ex alio
in alium locum &longs;i transferenda e&longs;&longs;et libra cum legitimis ponde
ribus tantæ gravitatis, nonne opus e&longs;&longs;et plau&longs;tro, ut tàm in
gens onus in de&longs;tinatum locum tran&longs;veheretur? Quare Statera
excogitata e&longs;t tanquam libra brachiorum inæqualium, in quâ
pondus minus longiori brachio adnexum æqualia habet mo
menta cum majori pondere, quod ex breviore brachio &longs;u&longs;pen
ditur. Sed ne varia pondera in promptu habere cogeremur,
quæ longioris brachij extremitati adnecterentur, pro variâ
oneris gravitate explorandâ, &longs;apienti&longs;&longs;imè à majoribus &longs;ta
tera con&longs;tructa e&longs;t quæ eodem æquipondio modò in majo
re, modò in minore di&longs;tantiâ à centro motûs, æquilibrium
con&longs;titueret. Ex quo fit &longs;tateram eandem vires &longs;ubire plu
rium librarum, prout plura longioris brachij puncta percur
rit æquipondium; mutantur &longs;iquidem Rationes di&longs;tantiarum
ponderum, manente eâdem mercium à &longs;parto di&longs;tantiâ, ac
merces inæquales.
Sunt autem &longs;tateræ partes Jugum, An&longs;a, Uncus aut lanx,
Æquipondium, quod aliis Sacoma, aliis Cur&longs;orium dicitur.
Jugum e&longs;t, quod in partes inæquales divi&longs;um ab axe, qui An
&longs;æ in&longs;eritur, definit Rationem ponderum, quæ momentis
æqualibus librantur. An&longs;a e&longs;t, ex quâ &longs;u&longs;penditur &longs;tatera, ut
liberè utramque in partem ver&longs;etur. Uncus, aut lanx, oneri
&longs;u&longs;tinendo de&longs;tinatur; quæ enim facilè molem unam efficiunt,
po&longs;&longs;unt ex Unco &longs;u&longs;pendi; &longs;ed quæ ex pluribus non facilè in
unam molem coëuntibus con&longs;tant, lance &longs;ubjectá recipi oportet. Æquipondium e&longs;t certæ gravitatis pondus, ex quo oppo&longs;itæ
gravitatis Ratio innote&longs;cit.
Sit AB jugum ab axe inæqualiter in C divi&longs;um, &longs;itque CA
brachium minùs, cujus extremitati A catena aut funis adnecti
tur cum unco aut lance E, & CB
brachium majus, cujus longitu
dinem pro opportunitate percurrit
æquipondium F. An&longs;a re&longs;pondens
lingulæ CD, ip&longs;ius axis extremi
tates recipit, ut facilè convolvi
po&longs;&longs;it. In minoribus & mediocri
bus &longs;tateris lingula cra&longs;&longs;iu&longs;cula ad
ditur, quæ an&longs;æ intercapedinem ita impleat, eíque congruat,
ut tamen nullo partium conflictu impediatur motus; in majori
bus & longioribus &longs;tateris aliquando lingula omittitur, vel quia
&longs;partum e&longs;t infrà rectam lineam jugi, quod non ni&longs;i horizonta
liter con&longs;i&longs;tit, vel quia &longs;i &longs;partum e&longs;t in &longs;uperiore loco, non
multùm à vero pondere aberrare permittit ip&longs;a brachij longitu
do, quæ facilè prodit paralleli&longs;mum aut inclinationem ad ho
rizontem; mediocris autem error in mercibus, quæ huju&longs;modi
magnis &longs;tateris expenduntur, neque emptori, neque venditori
incommodo e&longs;t; quapropter in iis &longs;ubtilitatem &longs;crupulosè per
&longs;equi inutile e&longs;t, & ineptum. Quæ in librâ circà Axem, lin
gulam, An&longs;am ob&longs;ervanda monuimus, &longs;tateræ pariter commu
nia &longs;unt, neque hîc iterum inculcanda.
