MACHINIS MOTA
DISSERTATIONES
PHYSICAE, HYDROSTATICAE
Multiplici Noua Methodo Terræ magnitudo &
Grauitas inue&longs;tigatur: ARCHIMEDES
terræ motionem &longs;pondens ab arrogantiæ
&longs;u&longs;picione vindicatur.
PAVLO CASATO
E SOCIETATE IESV
ROMÆ.
M.DC.LVIII.
D. IOANNI GEORGIO
PATRITIO VENETO
borat Archimedes, quòd dato, vbi ip
&longs;e con&longs;i&longs;teret, loco tellurem &longs;uis à fun
damentis conuellere &longs;e po&longs;&longs;e affirma
ret: id &longs;cilicet per &longs;ummam confiden
tiam dictum putant, quod cum reap
&longs;e ne tentari quidem, ne dum perfici,
queat, experimento refelli non pote&longs;t. Di&longs;cutienda fuit
conflata in bonum Senem inuidia criminis, quo &longs;oli erudi
tuli afflantur, qui &longs;cientijs leuiter a&longs;per&longs;i &longs;ibi &longs;apientes vi
dentur. Et quanquam me vindice non eget Archimedes,
&longs;ua &longs;apientia aduersùs calumniantium tela &longs;atis protectus;
illud forta&longs;&longs;e non inutile accidat, &longs;i vel minùs eruditi in
telligant nihil e&longs;&longs;e tam arduum, quod &longs;uperari non po&longs;&longs;it &longs;a
pientis indu&longs;tria. Tibi certè, Illu&longs;tri&longs;&longs;ime Domine, non
iniucundam fore hanc elucubratiunculam præ&longs;agit animus
pro ea humanitatis abundantia, qua literarios omnes cona-
e&longs;t, continuò ge&longs;tijt tuum conuolare in &longs;inum, in quo &longs;e be
nignè fouendam &longs;peraret. Allici
&longs;anguinis nobilitas auorum nominibus con&longs;picua, innutrita
virtutibus indoles, morum facilitas &longs;uaui&longs;&longs;ima, grauitas
que comitate condita, ingenij acies per&longs;picua, eruditio varia
atque præclara. Illud vnum ab&longs;terrere poterat properan
tem, quod de mouenda tellure di&longs;putans vix &longs;e &longs;u&longs;picari
debui&longs;&longs;et a&longs;piciendam ab homine, qui inter eos delectus,
quos aut ad proponendas dirimendasque partium cau&longs;as, aut
ad Bonum Regimen aduigilare Sapienti&longs;&longs;imus Princeps iu&longs;
&longs;it, intentis in Reipublicæ quietem, componendosque ciuium
motus oculis hæret. Sed cum nulla tibi pereani temporis
momenta, quienim potui&longs;&longs;es hunc iuuentutis florem matu
ris tot &longs;cientiarum fructibus coronare, naturæ rece&longs;&longs;us phi
lo&longs;ophando rimari, in infinitas Iuris ambages excurrere,
Theologicæ facultatis adyta penetrare, monumenta Eccle
&longs;ia&longs;ticæ vetu&longs;tatis euoluere, ni&longs;i velocis ingenij vigorem a&longs;
&longs;iduo &longs;tudio foui&longs;&longs;es? Cum, inquam, nulla tibi pereant
temporis momenta, de&longs;peran dum non fuit hi&longs;ce Di&longs;&longs;ertatio
nibus impetrari po&longs;&longs;e ea horarum re&longs;egmina, quæ aut amæ
nioribus Mu&longs;is, aut Mathematicæ contemplationi tribuere
&longs;oles, vt &longs;euerioris negotij laborem literato otio interrum
pas. Nihil hìc tua dignum eruditione, quæ e&longs;t Authoris
tenuitas, inuenies: meæ tamen ob&longs;eruantiæ ve&longs;tigia non ob
&longs;cura deprehendas, maximè velim. Multo autem notior
atquè illu&longs;trior meus erga te animus erit, vbi per tua iu&longs;&longs;a
licuerit mea in re &longs;tudia officijs vberioribus te&longs;tari. Tuæ
erit magnanimitatis exilem hanc ob&longs;equij mei te&longs;&longs;eram non
contemnere. Vale.
MACHINALIS & Hydro&longs;taticæ Philo&longs;ophiæ,
quam premo, &longs;pecimen aliquod exhibi
turus, ac Prodromum tanti&longs;per emi&longs;&longs;u
rus dum extrema manus operi accedat,
Archimedæum Problema &longs;elegi, quo tel
lurem moueri po&longs;&longs;e pro&longs;itebatur, ni&longs;i locus, vbi machi
na con&longs;i&longs;teret, defui&longs;&longs;et. Qua ille machinatione id per
&longs;icere moliretur, Hi&longs;toricis di&longs;putandum relinquo. Mul
tiplex &longs;uppetebat methodus; &longs;atis &longs;cio. Placuit tamen
poti&longs;&longs;imùm o&longs;tendere, quantum in hoc negotio machi
narum Compo&longs;itio præ&longs;tet earundem Augmento: id
que intrà eiu&longs;dem Facultatis genus; vt vel &longs;olos Vectes
adhibendo, vel &longs;olas Trochleas &c. quod verò in vno
genere explicatur, de cæteris dictum facilè intellig
Ne quis autem in motu i&longs;to per&longs;iciendo aut immen&longs;as,
aut innumeras requiri machinas exi&longs;timaret, tentaui to
tius globi terraquei grauitatem, quantum coniiciendo
a&longs;&longs;equi fas e&longs;t, explorare: vnde apertâ con&longs;ecutione con
&longs;icitur non adeò multis membris di&longs;tingui oportere ma
chinam hoc in opere nece&longs;&longs;ariam: vtinam de materi&etail;
ipsâ non &longs;atis &longs;irmâ dubitari non po&longs;&longs;et. Quoniam ve
rò grauitatis notitia pendet ex mole præcognitâ; vt ab
&longs;olutum e&longs;&longs;et Problema, methodos indicaui, quibus
terræ magnitudinem indagare po&longs;simus: vt videlicet ex
notâ mole pondus innote&longs;cat, & hinc de&longs;iniri po&longs;&longs;it ma
china quæ datæ grauitati mouendæ proportione re&longs;pon
deat. Sed quia in motu ip&longs;o aqua in partem &longs;ecederet,
motumque faciliorem efficeret; examinandum fuit,
potuit &longs;inè Hydro&longs;taticâ exercitatione, qua ignis terræ
vi&longs;ceribu, inclu&longs;i, aëris, & aquæ grauitates inuice&mtail;
conferrentur.
Duas in di&longs;&longs;ertationes tribueram hoc opu&longs;culum,
cum primùm problema hoc in Collegio Romano
&longs;ub au&longs;picijs Eminenti&longs;&longs;imi Principis Cardinalis Ha&longs;
&longs;iæ Lantgrauij explicatum e&longs;t ab Illu&longs;tri&longs;&longs;imo Co
mite Antonio de Mont&longs;ort. Sed quoniam di&longs;&longs;ertatio
nes illæ longiores erant, quàm vt facilè hominem ad
legendum allicerent, & per tempus non licuerat ad
marginem notas, qua&longs;i eorum, quæ dicuntur, indices,
apponere, placuit rem totam in quinque di&longs;&longs;ertationes
di&longs;pertiri, vt legentium commodo &longs;eruirem, additis ad
marginem notis. Ne verò pauculis ijs, ad quorum ma
nus olim veni
ctam cramben recoxi&longs;&longs;e, non prodeunt &longs;inè auctario hæ
di&longs;&longs;ertationes, quas plurium eruditorum virorum iu di
cia &longs;ubire de&longs;idero, vt doctior &longs;iam.
Fru&longs;tra quæras ex me, vt ea quæ di&longs;&longs;ertatione comple
xus &longs;um, aliorum authoritate &longs;irmentur: hæc enim &longs;i
legi&longs;&longs;em, nolui&longs;&longs;em ex&longs;cribere: ideo plura omi&longs;i, quæ
ab alijs dicta deprehendi. Non adeò tamen de&longs;ipui, vt
mihi vni Solem illuxi&longs;&longs;e cen&longs;eam: &longs;ieri potuit vt hæc ea
dem alijs occurrerent; &longs;ed quæcunque tandem illa &longs;int,
mihi primùm, nemine prælucente, in mentem vene
runt. Hæc autem eo tantùm con&longs;ilio dicta &longs;unt, n&etail;
plura, quæ in hanc &longs;ententiam afferri potui&longs;&longs;ent, omi&longs;&longs;a
calumnieris: neque enim omnia per&longs;equi otium &longs;uit.
Quod &longs;pectat ad &longs;criptionis methodum dialogicam;
Breuitatem cum per&longs;picuitate con&longs;ectanti methodus hec
magis arridebat. At quid opus erat calculorum progre&longs;
&longs;us, quibus numeri illi indagantur, quos in colloquio
di&longs;&longs;ertatores afferunt, ad fa&longs;tidium inculcare? Id enim
communiter periti Arithmetici non faciunt; &longs;ed calamo
in &longs;chedulâ taciti inue&longs;tigant: id quod ab huius dialogi
interlocutoribus factum ponimus. Quod &longs;i quis id mihi
culpæ vertat, &longs;ciat me peceare malui&longs;&longs;e omittendo,
quam tantâ numerorum vi lectorem onerando. Placuit
verò tres viros de Mathematicis di&longs;ciplinis optimè me
ritos (qui diem no&longs;tro æuo obierunt) Galilæum, Mer
&longs;ennum, Guldinum di&longs;&longs;ertatores exhibere; vt ex Italicâ
pariter, Gallicâ, atque Germanicâ Mathe&longs;i commen
tatiuncula hæc lucem mutuaretur, quam non potuit ab
authore recipere. Nec te pluribus volo.
Vale.
Machinarum vires inter &longs;e comparat.
DISSERT ATIO SECVNDA
Terræ grauitatem inue&longs;tigat.
DISSERT ATIO TERTIA
Methodos varias inueniendi terræ quantitatem proponit.
DISSERT ATIO QVARTA
Ex aquæ &longs;eparatione à terræ motus facilitatem infert.
DISSERT ATIO QVINTA
Minorem telluris grauitatem in aqua explicat.
In Prouincia Romana.
O
P. Paulo Ca&longs;ato Societ. No&longs;træ compo&longs;itum, tres viri
graues, ac docti Eiu&longs;dem No&longs;træ Societ. perlegerunt, &
in lucem edi po&longs;&longs;e iudicarunt: quarè facultate mihi conce&longs;&longs;a
ab Adm. Reu. Padre No&longs;tro Go&longs;vvino Ni
nerali, pote&longs;tatem facio vt imprimatur, &longs;i alijs, ad quos
&longs;pectat, ità vi&longs;um fuerit. Romæ die 2. Maij 1657.
Si videbitur Reuerendi&longs;s. Patri Sa
M. Oddus Vice&longs;g.
Fr. Vinoentius Maria Guini&longs;ius Magi&longs;ter, & Reuerendi&longs;simi
P. Fr. Raymundi Capi&longs;u
Præd.
PRIMA
comparat.
NVNQVAM minùs alieno tempo
re acce&longs;&longs;i&longs;tis, Amici, vt me
ve&longs;tro a&longs;pectu pariter ac fa
miliari&longs;&longs;imâ collocutione re
crearetis longo &longs;anè fa&longs;tidio
grauem.
Satis in tempore accedo, &longs;i ob&longs;eruan
tiam, qua te plurimos no&longs;tri æui Mathema
ticos &longs;apientiâ facilè antecedentem colo,
certis documentis liceat declarate.
Id nobis quoquè lucro futurum e&longs;t,
quod dolueris; quos nimirùm &longs;uaui&longs;&longs;im&atail;
illa tua dicendi facundia po&longs;t mole&longs;tas animi
curas luculentiùs beabit: nemo &longs;iquide&mtail;
pleniùs ex fonte bibit, quàm qui &longs;ubducto
recens &longs;i&longs;tulæ epi&longs;tomio aquam in libertatem
vindicat. Sed quæ demùm nebula &longs;ere na&mtail;
tranquillæ mentis diem valuit infu&longs;care?
Haud procul ab hi&longs;ce ædibus obuias
habui&longs;tis lappas, quæ nec à &longs;e inuicem, nec à
me, quamuis lite compo&longs;itâ, diuelli poterant.
Itane Verò, Galilæe?
auocatum à
mathematicis contemplationibus animum
(quod &longs;uperi omen obruant) ad fori conten
tiones tran&longs;tuli&longs;ti?
No&longs;tri &longs;ub&longs;ellij erat, quam detule
runt litem dirimendam. Hæc autem illos
controuer&longs;ia torquebat contentionis forta&longs;sè
cupidiores quàm veritatis: Qua videlicet
machinatione tellurem loco mouere decreui&longs;
&longs;et Archimedes, ni&longs;ilocus, vbi po&longs;&longs;et
defui&longs;&longs;et. Hic quidem aptâ quinque
coagmentatione rem totam fui&longs;&longs;e per&longs;icien
dam a&longs;&longs;erebat: Cont
iu&longs;modi vires vni Glo&longs;&longs;ocomo
tendebat. Nec planè nullius operæ fuit homi
nes Mathematicis leui&longs;simè a&longs;pe r&longs;os ad con
cordiam reuocare; cum alter
dum momenti &longs;atis frigidè cen&longs;eret, alter At
chimedem in multiplici tympanorum denta
torum acce&longs;sione facilitatis compendia quæ
&longs;i&longs;&longs;e affirmaret
Illud crediderim potiùs vocari
po&longs;&longs;e in controuer&longs;iam, vtrùm &longs;olâ tympano
rum dentatorum collabellatione, an verò
multiplicatâ helice in&longs;initâ mechanicu&mtail;
hoc miraculum fui&longs;&longs;et patraturus.
Ita lanè: modò inter nos conue
niat &longs;ieri po&longs;&longs;e, vt &longs;tatuamus, quibus mem
bris di&longs;tingueretur celeberrimum illud Ar
chimedis Inuentum Quadrage&longs;imum, quo
datâ potentiâ datum pondus moueri po&longs;&longs;e
iactabat; cui idcircò nonnulli nomen fecere
Pancratio: nullus &longs;iquidem dubitandi locus
relinquitur, quin hac machinâ telluri motum
conciliare moliretur. Cæterùm cum nihil
nobis ni&longs;i coniectura &longs;uppetat, & quide&mtail;
quæ varias duci pote&longs;t in partes, qua nihil
incertius (neque enim me Heronis Alexan
drini in Barulco Glo&longs;&longs;ocomum quicqua&mtail;
moratur) nihil facilè de Archimedis mente
au&longs;im affirmare, cùm dentatis rotis æquè at
que multiplici cochleâ in&longs;initâ idem a&longs;&longs;equi
potuerit, quod per&longs;icere meditabatur.
Athelicis vires inuentorem &longs;uum
latui&longs;&longs;e aute&mtail;
oporteat, &longs;i eam Glo&longs;&longs;ocomo, quod mera tym
pana dentata con&longs;tituant, po&longs;thabuerit; ne mo
enim &longs;apiens longioribus ambagibus id per&longs;e
quitur, quod po&longs;&longs;it breuiore
I
Glo&longs;&longs;ocomi
&longs;eu Pan
tij constru
ctio.
Sint tympana dentata quinque maior&atail;
A, B, C, D, E, totidemque minora F, G, H,
I, K; maximum A circa eundem axem cum
cylindro S, cui ductarius &longs;unis circumducitur,
con uertatur: quatuor minor a F, G, H, I, com
munem cum maioribus B, C, D, E, quibus
&longs;ingula in axe eodem cohærent, habeant con
uer&longs;ionem: minimum verò K, addito manu
brio LM, circùm &longs;e torqueatur, & ex illo
totius machinæ motus initium &longs;umat. Ma
nubrii autem flexus LM, ad tympani K &longs;emi
diametrum Rationem habeat quintupla&mtail;;
&longs;ibique pariter reliqua tympana, maiora vi
delicet cum minoribus &longs;ibi cohærentibus
comparata, pro portione re&longs;pondeant: nec
maximi tympani A Radius, atque cylindri
S illi infixi &longs;emidiameter, à Ratione hac quin
tuplâ di&longs;sideant.
II
Glo&longs;&longs;ocomi
vires e
His ita con&longs;titutis &longs;atis liquet potentiam
in M applicatam quintuplo velociorem e&longs;&longs;e
peripheriâ E, quæ ex mutuâ &longs;uorum denti
culorum ac tympani K collabellatione con
uertitur. At peripheria E quintuplo pariter
quàm B, & B quàm A, & A quàm S, hoc e&longs;t
pondus P illi adne xum. Igitur motus poten
tiæ in M (liceat in pagellâ rem ad calculos
reuocare) ad
ad 1. &longs;unt nimirum &longs;ex Rationes quintuplæ,
ex quibus Ratio motûs potentiæ ad motum
ponderis componitur. Quare potentia, quæ
ab&longs;que machinæ &longs;ub&longs;idio centum pondo mo
uere valeret, hac adhibitâ machinâ pondus
P librarum 1562500. attollet.
III
Cochleæ in
finitæ Com
po&longs;itæ uires
cum Glo&longs;&longs;o
como com
parantur.
Iam verò in minorum tympanorum FG
HIK locum reponantur helices maioru&mtail;
tympanorum ABCDE denticulis (quos 25 in
toto ambitu fui&longs;&longs;e exempli gratia &longs;tatuamus)
congruentes: ip&longs;aque tympana cum &longs;uis
axibus in &longs;piram deformatis in quadrato lo
culamento, vt helicis in&longs;initæ natura fert,
aptè di&longs;ponantur. Con&longs;tat tympani A pe
ripheriam quintuplo tantùm velociorem e&longs;&longs;e
pondere P; at peripheriam B vigequintuplo
velociorem quàm A: vnicus enim tympani A
denticulus promouetur ab integrâ ip&longs;ius B
conuer&longs;ione; ideoque vt &longs;emel A gyrum ab
&longs;oluat, vicies & quinquies tympanum B con
uertatur oportet. Eademque ratione tympani
C motus cum tympani B motu comparatur,
cuiuse&longs;t vigequintuplus: quemadmodum & Quare
Ratio motûs potentiæ M ad motum ponderis
P, ex vnicâ Ratione quintuplâ, & quinque
vigequintuplis componitur: E&longs;t igitur motus
potentiæ ad motum ponderis vt 48828125
ad 1: & potentia, quæ pondo centum valeat
attollere, pondus librarum 4882812500
mouebit.
Cum itaque tam immane pondus moueri
po&longs;&longs;it quinque tantum cochleis in&longs;initis,
quæ totidem dentatis tympanis congruant;
contra verò, reiectâ helice, decem maiora
totidemque minora tympana componi opus
&longs;it, vt pondus idem attollatur; liquidò con&longs;tat
longè
Glo&longs;&longs;ocomi v&longs;um; ac proinde quadrage&longs;i
mum Archimedis inuentum helicem fui&longs;&longs;e,
procliuius e&longs;t opinari.
Nec ego in&longs;icior, nec diffitetur Gul
dinus helicis in&longs;initæ vires cæteris machi
nationibus longè præ&longs;tare: &longs;ed quæ nos cogit
nece&longs;&longs;itas affirmandi Archimedem quadra
ge&longs;imo loco in inuentum planè facillimum
incidi&longs;&longs;e? Quis fuerit Archimedææ contem
plationis &longs;copus, in quo conquie&longs;ceret, me,
fateor, later Quid verò, &longs;i quis machinæ faci
litatem non in eo&longs;tatuendam cen&longs;eat, quod
illa paucioribus membris contineatur; &longs;ed in
Atqui tympanorum ambitum in denticulos
æquales di&longs;tribue
re, eosque &longs;atis &longs;it
mos, ne facilè vi
ponderis commi
nuantur, & i&ntail;
cylindro ver&longs;atili
helicem tympani
denticulis ritè
gruentem
re, haud &longs;anè o
&longs;citantem exigunt
arti&longs;icem. Porrò
a&longs;&longs;iduus ille toe
partium &longs;e inui
cem atterentium
affrictus mora&mtail;
infert non leuem.
Quare nec teme
re dixerit qui&longs;
piam, denticula
tis tympanis va
lere iu&longs;&longs;is, re&mtail;
totam faciliùs per. &longs;ici po&longs;&longs;e multiplici axe in peritrochio, qui
& leui negotio paratur, & moram recipit nul
lam ex mutuâ membrorum affrictione.
IV
Difficulta
tes in heli
cis constru
ctione &
A&longs;&longs;umantur quinque cylindri ABCDE
cra&longs;&longs;itudine inæquali (quo enim magis à
pondere di&longs;tant, graciliores e&longs;&longs;e po&longs;&longs;unt)
&longs;ingulisque rota canaliculum in ambitu ex
cauatum habens adijciatur, cuius d
diametri cylindri &longs;ibi cohærentis quintupla
&longs;it. Demùm cylindro F addatur manubrium
OP eandem Rationem quintuplam habens
ad illius &longs;emidiametrum. Hi verò cylindri
paralleli in &longs;uo conceptaculo facilè ver&longs;atiles
ita di&longs;ponantur, vt funis &longs;ingulas rotas am
biens ad &longs;uperiorem cylindrum ductus (&longs;i
fieri id commode po&longs;&longs;it) congruat lineæ,
quæ horizonti ad perpendiculum in&longs;i
Hìc pariter con&longs;tat Rationem motûs poten
tiæ P ad motum ponderis M ex rationibus
intermedijs, nimirum &longs;ex quintuplis, com
poni. Quarè & hìc motus potentiæ ad pon
deris motum e&longs;t vt 15625 ad 1, vt &longs;uperiùs,
Mer&longs;enne, ratiocinabaris. Maiore tame&ntail;
forta&longs;sè compendio, quod hìc rotæ cylindros
non atterant, nec vllum immineat pericu
lum, ne ex mutuâ illâ collabellatione den
tes aliquando excutiantur: quàm ægrè au
tem excu&longs;&longs;i dentis detrimento occurras, pa
làm e&longs;t; cum tamen di&longs;ruptum &longs;unem ite
rum facilè connectas.
V
Axes in Pe
ritrochio
po&longs;iti faci
liùs con&longs;tru
untur, &
idem præ&longs;ta
re po&longs;sunt.
Vbi licet obiter animum aduertere ad
multorum artificum imperitiam, qui ex ma
chinarum tantùm augmento vires ad mo
uenda pondera augeri autumantes, multo la
bore, nec modicâ argenti iacturâ immanes
aliquando machinas con&longs;truunt, quæ vel mo
le &longs;uâ prægrauatæ fati&longs;cunt, vel motam mo
tui inferunt non mediocrem, adeò vt &longs;olius
machinæ motio, etiam &longs;eclu
mouendo de&longs;tinatur machina, validioris po
tentiæ vires ab&longs;umat: cum tamen rem to
tam minore negotio, nec &longs;pernendo pecu
niæ atquè laboris compendio, perficere po&longs;
&longs;ent, &longs;i rectè intelligerent, quantum machi
narum compo&longs;itio præ&longs;tet earum augmento.
Id quod ex quinque his axibus in Peritrochio
inter &longs;e compo&longs;itis luculenti&longs;&longs;imè demon
&longs;tratur; &longs;i enim axi, cuius diameter palma
ris, rotam velis infigere, qua po&longs;&longs;is ide&mtail;
mouere pondus, quod tribus tantùm ex pro
po&longs;itis rotis A, B, C, mouetur, quàm imma
nis illa erit? Quippe quæ 125 palmos &longs;e
cundùm diametrum obtineat: quàm diffi
cilis ad conuet&longs;ionem? ob ingentem grauita
tis cardines prementis re&longs;i&longs;tentiam: quàm
multo argento parabilis? cum ea &longs;ola ad tres
A, B, C, &longs;imul &longs;umptas e&longs;&longs;et vt minimùm in
Ratione Quadrati &longs;uæ diametri 125 ad tri
plum Quadratum diametri rotæ A, hoc e&longs;t
li nece&longs;&longs;ariam nec modicâ pecuniâ nec &longs;inò
multis operis comparari po&longs;&longs;e & elaborari
palàm e&longs;t. Adde quòd illam ex alio in a
lium locum transferendi pro multiplici ne
ce&longs;&longs;itate, vix &longs;pes aliqua &longs;upere&longs;&longs;et.
Machina
rum Com
po&longs;itio me
lior e&longs;t,
earum Aug
mentum, &longs;e
cundùm ma
gnitudinem
Quamuis verò quinque alios in&longs;uper cy
lindros cum &longs;uis rotis adijcere opus e&longs;&longs;et, vt
æquale pondus attolleretur, atque tuâ illâ
quinque helicum coagmentatione; in diffi
cultates tamen longè maiores incurrat arti
fex, nece&longs;&longs;e e&longs;t, qui cochleas quinque cum
tympanis examu&longs;&longs;im congruentes formare
iubeatur, ac ille qui decem aut plures axes
in peritrochio effingere velit; in quibus &longs;i
guram exqui&longs;itè rotundam &longs;i de&longs;ideret, &longs;u
petuacuo &longs;e labore conficiat: neque enim
anguli, &longs;i qui &longs;int, obe&longs;&longs;e quicquam po&longs;&longs;unt,
modò poli, &longs;eu cardines, circa quos ver&longs;an
tur, læues &longs;int ac politi.
Sed quanto impendio &longs;taret tan
tam funium vim comparare, quæ &longs;atis e&longs;&longs;et;
vt pondus ad vnius decempedæ altitudinem
&longs;ubleuaretur? Primùm enim funis, cui pon
dus adnecteretur, decem pedes in longitudi
ne haberet; igitur qui rotam primam A am
biret, e&longs;&longs;et vt minimùm pedum 50. At rotæ
&longs;ecundæ motus e&longs;&longs;et prioris quintuplus, igi-
250, nec di&longs;pari argumentatione conficitur
rotam tertiam C exigere funem pedum
1250, quartam D 6250, quintam demùm
E 31250: ex quibus conflatur &longs;umma pe
dum 39060, quæ e&longs;&longs;et totius funis longitu
do. Liceat autem hìc di&longs;&longs;imulanter præte
rire immodicam rotarum E, D, & C cra&longs;&longs;i
tudinem, vt tàm longus funis earum am
bitûs terminis contineri po&longs;&longs;et. Nam &longs;i mul
tiplici &longs;pirarum &longs;erie ita duceretur funis, vt
ip&longs;e &longs;e circumplecteretur, motuum Ratio
nes, &, quod inde con&longs;equitur, potentiæ mo
menta, ip â euolutione mutarentur, auctis
&longs;cilicet aut diminutis ex &longs;ubiectâ &longs;pirarum &longs;e
rie rotarum ac cylindorum diametris.
Difficultasi
ex funis lox
gitudine.
hi pariter aliquandò nonnihil fecere negotij.
Et primùm quidem earum rotularum, qua
rum velocior e&longs;t motus, funes quoquè gra
ciliores e&longs;&longs;e debere ob&longs;eruabam, quippe qui
bus ponderis grauitas minùs reluctetur. Hinc
verò &longs;atis commodâ con&longs;ecutione conficie
bam fieri po&longs;&longs;e, vt exiguis &longs;patijs tanta funi
culi longitudo citrà incommodum compre
henderetur. Deinde quamuis non adeò lon
gus &longs;uppeteret funiculus, operæ pretium vi
debatur futurum, &longs;i illo euoluto rotas &longs;i-
infixum arctè cum &longs;uperiore cylindro n&etail;
pondus relaberetur, connectentes) explica
tum que funiculum, qui in conuer&longs;ione fui&longs;
&longs;et cylindrum &longs;uperiorem circumplexus, ite
rum rotæ ap&longs;idi circumduceremus.
Verùm cum vrbem obambulans oculos
fortè in officinam quandam conijcerem, vbi
obtu&longs;arum nouacularum acies re&longs;tituuntur,
En, inquam, funiculus in&longs;e&longs;e rediens maio
ri ac minori rotulæ ità circumplicatus, vt al
tera ex alterius conuer&longs;ione rotetur, nullos
motui terminos præ&longs;cribit: quidni igitur con
&longs;imili ratione funiculus vnâ aut alterâ &longs;pirâ
cylindrum &longs;uperiorem complectens &longs;ubie
ctum peritrochium ita apprehendere valeat,
vt pariter moueantur, nec conuer&longs;ionis &longs;i
nem inueniant, cum tantundem &longs;uccedere
po&longs;&longs;it funiculi, quantum dece&longs;&longs;erit? Id au
tem &longs;i fieri po&longs;&longs;e concedas, in &longs;uperioribus
&longs;altem FEDC (nam in inferioribus forta&longs;sè
ob ponderis nimiam grauitatem &longs;atius e&longs;&longs;et
funis extremum religare, ne excurreret)
difficultas omnis propo&longs;ita euane&longs;cit. Vti
nam & in trochleis &longs;imile funis compen
dium liceret inuenire.
An & trochleis tellurem ab Archi
mede fui&longs;&longs;e loco dimouendam exi&longs;timas?
do, operi non &longs;atis congrua negotium face&longs;
&longs;eret, &longs;ed & aptæ trochleæ con&longs;trui omninò
non po&longs;&longs;ent. Quis enim Chilio&longs;pa&longs;tum com
ponat? &longs;ed quantulum demùm e&longs;t Chilio&longs;pa
&longs;ti momentum? Neque &longs;i trochleas decem
orbiculorum millibus in&longs;trueres, quicquam
po&longs;&longs;es efficere. Nam &longs;i duas huiu&longs;modi tro
chleas ità componeres, vt altera ponderiad
necteretur, altera in &longs;uperiore loco firmare
tur, eumque adhiberes conatum, quo libras
centum ab&longs;que machinâ attolleres, conatu
eodem librarum vicies centena millia moue
re po&longs;&longs;es: &longs;ed quid hoc ad immen&longs;um terre
ni globi pondus?
Quæ fuerit Archimedi mens, planè
ignoro: potui&longs;&longs;e tamen fieri, vt ille propo&longs;i
rum trochleis adhibitis a&longs;&longs;equeretur, con
&longs;tanter a&longs;&longs;ero. Et quidem, quod ad trochleas
ip&longs;as attinet, ludum puto, modò funes, &
locus vbi con&longs;i&longs;terem, non dee&longs;&longs;ent. Cedo
mihi orbiculos quatuor aut &longs;ex &longs;upra cen
tum; &longs;atis habeo trochlearum, vt vnicâ ma
nu terram à fundamentis, quibus in&longs;r&longs;tit,
conuellam. Id quod vt minimè dubia de
mon&longs;tratione conficiam; illud primum, in
trochleatum v&longs;u extra controuer&longs;iam po&longs;i
tum accipio, quod, funis extremo trochleæ,
liquum extremum arreptum trahens plus
obtinet ad mouendum momenti, quàm &longs;i
funis alteri trochleæ à pondere remotæ ad
necteretur; in primo enim ca&longs;u motus po
tentiæ ad motum ponderis maiorem habet
Rationem, quàm in &longs;ecundo.
Orbiculi
pauci in plu
res minores
tro
stributiplus
po&longs;&longs;unt,
quàm duæ
tro
multis mil
libus orbi
culorum.
Id ego tibi lubens permitto.
Nam
&longs;i trochleas duas R &
mus binis orbiculis inftructas,
funis autem extremum A tro
chleæ S annulo alligetur, &
ducatur funis per ABCDEF
GHIK, con&longs;tat totius funis
longitudinem quadruplam e&longs;
&longs;e interualli, quo trochleæ à
&longs;e inuicem &longs;eiunguntur. Iam
verò plurimum intere&longs;t, vtri
trochlearum pondus adiunxe
ris: &longs;i enim pondus in R ad
nectatur, potentia K tamdiu
mouetur, ac ab
nec funistotus explicetur: per
currit igitur &longs;patium funis
longitudini æquale, videlicet
quadruplum interualli inter
R & S. At verò &longs;i pondus in
S alligetur, eadem potenti&atail;
tanto &longs;patio tran&longs;greditur, quanta e&longs;t funis
longitudo: igitur in toto motu percurrit &longs;pa
tium quintuplum eius, quod à pondere ver
sùs trochleam R moto perficitur. Quoniam
autem quò tardior e&longs;t ponderis motus cum
motu potentiæ comparatus, eò minùs pon
deris grauitas virtuti potentiæ trahentis ob
&longs;i&longs;tit; apertum e&longs;t ac manife&longs;tum faciliùs
trahi pondus, &longs;i trochleæ S (cui funis in A
adnectitur) quàm &longs;i trochleæ R alligetur: in
S nimirum motum obtinet motûs potentiæ
&longs;ubquintuplum, in R autem &longs;ubquadru
plum.
Non æqua
li facilitate
moueri pon
dus. vtrili
bet trochleæ
adnexu&mtail;,
demonstra
tur.
Quod præterea Ratione aliquâ mi
noris Inæqualitatis propo&longs;itâ, Antecedens
terminus ad Con&longs;equentem duplum maio
rem habeat Rationem, quàm ad alium Con
&longs;equentem, ad quem habeat Rationem pro
po&longs;itæ rationis duplicatam (&longs;i tamen &longs;ubdu
plam excipias, cùm eadem &longs;umma fiat ex
duplicis binarij additione, ac ex mutuâ illo
rum multiplicatione) nihil habet dubitatio
nis. Datâ &longs;iquidem Ratione &longs;ubtriplâ 3 ad
9, &longs;i Con&longs;equens 9 geminetur, & &longs;iat 18,
Ratio autem duplicetur inuento tertio con
tinuè proportionali 27, maior e&longs;t Ratio 3 ad
18, quàm 3 ad 27. Similiter maior erit
druplum, quàm ad alium Con&longs;equente&mtail;
terminum Rationis triplicatæ aut quadru
plicatæ, etiam &longs;i propo&longs;ita Ratio &longs;ubdupl&atail;
e&longs;&longs;et: &longs;ic &longs;i fuerit Ratio 2 ad 4, triplus Con
&longs;equens e&longs;t 12. Con&longs;equens verò Rationis tri
plicatæ e&longs;t 16: e&longs;t autem maior Ratio 2 ad
12, quàm 2 ad 16.
Maiorest
Ratio ad ter
minu&mtail;
Misltipli
cem, quàm
ad termi
num Ratio
nis &longs;imiliter
Multiplica
tæ.
Hinc infero maximum e&longs;&longs;e di&longs;crimen i&ntail;
augendo potentiæ momento, vtrùm trochleæ
augeantur orbiculis, an verò trochleæ mul
tiplices exii&longs;dem orbiculis con&longs;tituantur. Si
enim trochleæ duæ S & R, quas nuperrimè
de&longs;crip&longs;i&longs;ti, binis prætereà orbiculis augean
tur, ita vt &longs;ingulæ quaternos habeaut, mani
fe&longs;tum e&longs;t potentiam in K, quæ priùs mo
tum habebat quadruplum motûs ponderis in
R con&longs;tituti, factâ huiu&longs;modi orbiculorum
acce&longs;sione, motum habere octuplum, vel
quæ priùs quintuplò velocior erat pond re in
S adnexo, factam e&longs;&longs;e noncuplò velociorem.
At &longs;i quatuor ho&longs;ce orbiculos non adijcias
prioribus, &longs;ed duas alias trochleas ex illis com
ponas, iam mulrò maior e&longs;t potentiæ motus
cum ponderis motu comparatus.
Sint duæ trochleæ binos orbiculos haben
tes A & B: huic autem tum pondus P,
funis extremum adnectatur. Vtique poten-
tia in F motum ha
beret quintuplò velo
ciorem motu ponde
ris P. Ex quatuor a
lijs orbiculis duæ pa
riter trochleæ D & C
con&longs;tituantur: & tro
chleæ C adnectatur
prioris funis
Potentia E quintuplò
&longs;anè velociùs moue
tur quàm F, at F quin
tuplò velociùs quàm
P; igitur motus po
tentiæ E ad motum
ponderis P e&longs;t vt 25
ad 1. Quare potentia
vires habens decem
pondo trahendi &longs;inè
machinâ, in F trahe
ret libras 50, at in E
libras 250. Quod &longs;i
tam A quàm B qua
ternos haberent or
biculos, potentia i&ntail;
F tantum 90 libras
mouere po&longs;&longs;et.
Trochlea
rum coniu
gataru&mtail;
Compo&longs;itio,
quàm ma
gnas vires.
babeat.
Con&longs;tat itaque du-
ro in&longs;tructis, motum potentiæ in E haber&etail;
Rationem duplicatam Rationis, quam habet
motus potentiæ in F ad motum ponderis in
P: multiplicatis autem pari numero in ea
dem trochleâ orbiculis, ne duplicari quidem
motum ip&longs;ius potentiæ F. Quod &longs;i in E pa
riter duæ aliæ trochleæ &longs;imiles adiicerentur,
iam triplicaretur Ratio motûs in F ad mo
tum in P, & &longs;ic deinceps. Vno verbo di
cam: quot &longs;unt paria &longs;imilium trochlearum,
progre&longs;&longs;io fit tot Rationum &longs;imilium Ratio
ni, quam habet motus ponderis ad motum
potentiæ primis trochleis applicatæ. Sic &longs;i
e&longs;&longs;ent talium, quales exhibui, trochlearu&mtail;
paria decem, a&longs;&longs;umendæ e&longs;&longs;ent decem Ra
tiones quintuplæ; & motus potentiæ ad mo
tum ponderis Rationem haberet ex his com
po&longs;itam, quam &longs;cilicet habet vndecimus ter
minus in progre&longs;sione Rationis quintuplæ
ad vnitatem, hoc e&longs;t 9765625 ad 1.
Si igitur trochleæ omnes ABCD
tri&longs;pa&longs;ti e&longs;&longs;ent, vnicus equus in E idem pon
dus trahere po&longs;&longs;et, quod equi 49; cum ta
men in F, &longs;i A & B tripa&longs;ti fuerint, idem
valeat trahere quod equi &longs;eptem: &longs;i vero A
& B &longs;enis in&longs;truerentur orbiculis, æquiuale
ret equis tredecim. Quis ergo adeò debilis
mulriplicium ope, maius tormentum belli
cum trahere?
Vnicus e
quus f
pote&longs;t mo
uere ingens
vix po&longs;&longs;ent
plures equi.
Ita planè: &longs;ed ob&longs;eruandum in fu
nibus.
funium in
hac machi
natione re
quiratr.
Ne plura: &longs;atis animum ad hoc
aduerti. Monere volebas tantam requiri
longitudinem funis, qui &longs;ecundas trochleas
D & C circumplectitur, vt vnicus ex D in
C ductus æqualis &longs;it longitudini &longs;patij, quod
potentia in F totum explicans funem, & pon
dus P ad trochleam A v&longs;que deducens, per
curreret. Quarè &longs;i interuallum trochlearum
AB &longs;it pa&longs;&longs;uum decem, potentia ex F per
curreret pa&longs;&longs;us 50: funis itaque trochleas D
& C ambiens longitudinem habeat nece&longs;&longs;&etail;
e&longs;t pa&longs;&longs;uum vt minimum 200; trochle&atail;
enim D ibi firmanda e&longs;t, quò funis ABF
explicatus pertingere po&longs;&longs;it. Eademque d&etail;
cæteris, &longs;i plures fuerint trochleæ, ratio e&longs;to.
Non falsâ coniecturâ animum meum
pro&longs;pexi&longs;ti. Sed vt ad rem ip&longs;am propiùs
accedamus ob&longs;eruandum e&longs;t, propo&longs;ito quo
cunque orbiculorum numero, qui tame&ntail;
per 2, vel 4, vel 6, vel alium quemcunque
numerum parem diuidi po&longs;sit, longè maius
e&longs;&longs;e potentiæ momentum, &longs;i plures trochleæ
pauciorum orbiculorum, quàm &longs;i pauciores
tur. Exhibeantur enim, exempli gratiâ, orbi
culi 60, ex quibus &longs;i fiant 20 trochleæ trium
orbiculorum, iam &longs;unt decem trochlearum
paria, ac proinde decem Rationes vt &longs;um
mum &longs;eptuplæ; igitur momenrum potentiæ,
hoc e&longs;t vndecimus ab vnitate terminus, e&longs;t
vt 282475249. Si verò fiant binos haben
tes orbiculos, erunt 15 trochlearum pari&atail;
ideoq quindecim Rationes quintuplæ, ex qui
bus Ratio motûs potentiæ ad motum ponde
ris componitur; erit igitur vt 30517578125
ad 1. Con&longs;tat autem ne additis quidem ad
huc duobus tri&longs;pa&longs;torum paribus, vt &longs;int in
vniuer&longs;um orbiculi 72, po&longs;&longs;e adeò augeri po
tentiæ momentum: po&longs;itis &longs;iquidem 12 tri
&longs;pa&longs;torum paribus momentum potentiæ e&longs;t
&longs;olùm vt 13841287201. Hinc colligitur
plus ad mouendum momenti obtinere pau
ciores orbiculos in &longs;implicioribus trochleis,
quàm in trochleis maioribus plures orbicu
los: id quod alicui forta&longs;sè paradoxum ac
cidat.
Pauciores
orbiculos in
&longs;implicibus
trochleis
plus po&longs;&longs;&etail;,
quàm plu
res in mai&atail;
ribus o&longs;ten
ditur.
Nunc igitur &longs;i mihi orbiculos centum ex
hibeas, &longs;implices trochleas ex &longs;ingulis orbi
culis &longs;tatuo, fiuntquè 50 trochlearum &longs;im
plicium paria: adnexo autem pondere eidem
trochleæ, cui funis extremum alligatur, mo-
50 Rationes triplæ, ex quibus componitur
Ratio motûs potentiæ ad motum ponderis.
Quærendus iraque e&longs;t terminus in progre&longs;
&longs;ione datæ Rationis triplæ ab vnitate quin
quage&longs;imus primus.
Ratio autem tripla quintuplicata e&longs;t 243
ad 1. Ducatur 243 per &longs;e ip&longs;um, & e&longs;t Ra
tio decuplicata 59049 ad 1. Hic iterum ter
minus 59049 per &longs;e ip&longs;um ducatur, & Ratio
3486.784401 ad 1 e&longs;t ex viginti Rationi
bus triplis compo&longs;ita. Ducatur pariter 3486.
784401 per &longs;e ip&longs;um, & e&longs;t Ratio ex qua
draginta Rationibus triplis compo&longs;ita 12.
157665.459056.928801. ad 1. Hæc de
mùm Ratio ducatur per Rationem triplam
decuplicatam nimirum per 59049, & pro
ducitur Ratio, quæ ex 50 Rationibus tri
plis componitur 717897.987691.852588.
770249. ad 1.
Quod &longs;i placeat duas adhuc trochleas ter
nis orbiculis in&longs;tructas adijcere ip&longs;ique pon
deri immediatè adnectere, vt funium &longs;eptu
plici ductu faciliùs &longs;u&longs;tineri valeat, adhuc
Ratio &longs;eptupla addenda, vt ex hac & 50 tri
plis tota Ratio componatur: & quinquage&longs;i
mus primus terminus progre&longs;sionis Rationis
triplæ ducendus e&longs;t per 7, vt habeatur totum
potentiæ momentum 5025285.913842.
968121.391743. Quarè adhibito conatu,
quo libras decem ab&longs;què machinâ traheres,
mouere po&longs;&longs;es libras 50.252859.138429.
681213.917430. Quod pondus totius ter
reni globi grauitatem &longs;uperat. Orbiculis ita
que &longs;ex &longs;upra centum Archimedæum pro
blema de terræ motione ab&longs;olui po&longs;&longs;e o&longs;ten
di.
Orbiculis
106 po&longs;&longs;&etail;
tellure&mtail;
moueri
mom
Quid verò, &longs;i quis maiorem ad
huc grauitatem telluri tribuat?
Parum mihi face&longs;&longs;at hic negotij.
Addat duas præterea trochleas aut quatuor,
aut plures; & fortè non adeò multas adijcere
oportebit: Mihi &longs;atis e&longs;t chiliadas illas orbi
culorum atque myriadas, quas multi ad
&longs;truunt, tanquam minimè nece&longs;&longs;arias refu
ta&longs;&longs;e. Nunc quidem &longs;olâ coniecturâ terreni
globi grauitatem venari po&longs;&longs;umus: dato au
tem exrra terram loco, in quo con&longs;i&longs;teremus,
facile e&longs;&longs;et trochlearum numerum definire:
examinato nimirum &longs;taterâ telluris pondere;
quo demum cognito trochlearum, quibus
opus e&longs;&longs;et, numerus innote&longs;ceret.
At &longs;taterâ, cuius iugum aliquot
pa&longs;&longs;uum millia occuparet, &longs;acomatis autem
munere rupes non exigua &longs;ungeretur.
Mittamus i&longs;thæc; quæ à te reru&mtail;
causâ, in medium proferri &longs;atis video. La
nius vel cætarius opportunam &longs;tateram &longs;up
peditare po&longs;&longs;et; modò &longs;ingulæ libræ in &longs;uas
vncias ritè di&longs;tributæ iugum di&longs;tinguerent.
Erige enim ad perpendiculum trabes
æquali interuallo à &longs;e inuicem di&longs;tinctas: A&longs;
&longs;umaturque vectis AB, cuius pars decim&atail;
&longs;it) & extremum A &longs;it circa axem trabi AO
infixum ver&longs;atile, extremum verò B ita à
trabe ER di&longs;ter, vt a&longs;&longs;umpto &longs;imili vect&etail;
DE circùm E ver&longs;atili, & vtroque DE, AB
parallelo horizonti, perpendicularis BF &longs;e
cet pariter FE partem decimam totius DE.
Idquè in con&longs;equentibus vectibus factum in
telligatur. Porrò BF, DI &c.
&longs;int ex ma
teriâ &longs;olidâ, & circa clauiculos extrema B, F,
D &c. moueri po&longs;&longs;int; ac po&longs;tremus vectis
MR habeat propè V lingulam &longs;eu momen
tum, vt quandò trabi ER congruit, &longs;ignifi
cet vectes omnes con&longs;titutos e&longs;&longs;e horizonti
parallelos.
Statera
muni
ingens pon
dus exami
nari.
Tum applicatâ in M &longs;taterâ examinetur
omnium vectium &longs;imul momentum graui
tatis. Non grauitatem dico, &longs;ed momen
tum grauitatis; quia vectis AB in F &longs;u&longs;pen
&longs;us non &longs;ecundùm totam &longs;uam grauitate&mtail;
deor&longs;um nititur, quia & in A &longs;ultinetur, &longs;ed
tantùm &longs;emi&longs;&longs;em totius &longs;uæ grauitatis exer
cet. Quia verò DE e&longs;t longitudo decupla
ip&longs;ius FE, ideò grauitas totius AB non per
cipitur in D ni&longs;i &longs;ecundùm partem &longs;ui vige
&longs;imam, in Hverò &longs;ecundùm partem ducen
te&longs;imam, & &longs;ic deinceps; ita vt &longs;i fuerint in
vniuer&longs;um octo vectes, percipiatur in M &longs;o-
totius AB.
uitas percipitur in M &longs;ecundùm partem &longs;ui
bis millies mille&longs;imam. Tertius &longs;ecundùm
partem ducenties mille&longs;imam; & &longs;ic dein
ceps, donec vltimus MR grauitet in M &longs;e
cundùm &longs;uæ grauitatis &longs;emi&longs;&longs;em. Hinc e&longs;t
quod quamuis primus vectis AB valdè cra&longs;
&longs;us e&longs;&longs;e deberet, atque adeò grauis, vt pon
dus valeret &longs;u&longs;tinere, con&longs;equentes tame&ntail;
vectes minores, ac minores, quod ad cra&longs;&longs;i
tudinem &longs;pectat, requirerentur: ac proinde
eorum momenta in M &longs;imul &longs;umpta no&ntail;
e&longs;&longs;ent adeò multa. Ponamus itaque facoma
te in N exi&longs;tente fieri æquilibrium cum ve
ctibus horizonti parallelis.
His ita con&longs;titutis dimittantur vectes, vt
ferè quantùm po&longs;&longs;unt de&longs;cendant: & in Gad
nectatur pondus. Vel forta&longs;&longs;e opportunius
erit, &longs;i BF habeat in B annulum, cui in&longs;eri
po&longs;sit extremum vectis AB; vectis enim ex
annulo extractus &longs;olus deprimitur, quantum
pote&longs;t, & pondere in C adnexo, aliâ machi
nâ tractoriâ tanti&longs;per eleuatur, dum po&longs;sit
iterum annulo B in&longs;eri. Ni&longs;i fortè commo
dius alicui accidat ita machinam con&longs;truere,
vt iacens pondus illi adnectatur, deinde ma
china ip&longs;a æqualiter eleuetur, vnde fiat vt Tum &longs;acoma
in &longs;tateræ iugo ab aginâ remoueatur adeò,
vt vectis MR (atque adeò reliqui omnes)
horizonti parallelus con&longs;tituatur, & &longs;tatera
indicet æquilibrium ex. gr. in
Sumatur
igitur differentia SN, quot nimirum libras
aut vncias contineat: hæc autem multipli
cetur per momentum, quod habet potentia
in M applicata; id quod fit tot additis ci
phris, quot &longs;unt vectes, quos &longs;tatuamus e&longs;&longs;e
octo, SN verò indicare libras 3 vnc. 6. Sa
coma igitur in S o&longs;tendit pondus P e&longs;&longs;e libr.
300.000000. vnc.600.000000, hoc e&longs;t in
vniuer&longs;um libr.350.000000. Quare con&longs;tat
di&longs;po&longs;itis hac ratione 25. vectibus, po&longs;&longs;&etail;
vulgari &longs;taterâ examinari pondus libr. 250.
000000.000000.000000.000000. Vectium
autem huiu&longs;modi apta di&longs;po&longs;itio non ijs &longs;ca
teret difficultatibus, quæ &longs;uperari non po&longs;
&longs;ent.
Mihi quidem &longs;atis per&longs;ua&longs;um e&longs;t,
cas, de quibus hactenus di&longs;&longs;eruimus, machi
nationes ad tellurem loco dimouenda&mtail;,
eiusque pondus examinandum aptas e&longs;&longs;&etail;,
atquè ad potentiæ momenta ferè in immen
&longs;um
dem multiplicis quàm maioris v&longs;um; in ma
iori etenim augetur &longs;olùm Ratio, quæ i&ntail; Sic peritrochiu&mtail;
diametrum habens decuplam &longs;ui axis facit
potentiæ momentum decuplum: at duo pe
ritrochia Rationem quintuplam ad &longs;uos axes
habentia &longs;i componantur &longs;imul, potentiæ
momentum con&longs;tituunt vigequintuplu&mtail;.
Id quod & in cochlearum compo&longs;itione ma
nife&longs;tum e&longs;t, cum &longs;atius&longs;it duas cochleas cum
duobus tympanis componere, quam heli
cem vnam &longs;trictiorem vni tympano maiori
congruentem adhibere. Hoc in vectibus,
hoc in trochleis abundè e&longs;t demon&longs;tratum.
Sed adhuc rudioribus quibu&longs;dam eximen
da e&longs;t dubitatio, quæ ancipitem animu&mtail;
torquet, an videlicet ea &longs;it totius globi huius,
quem terram dicimus, grauitas, quæ ad
librarum numerum reuocata paucioribus
quàm triginta ciphris explicari queat. Quam
uis enim illam certis finibus circum&longs;cri
ptam, ac numero definitam exi&longs;timent, &longs;ibi
tamen facilè per&longs;uadent
vqmas me/non u/pa/rxei
medem arenæ multitudinem con&longs;iderantes
opinabantur: qua&longs;i Arithmeticæ facultatis
labor vltimus omnem poft &longs;e relinqueret no
menclaturam. Quantam igitur graui
globo huic, qui terras ac maria complecti
tur, tribuemus?
Vereor ne vobis grauis fiam, &longs;i ea
exponere voluero, quæ aliquando placuit in
hanc &longs;ententiam commentari.
Immò verò aures meæ ad iucun
dam hanc di&longs;putationem patent: ni&longs;i fortè
negotia habeas, Galilæe, quibus te nunc o
porteat intere&longs;&longs;e.
Sum planè vacuus: nec facilè patiar
tam citò abire amicos, quorum eruditâ con
&longs;uetudine tantoperè recreor. An a&longs;ymbo
lum te rece&longs;&longs;urum putas, Guldine; Vnum
habeo, de quo te pariter interrogem, qui a
lios ad dicendum excitas: vbi tamen Mer
&longs;ennus &longs;uas de terræ grauitate commentatio
nes in medium protulerit.
Si me audieritis, di&longs;putatione&mtail;
hanc in &longs;equentem diem transferemus; cum
enim numeris aliquot maioribus
beam, longè commodius accidet eos in &longs;che
dulâ domi priuatim adnotatos in promptu
habere, quàm illos inter colloquendum in
ue&longs;tigare non &longs;ine mole&longs;tâ &longs;ermonis interru
ptione, temporis iacturâ, & capitis defati
gatione: nam &longs;atis nos hodiè tor&longs;imus tot
Rationum compo&longs;itarum inuentione.
Fiat, vt Mer&longs;enno placet.
SECVNDA
&longs;tigat.
CVM ad ingentia onera loco
mouenda machinam mem
bris non adeò multis di&longs;tin
ctam, neque paratu valdè
difficilem con&longs;trui po&longs;&longs;e, &longs;atis
he&longs;ternâ di&longs;&longs;ertatione o&longs;tenderimus; illud
nunc ex te, Mer&longs;enne, audire expectamus,
quanta &longs;it terreni orbis grauitas, vt quà&mtail;
magnâ pariter ad eam &longs;uperandam machi-
gamus.
Principio, quoniam id ex me pla
cet audire, grauitatem ex mole inue&longs;tigan
dam intelligens, quotquot apud authores
extabant de terræ magnitudine opiniones,
cœpi ad examen reuocare, vt eas, quæ pro
babili alicui coniecturæ & rationi niterentur,
à temerè con&longs;titutis, &longs;ecernerem. Cum ve
rò in omuibus
rem, & in fingulis labem aliquam mihi vi
derer deprehendere, vulgati&longs;&longs;imæ demù&mtail;
opinioni acquie&longs;cendum cen&longs;ui, quæ 60 mil
liaria Italica mediocria &longs;ingulis gradibus tri
buit; vt à veritate minimùm recederem, vel
&longs;altem cos &longs;olùm haberem aduer&longs;arios, qui
nimio in antiquitatem &longs;tudio feruntur.
pulus tamen, fateor, animum quantumuis
di&longs;&longs;imulantem &longs;timulabat, quoties milliaria
huiu&longs;modi minoribus men&longs;uris di&longs;tinguere
atque ad pedes reuocare opus erat; que&mtail;
e
tus hærebam; cùm pro diuer&longs;a a&longs;&longs;umpti pe
dis longitudine moles tota terreni globi alia
atquè alia deprehenderetur, di&longs;crimine non
contemnendo.
Perpendens itaque corporum vmbras ideo
&longs;emper in oppo&longs;itam Soli plagam proijci,
quia nulla Solis particula ad perpendiculum
imminet corpori, quod luce afficitur, i&ntail;
&longs;pem erigebar aliquid deprehendendi ex &longs;pa
tio circa Syenen ab antiquis apud Cleome
dem ob&longs;eruato, in quo gnomones nullam
proijciun
picum Cancri percurrente. Si enim in pla
no AB erigatur &longs;tylus TI perpendicularis,
cui immineat Solis extremus limbus O, nul
la fit vmbra, quia quamuis ab alio extremo S
veniat radius SN vltimus, & intrà &longs;patium
TN nullus alius ve
niat radius ab eodem
puncto S radios in or
bem diffundente, in
trà illud tamen &longs;pa
tium TN veniunt
radij à cæteris om
nibus punctis inter
S & O intermediis:
quare nec vmbra vl
la aut penumbra ob&longs;eruari pote&longs;t. At &longs;i &longs;ty
lus in LM fuerit, proiicit vmbram LP, in
trà quod &longs;patium nullus cadit radius dire
ctus á punctis O & S aut intermediis; penum
bta verò ex P ad B procedit, donec in B me
ra lux incipiat. Cum itaque &longs;patio 300 &longs;ta
diorum gnomones vmbram non proiicerent,
tium illud e&longs;t,
cui Sol ad per
pendiculum im
minet.
Terræ ma
gnitudinem
eiu&longs;que ad
Solem pro
portione&mtail;
inue&longs;tigare
ex &longs;patio,
quod ombris
earet.
Sed
&longs;upererat, quo
ta totius circuli
maximi pars e&longs;
&longs;ent &longs;tadia 300.
Con&longs;tituto ita
que terræ centro
in C, & oculo
ob&longs;eruatoris i&ntail;
O, ductum
telligebam
tro S radium S
OC, qui cu&mtail;
radio OT Solem
contingente da
bat angulum S
OT &longs;emidiametrum apparentem Solis Apo
gæi gr.o.m.15., vt placet Tychoni, Longo
montano, Magino, Keplero. Item ex C du
ctus intelligatur radius Solem tangens, qui
terræ circulum maximum &longs;ecat in K; & e&longs;t
OK &longs;emi&longs;sis &longs;patij, quod caret vmbris meri
dianis die &longs;ol&longs;titij æ&longs;tiui; continet autem &longs;ta-
rem e&longs;&longs;e angulo
ducta terminaretur in T, tunc angulus SOT
externus maior e&longs;&longs;et interno oppo&longs;ito OCT:
quia verò, quod veritati magis e&longs;t con&longs;enta
neum, radius ex C Solem tangens e&longs;t CI vl
tra T, angulus ISC maior e&longs;t angulo T
ac proinde, cum anguli ad T & I &longs;int recti,
reliquus SCI e&longs;t reliquo SOT minor. E&longs;t igi
tur &longs;patium OK minus quàm min.15. Qua
re vnus gradus complectitur plura quàm
600 &longs;tadia, ideoquè totus telluris ambitus
maior e&longs;t &longs;tadijs 216000.