Poti&longs;&longs;imum, quod in &longs;taterâ ob&longs;ervandum e&longs;t, pertinet ad
divi&longs;ionem longioris brachij in minutiores particulas, ut exqui-
tur ab inci&longs;is in brachio notis indicantibus Rationem brachij
longioris ad brevius; e&longs;t &longs;cilicet minoris brachij longitudo
transferenda in alterum brachium, quoties fieri pote&longs;t; & quia
hoc longius produci pote&longs;t infinitè, proptereà &longs;tatera vocari
pote&longs;t libra qua&longs;i infinita brachiorum inæqualium. Sic di&longs;tan
tia AC tran&longs;lata in brachium CB ex. gr. quater, facit ut pon
dus in E po&longs;&longs;it e&longs;&longs;e quadruplum æquipondij F, &longs;i æquipondium
&longs;it in extremitate B: quia, ut dictum e&longs;t de librâ brachiorum
inæqualium, ut AC ad CB, ita pondus in B ad pondus in A:
& &longs; æquilibrium contingat &longs;acomate exi&longs;tente in G, erit ut
AC ad CG ita Sacoma in G ad pondus in E.
Hîc animadvertendum e&longs;t di&longs;tantiam AC, &longs;i &longs;it valdè nota
bilis, capacem e&longs;&longs;e multiplicis divi&longs;ionis, ac proptereà æqua
lem partem HG po&longs;&longs;e &longs;ubtiliùs dividi, ut non &longs;olùm uncias,
&longs;ed & unciæ quadrantes, aut etiam drachmas o&longs;tendat, &longs;i tran
&longs;itus ex H in G &longs;it nota unius libræ. Verum e&longs;t in brachio CB
huju&longs;modi majores partes minori brachio æquales non multas
e&longs;&longs;e po&longs;&longs;e: &longs;ed huic malo occurritur in adversâ parte jugi; con
ver&longs;a enim &longs;tatera aliam habet an&longs;am, puta SV, quæ minùs
di&longs;tat ab extremitate A; hæc autem di&longs;tantia &longs;æpiùs iterata plu
res exhibet partes, & factâ &longs;u&longs;pen&longs;ione VS, æquipondium in
extremitate B po&longs;itum æquilibratur cum majori pondere, quàm
cùm ex DC &longs;tatera &longs;u&longs;penditur; e&longs;t &longs;cilicet major Ratio BS ad
SA, quàm BC ad CA; nam ad eandem CA, majorem Ratio
nem habet BS major, quàm BC minor, & eadem BS majo
rem Rationem habet ad SA minorem, quàm ad CA majorem
ex 8 lib.
5. manife&longs;tum e&longs;t igitur majorem e&longs;&longs;e Rationem BS
ad SA, quàm BC ad CA. Si igitur pondera &longs;unt reciprocè ut
brachiorum longitudines, idem æquipondium in extremitate B
po&longs;itum minorem habet Rationem ad pondus in A, quando
brachia &longs;unt BS & SA, quàm cùm brachia &longs;unt BC & CA:
ac propterea tunc pondus in A e&longs;t majus.
Verùm hactenùs de &longs;taterâ perinde locutus &longs;um, ac &longs;i nulla
illi ine&longs;&longs;et gravitas; quæ tamen omninò contemnenda non e&longs;t,
quantumvis minuta &longs;it ip&longs;a &longs;tatera atque exilis, hac enim mi
norum ponderum gravitatem &longs;crupulo&longs;iùs exploramus: ideò
autem gravitatem à materiâ mente præcidere &longs;atius duxi, ut
net æquipondium; prout ad majorem, aut ad minorem motum
comparatè cum motu ponderis in A, e&longs;t di&longs;po&longs;itum. Cæterùm
pondus in A, quod æquilibrium facit cum &longs;acomate F, majus
e&longs;t quàm pro Ratione di&longs;tantiarum reciprocè &longs;umptâ; quia vi
delicet ip&longs;ius brachij longioris gravitas &longs;ua habet momenta ma
jora momentis brachij brevioris, ac propterea præter pondus,
quod Sacomati re&longs;pondet, addendum e&longs;t etiam pondus, quod
re&longs;pondeat exce&longs;&longs;ui momentorum brachij majoris &longs;uprà mo
menta brachij minoris. Cùm itaque ex dictis cap.2. hujus lib.