Stadij Græ
si quanti
tas.
Hìc. autem hærebam in &longs;tadio, ne cum
Suida errarem; hic enim primùm Milio &longs;ta
dia 7 1/2 tribuit; &longs;ed po&longs;tmodum Milijs de
cem, æqualia facit &longs;tadia 80: Cùm verò &longs;ta
dium pedibus 600 definierit, Milium dein
de in pedes 4200 di&longs;tribuit: Vnde conficitur
Milium & 7, & 7 1/2, & 8 &longs;tadia comprehen
dere. Quarè acquie&longs;cendum potius duxian
tiquo Scholia&longs;ti Græco in propo&longs;. 5. lib d&etail;
Gæod. Heronis Mechanici, qui &longs;tadium fui&longs;
&longs;e pa&longs;&longs;uum 100, pa&longs;&longs;um cubitorum 4, cu
bitum verò digitorum 24 &longs;cribit. Cum ita
que pa&longs;&longs;us Romanus pedibus quinque con
&longs;taret, &longs;inguli verò pedes palmis 4, hoc e&longs;t
digitis 16, pa&longs;&longs;us
96, &longs;c.4.cubitos, &longs;eu, quod idem e&longs;t, pedes
&longs;ex Romanos. Stadia igitur &longs;ingula pedes
Rom. antiquos 600 continebant, hoc e&longs;t pa&longs;
&longs;us 120. Hinc fit
&longs;tadijs 216000, maiorem quoquè e&longs;&longs;e pa&longs;si
bus
Rom. 25920. Quarè etiam gradibus &longs;in
gulis milliaria Rom. antiqua plura quàm 72
re&longs;pondent.
At anguli OCK quantitatem quot
&longs;crupulis definiebas minorem angulo
min. 15?
Quæ&longs;iui primùm quoties Solis &longs;e
midiameter ST contineretur à lineâ SO: dato
autem SOT gr.o.m.15. reliquus e&longs;t TSO gr.
89.m.45.cuius Secans SO
52745, quarum Radius ST e&longs;t 100000.
00000. Deinde po&longs;itâ terræ &longs;emidiametro
CO Algebricè 1 ℞, e&longs;t tota CS 1 ℞ † 22918384
52745, cui ex Tychone æquantur &longs;emidia
metriterre&longs;tres, 1182, hoc e&longs;t 1182 ℞. Qua
re
& 22918384.52745: & in&longs;titutâ diui&longs;ion&etail;
prouenit pretium vnius Radic
91407. &longs;emidiameter terræ CO, quarum So
lis &longs;emidiameter e&longs;t 100000.00000. E&longs;t igi
tur tota SC, &longs;i CO inuenta addatur ip&longs;i SO,
gr.89.m.45.&longs;ec.o.Ter.43. Quar. 53. &c.
hoc e&longs;t anguli ISC: atque adeò eius comple
mentum SCI erit gr.o.m.14.&longs;ec.59. Ter.
16. Quar. 7. quem metitur arcus OK &longs;tad.
150. Fiat demùm vt arcus OK inuentus
&longs;crupulorum Quart. 3237367 ad &longs;tadia 150,
ita totius circuli ambitus &longs;crupulorum Quart.
4665600000. ad &longs;tadia 216175 (67/100), hoc e&longs;t,
neglectâ fractione, milliaria Rom. antiqu&atail;
25941. Quapropter &longs;ingulis gradibus mill.
Rom. 72. pa&longs;&longs;.58 re&longs;pondent.
Porrò inuentâ peripheriâ, &longs;i fiat vt 22 ad 7
ita mill. 25941. ad mill. 8253 (21/22), habetur
diameter proximè minor verâ; & &longs;i fiat vt
223 ad 71 ita mill. 25941. ad mill. 8259 (54/223),
diametrum verâ proximè maiorem reperi
mus: & electo medio Arithmetico rectè &longs;ta
tuitur terræ diameter mill. Rom. ant. 8255.
pa&longs;&longs; 644, ideòque &longs;emidiameter mill. 4127.
pa&longs;&longs;. 822. Quod &longs;i libeat exactiùs operari,
quoniam circuli diametro po&longs;itâ 1, perime
ter polygoni laterum 2560 circulo circum
&longs;cripti minor e&longs;t quàm 3 (14160/100000), perimeter ve
rò in&longs;eripti maior e&longs;t quàm 3 (14159/100000); fiat pri
mò vt 3. 14160. ad 1, ita mill. 25941 ad
mill.8257.pa&longs;&longs;.257, quæ e&longs;t diameter mi-
numbrâ ægrè &longs;ecernas) &longs;iuè ex eclip&longs;ibus in
ferantur. Quod &longs;i a&longs;&longs;umpta Lunæ parallaxis
à vero aliquantulum ab&longs;it, vel quia à radio
rum refractione turbetur, vel quia motuum
periodi non &longs;atis accuratè ad calculos reuo
centur, interuallorum pariter, quæ inde col
liguntur, veritas nutat. Ad hæc fides adhi
benda e&longs;t antiquis &longs;patium, quod meridianis
vmbris caret, dimetientibus, atque &longs;tadijs
300 definientibus; quæ tamen ad notas no
bis men&longs;uras vt reuocemus, certioribus ad
huc argumentis indigemus. Illud tame&ntail;
hìc minimè di&longs;simulandum videtur, quod
eruditis non paucis placere video: non eam
&longs;cilicet &longs;tadij men&longs;uram, quam ex Gtæco
Heronis &longs;cholia&longs;te a&longs;&longs;um p&longs;i&longs;ti, adhibendam
e&longs;&longs;e, &longs;ed eam potiùs, quæ ex ip&longs;ius Heronis
Mechanici I&longs;agoge colligitur dicentis,
l
nas centum viginti, Cubitos octingentos, Pe
des Alexandrinos Philetæreos mille ducentos,
Italicos autem pedes mille quadringentos qua
draginta. Quare Stadium non Romanis pe
dibus 600 conltat, &longs;ed Alexandrinis 600, hoc
e&longs;t Italicis, pedibus 720, &longs;eu pa&longs;&longs;ibus 144.
Neque enim coniecturâ &longs;atis probabili caret
Erato&longs;thenem Alexandrinæ Bibliothecæ præ-
& Epiphane, v&longs;um fui&longs;&longs;e men&longs;urâ Ægyptijs
tunc familiari, pede nimirum Alexandrino,
atquè adeò etiam &longs;tadio, quod ille metieba
tur. Hinc fit &longs;tadia illa 300 re&longs;pondere pa&longs;
&longs;ibus Romanis 43200: arcum verò OK e&longs;&longs;e
pa&longs;&longs; Rom. 21600. lgitur &longs;i fiat vt arcus OK
à te nuper inuentus &longs;crupulorum Quart.
3237367 ad pa&longs;&longs;. Rom. 21600, ita tota pe
ripheria &longs;crup. Quart. 4665600000. ad pa&longs;&longs;.
Rom. 31.129297, erit multo maior terræ
ambitus, quàm à te fuerit deprehen&longs;us.
Terræ am
bitus, & dia
meter.
Stadium A
lexandri
num.
Sed & con&longs;titutam à Tychone Solis di
&longs;tantiam, nimis breuibus terminis de&longs;ini
tam exi&longs;timant Recentiores A&longs;tronomi, So
lemque altiùs promouendum cen&longs;ent, vt &longs;i
bi inuicem phœnomena omnia ritè re&longs;pon
deant. Certè P. Io. Bapti&longs;ta Ricciolius no
&longs;træ Societatis, qui iam annos plures in A
&longs;tronomiæ in &longs;tauratione feliciter de&longs;udat,
Solem Apogæum à telluris centro remouet
&longs;emidiam. terr.
7580., & Solis Apogæi dia
metrum apparentem &longs;tatuit Tychonicâ ma
iorem, nimirùm gr.o.m.30.&longs;ec.30. Qua
rè &longs;i angulus SOT &longs;it gr.o.m.15.&longs;ec.15.,
complementum TSO e&longs;t gr.89.min.44.
&longs;ec.45., cuius Secans, SO 22551105.10158,
quarum partium Radius ST e&longs;t 100000. terre&longs;t.
7580, &
SO &longs;emid.terr. 7579: & Secans SO diuida
tur per 7579, vt Quotiens 2975.47237,
prodat quantitatem &longs;emidiametri OC i&ntail;
partibus Radij ST. Additâ igitur CO ip&longs;i
OS, erit tota SC 22554080.57395, quæ
e&longs;t Secans anguli ISC gr.89.m.44.&longs;ec.45.
Ter. 7. Quar. 17. Quint. 28 1/2, atquè adeo
complementum SCI gr.o.m.15.&longs;ec.14.
Ter. 52. Quar. 42. Quint. 31 1/2. Duplicetur
itaque angulus SCI, & in Scrupula Quinta
reducatur, & &longs;crupula Quinta 395227503
re&longs;pondent &longs;tadijs Alexandrinis 300, hoc e&longs;t
pa&longs;sibus Rom. 43200. Si igitur arcus OK
duplicatus e&longs;t pa&longs;&longs;. 43200, totus circuli am
bitus &longs;crup. Quint. 279936.000000.erit pa&longs;&longs;.
Rom. 30598162.
Probabilior
terræ ma
gnitudo ex
eadem me
tbodo inue
stigatur.
Solis dictan
ti am à ter
ra, & pro
portione&mtail;
inue&longs;tiga
re ex da
ta terræ ma
gnitudine,
Solis Apog.
diametro ap.
parenti, &
&longs;patio, quod
vmbris ca
ret.
Plurimam, opinor, inires gratiam
ab Eruditi&longs;&longs;. Domino 10. Bapt. Baliano Pa
tritio Genuen&longs;i, &longs;i ille no&longs;tro huic colloquio
intere&longs;&longs;et; ab eo enim accepi &longs;e certi&longs;simo
experimento didici&longs;&longs;e, telluris ambitum non
minorem e&longs;&longs;e triginta millibus milliarium.
Tunc verò non poteram illi acquie&longs;cer&etail;,
cum viderem indè colligi Solem terræ it&atail;
vicinum, vt ab illâ non abe&longs;&longs;et &longs;emidiam.
terr.25; id quod e&longs;t manife&longs;tè fal&longs;um. Tri
buebam autem &longs;ingulis &longs;tadiis pa&longs;&longs;us Rom.
o.m.28.&longs;ec. 35. Ter. 12., dato ambitu
mill. 30000: & arcus OK cum e&longs;&longs;et gr.o.
m. 14.&longs;ec.17. Ter. 36., angulus ISC gr.89.
m. 45. &longs;ec.42. Ter. 24. dabat Secantem SC
24052069.35. in partibus Radij SI 100000.
00. Po&longs;itâ verò Tychonicâ &longs;emidiametro
apparente Solis SOT gr.o.m.15., Secans
SO erat 22918384. 52. Quare &longs;i per Se
eantium SO, SC, differentiam OC 1133684.
83. diuidatur Secans SC, Quotiens 21 dabit
di&longs;tantiam Solis à terræ centro in &longs;emidia
metris terre&longs;tribus; id quod aperti&longs;simè fal
&longs;um e&longs;&longs;e omnibus con&longs;tat.
At quoniam vis, Guldine, &longs;tadia illa 300
e&longs;&longs;e pa&longs;&longs;us Romanos 43200, tentemus a&ntail;
illa terræ magnitudo milliaribns 30000 de
finita &longs;ub&longs;i&longs;tat. Si ergo circuli totius peri
pheria ritè &longs;uas in partes tribuatur, con&longs;tat
pa&longs;&longs;. 43200 re&longs;pondere gr. o.m.
31. &longs;ec. 6.
Ter. 14. Quar. 24. præcisè: & angulus SCI
gr. o. m.
15.&longs;ec.33. Ter.7. Quar. 12. habet
52. Quar. 48., cuius Secans SC 22110369.
79050, quarum Radius SI e&longs;t 100000.00000.
Porrò angu'us SOT &longs;emidiameter apparens
Solis Apogæi nece&longs;&longs;ariò maior &longs;tatui debet
angulo SCI con&longs;tituto, cum &longs;ub maiore an-
quàm ex puncto C remotiore. Quare a&longs;&longs;u
mi non pote&longs;t Solis Apogæi &longs;emidiameter
apparens, quam ex Ricciolio a&longs;&longs;umebas, gr.
o m. 15. &longs;ec. 15., neque Tychonica gr.o.
m.15. multò minùs Vendelinica gr.o.m.14.
&longs;ec.45: Eam igitur ex no&longs;tro Bullialdo de
&longs;umamus gr.o.m.16.&longs;ec.9., & complemen
ti Secans SO e&longs;t 21288752.30947. Ablata
itaque SO ex SC relinquit terræ &longs;emidiame
trum OC 821617.48103. in partibus Radij
SI 100000.00000: id quod fieri omnino non
pote&longs;t, ni&longs;i tellus multò maior &longs;it Sole, & hic
ab illâ remoueatur tantùm &longs;emidiametros
terre&longs;tres ferè 27: quæ omnia à vero longi&longs;
&longs;imè ab&longs;unt.
At fortè Balianus non ita telluris
ambitum tricies millenis milliaribus circum
&longs;cribit, vt nihil in&longs;uper additum velit, ne
què facilè cum Bullialdo tam magnam &longs;ta
tuet apparentem &longs;olis Apogæi &longs;emidiame
trum, hæc enim &longs;ibi inuicem cohærere non
po&longs;&longs;unt. Et (quoniam hìc ex amicoru&mtail;
placitis aliquid vterque depromimus) quod
&longs;pectat ad apparentem Solis Apogæi diame
trum P. Ricciolius, qui eam labore improbo
plu&longs;quam trig inta diuer&longs;is methodis quadrin
genties, & co amplius, ob&longs;eruando venatus
hi aliquando affirmabat, illam nec mino
rem minutis 30. nec maiorem min. 31.
Fac igitur à Baliano mediam inter hæc ex
trema Solis Apogæi diametrum apparen
tem gr.o.m.30.&longs;ec.30. a&longs;&longs;umi, & telluris
ambitum ita augeri, vt ad illum pa&longs;&longs;us 43200
minorem habeant Rationem: & &longs;ibi omnia
re&longs;pondebunt. Statuatur itaque terræ peri
pheria pa&longs;&longs;.30598162; & pa&longs;&longs;us Rom.43200
&longs;unt gr.o.m.30.&longs;ec.29. Ter.45 Quar. 25.
Quint.4., adeoquè angulus SCI gr.o.min.
15.&longs;ec. 14. Ter. 52: Quar. 42. Quint. 32.
cuius Complementi Secans SC 22554080.
57395 in partibus Radij Iam anguli SOT
gr.o.m.15.&longs;ec. 15. Secans Complem. SO
22551105.10158. auferatur ex SC; & per
earum differentiam OC 2975.47237 diui
sâ totâ SC, prodibit di&longs;tantia SC &longs;emid. terr.
7580, vt P. Ricciolio placet.
Quarè exiis, quæ hactenus di&longs;putauimus,
illud infero, quod auctâ Solis Apogæi diame
tro apparente, minui debet terræ ambitus,
vt ad eum pa&longs;&longs;us 43200 maiorem habeant
Rationem, ne videlicet tam magno inter &longs;e
di&longs;crimine differant anguli SOT &
per Secantium SO & SC, differentiam OC
diuisâ di&longs;tantiâ SC
explicanda. Hinc fit retentâ eadem terræ
magnitudine non ita augeri &longs;emidiametrum
apparentem
&longs;it gr.o.m.15.&longs;ec.25.; Secans enim
22306254.81750; quæ ablata ex inuentâ
&longs;uperiùs
differentiam OC 247825.75645.atque adeò terr.
91. à terrâ remoue
retur. Minuenda itaque e&longs;t vel Solis appa
rens diameter, vel terræ magnitudo; ego ve
rò illam potiùs paucioribus &longs;crupulis &longs;ecun
dis definiendam cen&longs;erem, quàm terræ am
bitum ad pauciora milliaria reuocandum.
S
quorum &longs;ubtilior inue&longs;tigatio à no&longs;tro in
&longs;tituto aliena deprehenditur. Nemo t&etail;,
Mer&longs;enne, reprehendat cæle&longs;tes hypothe&longs;es
ex magno illo Atlante Tychone a&longs;&longs;umen
tem; & quamuis ego pariter in eâ &longs;im &longs;en
tentiâ, vt exi&longs;timem ab Eratho&longs;tene homi
ne Cyrenen&longs;i adhibitum &longs;tadium Alexandri
num, non autem Græcum, quod a&longs;&longs;umebas,
hanc tamen litem hìc agitare non e&longs;t operæ
pretium. Quapropter perge, &longs;i placet, qua
cæpi&longs;ti viâ, terreni globi grauitatem exp
rare.
Inuentam terræ &longs;emidiametrum
pedes quinque Romanos antiquos pa&longs;&longs;ibus
&longs;ingulis tribuens, & &longs;unt pedes 20639110.
Quia verò &longs;phæræ &longs;unt in triplicatâ Ratione
&longs;emidiametrorum, comparo terrenam &longs;e
midiametrum cum &longs;emidiametro pedali al
terius globi, &
v&longs;que, ad quartum terminum continuo, vt
&longs;it tertius terminus 425.972861.592100.
quartus autem 8791.700747.414127.
031000.
ad terrenam &longs;phæram
dem, quam vnitas ad quartum hunc termi
num. Vt autem &longs;phæræ argillaceæ, cuius
&longs;emidiameter pedalis &longs;it, grauitatem per
&longs;pectam haberem, argillam cum aquâ con
tuli, & experimento didici argillæ grauita
tem ad aquæ pondus e&longs;&longs;e vt 27 ad 16. At
&longs;tanni grauitas ad aquæ grauitatem apud
Ghetaldum in Archimede Promoto e&longs;t vt
100 ad (13 19/37), hoc e&longs;t, vt 37 ad 5.
Ratio &longs;tanni ad aquam, & aquæ ad argillam,
&longs;cilicet 37 ad 5. & 16 ad 27 in tribus termi
nis continuetur, ita vt &longs;int 592,80. 135,
Ratio grauitatis &longs;tanni ad argillæ pondus, da
tâ molis æqualitate, e&longs;t vt 592 ad 135.
Terreni glo
bi &longs;oliditas
inue &longs;tiga
tur.
Quoniam verò apud eundem Ghetaldum
&longs;tanneæ &longs;phæræ, cuius diameter &longs;it vnius pe-
& &longs;phæræ &longs;unt in triplicatâ Ratione diame
tro rum, &longs;phæra &longs;tannea &longs;emidi ametrum ha
bens pedalem, ac proinde diametrum bipe
dalem, e&longs;t octuplo grauior illâ, atque adeò
lib, 2432. Atqui &longs;tannum ad argillam, e&longs;t
vt 592 2d 135, igitur & eandem Rationem
habent æquales &longs;phæræ; ideoque &longs;i &longs;tannea
&longs;phæra &longs;emidiametrum pedalem habens nu
merat in grauitate libras 2432, argillace&atail;
æqualis erit lib. (554 22/37). Hæc autem ad terræ
globum e&longs;t vt vnitas ad quartum illum ter
minum continuè proportionalem in Ratione
&longs;emidiametrorum. Multiplicetur igitur quar
tus ille terminus 8791.700747.414127.
031000. per libras (554 22/37), & prodibit totius
terræ grauitas, &longs;i ex merâ argillâ con&longs;taret,
librarum 4.875829.711809.132072.
(327567 21/37).
Terreni glo
bi grauitas.
Immen&longs;um places, Mer&longs;enne: lu
culenti&longs;simè enim demon&longs;tra&longs;ti quatuor illis
aut &longs;ex &longs;upra centum orbiculis, de quibus
heri nobis &longs;ermo erat, tellurem totam facil
limè moueri po&longs;&longs;e, etiam &longs;i eius grauitati
quingenties millies millena librarum mil
lia adiiceres.
Ita planè, &longs;i mera e&longs;&longs;et argilla: &longs;ed
tot marmora ac lapides, ingentesque metal-
go &longs;uperant interuallo, non eam vim ad
dunt ponderis, quæ non facilè æ&longs;timari
queat?
Immò verò, ni&longs;i hæc pariter ad
mi&longs;cerentur, vererer plurimùm, ne mihi
tanquam prodigo &longs;uccen&longs;eretis, qui tantam
telluri grauitatem concederem. Metall&atail;,
fateor, ac marmora æqualem argillæ molem
pondere &longs;uperant: &longs;ed quota demum &longs;phæ
ræ huius pars illa &longs;unt? Quod &longs;i quis ill&atail;
grauiora e&longs;&longs;e inculcet, longè leuiorem a
quam cogitet, ex qua vici&longs;sim globus hic
terraqueus con&longs;tat. Qui&longs;quis enim &longs;e æquum
rerum æ&longs;timatorem præbuerit, non plus re
periri
immo illorum grauitatem ab huius leuitate
&longs;i non æquari, aut &longs;uperari, magnâ &longs;altem
ex parte compen&longs;ari facilè concedat. Iam
verò &longs;i ad aëris vim non modicam terræ ca
uernis atque cuniculis inclu&longs;am, &longs;eque intrà
&longs;iccorum corporum particulas non &longs;ibi om
ninò cohærentes in&longs;inuantem, animum ad
uertamus, apparebit illicò exuperantiam il
lam grauitatis hoc defectu vberrimè com
pen&longs;ari.
Terræ gra
uitatem pe
rinde &longs;e ba
bere proba
tur, ac &longs;i e&longs;
&longs;et mera ar
gilla.
Sed quoniam motus, cui tellus &longs;uo pon
dere ob&longs;i&longs;teret, in circum&longs;u&longs;o hoc aëre per-
tum ab halituum aëre leuiorum copiâ penè
infinitâ, quæ totam hanc molem peruadit.
Quemadmodum enim nauis aërem aquâ
leuiorem includens in aquâ non mergitur, &longs;i
tota moles compo&longs;ita æqualis aquæ grauita
tem non vincat; vel &longs;altem minore mo
mento de&longs;cendit pro inclu&longs;i aëris portione;
haud ab&longs;imili ratione fieri pote&longs;t, vt grauiori
corpori tot &longs;piritus aëre leuiores permi&longs;cean
tur, vt totius compo&longs;itæ molis grauitas non
mediocriter minuatur. Quantum verò hu
iu&longs;modi halituum metallicis lapid
mi&longs;ceatur, &longs;atis &longs;ciunt, quotquot &longs;odinarum
latebras penetrârunt. Et vt cæteras mi&longs;&longs;as
faciam, Hungaricæ aurifodinæ, mihi ma
gis notæ, omnem præcidunt dubitandi an
&longs;am. Cum enim duæ pateant ad de&longs;cen
dendum viæ, altera breui&longs;sima, vtpote re
cta, putei in modum (caminum rectiùs
forta&longs;&longs;e dixeris) cuius latera muniunt arctè
compacti arborum trunci: altera obliqu&atail;,
& longior per cuniculos: per hanc perpetuò
&longs;e frigidus aër magno impetu in ima fodinæ
vi&longs;cera in&longs;inuat, dum ex illâ calida pariter
atque graueolens expiratio erumpit. Ne
que indigent metallarii A&longs;trologorum næ
nijs, qui imminentem cæli mutationem
infernè a&longs;cendentium copiâ, quibus vix non
præfocantur, cœlum nubibus proximè ob
ducendum nec dubiè pronuncient. Quid?
quod aqua ip&longs;a (licet aëre grauior, &longs;i &longs;ibi ip&longs;a
relinquatur) leui&longs;simi vaporis &longs;peciem induit
calore &longs;ollicitata, quo interior terræ plag&atail;
abundat: Vbienim octoginta circiter hexa
podas, quibus extima hæc terræ regio frigi
di&longs;sima definitur, de&longs;cenderis, tepor primùm
grati&longs;simus ex frigidâ regione venientem ex
cipit ac recreat, paulatimque adeò augetur
calor, vt demùm metalli fo&longs;&longs;ores ve&longs;tem nul
lam ferant.
Quantum autem momenti ad pondus mi
nuendum obtineat grauium hæc atque le
uium mi&longs;cella, ille facilè intelliget, qui ob
&longs;eruauerit aliquando &longs;tibium ad v&longs;us medi
cos excoctum æquè graue reperiri, ac de
prehen&longs;um fui&longs;&longs;et, antequam igni commit
teretur; quamuis inde vi flammæ ingens va
porum ac fumi copia eruperit; &longs;piritus enim,
qui auolârunt, cum aërem leuitate vince
rent, cæteris partibus admi&longs;ti molem con
&longs;tituebant maiorem quidem, &longs;ed non grauio
rem, ac &longs;it deinde reliqua moles minor, fa
ctâ hac halituum &longs;ece&longs;sione. Perinde atque
&longs;i vas æreum aquâ iuxta ac aëre plenum in-
ra comprimantur, vt aquam omnem ac
rem
ris deteratur, adhuc æ qualibus momentis in
aquâ grauitare deprehenditur, &longs;i ad libræ
examen reuocetur; aeris nimirùm leuitas
æris momenta minuebat.
Ne quis verò ductam ex va&longs;e &longs;imilitudi
nem calumnietur; Liberum patrem coga
mus pauli&longs;per philo&longs;ophari. Dabitis, opinor
vini grauitatem ferè aquæ &longs;ub&longs;e&longs;qui-&longs;exage
cuplam, vel certè aquæ pondere non maio
rem; ita vt quodcunque aquæ grauitate præ
&longs;titerit, haud immeritò vino pariter grauius
cen&longs;eatur. Atqui Tartarum ex vino &longs;ub&longs;ide
re nemo ne&longs;cit; & quod inde elicitur oleum
Tartari, e&longs;t proximè ad aquam vt 3 ad 2: &longs;pi
titus autem vini ad aquam communem, Ra
tionem habet proximè, quam 3 ad 4. Qua
re &longs;i vinum eiu&longs;dem cum aquâ grauitatis
con&longs;tituatur, Oleum Tartari ad vinum e&longs;t
vt 6 ad 4, vinum autem ad &longs;piritum vini vt
4 ad 3.
Vini graui
tas medi&atail;
Harmonicè
inter Oleum
Tartari, &
Spiritu&mtail;
vini.
Quod &longs;i vini Tartarum &longs;imul ac &longs;piritum
complectentis grauitas medio loco &longs;e habet
(& quidem medietate Harmonicâ) inter id
quod grauius, & id quod leuius e&longs;t, quamuis
extrema illa non ita multo di&longs;criminentur in-
ui&longs;simis quidem metallis atque lapidibus,
&longs;ed & ex corporibus alijs argillâ leuioribus,
ex aquâ,
bus coagmentata mediocrem argillæ graui
tatem (quantum fas e&longs;t coniectutâ a&longs;&longs;equi)
obtinere dicatur?
Fallor, &longs;i Tartarus ip&longs;e ad exte
nuandam terræ grauitatem tibi &longs;uppetias
non tulerit. Ob oculos pones immen&longs;a &longs;pe
læa æternis flammarum globis redundantia,
& exaggeratâ longè latèque patentis impio
rum carceris magnitudine (quam penetra
bilis ignis implet, non modò nihil habens
proni, &longs;ed & &longs;ur&longs;um rectis lineis in cœle&longs;tem
locum &longs;ubuolare contendens) tantum ex tel
lure detrahes ponderis, quantum metall&atail;
omnia atque marmora re&longs;tituere non va
leant. Nec deerit docti&longs;simorum virorum
&longs;uffragium, qui cauernam hanc totius ter
reni globi partem quartam præci&longs;i&longs;&longs;imè &longs;ta
tuunt, cum qua pariter admirabilem illam
Lunaris motûs librationem con&longs;entire inge
niosè opinantur.
Et verò talia afferentem ca&longs;tigare
quis audeat? Ab&longs;tineo tamen, ne fortè ex
vobis audire cogerer iterum, quæ haud ita
pridem ne&longs;cio quis di&longs;&longs;erebat. Nihil e&longs;t,
aiebat ille, quod vim pati perpetuam cen
&longs;endum &longs;it. At &longs;i elementis omnibus terra
grauitare, ignis leuitate præ&longs;tet; an non. æ
ternum aberunt loco, quem &longs;ingulis naturâ
tributum e&longs;t vt velint con&longs;equi? Terra ete
nim, ni&longs;i columnis ba&longs;im in centro haben
tibus nixa fingatur, tota procul à centro con
quie&longs;cit; ignisiverò infimum locum tenet.
Qui&longs;quis autem ibi ignem à Deo perpetuis
vinculis eo tantùm con&longs;ilio coerceri exi&longs;ti
mat, vt &longs;celeratorum carnificinam exerce
diligenter per&longs;piciat velim, an non magis
pro Diuinæ &longs;apientiæ atque omnipotentiæ
dignitate locuturus ille &longs;it, qui, quemadmo
dum Iridem licèt naturæ penicillo in nubi
bus de&longs;criptam, in &longs;empiterni tamen fœde
ris te&longs;&longs;eram à Deo a&longs;&longs;umptam nouit, &longs;ic re
rum naturæ con&longs;entaneum ad&longs;truat hunc or
dinem, quo vniuer&longs;itatis rerum &longs;ubluna
rium elementa pro &longs;ui ponderis ratione ita
di&longs;ponantur, vt centrum omnium graui&longs;si
mus ignis impiorum carcer atque carnifex
con&longs;titutus obtineat, illumque minùs gra
uia elementa terra & aqua deinceps con&longs;e
quantur, v&longs;que eò dum leui&longs;simus aër cir
cumfu&longs;us reliqua complectatur.
Coniecturæ
pro adstru
enda ignis
inferni gra
uitate.
Fru&longs;tra a&longs;cendentem flammam aperti&longs;si
mum igneæ leuitatis argumentum obiicie-Nàm ille &longs;upremo ætheri terrenis ex
pirationibus immuni, ac lunari orbitæ fini
timo, quem Ari&longs;totele te&longs;te propter con&longs;ue
tudinem ignem dicimus, vix aliquid ignis
præter &longs;plendidum nomen reliquum facie
bat. A&longs;cendit in flammâ (&longs;ic ille) humido
vapori, quem calor eximius rarum fecit,
permi&longs;ta fuligo, aére &longs;anè non leuior: quidni
pariter à flammâ in cœlum &longs;ubuolante par
ticulæ igneæ abripiantur? quibus vel pru
narum vel candentis ferri incolis, nullum e&longs;t
cum æthereâ regione commercium. A&ntail;
fumeus vapor igniculorum coloniam in cœ
lum deduxi&longs;&longs;e cen&longs;endus e&longs;t, quia illi nobis
non videntibus in terram relabuntur? Sed
quis neget, incendio &longs;yluam depopulante,
immodicam fuliginum copiam in &longs;uperiora
rapi? quibus tamen ætheream &longs;edem ineptè
ad&longs;eriberemus, quia illas iterum de&longs;cenden
tes ob&longs;eruando notare non po&longs;&longs;umus. Quod
&longs;i ignibus &longs;ub dio &longs;emper excitatis, fuligo
nulla camino vnquam adhæ&longs;i&longs;&longs;et; nonnè &longs;a
tis habui&longs;&longs;emus argumenti, ex quo illius gra
uitas innotui&longs;&longs;et, &longs;i quando ingentem fuligi
nis ma&longs;&longs;am ex aëre decidentem licui&longs;&longs;et inNeque enim ideò grauitate &longs;poliamus
terre&longs;tres pului&longs;culi atomos, quia in aëre va
gantes ita &longs;en&longs;um di&longs;sipatæ fugiunt, vt eas
ad earum grauitatem ad&longs;truendam &longs;atis e&longs;&longs;e
putamus, quod particulæ illæ vinculum na
ctæ, quo lapidem con&longs;tituant, de&longs;cendentes
&longs;ub a&longs;pectum cadant. Cur igitur tanta&mtail;
ignium vim in fulmine delap&longs;am, &longs;eque in
ima terræ vi&longs;cera in&longs;inuantem cernentes,
perinde atque &longs;axum in aquam decidens
mergitur, igneæ grauitatis &longs;u&longs;picioni locum
non damus? Nemo &longs;iquidem facilè credat
accen&longs;um fulmen á &longs;uperioribus nubis parti
bus, quæ inferioribus tenuiores &longs;unt ac leuio
res, minùsque ad aqueam naturam vergen
tes, deor&longs;um reflecti: cùm nec ignis ex Auro,
quod ob impetûs ac efficaciæ &longs;imilitudinem
nomen à fulmine obtinuit, à quoquam re
flectatur, &longs;ed in&longs;itâ naturæ vi præceps deor
&longs;um feratur:
Ad hæc ignis genus vniuer&longs;um in &longs;pecies
certas, quibus iterum partes aliæ atquè aliæ
&longs;ube&longs;&longs;ent, partiebatur & di&longs;tinguebat: Alti
enim reperiuntur Ignes luce iuxta atque ca
lote con&longs;picui, quorum frequenti&longs;simus e&longs;t
& communis v&longs;us, atque vulgati&longs;sima noti
tia: Alii &longs;unt, qui, quoniam plus fulgoris
habentes, quàm caloris, o
diuntur, cum Fatuis Ignibus numerantur:
Alios demùm qua&longs;i latentes &longs;inu natura fo-
vulgus appellet, nimis tamen viuaci virtute
præditos vis cau&longs;tica &longs;atis prodit. Ad ter
tium hoc ignium genus, qui calidi cum &longs;int,
non tamen &longs;unt lucidi, reuocabat igniculos,
qui aquis acribus (Aquas Fortes vulgus ap
pellat) permi&longs;ti ac oleo Tartari &longs;eu Vitrioli
metalla &longs;oluunt, & in Chymicam calce&mtail;
redigunt breui temporis morâ, cum tamen
plurium dierum interuallo indigeat, qui&longs;
quis ea voluerit ardenti&longs;simæ fornacis igne in
calcem excoquere. Porrò aquas huiu&longs;modi
acres ex &longs;alibus extillari, &longs;ales autem graui
tate præditos non mediocri ob&longs;eruans illud
demum inferebat, quod ignes omnium ve
hementi
to a&longs;&longs;equi po&longs;&longs;umus, cum grauitate &longs;unt con
iuncti, vt in Aquis Fortibus, Oleo Tartari,
&. Vitrioli, in Auro pulueris pyrii &longs;peciem
nacto, & in Fulmine licet pa&longs;sim experiri.
Hæc ferè &longs;unt, &longs;ed paulò pre&longs;siùs atqu&etail;
&longs;ummatim expo&longs;ita, ex quibus ille conficie
bat ignium generi lati&longs;simè patenti &longs;ube&longs;&longs;&etail;
&longs;pecies qua&longs;dam graui&longs;simorum corporum
eximiâ vrendi facultate præditorum, quæ
mundanæ &longs;phæræ centrum meritò teneant,
ibique perpetuam efficiant noctem, ni&longs;i for
tè &longs;ublu&longs;trem, quantum &longs;atis &longs;it ad impio-
mitatum a&longs;pectu augendos. Quare Tarta
rum ille concipiebat qua&longs;i immen&longs;as Ther
mas ac balneas efficaci&longs;&longs;imis omnium cau
&longs;ticorum particulis plenas; quæ proptere&atail;
lacus & &longs;tagnum ignis, ob quietem &longs;cilicet,
à Diuinis literis dicuntur. Cumque plur&atail;
ijs quidem, quæ à Diuinis literis docemur,
con&longs;ona, &longs;ed præter eorum, qui aderant, opi
nionem, de ignis i&longs;tiu&longs;modi naturâ di&longs;pu
ra&longs;&longs;et; ne quid temerè prolatum videretur,
&longs;ermonem claudens Lactantij l. 7. diu. in&longs;t.
cap.
21. authoritate firmauit, vbi de igne
illo &longs;empiterno impiorum corpora crucian
te loquitur,
e&longs;t ab hoc no&longs;tro, quo ad vitæ nece&longs;&longs;aria vii
mur, qui ni&longs;i alicuius materiæ fomite alatur,
extinguitur. At ille Diuinus per &longs;e ip&longs;um
&longs;emper viuit ac viget &longs;ine vllis alimentis, nec
admi&longs;tum habet fumum, &longs;ed e&longs;t purus ac li
quidus, & in aquæ modum fluidus: non enim
vi aliqua &longs;ur&longs;um ver&longs;us vrgetur, &longs;icut no&longs;ter,
quem labes terreni corporis, quo tenetur, & fu
mus intermi&longs;tus exibire cogit &c. Cum ita
que &longs;u&longs;picarer, ne quis ve&longs;trum hæc eadem
obiiceret, &longs;i fortè ad extenuandum telluris
pondus ex inferorum ignibus argument&atail;
de&longs;ump&longs;i&longs;&longs;em, &longs;atius du xi ab&longs;tinere, ne co-
Haud ego &longs;anè ineptam dixeri&mtail;
philo&longs;ophiam illam, quæ rerum naturam
per experimenta ve&longs;tigat: &longs;ed nec aliquid te
merè pronunciandum, quo vetus opinio
tot &longs;apientum authoritate firmata de po&longs;&longs;e&longs;
&longs;ione deiiciatur. Quamuis autem &longs;ententia
hæc videatur adhuc enucleatiùs explicanda,
vt igni &longs;ummam grauitatem ad&longs;truat; non
is tamen ego &longs;um, cui &longs;tomachum moueant
quæcunque præter opinionem audire con
tigerit. Nec dubito quin, &longs;i rem penitiùs
intro&longs;picere vellemus, plura occurrerent à
no&longs;træ di&longs;putationis in&longs;tituto non alien&atail;.
Sed quoniam non placet his diutiùs immo
rari, ea commodiorem in locum reiiciamus.
Tibi interim datur, Mer&longs;enne, terreni glo
bi grauitatem, perpen&longs;is omnibus, haud
multum abe&longs;&longs;e ab eâ grauitate, quam æqua
lis moles argillacea obtineret. Cum verò
res tota ex telluris magnitudine pendere vi
deatur, ne quis &longs;upere&longs;&longs;et dubitandi locus,
opus e&longs;&longs;et Geometricè per&longs;pectam haber&etail;
telluris magnitudinem. Quare &longs;i quid ha
bes, Guldine, quo nos po&longs;sis, methodo non
adeò operosâ, in huius problematis cogni
tionem deducere, in medium proferre n&etail;
grau
Vnum præterea, quod non parùm
in rem tuam facit, Galilæe, audire placeat.
Illud autem e&longs;t, quod quamuis terram decu
plò grauiorem quis con&longs;titueret, ac ego de
prehenderim, nihilo tamen minùs tuis illis
orbiculis moueri facilè po&longs;&longs;et: non tantum
quia decuplex i&longs;ta grauitas non pertingeret
ad libras illas 50.000000.000000.000000.
000000. & eo amplius, verùm etiam qui&atail;
non totum illud pondus &longs;imul motui repu
gnaret. Sit eni&mtail;
totus terræ globus
HIKL, cuius cen
trum C congruat v
niuer&longs;i centro, i&ntail;
quo nullum habet
momentum ad de
&longs;cen&longs;um, &longs;ed in eo
quie&longs;cit. An non &longs;a
tis Archimedi fui&longs;
&longs;et, &longs;iterram vnum
aut alterum milliare aliò tran&longs;tuli&longs;&longs;et? Mo
ueri igitur intelligatur centrum ex C in T,
& &longs;it terra translata SORV. Huic motui
primùm, præter hemi&longs;phærium &longs;uperius
HLK, non repugnat totum hemi&longs;phærium
inferius HIK, cuius videlicet partes plurimæ
&longs;iunt centro C propiores, in quod &longs;uis nuti-Ponatur enim CT &longs;emidia
metri pars mille&longs;ima, hoc e&longs;t mill. 4. & eo
amplius; &longs;egmentorum MON & MVN Ra
tio inuenietur, ex ijs quæ Archimedes docet
l. 2. de &longs;ph. & eyl.
prop.
2. &longs;unt enim &longs;eg
menta illa æqualia conis eandem ba&longs;im MN
habentibus, hi autem ex Eucl. l. 12. prop.
14 inter &longs;e &longs;unt vt altitudines: Quapropter
inuentis conorum altitudinibus, quas Ar
chimedes docet, innote&longs;cit Ratio &longs;egmen
torum &longs;phæricorum conis illis æqualiu&mtail;.
Cum itaque CT &longs;it Radij (1/1000), &longs;egmenti
MVN altitudo e&longs;t 999, &longs;egmenti verò
MON altitudo e&longs;t 1001. Fiat igitur vt alti
tudo &longs;egmenti maioris 1001 ad &longs;ummam
ex eadem altitudine & Radio 2001, ita al
titudo &longs;egmenti &longs;phærici minoris 999, ad
coni æqualis altitudinem 1997. Similiter vt
altitudo minoris &longs;egmenti 999 ad &longs;ummam
ex eadem & Radio 1999, ita altitudo &longs;eg
menti &longs;phærici maioris 1001 ad coni æqua
lis altitudinem 2003. E&longs;t igitur MON ad
MVN vt 2003 ad 1997: atque adeò &longs;eg
mentum maius addit vltra hemi&longs;phærium
&longs;olùm totius globi (6/4000). Quare vt tellus ve
niret in T, &longs;olum &longs;egmentum &longs;phæricu&mtail;
MSORN vt &longs;ummum deor&longs;um versùs C
vrgeret; cui tamen, ne po&longs;tea de&longs;cende
deberet à centro remoueri illo de&longs;cendente:
ac proinde quamuis in motu &longs;emper auge
retur difficultas mouendi, nunquam tamen
tota grauitas, perciperetur, ni&longs;i quando I ve
ni&longs;&longs;et in C; tunc enim tota &longs;phæra deor&longs;um
niteretur. Præterea &longs;egmentum illud MON
non eadem obtineret ad de&longs;
menta in tantâ centri vicinitate, ac valdè
procul à centro: neque enim ex hoc, quòd
experimentis euincere non po&longs;&longs;imus grauia
centro propiora minùs vrgere deor&longs;u&mtail;
quàm remotiora (cum illa, quæ in experi
mentum a&longs;&longs;umuntur, non ea e&longs;&longs;e queant,
quæ di&longs;crimen inferant &longs;en&longs;u perceptibil&etail;)
repugnandum e&longs;t rationi mani&longs;e&longs;tæ id &longs;ua
denti, vbi & ponderis amplitudo & inter
ualli differentia in&longs;ignis e&longs;t atque con&longs;pi
cu&atail;.
Telluris
grauitas
tota re &longs;i &longs;te
ret Archi
medi traben
Hinc fit 24 axibus in peritrochio aut tym
panis dentatis, quorum partes e&longs;&longs;ent in Ra
tione decuplâ, fieri po&longs;&longs;e, vt virtute tres li
bras mouere valente terra per aliquod &longs;pa
tium moueretur; illa enim mouere po&longs;&longs;et
lib 3000000.000000.000000.000000; qui
numerus excedit &longs;emi&longs;&longs;em ponderis totius
terræ; ac proinde eou&longs;que moueret, du&mtail;
&longs;egmentum MON æquale e&longs;&longs;et toti ponderi,
hibitâ eadem machinâ, habitâ tamen ratio
ne &longs;egmenti MVN ex parte &longs;u&longs;tentantis &longs;eg
mentum &longs;uperius, ne tanto impetu deor&longs;um
vrgeat, quanto vrgeret, &longs;i &longs;egmentum MVN
non ade&longs;&longs;et. At verò compo&longs;itis tantùm 17
helicibus cum tympanis denticulos 25 ha
bentibus, & vltimo tympano ad axem, cui
funis ductarius circumuoluitur, Rationem
quintuplam habente, potentia vnius libræ
&longs;u&longs;tentatiua applicata manubrio æqualis lon
gitudinis cum tympanorum &longs;emidiametro,
po&longs;&longs;et mouere terram ex C in T v&longs;que eò
dum &longs;egmentum MON e&longs;&longs;et lib 2.910383.
045673.370361.328125. qui pariter libra
rum numerus excedit totius terrenæ graui
tatis &longs;emi&longs;&longs;em: at potentia decem libras lo
co transferre valens, decuplum pondus mo
uere po&longs;&longs;et, atque adeò totum globum ele
uare. Sed iam Guldinum audiamus, vt cer
tam telluris magnitudinem &longs;tatuamus.
Definitur
machin&atail;,
qua potui&longs;
&longs;et tellus mo
ueri ab Ar
chimede.
Fieri non pote&longs;t, vt paucis me ex
pediam, quia non vna tantùm, aut alter&atail;
&longs;uppetit methodus, &longs;ed plures aliquando ex
cogitaui, cum hac &longs;uper re animum diligen
tiùs aduerterem vehementer admirans ab
antiquis nihil ad nos veni&longs;&longs;e, in quo animus
Gemetricus po&longs;&longs;et conquie&longs;cere. Neque fa-
dus arrideat, cum pro diuersâ locorum op
portunitate aliâ atque aliâ methodo vti opor
teat. Quarè con&longs;ultius forta&longs;&longs;e fuerit, vel
omnes pariter &longs;ilentio obuoluere, vel in aliam
di&longs;&longs;ertationem reijcere. Quod &longs;i, quam he
ri Mer&longs;ennus temporis dilationem ad am
pliores numeros in pagellâ priuatim de&longs;cri
bendos impetrauit, mihi non denegetis, tæ
dio ve&longs;tro parcetur, & meo labori.
Rationi con&longs;entanea &longs;unt, quæ po
&longs;tulas; neque æquum e&longs;t præproperæ curio
&longs;itati ob&longs;ecundantes multiplicis methodi co
gnitione fraudari.
TERTIA
terræ quantitatem
proponit.
HAVD &longs;atis &longs;cio, an po&longs;
&longs;im aliquid proferr&etail;,
quod ve&longs;træ expectatio
ni faciat &longs;atis in eâ, quam
nobis hodiè examinan
dam &longs;tatuimus quæ&longs;tio
nem de terraquei huius
globi magnitudine inue&longs;tigandâ: vosidcircò
niam dabitis, &longs;i quid afferre contigerit ita
planum ac facilè, vt ve&longs;trorum ingeniorum
&longs;ublimitati non re&longs;pondeat. Nihil habe
certi quod &longs;tatuam de terræ quantitate, ne
que enim hactenus otium fuit, quæ conce
peram, in praxim deducere. Fieri autem
po&longs;&longs;e exi&longs;timo, vt operâ non longâ propo
&longs;itum &longs;copum a&longs;&longs;equamur: duo nimiru&mtail;
præcogno&longs;ci oportet, quibus notis ac certis
terrenam diametrum po&longs;&longs;umus inue&longs;tigar&etail;.
Primum e&longs;t altitudo oculi &longs;upra maris i&ntail;
immen&longs;um patentis &longs;uperficiem, quæ non
æquabili planitie explicatur, &longs;ed &longs;phæra&mtail;
æmulatur: hanc verò altitudinem certi&longs;&longs;i
mè nobis innote&longs;cere po&longs;&longs;e quis neget? cum
eam funiculo vel phy&longs;icâ aliâ men&longs;urâ &longs;æ
piùs dimetiri liceat. Alterum, quod in hoc
negotio requiritur, e&longs;t angulus, quem cum
lineâ perpendiculi ad terræ centrum ductâ
con&longs;tituit opticus radius extremum hori
zontem allambens: hunc &longs;i organo ad id
affabrè elaborato ob&longs;eruaueris, &longs;ereno c
tranquillo mari, cum minima refractionis
&longs;u&longs;picio &longs;ube&longs;&longs;e pote&longs;t, quid certius requi
ras? cum organo eidem in dimetiendis &longs;y
derum altitudinibus aut di&longs;tantijs acquie
&longs;cas. His autem duobus cognitis terræ quo-
mon&longs;tro.
Sit enim, exempli gratiâ, nota altitudo
BA Phari Genuen&longs;is (laternam vocant) cum
&longs;copulo palmorum Genuen&longs;ium 440, hoc
e&longs;t pedum Rom. ant. 480, ob&longs;eruatusque
angulus BAD &longs;it gr.89.m.36.&longs;ec.33. Quo
niam ergo linea AD circulum tangit in D,
angulus CDA e&longs;t rectus. Ex B ducta intel
ligatur BE parallela ip&longs;i CD; e&longs;tque triangu
lum ABE rectangulum ad E, cuius hypo
thenu&longs;a AB, & angulus adiacens innotue
runt. Quare fiat vt Radius 100000. 00000.
ad &longs;inum gr.89.m.36.&longs;ec.33.hoc e&longs;t 99997.
67348, ita AB ped. 480. ad BE ped. (479
98883.27040/100000.00000). Porrò triangula ABE, ACD æqui
angula &longs;unt propter linearum BE, CD pa
ralleli&longs;mum, adeoque & &longs;imilia, ac vt AB
ad BE, ita AC ad CD. Ponatur autem BC
vna Radix Algebricè. Igitur vt AB ped.480
ad BE ped. (479 98883.27040/10000000000), ita AC ped. 1 ℞ +
480 ad CD ped (47998883.27040/48000000.00000) ℞ + (479 47463969.79200/48000000.00000).
E&longs;t autem CD æqualis ip&longs;i CB 1 ℞. Quare
factâ Antithe&longs;i iuxta Algebræ regulas, ma
net æquatio inter hos terminos (111672960/48000000.00000) ℞ &
pedes (479 47463969.79200/48000000.00000). In&longs;titutâ denique di
ui&longs;ione prodir quotiens ped. 20631193.,
CD. E&longs;t ergo inuen
ta terræ &longs;emidiameter
mill. Rom. ant. 4126,
pa&longs;&longs;. 238. po&longs;ito angu
lo ad A gr.89.m.36.
&longs;ec 33. præcisè.
Prima me
tbodus in
ueniendi tel
luris &longs;emi
diametrum,
per Trigo
nometriam
& Alge
bram.
Quod &longs;i rem breuiùs
a&longs;&longs;equi velis, dato an
gulo ob&longs;eruato in A gr.
89.m.36.&longs;ec.33, &longs;umatur AC vt Radius, &
CD, vt &longs;inus dati anguli. Et a&longs;&longs;umptâ BC
1 ℞ fiat vt Radius 100000.00000. ad dati an
guli &longs;inum 99997.67348, ita AC ped. 1 ℞
+ 480 ad CD ped. (99997.67348℞+4799888327040/100000.00000). E&longs;t
CD ip&longs;i CB 1 ℞ æqualis, atque adeò vtrinque
demptâ &longs;ractione (99997.67348/100000.00000) ℞, remanet æqua
tio inter (232652/100000.00000) ℞ & ped. (4799888327040/100000.00000). Qua
rè factâ diui&longs;ione habetur pretium 1 ℞ ped.
20631193 vt priùs.
Idem aliter,
& breuiùs.
Methodus hæc plana e&longs;t atque fa
cilis cuiuis vel leuiter Analyticâ &longs;cientiâ a&longs;
per&longs;o: &longs;ed aliquos forta&longs;&longs;e deterreret, qui vel
ip&longs;um. Algebræ nomen horrent, quamuis
Mathematici audire velint. Alios angulo
rum ob&longs;eruatio, & linearum circulo ad&longs;cri
ptarum vel in&longs;eriptarum inue&longs;tigatio ex Ca-
A&longs;t ronomicas fractiones deducta exigat par
tis proportionalis inqui&longs;itionem.
Satis e&longs;t, &longs;i peritis Geometris hac
in re fiat &longs;atis. Verùm adhuc ab&longs;que Cano
ne Trigonometrico res tota perfici po&longs;&longs;et,
&longs;ed non &longs;ine Algebrâ. Habeatur enim qua
dratum Geometricum
vel etiam &longs;it rectangulum,
cuius latus AS maius &longs;it late
re AH, id quod aliquando
commodius accidet. Obuer
tatur latus AH ita, vt con
gruat radio optico terra&mtail;
tangenti AD, in &longs;uperior&etail;
figurâ: eritque triangulum
ABE &longs;imile triangulo OAS: Nam anguli
SAH & BEA &longs;unt recti, ac proinde lineæ
SA, BE parallelæ, intrà quas anguli alterni
SAO, EBA &longs;unt æquales; &longs;icut & alterni
EAB, SOA, intra parallelas SO, AE. Igi
tur vt OA ad AS, ita AB ad BE: &longs;ed vt AB
ad BE, ita in &longs;uperiore figurâ AC ad CD;
ergo vt OA ad AS, ita AC ad CD. Quare
&longs;i latus AS Quadrati vel rectanguli notum &longs;it
in particulis quibuslibet, facilè innote&longs;cet,
quot huiu&longs;modi particulas contineat AO.
Cum verò altitudo AB nota &longs;it ex.gr.ped.
læ OA ad particulas AS, ita ped. 1 ℞ + 480
ad aliud, & proueniet quartus terminus CD
æqualis ip&longs;i CB 1 ℞: atque adeò factâ Anti
the&longs;i, & in&longs;titutâ diui&longs;ione habetur quanti
tas ip&longs;ius CD.
Idem aliter
&longs;ine Trigo
nometria.
Quid?
quod &longs;inè Algebiâ res tam facilis
e&longs;t, vt penè non audeam dicere, ne vulgata
vile&longs;cat. Fiat vt OI, differentia inter OA &
AS, ad ip&longs;am AS, ita nota altitudo AB ad
aliud, & proueniet CD quæ&longs;ita terræ &longs;emi
diameter. E&longs;t nimirum AB differentia inter
AC & CD, & vt tota AC ad totam AO, ita
ablata CB ad ablatam AI; igitur & reliqua
AB ad reliquam IO vt tota AC ad totam
AO: ergo permutando, & diuidendo, vt
AB ad BC, hoc e&longs;t CD, ita OI ad IA, hoc
e&longs;t AS.
Idem &longs;ine
Trigonome
tria & &longs;ine
Algebra.
Et hæc quidem dicta &longs;int, &longs;i fortè angu
lum CAD præcogno&longs;cere non libeat. Cæ
terùm illo cognito rem aggredi licet &longs;ine
Algebrâ ope &longs;oliûs Trigonometriæ, quæ
multiplicem &longs;ubmini&longs;trare poterit metho
dum. Et primò datâ altitudine AB, & an
gulo ad A ob&longs;eruato ex. gr.gr.89.m.36.&longs;ec.
33., notus e&longs;t angulus ad centrum C gr.o.
m.23.&longs;ec.27, ac proinde eius &longs;ubten&longs;a i&ntail;
partibus Radij innote&longs;cit. Igitur ductâ BD,
æquales recto CDA vnâ cum angulo ob&longs;er
uato A. Cum verò triangulum CBD &longs;it
&longs;celes, vnu&longs;qui&longs;que an
gulorum ad ba&longs;im BD
e&longs;t æqualis &longs;emirecto &
&longs;emiangulo ob&longs;eruato.
Deme ergo &longs;emiangu
lum ob&longs;eruatum gr.44.
48. 16 1/2, ex angulo &longs;e
mirecto, hoc
45, remanet gr.o.m.11.
fec. 43 1/2 quantitas angu
li ADB: Hic autem angulus ADB &longs;emper
e&longs;t femi&longs;fis complementi anguli ad A ob&longs;er
uati; angulo enim ADB facto ad punctum
contactûs e&longs;t æqualis angulus in &longs;egmento
alterno, anguli verò ad peripheriam duplex
e&longs;t angulus ad centrum C. In triangulo ita
que ADB noti &longs;unt duo anguli ad A & ad D;
& latus BD notum e&longs;t in partibus 682.
13152.quarum Radius 100000.00000; ideo
que in ij&longs;dem Radij partibus inuenitur AB
partium 232657. Iam fiat vt AB partium
232657 ad Radium 100000.00000, ita AB
ped. 480 ad aliud, & prodibit quantitas BC
terrenæ &longs;emidiametri ped. 20631229, hoc
e&longs;t mill. 4126. pa&longs;&longs;. 246. Quod verò ali-
trum &longs;uperiùs inuentam, nil mirum, qui&atail;
vbi multiplex diui&longs;io intercedit, fractiones
aliquæ negligun
tur, vnde demum
aliqua oritur dif
ferentia.
Idem aliter
per Trigo
Sed placeat hìc
vnum prætere&atail;
ob&longs;eruare, quo mi
rificè &longs;um delecta
tus, cum primùm
animaduerti:
ctrinam
gonometricam illud idem exhibere po&longs;&longs;&etail;,
quod ab Algebrâ, in &longs;ecundâ methodo indi
catâ, po&longs;t omnes æquationes &longs;ubmini&longs;tra
tur. Fiat enim vt Sinus Ver&longs;us comple
menti anguli ob&longs;eruati, ad eiu&longs;dem anguli
ob&longs;eruati Sinum Rectum, ita nota altitudo
ad aliud, & habebitur quæ&longs;ita terræ &longs;emi
diameter. Sit enim CB terræ &longs;emidiame
ter, BA nota altitudo, AD linea optica tan
gens in D, per quod ex centro C ducatur re
cta CD, quæ producta in E occurrat peri
pheriæ circuli, interuallo CA, ex eodem
centro de&longs;cripti. Eadem ergo e&longs;t Ratio AB
ad BC, quæ e&longs;t ED ad DC. E&longs;t autem AD
DE. At CD e&longs;t æqualis Sinui Recto anguli
CAD ob&longs;eruati; e&longs;t &longs;iquidem CD æqualis
&longs;inui complementi anguli C. Igitur quæ
Ratio e&longs;t ip&longs;ius DE Sinus Ver&longs;i complemen
ti anguli ob&longs;eruati, ad DC Sinum anguli A
ob&longs;eruati, ea e&longs;t ip&longs;ius altitudinis notæ BA,
ad| quæ&longs;itam terræ &longs;emidiametrum BC.