momenta brachiorum &longs;ingulorum perinde &longs;e habeant, atque
&longs;i &longs;emi&longs;&longs;is gravitatis &longs;ingulorum e&longs;&longs;et in extremitatibus, po&longs;ito
jugo æquabilis cra&longs;&longs;itiei, &longs;i nota &longs;it totius jugi gravitas, & bra
chiorum Ratio, &longs;ingulorum quoque gravitas innote&longs;cit; cujus
&longs;emi&longs;&longs;is per &longs;ibi congruum terminum Rationis ductus exhibet
&longs;ingulorum momenta. Sit AB jugum lib.5. unc.10, hoc e&longs;t
omninò unc.70: Ratio AC ad CB &longs;it ut 2 ad 5; igitur gravi
tas AC e&longs;t unc. 20, & CB unc.50: &longs;emi&longs;&longs;is AC unc.10 ductus
per 2 (qui e&longs;t terminus Rationis illi congruens) dat momen
tum 20: &longs;emi&longs;&longs;is CB unc. 25 ductus per 5, dat momentum 125:
differentia momentorum e&longs;t 105 dividenda per terminum Ra
tionis congruum di&longs;tantiæ AC, videlicet per 2: Quare ut fiat
æquilibrium cum &longs;olâ gravitate brachij longioris, addendæ
&longs;unt extremitati A unciæ 52 1/2: igitur addito &longs;emi&longs;&longs;e gravita
tis AC, intelliguntur in A unciæ 62 1/2; & in B unciæ 25: &longs;unt
autem 62 1/2 ad 25, ut 5 ad 2, quæ e&longs;t Ratio reciproca brachio
rum. Quare &longs;i jugum AB æquabile &longs;it, ut fert hypothe&longs;is, &
in extremitate B &longs;it Sacoma lib.2, pondus in A (computatâ
etiam gravitate catenæ & unci AE) non erit &longs;olùm lib.5. ut
exigit Ratio longitudinis brachiorum, &longs;ed prætereà unc.52 1/2,
hoc e&longs;t omnino lib.9. unc.4 1/2.
Quia verò aliquando accidit properatâ ad &longs;ubitum u&longs;um &longs;ta
terâ uti, videlicet cra&longs;&longs;iore tigillo, cujus gravitas non e&longs;t planè
contemnenda, &longs;ed valdè notabilis; proptereà hîc brevem
praxim adjicere placet, quæ etiam minùs peritis u&longs;ui e&longs;&longs;e po&longs;&longs;it,
ut &longs;tatim inveniant gravitatis quantitatem, quæ &longs;oli gravitati
brachij longioris re&longs;pondet. Sit tigillus AB, in quo intelliga-
chiorum differentia HB. Ponamus totam jugi longitudinem
e&longs;&longs;e di&longs;tinctam in partes 22, quarum AC &longs;it 4, CB 18, ac dif
ferentia HB 14. Sit verò tigilli pondus lib.84, cujus &longs;emi&longs;&longs;em
lib.42 accipio. Tum fiat ut longitudo brachij minoris 4 ad dif
ferentiam brachiorum 14, ita &longs;emi&longs;&longs;is gravitatis jugi lib.42 ad
aliud, & provenient lib.147 addendæ brachio minori, ut fiat
æquilibrium cum &longs;olâ gravitate longioris. Sic in &longs;uperiore
exemplo, ubi brachia erant ut 2 ad 5, differentia 3, pondus ju
gi unc.70, cujus &longs;emi&longs;&longs;is unc.35; fiat ut 2 ad 3, ita unc.35 ad
uncias 52 1/2, quod e&longs;t pondus ibi inventum pluribus calculis.
Ex his infertur jugum æquabilis cra&longs;&longs;itiei &longs;i &longs;u&longs;pendatur ex
quartâ parte &longs;uæ longitudinis, &longs;u&longs;tinere &longs;inè æquipondio pon
dus additum minori brachio, cujus gravitas æqualis &longs;it gravita
ti totius jugi. Si ex &longs;extâ parte &longs;u&longs;pendatur, &longs;u&longs;tinet pondus
duplex gravitatis ip&longs;ius jugi: &longs;i ex octavâ parte, &longs;u&longs;tinet pon
dus triplex gravitatis jugi; &longs;i ex decima parte, &longs;u&longs;tinet pondus
quadruplex; &longs;i ex duodecimâ, &longs;u&longs;tinet pondus quintuplex, &
&longs;ic deinceps.