Quarè &longs;inum anguli ob&longs;eruati 99997.67348
deme ex Radio, & remanet 232652. Sinus
Ver&longs;us complementi.Iam &longs;i fiat vt 232652.
ad 99997.67348, ita nota altitudo ped.
480. ad ped. 20631193, eadem inuenitur
&longs;emidiameter, quæ &longs;uperiùs per Algebram
innotuit.
Idem aliter
et breui&longs;&longs;i
mè.
Si perficires commodè po&longs;&longs;it &longs;inè
inue&longs;tigatione anguli, quem cum perpendi
culo facit opticus radius &longs;phæricam terræ &longs;u
perficiem tangens, angulum illum libens
prætermitterem: vix enim &longs;cio, quàm exa
ctè deprehendi queat, quamuis organis di
ligenti&longs;&longs;imè elaboratis non caream.
Quam exa
ctè ob&longs;erua.
ri po&longs;sint
anguli.
Omnem mihi eximo &longs;crupulum,
&longs;i quando contingat in Quadrante lineas, in
quibus minuta di&longs;tinguuntur, ita obliquè &longs;e
cari â perpendiculo, vt anceps hæream, in
quam minuti partem cadat, cum tame&ntail;
etiam ip&longs;as minutorum minutias per&longs;equLatus vnum Quadran
tis ita duplico, vt iam perpendiculum non
ex circuli centro, &longs;ed ab extremâ diametro
pendeat, & ob&longs;eruationem in&longs;tituo: hinc
enim &longs;æpiùs fit, vt &longs;ilum &longs;ericum crudum,
ex quo pilula plumbea &longs;u&longs;penditur, minùs
obliquè lineam minutorum &longs;ecet, quàm &longs;i
ex centro penderet: Et quoniam angulus ad
peripheriam &longs;ubduplus e&longs;t anguli ad cen
trum, notæ verò graduum & minutoru&mtail;
Quadrantis limbo appo&longs;itæ indicant angulos
ad centrum, angulus à perpendiculo nota
tus, ex.gr.1. 45 1/2, bifariam diuiditur, &
quæ&longs;itus angulus e&longs;t, gr.o.m.52.&longs;ec.45.
Quare &longs;i qua labes ob&longs;eruationi adhæreat,
pauculis &longs;crupulis &longs;ecundis definitur.
Dari id quidem facilè pote&longs;t &longs;y
derum ob&longs;eruatori; in tantâ nimirum di&longs;tan
tiâ ip&longs;ius organi magnitudo, quantacunque
demùm illa &longs;it, planè euane&longs;cit; ac proinde
i&longs;ta, vt ita dicam, virtualis duplicatio Qua
drantis nihil officit. At verò telluris ma
gnitudinem indaganti ex radio optico, res
non ita in plano e&longs;t: &longs;i enim oculum centro
Quadrantis admoueas ita, vt latus alterum
in&longs;trumenti congruat lineæ visûs, reliquum
verò latus fuerit, vt dicebas, duplicatum;
angulus à perpendiculo & lineâ visûs ex cen-
& lineâ visûs ex extremâ diametro, quia pa
rallelæ e&longs;&longs;e non po&longs;&longs;unt duæ lineæ eunde&mtail;
circuli quadrantem tangentes. Oculus au
tem in extremâ diametro po&longs;itus, vtpotè
magis à tellure remotus, plus terræ videt &longs;ub
minore angulo: neque omninò contemnen
da e&longs;t altitudinum differentia, &longs;i Quadrantis
Radius pedes quinque &longs;ecundùm longitudi
nem habere ponatur; cum circuli in terrâ
maximi quantitas, quæ patet oculo decem
pedes alto, &longs;itferè &longs;e&longs;quialtera eius, quæ pro
&longs;picitur ab oculo quinque tantùm pedes à
terrâ remoto.
Non eadem e&longs;t ratio; &longs;i in arenâ
litoreâ, ac in editâ turri, aut in colle ob&longs;er
uationem, vt hìc &longs;upponimus, in&longs;tituas; cum
enim terræ &longs;emidiameter aucta oculi altitu
dine &longs;it Secans Complementi anguli ob&longs;er
uati, &longs;ecantes autem angulorum æqualiter
cre&longs;centium addant &longs;emper maiorem, &
maiorem differentiam, patet quàm modi
cum anguli di&longs;crimen intercederet, &longs;i non
ad centrum Quadrantis, &longs;ed ad extremam
circuli diametrum applicaretur oculus i&ntail;
vltimum Horizontem directus &longs;ecundùm
regulam lateri Quadrantis parallelam. Sed
quicquid de hoc e&longs;&longs;e contingat, certum e&longs;t
indè &longs;u&longs;pen&longs;um cadere parallelum perpendi
culo, quod ex extremâ diametro pendet, &
con&longs;tituere angulum cum diametro æqua
lem illi, quem cum eâdem diametro facit
perpendiculum ex eius extremitate &longs;u&longs;pen
&longs;um. Quare &longs;ola centri altitudo &longs;upra terræ
&longs;uperficiem, a&longs;&longs;umenda e&longs;t tanquam oculi
ob&longs;eruatoris altitudo.
Vel &longs;i minùs placeat perpendiculo
vti, Quadrantis latus horizontaliter iacens
duplicetur, & in extremâ diametro fiat cen
trum, circa quod conuer&longs;a dioptra tanti&longs;per
eleuetur, dum linea fiduciæ congruat radio
optico terram tangenti; nam &longs;emi&longs;sis anguli
ad centrum facti cum latere horizontaliter
iacente in&longs;i&longs;tentis eidem arcui, e&longs;t angulus
depre&longs;sionis infra horizontem, æqualis an
gulo ACD facto in centro terræ; cuius com
plementum e&longs;t quæ&longs;itus angulus CAD.
An non operæ pretium facturus e&longs;
&longs;et, qui illud demum in praxim deduceret,
quod ego olim aliud ne&longs;cio quid meditans
perficiendum iu&longs;si, &longs;ed aliis curis di&longs;tractus
ad exitum non perduxi? Duxeram in plano
verticali lineam horizontalem longam pe
des Rom. ferè quinquaginta, & in eius ex
tremo puncto, quod eam cæli plagam re-
uandum, axiculum ritè infigi curaui, circa
quem po&longs;&longs;et dioptra conuerti. Erat autem
animus parare regulam longam pedes 42,
cuius latitudo 4. digitos, cra&longs;&longs;ities aute&mtail;
duos digitos obtineret; &longs;ic enim fieri po&longs;&longs;e
&longs;perabam, vt latitudine in altitudinem con
uersâ, regula non adeò grauis horizontali
ter con&longs;tituta, & &longs;ecundum alteram extremi
tatem axiculo in&longs;erta, non &longs;inuaretur, nec
à &longs;uâ rectitudine deflecteret: cùm maximè
decreui&longs;&longs;em ita illi aliam regulam breuiorem
&longs;ubiicere, in modum vectis primi generis, vt
inferiorem regulam premens eleuarem &longs;u
periorem, quæ circiter duas tertias &longs;uæ lon
gitudinis partes (hoc e&longs;t circirer pedem ab
axiculo, circa quem volueretur, trige&longs;imum)
&longs;u&longs;tentata à &longs;uppo&longs;ito vecte non e&longs;&longs;et adeò
obnoxia inflexioni. Tum regulæ longitudi
ne in pedes di&longs;tinctâ; accipiendi erant pedes
41 2/3, vt e&longs;&longs;ent in vniuer&longs;um vnciæ 500; cum
enim pedis vncia ita &longs;en&longs;ibiliter in laminâ
metallicâ diuidi queat in particulas 50 vt ha
beantur vnciæ particulæ cente&longs;imæ, erat Ra
dius circuli de&longs;cribendi partium 50000, &longs;ub
duplus numeri, qui habetur in Canone Tri
gonometrico communi. Quare &longs;i hoc Ra
dio in plano verticali de&longs;cribatur arcus, &
cularis æqualis Tangenti gr. 1. (accepto &longs;e. &longs;emi&longs;&longs;e numeri Canonis in partibus vnciæ
cente&longs;imis) ab&longs;cindi poterit ex arcu gradus
vnus, cuius &longs;inus parum di&longs;tabit à palmo
Romano architectonico. Quare facilè pote
rit arcus in 60 minuta diuidi, & ex eodem
centro interuallo maiore de&longs;cripto alio arcu,
poterunt duci lineæ tran&longs;uer&longs;æ, in quibus
minutorum partes &longs;ex age&longs;imæ di&longs;tingui po
terunt, prout moris e&longs;t. Hoc in&longs;trumento pa
rato angulum depre&longs;sionis in&longs;ra horizontem,
&longs;eu potiùs infra lineam horizonti parallelam,
tam clarè po&longs;&longs;umus deprehendere, vt nihil
reliqunm
Nec omnibus nec vbique locorum
commodum erit tuum hoc in&longs;trum
fabricari, quod vel vni loco affixum &longs;it opor
tetNe&longs;cio
quo fato adhu
gellâ dura
&longs;es aliquot cuidam, qui &longs;e in Geometriæ pra
xi exercens dolebat nullum hactenus à &longs;e ad
hibitum in&longs;trumentum, cui citrà in&longs;ignis
erroris &longs;u&longs;picionem po&longs;&longs;et fidere, organum
propo&longs;ui, quod & facilè con&longs;truitur, & vix
vllum relinquit errandi periculum, prout ip&longs;a
docuit experientia. In triangulum còag-
vnciarum 10. ex. gr. & &longs;uper eâ fiat triangu
lum æquilaterum AEB. Quod &longs;i tigilli ni
mis longi e&longs;&longs;ent, interuallo A
duo arcus &longs;e inuicem &longs;ecantes in
uallo A
puncta ductâ lineâ ex A, accipi poterit AE
æqualis ip&longs;i AB, & erit angulus BAE gr. 60.
Relinquitur autem tigillus EF longior, vt in
FA &longs;ecundùm rectam EA productam in F
con&longs;tituatur tubulus H infundibuli in mo
rem excauatus, per quem vi&longs;us rectâ traii
ciatur in de&longs;tinatum &longs;copum per aciculam
in E perpendiculariter erectam, ex qua de
pendet perpendiculum; hoc enim facit cum
lineâ EA angulum di&longs;tantiæ obiecti à verti
ce ob&longs;eruatoris. Quantitas verò huius an
guli per Trigonometriam reperitur ex datis
lateribus AE, & A
hen&longs;o gr. 60. ex con&longs;tructione. Vel &longs;i fortè
breuiùs operari placeat, fingatur ex E puncto
in &longs;emi&longs;&longs;em lateris AB cadere linea perpen
dicularis, cuius quantitas facilè innote&longs;cit:
deinde illâ a&longs;&longs;umptâ vt Radio, vt Tangent&etail;
verò di&longs;tantiâ perpendiculi à medio lineæ
AB, angulus quæratur, quem facit perpen
diculum cum fictâ illâ perpendiculari; hic
enim angulus additus gradibus 30. &longs;i perpen-
demptus autem ex gr. 30. &longs;i cadat citrà &longs;e
mi&longs;&longs;em, dabit angulum di&longs;tantiæ à vertice
quæ&longs;itum. Quod &longs;i di&longs;tantia à vertice exce&longs;
&longs;erit gr. 60 perpendiculum cadet extrà latus
BE; idcirco &longs;u&longs;pendendum èrit ex I puncto,
quod bifariam diuidit rectam AE; & &longs;iqui
dem obiectum in Horizonte fuerit, perpen
diculum ex I cadet in B, &longs;i verò &longs;upra Hori
zontem, cadet in latus AB, &longs;i demum infra
lineam Horizonti parallelam depre&longs;&longs;um fue
rit, cadet perpendiculum in latus BE: &longs;em
per autem in note&longs;cet angulus &longs;actus à perpen
diculo & lineâ FI visûs, dantur enim vel la
tera IA & AB, vel IE & EB cum angulo
comprehen&longs;o: vel &longs;altem in vtroque latere
de&longs;ignari poterit punctum, in quod ad rectos
angulos cadit linea ex I.
V&longs;us Tri
goni æqui
lateri ad
ob&longs;eruandos
angulos, in
quo vix er
rari po&longs;sit à
Geometra.
Vt verò in quàm minimas particulas di
ui&longs;am haberet rectam AB, iubebam duci
AC æqualem vni ex ijs partibus, in quas pri
mò di&longs;tinctam eam po&longs;uimus, nempe i&ntail;
vncias pedis, & perfici parallelogrammum
ABDC. Tum diuisâ AC, & BD in quinque
æquales partes, & ductis parallelis ip&longs;i AB,
inter po&longs;trem as
li; C
quinque partes, vt factum hìc e&longs;t in vnciâ
donec diagonalem attingant: & his paratis
habetur vncia AC diui&longs;a in partes 500, vel
&longs;altem in 250, &longs;i differentia inter duas pro
xim as lineolas &longs;en&longs;ibilis non &longs;it.
Quamuis autem tam exiguum triangu
lum AEB non &longs;it aptum ad acutiores angu
los inueniendos, con&longs;tat tamen po&longs;&longs;e illud
&longs;inè magno incommodo con&longs;trui ita, vt pe
des aliquot &longs;ingula latera obtineant; & tunc
etiam minores angulorum particulæ innote
&longs;cent. Quod &longs;i quando res &longs;it quàm exa
cti&longs;simè perficienda, poterit lateri FE addi
regula, ex cuius extremo &longs;ecundùm rectam
AE productam &longs;u&longs;pendatur perpendiculum,
perinde enim erit ac &longs;i totum triangulum
augeretur. At &longs;i ex editiore loco deor&longs;um
&longs;pectandum &longs;it, & perpendiculum ex I &longs;u
&longs;pen&longs;um cadat in latus EB tam propè ip&longs;um
B, vt non facilè po&longs;sint particulæ numera
ri, erigatur ad latus AE regula perpendicu
laris, in qua recta ex B per I ducta faciat an
gulum EIG rectum, & &longs;umatur IG æqualis
ip&longs;i IE, & ex G &longs;u&longs;pendatur perpendiculum,
quod intelligatur cadere ex. gr. in R.
Si
enim concipiatur ex I aliud perpendiculum
IS, vtique linea IG incidens in duas paralle
las IS, & GR facit angulum SIB externum
innote&longs;cit angulus, quem cum perpendicu
lo facit linea vi&longs;ualis AE. Cum itaque IE &
IG æquales &longs;int & ad angulum rectum, con
ceptâ rectâ EG angulus EGI e&longs;t gr. 45, cui
æqualis reliquus GEI additus angulo BEA gr.
60. fit totus angulus GER gr. 105; latus au
tem EG &longs;ubtendens angulum rectum, no
tum e&longs;t, cum nota &longs;int latera IG, & IE; de
mum notum e&longs;t latus ER. Quare ex notis
lateribus EG & ER vnâ cum angulo com
prehen&longs;o inuenitur angulus EGR, qui dem
ptus ex EGI gr. 45. relinquit notum RGI
æqualem angulo SIB; & &longs;ic innote&longs;cit angu
lus SIE æqualis angulo, quem facit linea vi
&longs;ualis EA
&longs;eruatoris cum centro terræ. Cum igitur &longs;atis
amplum huiu&longs;modi triangulum facilè po&longs;si
mus con&longs;truere, etiam additâ regulâ IG, &
alibi diui&longs;am vnciam habere po&longs;simus i&ntail;
partes &longs;altem cente&longs;imas eâ methodo, quam
indicat Adrianus Metius in &longs;uâ Geometriâ,
patet fieri po&longs;&longs;e, vt quàm euidenti&longs;simè in
notefcat angulus ille, quem ob&longs;eruatum &longs;up
ponebam ad inue&longs;tigandam terræ magnitu
dinem. Sed mi&longs;sis organis ad in&longs;titutum
redeamus.
Audiui haud ita pridem hominem,
zontis Phy&longs;ici &longs;emidiametro, totius telluris
&longs;emidiametrum colligere &longs;e po&longs;&longs;e cen&longs;eban.
Primùm verò &longs;tatuebat, quod certum e&longs;t, ab
iis qui de maximâ visûs di&longs;tantiâ di&longs;putant,
non eam inquiri, quæ vel ab oculis variè af
fectis, vel à diuersâ corporum videndorum
magnitudine, vel ab inæquali &longs;pecierum,
quas vocant, intentionalium, quibus obie
ctum repræ&longs;entandi vis ine&longs;t, diffu&longs;ione ea
rumue inten&longs;ione petenda e&longs;t. Nemo enim
ignorat, &longs;i i&longs;ta &longs;pectentur, fieri non po&longs;&longs;&etail;,
vt certa visûs di&longs;tantia vniuer&longs;im præ&longs;criba
tur. Acribus &longs;iquidem & acutis oculis longè
remotiora patere con&longs;tat, quàm hebetiori
bus, quos languidiores radii procul immi&longs;si
ad vi&longs;um non excitant. Vbi verò obiectum
corpus non oculorum vitio latet, &longs;æpè &longs;uâ
&longs;e paruitate ita protegic, vt admi&longs;si à pupillâ
radii in tenui&longs;simum angulum coale&longs;centes
eam demùm afficiant Retinæ particulam,
quæ eum &longs;en&longs;um omnem effugiat, &longs;entiendi
quoquè munere &longs;olitaria fungi non pote&longs;t:
vt proinde Dioptrica &longs;ub&longs;idiarias lentes vi
treas in tubo&longs;picillo ritè di&longs;po&longs;itas mortali
bus tran&longs;mi&longs;erit, quarum ope inflexi Radii
ampliorem angulum con&longs;tituant, ideòqu&etail;
maiorem Retinæ particulam ad videndum Sed & illud maximè ambiguam
facit visûs di&longs;tantiam, quod pro di&longs;pari luce,
qua corpora imbuuntur, di&longs;par quoque exi
gunt interuallum, vt &longs;ub a&longs;pectum cadant:
&longs;ic aliquando Perigæam Lunam, quæ ant&etail;
pauculas horas ferè pleno orbe immen&longs;&atail;
collucebat, &longs;ereno cœlo ami&longs;&longs;am quærimus,
&longs;i fortè eius centrum in terre&longs;tris vmbræ a
xem inciderit; cum tamen eadem tenebra
rum immunis, quamuis Apogæa Soli pa
riter Apogæo oppo&longs;ita, &longs;ponte in oculos
incurrat. Ea nimirum quamuis remotior&atail;
con&longs;piciuntur, quæ vberiori lumine &longs;iuè in
nato, & in&longs;ito, &longs;iuè extrin&longs;ecùs mutuato per
funduntur.
Maxima vi
&longs;us dictan
tia ex qui
bus debeai
definiri.
Semotis igitur omnibus hi&longs;ce impedi
mentis, per quæ fieret, quominùs certi ali
quid de visûs di&longs;tantiâ &longs;tatui po&longs;&longs;et; phy&longs;ici
horizontis &longs;emidiameter ex ipsâ terreni glo
bi configuratione petenda e&longs;t. Cum enim
vetus ille error opinantium terræ faciem æ
quabili planitie diffu&longs;am iam dudum exta
buerit, & nemo &longs;it, qui pilæ in &longs;peciem ter
ras ac maria vndique in &longs;e&longs;e nutibus &longs;uis con
globata non intelligat; nemo pariter ambi
gat, quin ob conuexam huiu&longs;ce globi &longs;uper
&longs;iciem, inclinatis nimirùm partibus, &longs;eque
ocuio &longs;ubducentibus, a&longs;pectus omnis quan-
&longs;cribatur. Hinc circulus partem hanc a&longs;pe
ctabilem à latente di&longs;pe&longs;cens, Horizontis
Phy&longs;ici nomine donatus e&longs;t, vt ab eo &longs;ecer
natur Horizonte, qui tellurem in duo hemi
&longs;phæria &longs;egregans, quia non oculo, &longs;ed
ratione comprehendi pote&longs;t, Rationalis di
citur.
Quamuis autem Phy&longs;ici Horizontis dia
meter tota intrà terræ cra&longs;situdinem deliteat,
quippe quæ recta e&longs;t linea arcui illi &longs;ubten&longs;a,
qui extremis terminis oculi Ortum Occa
&longs;umque circum&longs;picientis intercluditur: quia
tamen arcus huiu&longs;modi exiguo di&longs;crimin&etail;
quod vix &longs;ub &longs;en&longs;um ca
dat, &longs;ubten&longs;æ rectæ li
neæ longitudinem &longs;upe
rat; ideò non abs re ar
cus ip&longs;e pro phy&longs;ici hori
zontis &longs;emidiametro in
di&longs;criminatim v&longs;urpatur;
huiu&longs;ce &longs;emi&longs;sis maxi
mam obiecti a&longs;pectabilis
di&longs;tantiam metitur.
Ex his &longs;ic ille argumentabatur.
Sit arcus
AB men&longs;ura di&longs;tantiæ vi&longs;us, qui non à rectâ
lineâ &longs;en&longs;ibiliter deflectat
perpendiculariter in&longs;i&longs;tens rectæ DC: radius
&longs;emidiametro angulum DBC rectum. Igi
tur recta AB ab angulo recto ad B cadens i&ntail;
ba&longs;im perpendicularis, e&longs;t medio loco pro
portionalis inter &longs;egmenta DA & AC. Qua
rè diui&longs;o quadrato maximæ visûs di&longs;tantiæ
AB per altitudinem AD, prouenit quæ&longs;it&atail;
terræ &longs;emidiameter AC. Verùm arcum
pro rectâ lineâ a&longs;&longs;umere parùm Geometri
cum e&longs;t.
Abu&longs;us li
neæ curuæ
pro rect&atail;,
inutilis.
Ideò paruiperrdendendam cen&longs;ui
methodum, quæ mihi aliquando occurrit
per circulos Azimuthales; &longs;i nimirùm duo
rum locorum, quorum alteruter ex alterius
editâ turri con&longs;pici queat, nota fuerit di&longs;tan
tia, & poli altitudo. In &longs;ummâ enim turri
planum horizontale con&longs;tituatur, in eoque
meridiana linea de&longs;cribatur: tum ob&longs;eruetur
&longs;ub quo Azimutho locus alter con&longs;picuus
appareat: & ex his datis quæ&longs;ita eruantur.
Sit PAC loci Meridianus, vbi ob&longs;eruatio in
&longs;tituitur
tran&longs;iens AB, cum Meridiano faciat angu
lum CAB ob&longs;eruatum; P verò &longs;it Polus, &
AP, BP &longs;int complementa nota datarum
poli eleuationum. Cum itaque in triangu
lo &longs;phærico BAP nota &longs;int duo latera AP,
BP, & angulus BAP complementum angu-
AB in gradibus &longs;eu
graduum particulis.
Tum fiat vt AB pars,
inuenta ad totum cir
culum, hoc e&longs;t gr.
360, ita data duorum
locorum di&longs;tantia ad
aliud, & prodibunt
milliaria toti circulo
in terrâ maximo re&longs;pondentia. Notâ autem
circularis peripheriæ quantitate neminem
Geometram diameter quæ&longs;ita latere poterit.
Sed quoniam di&longs;tantia illa non adeò exigua
e&longs;&longs;e pote&longs;t, quæ careat omni &longs;u&longs;picione abu
sûs rectæ lineæ pro curuâ, ideò methodum
hanc inter qui&longs;quilias, à quibus parùm di
&longs;tat, reieci.
Secunda me
thodus in
uestigandi
telluris am
bitum.
Id tamen per &longs;ummam iniuriam
factum: neque enim magnum intercedere
pote&longs;t di&longs;crimen, quod propo&longs;ito officiat: e&longs;t
&longs;cilicet di&longs;crimen minus datâ altitudine. Ex
altitudine enim BA pro&longs;piciénti pateat ar
cus BD, quem &longs;ubtendit recta BD, Tan
gens autem e&longs;t AD. Con&longs;tat ex Archimede
lib.
1. de &longs;phær & cyl. Tangentem AD ma
iorem e&longs;&longs;e arcu BD, arcum autem BD maio
rem rectâ BD &longs;ubtensâ: At duæ rectæ BD,
igitur exce&longs;&longs;us ip&longs;ius
AD &longs;upra rectam BD
minor e&longs;t, quàm &longs;it
data altitudo BA; er
go multò minor e&longs;t
exce&longs;&longs;us rectæ AD
tangentis &longs;upra ar
cum BD, vel arcûs
BD &longs;upra rectam BD
&longs;ubten&longs;am. Ne verò
po&longs;itâ altitudine BA notabili, cen&longs;eret quis
in&longs;ignem quoque e&longs;&longs;e differentiam inter cur
uam & rectam lineam, quæ locorum inter
ualla metiatur, animaduertere oportet ar
cum BD maiorem e&longs;&longs;e latere BD polygoni
in&longs;cripti, minorem autem latere EF polygo
ni circum&longs;cripti: at exce&longs;&longs;us lateris EF ad la
tus BD habet Rationem, quam EB ad BC.
Vnde liquet manife&longs;tè, quàm modico di
&longs;crimine differat arcus à rectâ vel &longs;ubtensâ
BD, vel tangente EF.
Differentia
longitudinis
inter arcum
et eius Tan
gentem vel
&longs;ubten&longs;am.
Memini me aliquando calculos &longs;ubducen
tem, quàm longè pateat vi&longs;us Romæ è &longs;um
mo crucis apice, quæ ædis Apo&longs;tolorum
Principi &longs;acræ tholo incumbit. deprehendi&longs;
&longs;e arcum DB gr. o. m.
23. &longs;tatuebam autem
in &longs;ingulos gradus milliaria Italica 60, vt
dinem &longs;upra maris Mediterranci &longs;uperfi
ciem ponebam palmorum, quorum v&longs;us ho
diè e&longs;t apud Romanos architectos, circiter
700. Quare arcus BD erat mill. 23. & Tan
gens AD mill. (23 441/100000), di&longs;crimine vix pa&longs;
&longs;uum 4 1/2. Quid igitur officiat, &longs;i quis pro
arcu BD a&longs;&longs;umat, aut &longs;ubten&longs;am BD, aut
tangentem EF, aut aliquam ex intermediis?
nullus &longs;iquidem oriri pote&longs;t error, qui &longs;ub
&longs;en&longs;um cadat. Quod &longs;i, vt &longs;æpè fieri pote&longs;t,
di&longs;tantiam BD decempedâ dimetiamur, ni
hil erit quod de tuâ illâ methodo dubites,
Mer&longs;enne, non enim arcu pro rectâ abu
teris.
Su&longs;picor Mer&longs;enne (detur hoc Ger
mano candori) non tibi rectam pro arcûs
men&longs;urâ a&longs;&longs;umptam di&longs;plicuiffe; &longs;ed metho
dum illam, quæ tibi nihil de Ptolemæo co
gitanti occurrerat, demùm di&longs;plicui&longs;&longs;e faci
lè crediderim, vbi eam veteribus quoquè
innotui&longs;&longs;e deprehendi&longs;ti: ideo illam inter
&longs;cruta reieci&longs;ti. Ego quoque, vt vera nat
rem, &longs;æpiùs doleo, quod veritatem tan
quam peregrinam exceperim, qua&longs;i ad m&etail;
primum diuerti&longs;&longs;et; quam po&longs;tea alienum
limentriui&longs;&longs;e comperio: nec &longs;anè me pro
bro&longs;is &longs;uis &longs;timulis vrget inuidentia, qua cæ-
calumnia, qua &longs;æpè apud iniquos iudices fur
ti reus laboras, ni&longs;i id aliundè acceptum
&longs;ponte profitens mentiaris: perinde atque &longs;i
nemo &longs;olem po&longs;&longs;et intueri, qui alios de eius
luce narrantes non audierit. Simile quid in
hoc eodem Problemate mihi contigit expe
riri. Cum enim locorum duorum AB di
&longs;tantiam notam ponerem, ac complemen
ta eleuationum poli; ob&longs;eruabam Solis di
&longs;tantiam à vertice SA, quam per Tabulas
Anacla&longs;ticas, & Parallacticas corrigebam;
Tum ex complemento declinationis Solis
SP, complemento altitudinis Poli AP: & di
&longs;tantia Solis à vertice SA, inquirebam angu
lum ASP; ex quo inuento, vnâ cum latere
SP, & complemento altitudinis poli BP,
inueniebam SB, quod demptum ex SA re
linquebat arcum BA quæ&longs;itum. Hæc ta
men methodus po&longs;t dies aliquos di&longs;plicuit,
cum animaduerterem ingenio&longs;um A&longs;trono
mum eâdem viâ ince&longs;si&longs;&longs;e, & quidem fe
liciùs non Solem &longs;ed &longs;ydera fixa vertici pro
xima ob&longs;eruando, quæ nec parallaxi nec re,
fractioni &longs;unt obnoxia.
Tertia me
thodus in
uestigandi
ambitu&mtail;
terræ.
Cæterùm ne ab in&longs;titutâ quæ&longs;tione de
flectamus, illud e&longs;t ob&longs;eruandum, quod &longs;i
cognitâ altitudine BA, & dato angulo ad A,
B, vt notam faceret
di&longs;tantiam inter B &
D, &longs;ibique per&longs;uaderet
aut à ba&longs;i aut ab hy
pothenusâ illius
guli
tiam BD, longè di
&longs;taret à veritate. Du
cta &longs;iquidem perpen
dicularis BH longè
minor e&longs;t arcu BD, cum &longs;it Tangens &longs;e
mi&longs;sis illius arcûs: hypothenu&longs;a verò AH
minor e&longs;t quàm tota Tangens AD, de
fectu æquali ip &longs;i ba&longs;i BH; &longs;unt enim HB,
HD æquales, cùm vtraque &longs;it circulum
tangens ab eodem puncto H ducta. Qua
propter deberet aggregatum ex ba&longs;i BH
& hypothenusâ AH accipere, vt totam
AD haberet, quæ non multùm differt ab ar
cu BD, ni&longs;i altitudo BA fuerit mons aut &longs;co
pulus.
Quod &longs;i tuus ille terrenæ magnicudinis
inue&longs;tigator lineam BD pro maximâ vi
sûs di&longs;tantiâ a&longs;&longs;ump&longs;i&longs;&longs;et, illa vtique media
e&longs;t proportionalis, ex cuius quadrato per no
ram altitudinem DA diui&longs;o prouenit tertius
&longs;emidiameter cogno&longs;ci
tur, non tamen ip&longs;a &longs;e
midiameter AG e&longs;t ter
tius analogiæ terminus,
&longs;ed potius conflatum ex
DC, CB: latus enim v
num trianguli rectan
guli e&longs;t medium propor
tionale inter differentiam
hypothenu&longs;æ ac reliqui lateris, & eorum ag
gregatum: ac proinde vt AD ad DB, it&atail;
DB ad DCB. Quare ex inuento tertio ter
mino debui&longs;&longs;et datam altitudinem DA de
mere, & re&longs;iduum bifariam diui&longs;um dedi&longs;
&longs;et quæ&longs;itam &longs;emidiametrum AC. Hinc
rectè dicebas arcum illum pro rectâ lineâ
parùm Geometricè a&longs;&longs;umptum.
Quarta me
thodus in
neniendi ter
ra &longs;emidia
metrum.
Nunquam hominem potui à con
ceptâ &longs;ententiâ reuocare, ni&longs;i vbi eum iu&longs;&longs;i
rem rotam in praxim deducere. Statuimus
primùm extra omnem controuer&longs;iam po&longs;i
tum videri, in decernendâ Phy&longs;ici horizon
tis amplitudine non incertam ædium aut
montium altitudinem e&longs;&longs;e &longs;pectandam, &longs;ed
eam ex communi vulgarique hominum ma
gnitudine definiendam. Quandoquidem
humiliori autem breuioribus terminis coer
ceatur, quis non videt certam &longs;tatui non po&longs;
&longs;e visûs di&longs;tantiam, quin certa pariter alti
tudo, ex qua oculus circum&longs;picere valeat,
con&longs;tituta intelligatur? præter eam verò,
quam humano corpori natura plerunqu&etail;
conce&longs;&longs;it, cur hanc præ aliâ eligas altitudi
nem, nulla &longs;uppetit ratio. Nec ambigi vllo
pacto pote&longs;t, an Veteres horizontis phy&longs;ici
amplitudinem indagantes, altitudinem vl
lam humanâ maiorem a&longs;&longs;ump&longs;erint; eam
&longs;iquidem horizonti tribuendam cen&longs;uer&etail;
magnitudinem, quæ cum &longs;phæricæ &longs;uperfi
ciei pars &longs;it, à planâ tamen minimùm differ
re po&longs;&longs;it. Hinc Macrobius Saturnal. lib.
7.
cap.
14.
re; & hoc e&longs;t quod Horizontem veteres voca
uerunt: quorum indago fideliter deprehendit,
directam ab oculis aciem per planum contrà
a&longs;picientibus non pergere vltra CLXXX &longs;ta
dia, & inde in orbem iam recuruari. Per pla
num, ideò adieci, quia altitudines longi&longs;&longs;imè
a&longs;picimus; quippe qui & cælum videmus.
Ergo in omni horizontis orbe ip&longs;e qui intuetur,
centron e&longs;t. Et quia diximus quantum à cen-
nè dubio in horizonte diametros orbis CCCLX
&longs;tadiorum e&longs;t: & &longs;i vlt eriùs qui intuetur ac
ce&longs;&longs;erit, &longs;eu retror&longs;um rece&longs;&longs;erit, &longs;imilem cir
ca &longs;e orbem videbit.
Rogaui deinde vtrùm Macrobio potiùs
&longs;tadijs 180 Phy&longs;ici Horizontis &longs;emidiame
trum de&longs;inienti acquie&longs;ceret, an verò &longs;ibi
cum Recentioribus quibu&longs;dam conueniret,
qui cum Clauio (cui facilè adh æ&longs;it Blanca
nus) in cap.2. &longs;phær. tanquam veritati ma
ximè con&longs;entaneam ad mittunt eorum &longs;en
tentiam, qui a&longs;pectum ad milliaria 62 1/2 pro
trahi opinantur. Neque enim illum aut cum
Alberto Magno aut cum Proclo &longs;entire cen
&longs;ebam, quorum prior &longs;tadia mille, po&longs;te
rior bis mille horizontis phy&longs;ici &longs;emidiame
tro dedit.
Ostenditur
error ex ab
curuæ pro
rect a.
Ille quidem con&longs;e&longs;tim Neotericorum
&longs;ententiam arripuit: &longs;ed vbi iu&longs;&longs;us e&longs;t AB
mill. 62 1/2 ad pedes 312500 reuocare, &
huius numeri quadratum 97656.250000.
diuidere per DA ped. 6, quanta e&longs;t hominis
mediocris altitudo, videns AC prouenir&etail;
maiorem milliarijs 3.000000, ad Macro
bium con&longs;ugit, & longitudinem AB &longs;tadiis
180 circum&longs;crip&longs;it: nec &longs;tadia Romana ped.
bat, cum ex Erato&longs;thene men&longs;ura illa de
&longs;umpta fui&longs;&longs;et. Sed hìc pariter Syrtes inue
nit, cum reuocato &longs;tadiorum 180 numero
ad pedes 108000, eius quadratum 11664.
000000. diuidere tentans per pedes 6, quo
tientem pariter ampli&longs;&longs;imum deprehendit.
Quid &longs;i volui&longs;&longs;em contentio&longs;iùs agene, affir
mando Erato&longs;thenem nec Romanis, nec
Græcis, &longs;ed Alexandrinis &longs;tadijs longioribus
v&longs;um fui&longs;&longs;e? Quantò longiùs aberrâ&longs;&longs;et?
Sed
placuit miti&longs;&longs;imè agere. Quapropter eum
rogaui, vt &longs;altem AB milliarium trium, hoc
e&longs;t ped. 15000 con&longs;tituens tentaret, an ex
a&longs;&longs;umptâ priùs iu&longs;to maiore &longs;emidiametro
horizontis phy&longs;ici, an verò etiam ex metho
di ageometriâ enormi
retur: Quadratum itaque ped. 225.000000
diuidens per DA ped.6, inueniebat AC ped.
37.500000, hoc e&longs;t mill. 7500; vnde oritur
terræ ambitus mill. 47 142 6/7 duplo maior,
quàm communiter concedatur.
Mirabatur ille vehementer, quòd præter
&longs;pem tam procul à vero rece&longs;&longs;i&longs;&longs;et: &longs;ed nec
conceptam de maiore, iuxta communem
errorem, horizontis phy&longs;ici &longs;emidiametro
opinionem deponere audebat: hærebant &longs;ci
licet animo altè impre&longs;&longs;a plurium authorum
mediocris visûs di&longs;tantia: quare multò pro
babilior ei apparebat Macrobrii atque Era
to&longs;thenis &longs;ententia di&longs;tantiam huiu&longs;modi &longs;ta,
&longs;tijs 180 de&longs;inientium. Nos igitur ad exa
minandum contulimus telluris ambitum ab
Erato&longs;thene con&longs;titutum, an cum illa &longs;emi
diametto &longs;tadiorum 180 cohæreret:
1. in &longs;omn. Scip. cap.
20.
dimen&longs;ionibus con&longs;tat vniuer&longs;æ terræ ambitum,
quæ vbicunque vel incolitur, vel inhabitabilis
iacet, habere &longs;tadiorum millia ducenta quin
quaginta duo. Cum verò huius peripheriæ
pars mille&longs;ima quadringente&longs;ima &longs;int &longs;tadia
180, per quæ directa ab oculis acies pergit,
arcus BA gr.o.m.15.&longs;ec.25. Ter. 42. &c.
metitur angulum BCD in centro factum.
At quoniam peripheria ponitur &longs;tad 252000
e&longs;t diameter minor verâ &longs;tad. (80181 9/11), maior
autem verâ (80233 41/223): quarè &longs;umpto medio
Arithmetico e80207 1/2,
& &longs;emidiameter CA &longs;tad. 40103 3/4. Fiat igitur
vt CB 99998.99293. &longs;inus complementi
gr.o.m.15.&longs;ec. 25. Ter.42. ad CD Radium
100000.00000, ita CB &longs;tad.40103 3/4 ad CD
&longs;tad, 40104, & ped.92. Demptâ autem CA
tudo, ex qua pro&longs;pici pote&longs;t in B ita, vt a&longs;pe
ctui pateant &longs;tadia 180. Quî autem &longs;ieri po&longs;
&longs;it ab intuente &longs;imilem &longs;emper orbem vide
ri, &longs;iuè vlteriùs acce&longs;&longs;erit, &longs;iue retror&longs;um re
ce&longs;&longs;erit, Macrobius ip&longs;e viderit: an &longs;emper
eam inueniat altinudinem, cui in&longs;i&longs;tat, vt
oculus ab extimâ telluris facie pedibus 242
&longs;emoueatur? Horizontis igitur &longs;emidiame
trum aut &longs;tadijs 180 minorem e&longs;&longs;e, aut non
ex humani corporis altitudine de&longs;iniri ne
ce&longs;&longs;e e&longs;t.
Authorum
aliquorum
lap&longs;us i&ntail;
statuend&atail;
nimia vi&longs;us
distantia.
Cum verò ille labantem Macrobij &longs;en
tentiam &longs;uffulcire aliquatenus &longs;e po&longs;&longs;e con
&longs;ideret, &longs;tadia 180 tribuendo non arcui AB,
&longs;ed rectæ DB terram tangenti, quæ & radij
optici ex oculo D prodeuntis men&longs;ura e&longs;t, &
arcu AB longior; operam demum &longs;e ludere
&longs;en&longs;it, cum re ad Geometricam normam
reuocatâ tantam opticæ lineæ longitudinem
terrenæ magnitudini minimè congruere ma
ni&longs;e&longs;tè deprehendit, etiam &longs;i oculus à terrâ
pedum 50. interuallo AD &longs;eiunctus intelliga
tur. Quoniam enim quadratum Tangentis
BD, quæ &longs;tad. 180 ponitur, æquale e&longs;t re
ctangulo ADE, diuidatur quadratum ped.
11664.000000 per 50, & erit DE ped. 233
280000, hoce&longs;t &longs;tadiorum Græcorum (quæ
AE, demptis pedibus 50, erit &longs;tad. 388799
ped. 550. Ex qua diametro colligitur am
bitus longè maior eo, quem Macrobius cum
Erato&longs;thene con&longs;tituit.
Sed lap&longs;is in ageometriam Erato
&longs;thene ac Macrobio, quod &longs;tadia 180 tribue
rint horizontis phy&longs;ici &longs;emidiametro, cum
totum telluris ambitum &longs;tadijs 252000 de
&longs;inierint; quid de ijs dicendum, qui terram
paucioribus &longs;tadijs circum&longs;eribentes (mil
liarijs nimirum 21600, quæ ad &longs;tadia Græca
180000 reuocantur, quot Ptolemæus toti
terrenæ peripheriæ conce&longs;sit) oculum tamen
ad maiora &longs;patia longo limite ducunt? Hi
&longs;anè longi&longs;simè ab&longs;unt à veritate, cum lineæ
opticæ tribuunt longitudinem &longs;tadiorum
vt minimum 500. Nam quadratum lineæ
opticæ e&longs;t æquale rectangulo &longs;ub altitudin&etail;
oculi, & terræ diametro auctâ eâdem altitu
dine. Sit ergo D linea optica, & B &longs;it terræ
diameter, & A &longs;it altitudo: igitur DQ æqua
tur AQ+A in B. Quare ad inueniendam A
&longs;iat expurgatio per vncias couditionarias qua
drati, &longs;c. per &longs;emi&longs;&longs;em coëfficientis B.
Sit er
go E æqualis A+B 1/2: igitur E——B 1/2 e&longs;t æqua
lis ip&longs;i A. Factâ itaque prioris æquationis in
terpretatione erit æquatio inter EQ——BQ 1/4
DQ+BQ 1/4
Cum igitur data &longs;it peripheria &longs;tad.
180000,
erit diameter B &longs;tad. 57291 2/3 proximè: atquè
adeò BQ 1/4 e&longs;t 820583766 2/3; ip&longs;ius verò D
&longs;tad. 500, quadratum 250000. ex quotum
&longs;ummâ 820833766 2/3, quæ æqualis e&longs;t EQ,
&longs;i eruatur radix, erit E, hoc e&longs;t A+B 1/2 &longs;tad.
(28650 196/1000), ablatâ autem B 1/2, quæ e&longs;t &longs;tad.
28645 5/6, remanet A &longs;tad. (
tudo oculi: ex qua altitudine con&longs;tat no&ntail;
e&longs;&longs;e de&longs;iniendam horizontis phy&longs;ici amplitu
dinem. Quod &longs;
mus vnius &longs;tadij, retentâ eâdem opticæ li
neæ longitudine, proueniret terræ diameter
&longs;tad. 249999, quæ maior e&longs;t totâ periphe
riâ ab ip&longs;is con&longs;titutâ.
Hinc pariter Blancano fucum factu&mtail;
comperiemus (nam & quandoque bonus
dormitat Homerus) qui maximam mon
tium altitudinem &longs;e&longs;quimilliari aut duobus
vt &longs;ummum milliaribus de&longs;iniens, Ætnæ in
Siciliâ duo tantùm milliaria conce&longs;sit, ad
mi&longs;ittamen ex Maurolyco dial. 3. Co&longs;mogr
pag. 75.indè pro&longs;pici in mare vltra ducenta
pa&longs;&longs;uum millia. Namque a&longs;&longs;umptâ lineæ
opticæ terrenum globum tangenti
per Ætnæ altitudinem con&longs;titutam mill. 2
diui&longs;o, & ex Quotiente 20000 demptâ mon
tis altitudine, ne&longs;cio qua ratione re&longs;iduum
mill. 19998 terrenæ diametro tribuendum,
idem Blancanus Sphæræ part. 3.c.5.pag.93.
&longs;ibi gratulatur proximè conuenire cum am
bitu mill. 21600 ab aliis po&longs;ito, & à &longs;e ad
mi&longs;&longs;o. Cum tamen hinc debui&longs;&longs;et potiùs
&longs;u&longs;picari montium altitudinem à &longs;e breuiori
bus, quàm par &longs;it, terminis de&longs;initam, ex
qua tanta ferè colligitur diameter, quant&atail;
e&longs;&longs;e deberet peripheria.
An igitur cos quoque mendacii
manife&longs;tos redarguas, qui Alexandrina&mtail;
&longs;peculam ab in&longs;ulâ, in qua extructa fuit, Pha
ron dictam ad eam prouehunt altitudinem,
vt indè naues &longs;excentorum milliarium in
teruallo di&longs;sitas videri potui&longs;&longs;e per &longs;ummam
con&longs;identiam â ffirment?
Vnde fabulam i&longs;tam Io.Bapt. Por
ta lib.17. Mag. nat.in proëm. hau&longs;erit, pror
&longs;us ignoro. Neque enim Diodorus Siculus,
aut Strabo, aut Cæ&longs;ar, aut A. Hirtius, aut
Plinius, aut Lucianus, aut Solinus, aut Am
mianus Marcellinus, aut alius ex ijs, quos le
gerim, cum turris illius Alexandrinæ me
minerint, adeò in&longs;anam altitudinem nobis Quod autem Porta a&longs;&longs;erit in eâ
turri à Ptolemæo con&longs;titutum &longs;peculu&mtail;,
quod deinde cap.
11. ip&longs;e &longs;pecillum potiùs
quàm &longs;peculum vocat, vt ad 600 pa&longs;&longs;uum
millia ho&longs;tium naues, quæ eius regiones in
uaderent, con&longs;piceret, occa&longs;io fuit aliquan
do nonnemini opinandi ad ea v&longs;que tempo
ra referendum v&longs;um Tubi optici. Huiu&longs;ce
tamen &longs;peculi, &longs;iue &longs;pecilli, ve&longs;tigium nul
lum apud antiquos &longs;criptores deprehender&etail;
potui; quamuis eorum libros multâ diligen
tiâ per&longs;crutatus. Quapropter ea mihi mens
incidit, vt exi&longs;timarem Portæ, viro cætero
qui erudito, fucum factum ab inepto quo
piam Typographo, vel ignaro &longs;criptore, qui
pro &longs;peculâ in Pharo (in&longs;ulâ) extructâ, &longs;pe
culum in Pharo (turri) con&longs;titutum &longs;uppo
&longs;uerit.
Fabula de
altitudin&etail;
Phari Ale
xandrinæ
reijcitur.
Cæterùm cum tubus opticus vitreis lenti
bus ritè in&longs;tructus ea &longs;olum corpora di&longs;tinctè
articulatimque videnda proponar, ad quæ
recta oculorum acies pergit, tota hæc fabu
la euane&longs;cet, vbi tantam altitudinem, quæ
vi&longs;um ad milliaria 600 producat, probab
tate omni carere con&longs;titerit. Neque opus
erat altitudine illâ imman
illa à Ptolemæo Philadelpho (quamuis Amm.
Marcell. lib.
22. à Cleopatrâ, quæ &longs;oror, &
ny&longs;ij turrim illam excel&longs;am excogitatam &longs;cri
bat) So&longs;trati Cnidij architecti operâ extructa
&longs;uerit, vt noctu accen&longs;æ faces indè præluce
rent nauigantibus, quò breuia & &longs;yrtes de
clinarent, quibus ora illa fallacibus & in&longs;i
dio&longs;is acce&longs;sibus importuo&longs;a &longs;catebat, & di
&longs;criminibus plurimis incautos nautas affli
gebat.
Sed iam examinemus quantâ altitudine è
mari eminuerit Pharos illa Alexandrina, &longs;i
inde ad milliaria 600 vi&longs;us excurrere potuit.
Po&longs;ito terræ ambitu mill. 21600, arcus BA,
in &longs;uperiori &longs;igurâ mill.600 complectitur gra
dus 10. Fiat igitur vt CB Radius 100000.
00000. ad CD 101542.66119 &longs;ecante&mtail;
grad. 10, ita &longs;emidia
metet CB mill. 3437 1/2
proximè ad CD mill.
3490 1/2. Ablatâ autem
femidiametro CA, re
manet AD mill. 53. al
titudo turris Alexandri
næ: quanta videlicet
communiter tribui &longs;olet
halitibus illis, quibus
tum prima illuce&longs;centis diei, tum po&longs;trema
aduentantis noctis crepu&longs;cula debId
pluribus opus e&longs;t explicare? quotus enim
qui&longs;que e&longs;t, qui turri octingentis tale ntis ex
citatæ duûm triumue milliarium altitudi
nem concedat? Pharos igitur In&longs;ula imma
nis &longs;copulus fuit 50 &longs;erè milliaria iuxta per
pendiculum numerans; qua de re mirum
apud &longs;criptores omnes &longs;ilentium: ac proin
de tantæ altitudini parùm vtilis accidi&longs;&longs;et &longs;pe
culæ acce&longs;sio. At &longs;tatuamus cum Erato&longs;the
ne telluris ambitum &longs;tadiorum 252000, &
&longs;tadia &longs;ingula &longs;int Alexandrina |ped. Rom.
720., totus ambitus e&longs;&longs;et mill.Rom. 36288;
igitur milliaria 600 e&longs;&longs;ent gr. 5. m. 57. Qua
re vt Radius 100000.00000. ad 100541.
64449. &longs;ecantem gr.5.m. 57., ita &longs;emidia
meter CB mill. 5775. pa&longs;&longs;. 419. ad CD
mill. 5806. pa&longs;&longs;. 701: ex qua &longs;i au&longs;eratur &longs;e
midiameter CA, remaneret altitudo AD
mill. 31 & pa&longs;&longs;. 282; quæ altitudo adhuc e&longs;t
immanis: & maior hac e&longs;&longs;et, &longs;i &longs;tadia ill&atail;
252000. non. Alexandrina &longs;ed Græca a&longs;&longs;um
pta fui&longs;&longs;ent; e&longs;&longs;et enim CD &longs;ecans anguli
DCB gr.7.m.8.&longs;ec.38. Non itaque &longs;ieri po
tuit, vt ex Alexandrinâ turri ad &longs;excenta pa&longs;
&longs;uum millia pro&longs;pectus pateret.
Longè minor e&longs;t Phy&longs;ici Horizontis &longs;e
midiameter, quàm vulgus cen&longs;eat, &longs;i res Geo-
plitudo pro ma
Hinc quamuis nauclerorum plurimos Geo.
metriâ non &longs;atis in&longs;tructos &longs;æpè fugiat ratio,
eorum tamen varia in determinandâ visûs
di&longs;tantiâ &longs;ententias facilè po&longs;&longs;umus concilia
re; alijs &longs;iquidem ex altiori, alijs ex humilio
ri &longs;peculâ pro&longs; picientibus obiectum idem,
illis quidem longiùs, his verò propiùs &longs;e ob
tulit contemplandum. Quocirca ex huiu&longs;
modi hominum effatis nihil timendum,
quod datâ telluris magnitudine con&longs;titutam
Geometricisque rationibus &longs;olidatam de vi
sûs di&longs;tantiâ &longs;ententiam labefactare po&longs;sit:
duo enim hæc inuicem perpetuo vinculo
colligantur.
At qui&longs;quis ex eorum numero &longs;ue
rit, quos pudet aliquando &longs;apere, & antiquos
errores dedi&longs;cere, vbi visûs di&longs;tantiam eius
opinione minorem demon&longs;traueris, illi
terræ magnitudinem non ritè con&longs;titutam
calumniabitur, & ad Ari&longs;totelem prouoca
bit, qui lib.
2. de Cœlo text. vlt.
&longs;cribit terræ
circuitum patere quadraginta &longs;tadioru&mtail;
myriadibus, hoc e&longs;t &longs;tadiis 400000. aut ad
Archimedem, qui in Arenario telluris am
bitum ter mille &longs;tadiorum millia & eo am
pliù
Prouocet: per me licet.
Eum ta
men monitum velim parum ex Archimede
&longs;perandum, cum eam
iHe magnitudinem da
tâ operâ ponere volue
rit, quam nemo eo
rum, quibu&longs;cum di
&longs;putabat, iu&longs;to mino
rem calumniari po&longs;&longs;et.
Quodverò ad Ari&longs;tote
lem &longs;pectat, non mul
tum habet momen
ti Peripatetica autho
ritas, cui apodictica ratio aduer&longs;atur. Porrò
longè melius e&longs;t opticæ lineæ longitudinem
præcogno&longs;cere, & ex illâ terræ magnitudi
nem inue&longs;tigare, quàm incertis coniecturis
telluris ambitum &longs;tatuere; & infirmo huic
&longs;undamento æquè nutantem de a&longs;pectûs lon
gitudine &longs;ententiam &longs;uper&longs;Sit enim
nota altitudo BA, & ob&longs;eruatus fuerit angu
lus BAD; notus e&longs;t angulus complementi,
qui e&longs;t ad C: ducta autem recta BD facit
angulum BDA, qui e&longs;t &longs;emi&longs;&longs;is noti anguli
ad C, vt paulò antè dicebam. Cum itaque
in triangulo BAD datum &longs;it latus AB, & duo
anguli ad A & ad D innotuerint, inueniri
poterit quantitas lineæ opticæ AD. Tum
& in triangulo rectangulo ABE, datis angu
lo A & latere AB, inueniatur latus BE. De
mum quia triangula ABE, ADC rectangula
habentia communem angulum ad A &longs;unt &longs;i
milia, fiat vt AB data ad BE inuentam, ita
AD lineæ opticæ longitudo inuenta ad DC
quæ&longs;itam terræ &longs;emidiametrum.
Inuenir&etail;
vi&longs;us: & ex
ea Quint&atail;
methodus
vestigandi
terræ &longs;emi
diametrum,
In ijs, quæ hactenus attuli&longs;ti ad
terræ &longs;emidiametrum inueniendam, illud
accidit incommodum, quod totam altitu
dinem &longs;upra maris &longs;uper&longs;iciem innote&longs;cere
oportet: id quod haberi non pote&longs;t, ni&longs;i aut
prærupta rupes mari immineat, aut turris in
litore &longs;it con&longs;tituta. Quin methodum ali
quam excogitas, qua etiam ex turri procul ab
æquore in colle po&longs;itâ explorare po&longs;&longs;imus,
quanta &longs;it terræ magnitudo?
Tentemus pariter, quid po&longs;&longs;imus:
tertius ade&longs;t Galilæus; ni&longs;i aliquid inueneri
mus, dicam nos iratis Mu&longs;is conueni&longs;&longs;e. Sit
igitur in perpendiculari AC, nota in&longs;ignis
aliqua altitudo BA, at non talis, vt eius hu
millimum punctum B &longs;phæricæ &longs;uper&longs;iciei
adhæreat, cum potiùs ex illâ emineat i&ntail;
colle DB, cuius altitudo ignota e&longs;t: &longs;ieri au
tem po&longs;&longs;it, vt liber pro&longs;pectus in Horizon
tem pateat, &longs;iue in &longs;ummo A, &longs;iue in imo B
Quibus angulis ob&longs;er
uatis intellige rectam
BF occurrere Tangen
ti AE in G. In triangu
lo itaque ABG angu
lus AGB e&longs;t notus, vt
pote differentia duo
rum ob&longs;eruatorum C
BG, CAG: angulus A
e&longs;t ob&longs;eruatus, & data
e&longs;t altitudo BA: ergo
inueniri pote&longs;t quantitas rectæ BG. Iam du
cantur rectæ CF, CE, & &longs;unt duo triangula
AEC, BFC rectangula, in quibus duo an
guli EAC, ECA &longs;imul &longs;unt æquales duo
bus FBC, FCB: Atqui angulus ECA e&longs;t æ
qualis duobus ECF, FCB; ergo tres ECF,
FCB, CAE &longs;unt æquales duobus FBC, FCB;
& dempto communi FCB, remanet FBC
æqualis duobus ECF, EAC. E&longs;t igitur ECF
differentia nota duorum ob&longs;eruatorum CAE,
CBF. Ducatur demùm recta CG.
Et quo
niam GF, GE &longs;unt tangentes circulum ab
eodem puncto exeuntes, inter &longs;e æquales
&longs;unt, &longs;icut & CF, CE ex centro ductæ; CG
verò e&longs;t vtrique triangulo FCG, ECG com
munis; ergo angulus ECF notus diuiditur à Quare &longs;i angulorum ob
&longs;eruatorum &longs;emidi&longs;&longs;erentiam GCF addas an
gulo FCB complemento noto anguli ob&longs;er
uati in B, notus e&longs;t etiam angulus BCG; ex
quo, vnâ cum angulo CBG ob&longs;eruato, & la
tere BG inuento, inuenitur latus CB: Cui &longs;i
addatur data altitudo BA, notum erit latus
CA vnâ cum angulo ad A ob&longs;eruato in trian
gulo AEC rectangulo: quare & inuenitur CE
&longs;emidiameter quæ&longs;ita, quæ e&longs;t ip&longs;i CD æqua
lis, vnde innote&longs;cit altitudo collis BD, &
lineæ opticæ AE longitudo cogno&longs;ci pote&longs;t.
Sexta me
tbodus ter
ræ &longs;emidia
metrum in
quirendi, &
montis alti
tudinem co
gno&longs;cendi.
At contingat ex edito quidem monte pro
&longs;pici po&longs;&longs;e in
horizontem, &longs;ed planè
ignotam e&longs;&longs;e montis al
titudnem. Eligatur lo
cus aliquis con&longs;picuus,
qui ita di&longs;tare cen&longs;ea
tur, vt perpendicula
res ex vtroque loco ad
centrum ductæ à pa
ralleli&longs;mo deffectentes
&longs;en&longs;u digno&longs;ci queant.