Ut igitur ex ratione & certâ methodo con&longs;trueretur &longs;tatera
exqui&longs;itè di&longs;tincta in &longs;uas particulas, oporteret brachium mi
nus cum adnexis appendiculis, catenâ, unco, &longs;eu lance, tantæ
gravitatis e&longs;&longs;e, ut cum &longs;olâ longioris brachij gravitate æquili
brium con&longs;titueretur: tùm di&longs;tantia inter punctum, ex quo
onus &longs;u&longs;penditur, & centrum motûs transferenda e&longs;&longs;et ex eo
dem centro motûs in brachium longius, quoties fieri po&longs;&longs;et, &
&longs;ingula intervalla in certas partes minores dividenda, vel pro
libito vel (quod magis rationi congruum e&longs;t) in partes pro
prias men&longs;uræ, quæ adhibetur, ut &longs;i libra &longs;it in uncias, &longs;i un
cia, in drachmas. Hoc autem pendet ex gravitate &longs;acomatis,
quod eligitur: nam &longs;i libram unam pendat unà cum &longs;uo annu
lo æquipondium, tot erunt ponderis libræ, quot partes minori
brachio æquales intercipiuntur inter &longs;partum & ip&longs;um æqui
pondium: at &longs;i bilibre &longs;it &longs;acoma, jam partes illæ a&longs;&longs;umptæ
æquales minori brachio &longs;unt bifariam dividendæ, ut &longs;ingula
rum librarum notæ in jugo habeantur. Quod &longs;i con&longs;tructá jam
hoc modo &longs;taterâ, & majoribus partibus di&longs;tinctis in particulas
ex libito a&longs;&longs;umptas, velis apponere æquipondium majus, quàm
procam e&longs;&longs;e di&longs;tantiarum Rationem & ponderum, quæ in æqui
librio &longs;unt.
At &longs;i contigerit ea omnia, quæ breviori brachio adhærent,
non con&longs;tituere æquilibrium cum brachio longiore &longs;eor&longs;im
&longs;umpto ab&longs;que &longs;acomate, vel quia graviora &longs;unt, vel quia mi
nùs gravia; &longs;atis apparet æquipondium in di&longs;tantia à &longs;parto du
plà brachij minoris non habere duplum momentum, &longs;ed inve
niendum e&longs;&longs;e aliud punctum, à quo di&longs;tantiæ men&longs;ura de&longs;u
matur.
Sit &longs;tatera ACB, quæ in C &longs;u&longs;pendatur: gravitas brachio
rum ita &longs;e habet, ac &longs;i illius &longs;emi&longs;&longs;is in &longs;ua cuju&longs;que brachij
extremitate poneretur. Huju&longs;modi &longs;e
mi&longs;&longs;es gravitatum repræ&longs;ententur à li
neis BD & AE, quæ &longs;unt utique invi
cem in Ratione brachiorum (quoniam ju
gum æquabile & uniforme ponitur) & ut
AC ad CB, ita AE ad BD. Sed ut fiat
æquilibrium debet e&longs;&longs;e vici&longs;&longs;im ut AC
ad CB, ita BD gravitas in B ad AF gra
vitatem in A: E&longs;t igitur AE ad AF in
duplicatâ Ratione brachiorum AC ad
CB, hoc e&longs;t ut Quadratum AC ad Qua
dratum CB: Ergo etiam dividendo, per 17. lib.5. ut Quadra
tum CB minus Quadrato AC ad Quadratum AC, ita AF
minùs AE ad AE; hoc e&longs;t ut, differentia Quadratorum utriu&longs;
que brachij ad Quadratum brachij minoris, ita FE pondus ad
dendum, ad AE &longs;emi&longs;&longs;em gravitatis brachij minoris, ut fiat
æquilibrium cum &longs;emi&longs;&longs;e gravitatis, & momento brachij CB
longioris. Id &longs;i factum fuerit, a&longs;&longs;umantur in CB, incipiendo à
puncto C, partes æquales ip&longs;i CA, & tunc ad mercem addi
tam in F habebit gravitas &longs;acomatis H eam Rationem, quam
habuerit AC ad di&longs;tantiam eju&longs;dem &longs;acomatis à puncto C, ut
&longs;uperiùs dicebatur.