Hinc enim terræ &longs;emidiametrum eruer&etail;
po&longs;&longs;umus. Ex A igitur &longs;it linea AD terram
tangens: ob&longs;eruetur angulus CAD. Tum
ad ea&longs;dem vel alias partes eligatur locus B
CAB. Di&longs;tantia autem BA vel &longs;it præcogni
ta, vel ex tertio loco ob&longs;eruetur, vt fieri com
muniter &longs;olet. Demum ex B ob&longs;eruetur an
gulus ABC; cognitâ &longs;cilicet di&longs;tantiâ ip&longs;ius
A puncti à vertice ob&longs;eruatoris in B, com
plementum ad duos rectos dat angulum
ABC: erit autem indicium di&longs;tantiæ AB
&longs;ufficientis, &longs;i anguli CAB, CBA &longs;imul &longs;um
pti minores fuerint duobus rectis. Quare in
triangulo ABC dato latere BA & angulis
adiacentibus inuenitur latus AC. Inuento
autem latere AC & ob&longs;eruato angulo CAD
in triangulo CDA rectangulo, inuenitur CD
quæ&longs;ita terræ &longs;emidiameter, nec latebit
montis altitudo.
Septima me
thodus in
uestigandi
&longs;emidiame
trum terræ,
& montis
altitudin
Quod &longs;i locorum opportunitas ferat, vt
detur altitudo FE nota, ex qua ob&longs;eruari
queat angulus CED, & in monte procul po&longs;i
to liber &longs;it a&longs;cen&longs;us, donec ex A puncto per
E in extremum horizontem D productus ra
dius efficiat angulum CAD, qui ob&longs;eruatio
ne cogno&longs;catur; ea habemus, quæ ad inue
niendam terræ &longs;emidiametrum, vel ad exa
minandamiam inuentam &longs;ufficiant. Intel
ligatur enim ex F exire recta FG parallela ip&longs;i
CA (perpendiculares &longs;iquidem EC, AG á
paralleli&longs;mo deflectere ponimus) e&longs;&longs;ormatur
gulus EGF æqualis angulo EAC ob&longs;eruato
innote&longs;cit propter linearum CA, FG paral
leli&longs;mum, & GEF e&longs;t complementum ad
duos rectos anguli FED ob&longs;eruati. Inuenia
tur itaque latus EG; quod ablatum ex EA
di&longs;tantiâ iam notâ duorum locorum, i&ntail;
quibus in&longs;titutæ &longs;unt ob&longs;eruationes, relin
quit GA. At in triangulo EAC, lateri AC
parallela e&longs;t GF, ergo vt EG ad GA, ita EF
data altitudo ad FC quæ&longs;itam &longs;emidiame
trum. Vel etiam ij&longs;dem po&longs;itis, & ob&longs;erua
tis angulis CED, GAE, atque di&longs;tantiâ AE,
in triangulo CAE noti &longs;unt duo anguli, (an
gulus &longs;iquidem CEA e&longs;t complementum ad
duos rectos anguli CED ob&longs;eruati) & latus
adiacens AE: inueniatur igitur latus EC, ex
quo dempta nota altitudo FE relinquit quæ
&longs;itam terræ &longs;emidiametrum. Erit autem
indicium &longs;ufficientis di&longs;tantiæ inter A & E, &longs;i
angulus CED ob&longs;eruatus fuerit maior angu
lo CAE.
Octau a me
thodus in
ueniendi ter
ræ &longs;emidia
metrum.
Idem aliter.
Ea profectò funt, quæ hactenus di
&longs;putata &longs;nnt, vt vix cen&longs;eam fieri po&longs;&longs;e, vt
alicui nulla ex his methodis arrideat. Ve
rùm &longs;crupulus e&longs;t, quem forta&longs;&longs;e ex multo
rum animis non facilè eximas: exi&longs;tima
bunt &longs;iquidem angulum, quem cum per-
quam minimè dubiâ ob&longs;eruatione inue&longs;tiga
ri po&longs;&longs;e. Nam vltimum visûs terminum &longs;i
in terrâ &longs;pectes, quamuis plani&longs;&longs;ima facies
videatur, quî fiat, vt nullus pateat dubita
tionilocus, an molli inclinatione indè per
longa terrarum &longs;patia in mare de&longs;cendatur?
ac proinde linea illa ab oculo exiens non cir
culum in &longs;phæricâ &longs;uperficie contingeret, vt
exigitur. Si verò in immen&longs;um æquor vi
&longs;us excurrat, et&longs;i &longs;opitis ftuctibus otia agat,
nulloque æ&longs;tu intume&longs;cat, quis ne&longs;ciat At
mo&longs;phæram vaporibus non adeò paucis te
nuibu&longs;ue &longs;catere, vt nulla refractionis, qua
Oceani partes infra horizontem depre&longs;&longs;æ
quæ telluris &longs;emidiametro his methodis in
uentæ officere po&longs;&longs;int: Si enim aliâ atque
aliâ methodo inue&longs;tigetur, nec valdè in&longs;i
gni di&longs;crimine differant, quæ inueniuntur,
medium Arithmeticum inter extrema in
uenta dabit quantitatem quæ&longs;itam &longs;emidia
metro terræ tribuendam. Tellus quipp&etail;
toreuma non e&longs;t vndequaque expolitum; &longs;ed
cum eius partes à centro di&longs;paribus interual
lis ab&longs;int, ita tamen vt proximè &longs;phæra&mtail;
æmuletus, facis e&longs;t &longs;i mediocris à centro di-Nihilominus tamen aliam
placet tentare viam nullâ habitâ ratione radii
optici terram tangentis, modò ea &longs;it duo
rum locorum di&longs;tantia, vt perpendiculares
ad centrum notabiliter deflectant à paralle
li&longs;mo.
Nona me
thodus tel
luris &longs;emi
diametrum
inquir ndi.
Eligatur itaque alti
tudo quæpiam in&longs;ignis
& nota AB, ex cuius
vtroque extremo vi
deatur in monte ad
plura milliaria procul
po&longs;ito locus D: loca ve
rò huiu&longs;modi di&longs;tantia
po&longs;&longs;unt ob&longs;eruari poti&longs;
&longs;imùm noctu accen&longs;o
ibi igné ab amico, ni&longs;i
ad&longs;it nota aliqua peculiaris, vt ædi&longs;icium,
turris &c. Tum ex A ob&longs;eruetur angulus
BAD, & ex B angulus ABD, notus enim fiet
reliquus angulus BDA. Ex puncto autem
D ob&longs;eruetur angulus BDC, vnde ablato an
gulo BDA noto, reliquus ADC innote&longs;cit.
His paratis in triangulo ABD dantur duo an
guli A & B cum latere adiacente BA, inue
niatur igitur latus AD. Ex hoc autem late
re AD inuento vnâ cum angulo ADC, qu
ex ob&longs;eruatione innotuit, & angulo DAC
DAB ad duos rectos, inueniri pote&longs;t latus
AC quæ&longs;ita terræ &longs;emidiameter.
At non vacat montem con&longs;cendere?
alia
&longs;uppetit via, qua leuiori labore propo&longs;itum
a&longs;&longs;equamur. Sit nota altitudo IG, vnde vi
deatur locus aliquis in &longs;uperficie terræ F: ob
&longs;erueturque angulus IGF. Quod &longs;i di&longs;tantia
GF iam nota non
fuerit, ex tertio
quopiam loco e&atail;
ob&longs;eruetur,
niaturque
gonometriæ præ
cepta. His com
paratis producatur
GI per terræ cen
trum T v&longs;qu&etail;
in O, vt IO &longs;it
terræ diameter: ex
F cadat in dia
metrum perpendicularis FC. Quonia&mtail;
igitur in triangulo GCF rectangulo datur hy
pothenu&longs;a GF, & angulus CGF, inueniantur
reliqua latera FC, CG. Auferatur autem
IG data altitudo ex GC, & remanet IC.
Quia verò FC à puncto peripheriæ F cadit
perpendicularis in diametrum IO, e&longs;t me-
menta IC, CO; ac proinde quadratum me
diæ CF æquale e&longs;t rectangu lo &longs;ub extremis.
Itaque quadrato ip&longs;ius CF diui&longs;o per IC &longs;eg
mentum notum, Quotiens dabit CO, cui
addatur CI, & habetur tota diameter IO
quæ&longs;ita.
Decima me
thodus, qua
terræ dia
meter inue
nitur.
Idem aliter.
Hæc mihi methodus magis arridet vtpo
te breuior, qua quadratum CF diuido per
IC: cæterùm &longs;i rem merè Trigonome
tricè perficere quis malit, inuentis CF
& IC, quærat angulum IFC; quem du
plicet, & dupli &longs;inum ex Tabulis inue
niat: tum fiat vt &longs;inus inuentus ad Ra
dium ita inuenta CF ad quæ&longs;itam &longs;emidia
metrum FT. Huius operationis ratio pa
tet, quia cum angulus IFO in &longs;emicircu
lo &longs;it rectus, triângulum ICF e&longs;t &longs;imile trian
gulo IFO, igitur angulus inuentus IFC æqua
lis e&longs;t angulo FOC; huius autem duplex e&longs;t
angulus ITF ad centrum, qui proinde inno
te&longs;cit, vnâ cum &longs;inu FC in partibus Radii
hæc verò linea cum nota &longs;it etiam in men
&longs;urâ homogeneâ altitudini datæ IG, manife
&longs;tabit pariter in eâdem men&longs;urâ Radium TF.
Placetne aliam adhuc inire viam?
nec
Diony&longs;iodori Geometræ protritam ve&longs;tigiis;
cuius in &longs;epulchro inuenta e&longs;t epi&longs;tola ad
terram de&longs;cendi&longs;&longs;e &longs;ignificabat, illudque &longs;pa
tium &longs;tadia 42000. complecti. Sint datæ al
titudo eadem IG, ac di&longs;tantia GF, & angu
lus IGF ob&longs;eruetur. Ducatur ex I tangens
IH; quæ &longs;ecet GF in H. Quæratur ergo i&ntail;
triangulo GIH rectangulo, ex dato latere GI
& angulo G, latus GH, quod ex GF ablatum
relinquit HF, & quæratur latus IH. Deinde du
ctâ rectâ IF, in triangulo IHF nota &longs;unt la
tera IH & HF, angulus autem compræhen
&longs;us IHF æqualis e&longs;t duobus internis notis, &longs;ci
licet recto GIH, & IGH ob&longs;eruato: quar&etail;
inueniri pote&longs;t tum latus IF, tum angulus
HIF: cui æqualis e&longs;t angulus IOF in alterno
&longs;egmento: huius autem duplus e&longs;t angulus
ITF ad centrum. Fiat igitur vt &longs;inus &longs;emian
guli inuenti ITF, hoc e&longs;t&longs;inus angult HIF,
ad Radium, ita &longs;emi&longs;sis inuenti lateris IF
ad quæ&longs;itam &longs;emidiametrum TF.
Vndecim&atail;
methodus
inueniendi
terræ &longs;emi
diametrum
Compendio&longs;iùs forta&longs;&longs;e operabimur, &longs;i da
tis IG & GF cum angulo G compræhen&longs;o,
inueniatur ba&longs;is IF & angulus GIF, qui e&longs;t
nece&longs;&longs;ariò obtu&longs;us. Fiat ergo vt &longs;inus exce&longs;
sûs anguli GIF &longs;upra rectum ad Radium,
ita &longs;emi&longs;sis inuentæ ba&longs;is IF ad quæ&longs;itam
terræ &longs;emidiametrum.
Id
Verùm omi&longs;&longs;o tot linearum apparatu res
facillimè conficitur, e
tiam&longs;i nulla &longs;it data al
titudo nota. Ob&longs;erue
tur di&longs;tantia AD: tum
in A accipiatur angulus
CAD, & in D ob&longs;erue
tur angulus, quem cum
verticali lineâ ex D ad
Zenith productâ facit
radius opticus DA; hu
ius enim complementum ad duos rectos
e&longs;t angulus ADC: dato autem latere AD
cum angulis adiacentibus, latere non po&longs;
&longs;unt reliqua latera CD quæ&longs;ita terræ &longs;emidia
meter, & CA &longs;emidiameter aucta montis
AB altitudine.
Duodecima
methodus
&longs;emidiame
trum terræ
inuestigan
di, & montis
altitudini.
Sed iam, Mer&longs;enne, cadunt altis d&etail;
montibus vmbræ; nec Galilæum diutur
niore colloquio di&longs;tinere &longs;as e&longs;t.
Rectè mones: præceps in noctem
ruit dies: &longs;ed huiu&longs;modi colloquia cum in
&longs;tituuntur, omnis hora momentum e&longs;t. Va
le Galilæe.
Valete amici; & quas debeo gratias
pro humani&longs;simâ con&longs;uetudine, nunc planè
non reddo, &longs;ed apud me &longs;eruo, vt iterum
redire ccgamini depo&longs;itum repetituri.
QVARTA
ra, motus facilitatem
infert.
PVRGARE ne tibi m&etail;
debeo, Galilæe, quòd con
dictam Mer&longs;enno horam te
adeundi importunus præ
uenetim?
Si putas Batauis in nouâ Zem
blâ po&longs;t longas dierum 84 tenebras lu-
pantem accidi&longs;&longs;e importunum, ac purgatio
ne opus habui&longs;&longs;e, quòd legitimum exoriendi
tem pus anteuerterit; præ&longs;tò &longs;um, vt intelli
gam, qua te demum excu&longs;atione ab imma
ni hac culpâ eximas. Sed cum expectatus
adueneris, quam tuorum in me meritorum
partem referendâ gratiâ con&longs;equi potero?
Culpam hanc omnem, quantæ
cunque e&longs;t, in te transfero: id &longs;cilicet profu
sâ tuæ humanitatis &longs;ignificatione effeci&longs;ti, vt
mihi liceat e&longs;&longs;e temerario; nec ab&longs;urdu&mtail;
duxerim &longs;tatim, ac me tui de&longs;iderium ce
pit, aduolare, & immaturum fœtum, ve
riùs dixerim víx dum conceptum, ante t&etail;
ponere, vt vitæ igniculos illi tuâ luce imper
tias.
Siccine iuuat fe&longs;tiuis inanium offi
ciorum argutijs iocari? Mittamus i&longs;thæc: &
quam primum edi&longs;&longs;ere, quæ te benigna ege
rit Minerua, vt frigidos no&longs;tros cineres fodi
catum venires, ignem ætheriâ vtique domo
&longs;ubductum depo&longs;iturus.
bebam; cumque aliud meditans pagellas te
merè oculo percurrerem, incidi in textum
78. lib.
2. de Cœlo, vbi Thaletis Mile&longs;ii d&etail;
causâ terræ quie&longs;centis &longs;ententiam exponit,
neat, quemadmodum lignum vel aliquid tale
aliud; etenim horum &longs;uper aerem quidem ni
hil natura aptum e&longs;t manere, &longs;ed &longs;uper aquam.
Id quod mihi in mentem reuocauit eadem
pror&longs;usapud Senecam à me iam pridem le
cta lib.
6. Nat. quæ&longs;t. cap.
6. terram videli
cet totam, Thaletis opinione, &longs;ubiecto hu
more portari, & innatare, ita vt vndâ &longs;u&longs;ti
neatur orbis velut aliquod grande nauigium.
Ex quo ille ab eodem Senecâ lib.
3. cap.
13.
reiectus non &longs;atis aptèterræ motuum cau&longs;am
inferebat, perindè atque &longs;i nauigium hoc in
natans concuteretur. Tum, quæ e&longs;t phan
ta&longs;matum atque formarum menti inhæren
tium mira connexio, vix cæperam tacitus
ridere commentitium ingentis huius nauigii
nullos in &longs;copulos impacti tremorem, cum
in eius grauitatis con&longs;iderationem delap&longs;us
&longs;um: tenuique hac vellicatione excitatæ he
&longs;terni no&longs;tri congre&longs;&longs;us reliquiæ animo in&longs;i
dentes me protinus abripuerunt, & &longs;en&longs;im
nec aduertentem deduxerunt ad eam cogi
tationem, vt &longs;u&longs;picarer ex aquarum naturâ,
ad tellurem Archimedæis machinationibus
mouendam, &longs;ub&longs;idium aliquod companari
po&longs;&longs;e. Plura illicò huic cogitato a&longs;&longs;inia i&ntail;
mentem confusè & permixtè irruperunt,
quere viderentur; &longs;ed quoniam, vbi multa
&longs;unt, quæ &longs;ibi aptis nexibus cohærere de
beant, ne veritatis compages luxata pereat,
periculo&longs;um & lubricum e&longs;t facilè a&longs;&longs;entiri,
nolui me in præcipitem locum committere;
&longs;u&longs;tinendam potiùs tanti&longs;per omnem a&longs;&longs;en
&longs;ionem duxi, dum rem totam di&longs;cretè &
electè &longs;uas in partes, te pro tuâ &longs;apientiâ
opem ferente, digererem. En habes, qui
bus &longs;timulis actus mihi imperare non potue
rim, vt à te diutiùs abe&longs;&longs;em.
Gratias tibi habeo, mi Guldine, im
mortales; quod &longs;pem iniicias audiendi ex te
hodie, quid de iis &longs;entias, quæ iam tum ab
anno huius &longs;æculi duodecimo con&longs;crip&longs;i de
Innatantibus.
Librum legi tuo dignum ingenio;
nec potuit feliciùs enodari, quam con&longs;titue
ras examinandam quæ&longs;tionem de &longs;olidis in
natantibus, quamuis humido &longs;ecundùm &longs;pe
ciem grauioribus, &longs;i quidem &longs;olitaria &longs;uman
tur, componentibus tamen vnâ cum aër&etail;
&longs;ibi adhære&longs;cente molem aquâ non grauio
rem. Illud maximè dolui, quod nactus &longs;im
exemplar &longs;iue Bibliopolæ &longs;ine Bibliopegi in
curiâ mutilarum integro folio, & quidem il
lo ip&longs;o, in quo totius futuræ di&longs;putationis
e&longs;t fortuna, nu&longs;quam licuit integrum librum
reperire, ex quo no&longs;tri exemplaris hiatum
&longs;upplerem, Id quod te præmonni&longs;&longs;e oppor
tunum fuit, ne, &longs;i fortè nobis non conuene
rit, me contradicendi &longs;tudio actum putes:
ni&longs;i me tamen mea fallit opinio, in minimis
di&longs;&longs;entiemus.
Libens audio, qui contra &longs;en&longs;erint.
Sed ne te longiùs ab in&longs;tituto tuo, veriùs di
cam, no&longs;tro, abducam; quidnam ex aquâ
emolumenti &longs;peras ad facilem terræ motio
nem, de qua nobis fuit di&longs;putatio?
Futurum puto, vt plurimum pon
deris de terreno hoc orbe, qui aquam pariter
ac terram complectitur, deduceretur. Illud
enim extra omnem controuer&longs;iam po&longs;itum
accipio, quod aqua aëre grauior e&longs;t; & fluida
cùm &longs;it, &longs;emper ad inferiora delabitur, vt
infra aërem vniuer&longs;icentro vicinior con&longs;i&longs;tat.
Ex quo fit, vt nu&longs;quam quie&longs;cat, ni&longs;i vbi
nullus patet locus, in quem de&longs;cendat. Qua
rè cum &longs;ola &longs;uperficies &longs;phærica paribus ra
diis à centro remoueatur, aquæ quie&longs;centis
&longs;uperficiem &longs;phæricam e&longs;&longs;e nece&longs;&longs;e e&longs;t: quan
doquidem &longs;iquæ &longs;uperficiei partes à centro
longiùs abe&longs;&longs;ent, vtpote altiores ad motum
procliues non &longs;ub&longs;i&longs;terent, &longs;ed humiliorem
quam &longs;ponte a&longs;cendere dixerim, quòd ea in
vitreis fi&longs;tulis immer&longs;is aliquantulum a&longs;cen
dat.
Aquæ &longs;u
perficies est
&longs;phærica.
Nemo id facilè inficietur: immo, &longs;i
id quidem in rem tuam faciat, vltro dabo
maria omnia, quæ aquarum communion&etail;
iunguntur (&longs;i æ&longs;tum omnem &longs;ublatum, ven
tosque &longs;ilentes animo fingamus) non e&longs;&longs;&etail;
alia alijs &longs;ecundùm &longs;uperficiem altiora: qui
bus enim frænis cohiberentur Sinûs Arabici
aquæ, &longs;i altiores e&longs;&longs;ent, ne in Erytræum ma
re in&longs;luerent? aut quibus aggeribus ob&longs;true
retur fretum Herculeum, ne Oceanus Atlan
ticus ac Mediterraneum mare in vnâ &longs;u
perficie aquarum libramentum &longs;u&longs;ciperent?
Quod verò de aquâ in immer&longs;is fi&longs;tulis
vtrinque hiantibus a&longs;cendente addis, nihil
planè officit naturæ aquarum &longs;e in &longs;phæram
circa terræ centrum conglobantium; neque
illicò po&longs;&longs;e aquam ad digitivaltitudine&mtail;
&longs;ponte a&longs;cendere affirmandum e&longs;t, quòd il
lam in tenui&longs;&longs;imis fi&longs;tulis eò pertingere ali
quando videamus; quò enim ampliores &longs;unt
fi&longs;tulæ, eò minùs in iis aquam af&longs;urger&etail;
con&longs;tat, nec fortè &longs;ine &longs;u&longs;picione minoris,
quàm appareat, altitudinis, propter &longs;pecie
rum vi&longs;ibilium ex vitro refractionem, i&ntail;
quantùm illa a&longs;cendat, quod &longs;phæricam &longs;u
perficiem corrumpat? Sed nec aquam om
ninò &longs;ponte dixerim in fi&longs;tulâ a&longs;cender&etail;
Quando enim fi&longs;tula deprimitur, vt aquæ
immergatur, vtique &longs;ubiectus aër premitur,
& locum &longs;ubeunti fi&longs;tulæ concedens, quâ
patet via, recedit, vt locum &longs;uppleat à fi&longs;tu
lâ deor&longs;um motâ relictum. Sicut autem
aër, cui corpus in motu occutrit, comprimi
tur, ita is, qui ponè e&longs;t, paululum di&longs;trahi
tur ac rare&longs;cit; hic verò ad ingenium rediens
proximum aërem attrahit ad &longs;upplendum
locum à fi&longs;tulâ relictum; cumque nullus
&longs;ubiectus aër tam in promptu &longs;it, quàm is
qui fi&longs;tulæ cauitatem implet, hic ex&longs;ugitur;
atque adeò cum aér fi&longs;tulæ &longs;ur&longs;um mouea
tur, &longs;ubiectus aër compre&longs;&longs;us in fi&longs;tulæ caui
tatem &longs;uccedit, per quam faciliùs elabitur,
ac per impul&longs;ionem, aut compre&longs;&longs;ionem
contigui aëris, qui lateribus adiacet. Por
rò ea e&longs;t fluidorum corporum natura, vt
conceptum ex motu impetum, etiam intrà
homogeneum corpus, non ita facilè remit
tant; quemadmodum in aquâ colore aliquo
infectâ intrà aliam aquam leui&longs;&longs;imè effusâ
videre e&longs;t. Hinc e&longs;t aërem &longs;ur&longs;um in &longs;i&longs;tulâ
incitatum ex concepto impetu tanti&longs;per per-
rem, vt hic eum qui fi&longs;tulæ latera extrin&longs;e
cùs ob&longs;idet, propellat, fiatque per inferius
hiantis fi&longs;tulæ o&longs;culum illa circumpul&longs;io, de
qua Pla
per fi&longs;tulam liberrimè ab&longs;que vllo alterius
aëris intercurrentis obice, attrahitur aër in
ferior.
Cur aqu&atail;
in tubo v
trinque bi
ante a&longs;een
dat aliquan
tulum.
Veriùs &longs;orta&longs;&longs;e dixeris & attrahi
pariter & circumpelli:
Vbi igitur inferioris o&longs;
biectam aquam ita attigerint, vt aer &longs;uece
dere nequeat, illud nece&longs;&longs;ariò fit, vt a&longs;cen
dentem aetem aqua &longs;ubiecta con&longs;equatur
&longs;iue attracta, &longs;iue ex circumpul&longs;ione propul
&longs;a. Cum autem hoc à naturâ liquoribus
comparatum &longs;it, quod &longs;olidis corporibus ad
hære&longs;cant; vbi aliquid aquæ &longs;ur&longs;um ab aere
a&longs;cendente eleuatæ, internis fi&longs;tulæ lateribus
adhæ&longs;eri
vim aéris a&longs;cendentis, qui propterea in reli
quam aquam non adhære&longs;centem vires &longs;uas
omnes exercet. At quia in tenui&longs;simâ fi&longs;tu
lâ cylindrulus aquæ primùm eleuatus ita ferè
totu
quat capillarem medullam ab aëre &longs;u&longs;tenta
tam, multò faciliùs pergic in a&longs;cen&longs;u, & ali
demum præter aquam &longs;ponte adhærentem
lateribus, tantum aquæ ab aere &longs;u&longs;tineatur,
vt eam dimittere non po&longs;&longs;it, quin ip&longs;e magis
di&longs;trahatur & rare&longs;cat: id quod natura potiùs
refugit, quàm permittere tantillulæ aquæ
&longs;u&longs;pen&longs;ionem. Ni&longs;i fortè malles dicere, a
quam illam medullarem con&longs;titutam intrà
aquam lateribus adhærentem iam non conari
deor&longs;um. Quando verò amplior e&longs;t fi&longs;tulæ
capacitas, con&longs;tat
te per eam ferri &longs;ur&longs;um cæteris paribus, ac per
fi&longs;tulam tenuiorem, & præterea cylindrulus
aquæ &longs;ur&longs;um attractus ampliorem habet ba
&longs;im, & in minori altitudine habetur tota ea
quantitas aquæ, quæ valet ab aere &longs;u&longs;tentari:
hinc fit eò minùs aquam attolli, quò am
plior e&longs;t fi&longs;tula. Mitto hìc di&longs;putare an in
&longs;en&longs;ibilis aquæ expiratio adhærens fi&longs;tulæ,
vel in eam incurrens, excitet electricam
vitri expirationem; an verò ea fi&longs;tulam de
ui&longs;&longs;imè humectans, dum attactu illo &longs;tatim
concre&longs;cit, &longs;tatim &longs;ubiectam aquam attrahat
vt &longs;ibi vniat, quemadmodum in calamo
&longs;criptorio &longs;æpe ob&longs;eruare e&longs;t, quando atra
mento valdè diluto vtimur; vix enim hu
mens calamus &longs;ubiectum atramentum con
tingit, cum co confe&longs;tim imbuitur. Hæc
cerent. Illud certum e&longs;t, quod aqua &longs;ponte
fluens ita &longs;emper ad humiliora loca delabi
tur, vt à &longs;phæricâ &longs;uperficie non recedat, &longs;i
res Phy&longs;icè &longs;altem con&longs;ideretur. Atque adeò
id tibi vltrò concedens audire expecto, quid
indè conficias.
Non eandem e&longs;&longs;e &longs;emper aquæ &longs;u
perficiem; quò enim maioribus à centro in
teruallis &longs;emouetur, eò propiùs æquata&mtail;
planitiem æmulatur; at centro vicinior ma
iori conuexitate inflectitur; hìc &longs;cilicet mi
noris, ibi maioris &longs;phæræ portio e&longs;t. Iam
verò vt pla
niùs & aper
tiùs verbis
complectar,
quod volo,
ex Graphi
de Sit pro
globo terra
queo circu
lus ABCD,
cuius cen
trum T
gruat
tiatur appellationem. Ambigi non pote&longs;t,
quin maria hæc in eadem &longs;phæricâ &longs;uperficie
exi&longs;tant, quandoquidem ab vniuer&longs;i centro
T paribus interuallis di&longs;iunguntur. At &longs;i
tellurem ab vniuer&longs;i centro (quod, ne i&ntail;
vocabulis laboremus, centrum grauiu&mtail;
liceat appellare, cum eius rationem habue
rit natura &longs;uum cuique corpori locum tri
buens) remotam intelligamus, ita vt illius
quidem centrum &longs;it T, centrum verò gra
uiu
triu&longs;que maris &longs;uper&longs;icies; &longs;ed Hyperboreum
&longs;ub&longs;idere magis & explicari debet, Indicum
verò a&longs;&longs;urgere, Cum enim aqua A remo
tior &longs;it quàm B & E à centro V, pote&longs;t de
&longs;cendere, nec con&longs;i&longs;tet, ni&longs;i vbi fuerit &longs;u
per&longs;icies BIE. Contra autem aqua F & D
remotior e&longs;t quàm C à centro V; pote&longs;t
igitur ver&longs;us C de&longs;cendere; & relicto loco
ad F & ad D, a&longs;&longs;urget
ficies FHD maiorem habens conuexita
te&mtail;.
Si tellus a
liò trabere
tur, aqu&atail;
mutaret fi
guram.