Verùm &longs;i præter AE gravitatem re&longs;pondentem minori bra
chio AC, pendere intelligatur ex A non &longs;olùm gravitas EF,
quæ &longs;ufficiat ad æquilibrium cum longiore brachio CB, &longs;ed
præterea &longs;it etiam gravitas FG, ita ut tota gravitas addita &longs;it
ut pondus H ad pondus FG exce&longs;&longs;um &longs;uprà id, quod requiri
tur ad æquilibrium, ita di&longs;tantia AC ad aliud ex. gr. CI: &
ex I initium &longs;umere debet divi&longs;io transferendo in longius bra
chium, & iterando di&longs;tantiam CA ita, ut AC æqualis &longs;it ip&longs;i
IN: &longs;i enim in G addatur tantum mercis, cujus gravitas GM
&longs;it ad æquipondium H, ut IN ad AC, fiet in N æquilibrium.
Quia &longs;cilicet ut FG gravitas ad gravitatem H, ita IC di&longs;tan
tia ad di&longs;tantiam CA ex con&longs;tructione; & ut gravitas H ad
gravitatem GM, ita CA di&longs;tantia ad di&longs;tantiam IN; erit ex
æqualitate per 22. lib.5. ut gravitas FG ad gravitatem GM,
ita di&longs;tantia CI ad di&longs;tantiam IN; Ergo componendo, per 18.
lib.5. ut FM ad GM, ita CN ad IN; &longs;ed ut GM ad H, ita
IN ad CA ex hypothe&longs;i; igitur ex æqualitate ut FM gravitas
ad gravitatem H, ita CN di&longs;tantia ad di&longs;tantiam CA. Cùm
itaque pondera addita ultrà æquilibrium, quod addità gravita
te EF fit in C puncto &longs;u&longs;pen&longs;ionis, &longs;int in Ratione reciprocâ
di&longs;tantiarum à &longs;parto C, nece&longs;&longs;ariò &longs;equitur æquilibrium in N.
Idem dicendum de cæteris deinceps punctis iterando di&longs;tan
tiam IN, prout brachij longitudo ferre pote&longs;t, nam duplicatâ
di&longs;tantiâ IN, poterit in G addi gravitas dupla gravitatis æqui
pondij H.
Quod &longs;i demùm partes minori brachio CA adjacentes non
e&longs;&longs;ent tantæ gravitatis, ut fieret cum longiore brachio CB
æquilibrium, quemadmodum &longs;i e&longs;&longs;ent ut OE ad EA &longs;emi&longs;&longs;em
gravitatis brachij minoris; primò ob&longs;erva, quantum de&longs;it gra
vitatis, ut fiat æquilibrium, &longs;cilicet &longs;it quantitas OF, quæ po
natur minor gravitate æquipondij H: intelligatur itaque gravi
tas æqualis gravitati æquipondij H, & &longs;it exce&longs;&longs;us FG. Quare
&longs;icuti paulò antè dicebatur, fiat ut pondus H ad gravitatem
FG, ita AC ad CI, & erit I punctum à quo incipienda e&longs;t di
vi&longs;io jugi, ita tamen ut facto æquilibrio in I intelligatur addita
merx æqualis gravitatis cum æquipondio H, & erit ex. gr. pri
ma libra. At verò &longs;i OE tam modica gravitas e&longs;&longs;et, ut etiam
addita gravitas æqualis gravitati &longs;acomatis H, nondum adæ
quaret gravitatem EF, addatur duplex, triplex, quadruplex
gravitas &longs;acomatis H ita, ut demum excedat gravitatem EF
nece&longs;&longs;ariam ad æquilibrium cum &longs;olo brachio longiore; tum fiat
ita AC ad CI, & e&longs;t I punctum quæ&longs;itum, ex quo incipit divi
&longs;io, & in quo &longs;i fiat æquilibrium mercis cum &longs;acomate, indicat
mercis gravitatem e&longs;&longs;e duplam, triplam, quadruplam gravita
tis &longs;acomatis H, prout hanc duplicare oportuit, aut triplicare.
Sed quas habemus communes &longs;tateras ab hác &longs;edulitate pro
cul remotas e&longs;&longs;e omnibus con&longs;tabit, &longs;i ob&longs;ervaverint amplitu
dines priorum divi&longs;ionum non omninò re&longs;pondere brachij mi
noris longitudini, hoc e&longs;t, intervallo, quo pondus di&longs;tat à &longs;par
to; neque id &longs;olùm, quia artifices tantam adhibere diligentiam
recu&longs;ant pro tenui mercede; verùm etiam ne adeò graves
exi&longs;tant majores &longs;tateræ, &longs;