Quod &longs;i ad latus iaceat aqua, vt MNO,
factâ translatione centri ex Vin T, vtique
ex M versùs O de&longs;cendet; &longs;ed &longs;i mons OS
prohibeat, demùm con&longs;i&longs;tet aqua circa vni
uer&longs;i centrum V in &longs;uper&longs;icie &longs;phæricâ RNS.
cæteris e&longs;to iu
dicium; nulla
&longs;iquidem repe
riri po&longs;&longs;et a
quarum con
gregatio, quæ
tellure transla
tâ, &longs;e aliam in
conglobaret,
altioribus par
tibus in inferiorem locum delap&longs;is.
Noua bypo
the&longs;is mari
ni æstus in
di
non proba
tur.
Igno&longs;ce, quæ&longs;o, interpellanti.
Ne
&longs;cio quam mihi inijcis &longs;u&longs;picionem no
hypothe&longs;is, qua citra omnem
ginem &longs;iuè in orbe annuo, &longs;iuè circa &longs;uum
axem, marinus æ&longs;tus explicari po&longs;&longs;et: &longs;i ni
mirum terræ centrum lento ac tenui motu
vltro citroque commeans centro grauiun
congrueret &longs;olùm in medio fluxu aut reflu
xu. Si enim T accedat ad V, aqua incipit
ex S refluere ver&longs;us M, vbi aqua &longs;emper au
getur, quò magis centrum T recedit ab V
in X: iterumque fluit ex M in O, cum cen
trum ab X recedens in V & in T &longs;ua per
ve&longs;tigia eò remeat, vnde di&longs;ce&longs;&longs;it, &longs;eruatâ
que in motu reciprocando con&longs;tantiâ, al-
uer&longs;is litoribus. Quod &longs;i in oppo&longs;itis eiu&longs;
dem aluei ripis eodem tempore fluxus con
tingat aut refluxus, vt in B & E, aut in F &
D; tunc opinari quis po&longs;&longs;et mare illud eam
habere po&longs;itionem, vt in illud incidat linea
motûs, quæ ex centro grauium per terræ
centrum ducitur: dum enim aqua deprimitur
ex A in I, augetur in B & E, &longs;i litus fluxui
ob&longs;tet, aut vlteriora &longs;patia occupat in P & K:
dum verò a&longs;&longs;urgit ex C in H, minuitur in F &
D. Porrò alternâ hac nutantis terræ mo
tione non magis eius &longs;tabilitas vacillaret,
quàm illiùs firmitati ob&longs;it trepidationis mo
tus à pleri&longs;que admi&longs;&longs;us ob variam centri
grauitatis po&longs;itionem: &longs;tare &longs;iquidem terra
dicitur, quæ &longs;uo ex loco non decedit, quam
quam in eo &longs;uæ diametri particulam (1/4000000)
percurrat. Finge enim motûs extremos
terminos T & X non ampliùs pa&longs;&longs;um vnum
di&longs;tare à grauium centro V: tanta e&longs;t aqua
rum in immen&longs;um patentium copia, vt èxi
guâ inclinatione, quam motu illo acquire
rent, dilabentes æ&longs;tum non exiguum effi
cerent: qui tamen in lacubus, angu&longs;tiori
busque alueis ob minorem aquarum copiam
non perciperetur.
Blandiuntur i&longs;thæc facilè mentis
otium liceret, haud &longs;atis &longs;cio, quàm aptè
commentum hoc cum marini æ&longs;tûs phæno
menis cohæreret. Nec pauca in hanc &longs;en
tentiam afferri po&longs;&longs;ent: &longs;ed non vacat his im
morari, ne longiùs ab in&longs;tituto digrediar,
aut &longs;ortè, quæ e&longs;t fugacis memoriæ incon
&longs;tantia, excidant, quæ nunc animo obuer
&longs;antur. De mari certè, quod Kiùn, in In
&longs;ulâ Hainan, alluit, illud notatu dignum
&longs;cribunt Sinæ Geographi, quod maris æ&longs;tum
Diurnum non &longs;entiat, &longs;ed per dimidiam
men&longs;is partem versùs Ortum, per reliquam
dimidiam versùs Occa&longs;um fluat. Quid?
Quod in vertice montis Hucùng in Fokien
Prouinciâ puteus e&longs;t (Hiai nomen e&longs;t) cuius
aqua æ&longs;tum marinum &longs;uo acce&longs;&longs;u & rece&longs;&longs;u
refert. Adde huic fontem prope Nuikiang
in Suchuen Prouinciâ, quem modò a&longs;cen
dentem modò de&longs;cendentem æ&longs;tûs marini
horas adeò procul à mari &longs;equi ob&longs;eruatum
e&longs;t. Hos autem æ&longs;tus ex motu illo orbis,
quem innuebas, non oriri palàm e&longs;t. Quòd
&longs;i ex tam remotâ Sinatum regione petitum
argumentum reijcis, mihi &longs;anè con&longs;tat non
hanc e&longs;&longs;e æ&longs;tûs effectricem cau&longs;am; nam
ip&longs;a quoque flumina, terræ centro accedente
ad centrum grauitatis aut recedente, diebus
mitterentque, aut etiam &longs;uum in caput re
laberentur; id quod nondum licuit ob&longs;erua
re. Præterquam quod vix dixeris, quo mo
uente tellus vici&longs;&longs;im commearet, citrà fabu
larum figmenta.
At &longs;i contingere po&longs;&longs;et, vt machinarum
ope telluris centrum ex V in T transferretur,
fluminis GL aqua ex G reflueret in L, & in
eundem alueum &longs;e in&longs;inuaret mare v&longs;qu&etail;
dum tota aquæ &longs;uperficies continua in &longs;phæ
ram inflexa con&longs;i&longs;teret, cum non haberet,
quò de&longs;cenderet. Hinc illud fit, quod cum
maria omnia (ni&longs;i fortè Ca&longs;pium velis exci
pere, cui tamen per &longs;ubterraneos cuniculos
cum Ponto Euxino communio e&longs;t) &longs;ibi in
uicem continuata iunctaque &longs;int, terrâ in
partem vnam translatâ, aquæ ferè omnes in
oppo&longs;itam recederent, vel per patentem al
ueum dilabentes, vel etiam exundantes:
idem quippe tunc aquarum conceptaculis
contingeret, ac &longs;i nunc vas liquore qua&longs;i
plenum magis & magis inclinaretur, totus
enim demùm liquor effunderetur. Quarè
vniuer&longs;a ferè aqua &longs;uperiorem locum relin
quens in H conflueret eò velociùs, quò lon
giùs telluris centrum T recederet ab V cen
tro grauium &longs;eu vniuer&longs;i.
Tellur&etail;
translata fe
rè tota aqua
ab ea &longs;eiun
geretur.
In eo igitur, &longs;i quid video, machi
nalis motionis, qua terra transferretur, faci
litatem con&longs;tituis, quod eo ip&longs;o tempore,
quo terra &longs;upra FVD planum (quod ho
rizontale vocetur) attolleretur, aqua de
&longs;cenderet; ac proinde &longs;uperioris partis pon
dus minueretur, donee demum terra pro
cul à centro translata totius ferè aquæ ponde
re leuaretur, quæ cirea vniuer&longs;i centrum V
aqueum in globum &longs;uis &longs;e nutibus confor
maret.
Ita planè: nec illud quidem con
temnendum e&longs;&longs;et compendium, &longs;i ponderis
aquæ rationes ineamus. Quotam enim ter
raquei huius globi partem con&longs;tituendam
e&longs;&longs;e aquam cen&longs;es?
Res e&longs;t, in qua Geometriæ apices
per&longs;equi non po&longs;&longs;umus, cum exactâ ma
rium omnium notitiâ careamus, & planè
varia &longs;it aquarum profunditas: qua propter
coniecturis contentos nos e&longs;&longs;e oporter. Et
quidem quod ad aquarum &longs;uperficiem per
tinet, eas arbitror æquis cum terrâ portioni
bus globi conuexitatem di&longs;pertire: altitudi
nem verò adeò incon&longs;tantem reperio, vt &longs;i
profunditates maiores cum minoribus com
pen&longs;emus, vix vltra milliaris dodrantem aut
integrum milliare altitudini in vniuer&longs;um
&longs;iquidem illæ altitudines & aby&longs;&longs;i aquarum
rari&longs;&longs;imæ &longs;unt, quæ bolide nullâ explorari
po&longs;&longs;int. Idcircò libentiùs vuiuersè altitu
dinem &longs;olùm dodrantalem aquis tribuo, mil
liari a&longs;&longs;umpto pro a&longs;&longs;e. Iam, &longs;i placet,
aquæ grauitatem ad calculos reuocemus, &
globi perimetrum accipiamus, quam olim
Mer&longs;ennus, nobis non repugnantibus, con
&longs;tituebat mill. Rom. ant. 25941. Perime
tro con&longs;titutâ, diametrum &longs;ic inquiro ex ra
tionibus Vietæ: Vt 31415.926536 ad
10000.000000, ita maximi circuli periphe
ria mill. 25941 ad diametrum mill. 8257
pa&longs;&longs;. 276. Igitur &longs;idiameter in peripheriam
ducatur, producetur &longs;phærica &longs;uperficies
mill. quadrat. 214.201996. pa&longs;&longs;.
quad.
716000: Huius autem &longs;emi&longs;sis mill. 107.
100998. pa&longs;&longs;. 358000. erit &longs;uper&longs;icies a
quæ.
Totius a
quæ quanti
tas, & gra
uitas in qui
ritur.
Et vt breuitati &longs;eruiam, duco inuentam
aquæ &longs;uperficiem in altitudinem con&longs;titutam
mill.3/4: ne &longs;cilicet longioribus ambagibus in
quiram totius globi &longs;oliditatem. Deinde in
&longs;titutâ analogiâ,
ad cubum eiu&longs;dem diametri mulctatæ dodran
te milliaris, ita &longs;oliditas &longs;phæræ ex inuenta dia-
niam harum &longs;phærarum differentiam, cuius
differentiæ &longs;emi&longs;&longs;is aquarum &longs;oliditati tri
buatur. Neque enim e&longs;t operæ pretium
nos ip&longs;os hoc labore conficere; quandoqui
dem vix di&longs;creparet inuentus numerus ab eo,
qui ex ductu &longs;uperficiei in altitudinem pro
dibit. Duco igitur aquæ &longs;uperficiem mill.
107.100998 pa&longs;&longs;. 358000 in altitudinem
mill. 3/4, & prodit &longs;oliditas milliarium cubi
corum 80.325748, & pa&longs;&longs;. cubic.
768.
500000.
Inuentâ &longs;oliditate grauitatem inue&longs;tigo:
& quamuis &longs;ciam marinam aquam ob ad
mixtam &longs;alis copiam grauiorem e&longs;&longs;e aquâ
communi; vt tamen compen&longs;etur, &longs;i quid
plus æquo tributum e&longs;t &longs;uperficiei, aut pro
funditati, communem aquæ grauitatem ac
cipio; Et quoniam milliaribus Romanis an
tiquis vtimur, pondus pedis cubici antiqui,
hoc e&longs;t amphoræ aquâ plenæ, e&longs;t lib.
80.
Igitur quia milliare cubicum con&longs;tat pedi
bus &longs;olidis 125000.000000, hic numerus per
lib.
80 ductus dabit libras 10.000000.000000
grauitatem &longs;ingularium milliarium cubico
rum aquæ. Iam &longs;i &longs;oliditas mill. 80.325748
pa&longs;&longs;. 768.500000. ducatur per libras 10.
000000.000000. erit totius aquæ pondus lib.
Pondus ter
reni globi
quàm nota
biliter mi
nueretur ex
aquæ &longs;ece&longs;
&longs;iane.
Hoc autem pondus ex totius globi graui
tate demptum faciliorem efficeret terræ
motionem, vbi iam tota terra aliquou&longs;que
ab vniuer&longs;i centro rece&longs;&longs;i&longs;&longs;et, ibique aquam
ferè totam, quæ nunc in globi &longs;uperfici&etail;
fluit, reliqui&longs;&longs;et. Sedquid &longs;i in maris fundo
amplæ pateant voragines, per quas in tellu
ris cuniculos aqua &longs;e in&longs;inuer, ingentesque
aby&longs;&longs;os crect? An non per hiatus eo&longs;dem
aqua &longs;e exoneraret tellure in altum &longs;ublatâ
Ex quo & illud con&longs;equeretur, quod in im
men&longs;a illa hydrophylacia aër tenuisque va
por defluenti aquæ &longs;uccederet, fieretque to
tius compo&longs;itæ molis grauitas &longs;ecundùm
&longs;peciem minor. Verùm terræ vi&longs;cera n&etail;
&longs;crutemur; aquasque illas hypogæas &longs;uis in
concepraculis &longs;tagnantes relinquamus. A
qua certè, quæ terræ faciem nunc alluit,
tanta e&longs;t, vt &longs;i eam di&longs;ce&longs;&longs;i&longs;&longs;e in locum alium
à rellure animo concipiamus, continuo non
exigua &longs;anè momenta ex globo terraqueo
dempta intelligamus. Et &longs;i ad manum e&longs;&longs;et
&longs;chedula, in qua Mer&longs;ennus rationes dige&longs;&longs;it,
quibus telluris grauitatem nudius tertius in
ue&longs;tigabat, numerum hunc librarum ex illo
&longs;ubducentibus con&longs;taret non planè contem
nendum e&longs;&longs;e hoc laboris compendium.
Secum illam tulit Mer&longs;ennus: &longs;ed
non e&longs;t opus grauitatem hanc aquæ cum to
tius globi pondere comparare, &longs;atis enim pet
&longs;e patet, quàm in&longs;ignis foret i&longs;ta ponderis
dece&longs;&longs;io. Vnum autem hìc ob&longs;eruo, quod
nimirum, quamuis ingens hæc ponderis di
minutio tunc &longs;olùm contingeret, quando
terra e&longs;&longs;et ab aquis diuul&longs;a, & ab vniuer&longs;i
centro V tota abe&longs;&longs;et, emolumentum tamen
non paruum faceret in motione a&longs;&longs;iduus a
quarum fluxus in partem oppo&longs;itam. Si enim
globus plano FD &longs;ectus eò transferatur, vt
vniuer&longs;i
trum
no, non &longs;o
lùm
cipitur toti
us globi, &longs;ed
netotius qui
dem
ti
uitas:
doquidem
mentum FA
D de or&longs;um non conatur contra vim &longs;ur&longs;um
mouentem aut &longs;u&longs;tinentem, ni&longs;i iuxta mo
menta ponderis, quibus &longs;uperat moment&atail;
trum V nitentis, ne indè remoueatur. At
qui aquæ pars aliqua &longs;upra planum FD exi
&longs;tens infra illud de&longs;cendit; igitur in &longs;egmen
to FAE minuitur pondus, & fit ponderis ac
ce&longs;&longs;io &longs;egmento FCD; quarè multò minor
e&longs;t differentia grauitatum inter &longs;egmenta, ac
proinde minor in mouendo labor, aut in &longs;u
&longs;tinendo. Quemad modum enim &longs;i in libræ
lancibus po&longs;ita fuerint pondera inæqualia, &
exlance grauiore pars ponderis transferatur
in alteram lancem, propiùs accedunt ad
æquilibrium, & faciliùs &longs;u&longs;tinetur lanx gra
uior; Sic etiam pondus aliquod aquæ ex
&longs;egmento maiore FAD tran&longs;latum in &longs;eg
mentum minus FCD, efficit minorem
ponderum inæqualitatem, ac proinde minor
grauitas percipitur à mouente globum, vel
&longs;u&longs;tinente.
Facilitas
mouendi ter
ram ex de
fluxu aqua
rum.
Opportunè &longs;ugge&longs;&longs;i&longs;ti, quæ m&etail;
aliò ab&longs;tractum pror&longs;us effugerant. Thales
Mile&longs;ius &longs;uo illo ingenti nauigio me longtùs
abduxerat.
Quid illud e&longs;t?
an putas terra&mtail;
aquis innatare, aut po&longs;&longs;e pari facilitate atque
nauigium trahi &longs;eu impelli?
Minimè omnium: &longs;ed ne&longs;cio quæ
mens mihi incidit, vt inciperem &longs;u&longs;picari,
bus translato, & aquâ in oppo&longs;itam partem
confluente, fieri po&longs;&longs;et, vt aquæ terra inna
taret, aut &longs;altem aliquid de grauitare remit
teret. Id quod opportuniore &longs;chemate ex
hibeo. Sit idem terræ globus ABCE, cu
ius centrum T non congruat ampliùs cum
centro V.
At ecce Mer&longs;ennum in ip&longs;o tem
pore.
In &longs;acinore manife&longs;to deprehen&longs;i
pœnas dabitis, boni viri. Nullus e&longs;t infi
ciandi locus. Graphium adhuc tenet Gul
dinus; Galilæus cubito incumbens attentus
operam dabat.
Salue amicum caput.
Quicquid in
nobis e&longs;t criminis, animaduertatur; &longs;ed te
iudice. Dabimus, quas volueris pœnas.
Vtique leues, &longs;i fuerint peccato
pares.
Vos nunquam &longs;atis de hoc apud
me purgabitis, quod ante condictam mihi
horam conueneritis, amæna &longs;cientiarum
&longs;patia, me prætermi&longs;&longs;o, percurrentes. Ve
niam non impetrabitis, ni&longs;i me illicò in &longs;ua
ui&longs;&longs;imi docti&longs;&longs;imique veftri &longs;ermonis con
&longs;ortium admittentes probaueritis vo
ve&longs;tra cogitata non
Præclarè nobi&longs;cum agitur, te iudicè.
Culpâ vacamus: nam præmium nobis, non
pœna decernitur. Quarè rumpe moras, Gul
dine, & quæ nobis e&longs;&longs;et di&longs;putatio, edi&longs;
&longs;ere.
Ex ingenitâ aquarum propen&longs;io
ne, qua &longs;emper ad ima delabuntur, ani mad
uertebamus oriri po&longs;&longs;e, vt &longs;i telluris globus
Archimedæis machinationibus extra vniuer&longs;i
centrum transferretur, aquæ in oppo&longs;itam
motui partem delabentes, vt fierent centro
viciniores, minùs grauem relinquerent &longs;u
periorem terræ portionem. Si enim e&longs;&longs;et
ABCE terræ globus, cuius centrum T non
congrueret centro V, quod vniuer&longs;i, aut &longs;al
tem
centrum &longs;ta.
tuitur, aquæ
omnes, qui
bus liber&atail;
pateret ad
ver&longs;us C, vt
ad centrum
V propiùs
accederent,
bus radijs à centro V remota con&longs;i&longs;terent.
Cum itaque &longs;uperiori &longs;egmento &longs;ieret pon
deris aquæ dece&longs;&longs;io, inferiori autem acce&longs;
&longs;io, aliquod haberetur in perficiendâ mo
tione compendium: Hæc verò di&longs;&longs;erenti
bus nobis incidit &longs;u&longs;picio, an fortè continge
re po&longs;&longs;et, aquis in partem vnam delap&longs;is
terram innatare.
Sit enim globus terraqueus ABCE, cu
ius centrum T ad illud interuallum à centro
V venerit machinarum vi, vt aqua deor&longs;um
delap&longs;a &longs;it B
HECB, &
portionem B
CE circum
plectatur.
verò dimit
tatur tellus à
&longs;u&longs;pendente:
finirem, v
trùm terr&atail;
con&longs;i&longs;teret,
an potius de
&longs;cenderet ex T in V, an verò etiam &longs;ponte
a&longs;cenderet longiùs recedens ab V. Verùm
&longs;i &longs;ponte &longs;uâ aliquou&longs;que a&longs;cenderet, ia&mtail;
lam deduceret: &longs;i con&longs;i&longs;teret, iam innataret
aquis, ac proinde non multo labore fui&longs;&longs;et
opus, vt ex V in T transferretur: &longs;i demùm
de&longs;cenderet, illud &longs;altem haberetur emolu
menti, quod intrà aquam exi&longs;tenti multum
ponderis decederet, & motio faciliùs perfi
ceretur.
Facilitas
motus terræ,
antequa&mtail;
ab aqua &longs;e
iungeretur.
Lepidum &longs;anè inuentum ad vin
dicandum à calumniâ Archimedem, &longs;i cui
fortè audaciùs locutus videatur: neque enim
ex machinis &longs;olùm, de quibus abundè Gali
læus di&longs;putabat, verùm etiam ex ipsâ aquâ
&longs;ub&longs;idium non leue peti po&longs;&longs;et ad tellure&mtail;
loco dimouendam. Sed quid potiùs dicen
dum exi&longs;timas? innataret-ne?
an verò de
&longs;cenderet?
Ex his fluctibus enauigare &longs;olus
non po&longs;&longs;um: vobis pariter adremigandum
e&longs;t. Illud primum &longs;tatuere oportet, vtrùm
terrenus globus &longs;it aquâ leuior &longs;ecundùm
&longs;peciem; deinde quota pars ex aquis &longs;ponte
emergeret: vt hinc innote&longs;cat, quantu&mtail;
fui&longs;&longs;et Archimedi laborandum. Et quod
ad primum &longs;pectat, certum e&longs;t aërem vni
uer&longs;um in terræ cauernis delite&longs;cente&mtail;,
omnesque halitus, & corpora aquis leuiora
plurimum po&longs;&longs;e demere de grauitate: aër
grauitatem addit, nec leuitatem: at &longs;i cor
pus aquæ in&longs;i&longs;tat, ex aëre recipit leuita
tem.
Pondus tel
luris in a
qua minue
retur, ob in
clu&longs;os bali
tus.
Sit enim vas AR æreum
parallelepipedum
cuius moles &longs;olida, hoc e&longs;t
vnâ cum aëre inclu&longs;o, &longs;it
pedalis: expendatur i&ntail;
aëre, & &longs;it lib.
12. Vtique
&longs;i aquæ imponatur, nata
bit, cum pes cubicus a
quæ &longs;it lib. Rom. 80; e&longs;t
enim grauitas molis compo&longs;itæ ex va&longs;is ma
tèriâ & aëre ad grauitatem aquæ, vt;. 3. ad 20.
At &longs;i &longs;eruata eadem materia, & æris den&longs;i
tate eâdem manente, confletur in ma&longs;&longs;am,
aut ex eâ fiat vas minoris capacitatis, erit
quidem in aëre idem pondus, at non item
in aquâ. Haud di&longs;pari ratione aër terræ vi
&longs;ceribus inclu&longs;us, qui ad extenuandum orbis
pondus in aëre nihil iuuat, ni&longs;i quatenus lo
cum occupat cæteroqui grauioribus corpo
ribus replendum, intrà aquam conferret in
&longs;uper leuitatem, & de grauiorum corporum
pondere aliquid demeret. Quod &longs;i ignem
aëre multò leuiorem addamus, qui plurima
globi huius &longs;patia implet, incrementum ac-
ti&longs;&longs;imùm ex globi totius grauitate demen
dum &longs;it huius aquæ terram circumplectentis
pondus: iam enim non pars e&longs;&longs;et oneris mo
uendi, &longs;ed e&longs;&longs;et medium, in quo motus per
&longs;iceretur.
lite&longs;centem certis men&longs;uris definire non po&longs;
&longs;umus, &longs;ed coniecturas tantùm per&longs;equi ne
ce&longs;&longs;e e&longs;t: illud potius à Philo&longs;opho exigi po&longs;
&longs;e videtur, vt aëris grauitatem cum aquâ
comparatam determinet. Quamuis autem
con&longs;tipato intrà vas aëre aliquando depre
henderim aërem quadringenties aquâ leuio
rem, mihi tamen omninò non &longs;atis facio: ex
illo enim experimento hoc certè euincitur,
quod aër con&longs;tipatus in aere communi non
con&longs;tipato grauitat: &longs;ed cum varia &longs;it aëris
con&longs;tipatio, incon&longs;tans pariter e&longs;t grauitatis
men&longs;ura, quæ ex illâ colligitur. Quare aëris
grauitatem explorare oporteret in medio le
uiore, quemad modum aqua non intrà aquam
&longs;ed in aëre expenditur. Et quidem &longs;ubiit ani
mum aliquando hæc cogitatio, vt &longs;ubiectis
pruni aërem vehementer calefacerem, in
quo aërem
penderem; vt exploratâ deinde eiu&longs;dem
ro, innote&longs;ceret aëris pondus: &longs;ed cum per
&longs;pectum haberem ex prunis vaporem cali
dum a&longs;cendere, timui, ne motus a&longs;cenden
tis vaporis ac medii commoti mihi fucum
faceret, vt detractum de grauitate cen&longs;erem,
quicquid virium ad de&longs;cendendum occur
rentis vaporis motus impediret. Præter
quamquod ex variâ aeris calefacti raritat&etail;,
varia pariter deprehenderetur aëris commu
nis grauitas.
Aquæ & aë
ris grauita
tes compa
rantur, &
inquirun
tur.
De aëris con&longs;tipati grauitate nullus
dubito, nequè de eiu&longs;dem dilatati leuitate; Id
quod ex hac poti&longs;&longs;imùm occa&longs;ione depre
hendi. Mirabar maris æ&longs;tum Lunæ moti
bus adeò con&longs;tanter ob&longs;equentem, cau&longs;am
que curio&longs;iùs inue&longs;tigans animus nunquam
quieuit, ni&longs;i vbi &longs;u&longs;picari cœpit latere in ma
ris fundo corpora, quæ a&longs;cendente ad Me
ridianum Lunâ intume&longs;cerent, aquasque
attollerent, Lunâ verò ad Occa&longs;um de&longs;cen
dente &longs;ub&longs;iderent cum aquis. Hinc pro
maiore huiu&longs;modi corporum copiâ, aut mi
nore, aut pro inæquali eorum contumaciâ,
aut facilitate ad intume&longs;cendum, æ&longs;tuum
inæqualitas &longs;atis explicata videbatur. Vtau
tem aliquod mariniæ&longs;tûs, ex corpore ad Lu-
uarem, Bi&longs;emuti glebam nullum ignem pa&longs;
&longs;am mihi comparaui (Bi&longs;emutum no&longs;tri ho
mines vocant plumbum cinereum, quod in
ter plumbi albi & nigri &longs;peciem medium
e&longs;t) congruæ retortæ impo&longs;ui, ignem &longs;e
cundùm gradus admini&longs;traui horas duode
cim, & qui extillauit humor candidus ac dul
cis, amplo capacique Recipiente excepi:
hunc iterum ac tertiò, vt artifices loquuntur,
rectificaui, vt purior euaderet ac dulcior.
Huius liquoris libram vnam conieci in vi
treum va&longs;culum tantæ capacitatis, vt liquor
phialæ be&longs;&longs;em impleret, reliquum trientem
aër occuparet. Vitrum optimè clau&longs;um, ne
quid expirare po&longs;&longs;et, in loco, vbi quie&longs;ceret,
&longs;tatui; nec &longs;inè animi voluptate licebat in
Plenilunijs manife&longs;ta inclu&longs;i liquoris incre
menta ob&longs;eruare, in Nouilunijs verò decre
menta, neque ea tantùm, quæ prioribus in
crementis re&longs;ponderent, &longs;ed vt aliquid de
ce&longs;&longs;i&longs;&longs;e videretur ex eâ liquoris quantitate,
quam primùm infuderam. Contigit au
tem, vt, curio&longs;itate animum vellicant&etail;,
phialam tùm in Pleniluniis, tum in Noui
luniis expenderem, &longs;emperque paria depre
hendi grauitatis momenta, perinde atque &longs;i
tunc primùm in phialam liquorem inieci&longs;-Ex quibus intellexi, quantum grauita
tis &longs;ecundùm &longs;peciem decederet liquori in
tume&longs;centi, tantum accedere aëri intrà phia
lam apprimè occlu&longs;am con&longs;tipato: contrà
verò liquore ad minora &longs;patia coangu&longs;tato
aërem quidem inclu&longs;um dilatari
rare&longs;centis leuitate auctâ, maiorem illius
conden&longs;ati grauitatem compen&longs;ari.
Nec di&longs;similem grauitatis con&longs;tantiam
ob&longs;eruabimus, &longs;i vitreum globum, cui lon
giu&longs;eulum collum & gracile adhæreat, vini
&longs;piritu ex multiplici di&longs;tillatione tenui&longs;&longs;imo
repleamus, aëre collum occupante; o&longs;cu
lum autem Hermetis &longs;igillo claudatur. Ex
calore enim rare&longs;cet vini &longs;piritus, aëremque
con&longs;tipabit, & frigore &longs;ub&longs;equente &longs;ub&longs;idet,
aëri locum relinquens: neque tamen Ther
mo&longs;copii huius grauitas variabitur, cu&mtail;
vnius corporis con&longs;tipati grauitas alterius rari
leuitate compen&longs;etur. Hinc tamen cert&atail;
definiri non po&longs;&longs;e momenta, quibus aër
compre&longs;&longs;us deor&longs;um nitarur, palam e&longs;t.
Rem ego quantâ potui diligentiâ
aliter inue&longs;tigaui, & aërem non quadrin
genties tantùm, &longs;ed etiam adhuc triplo le
uiorem inueni, ita vt grauitas aquæ ad gra
uitatem aëris non &longs;it in minori Ratione quàm
1200 ad 1, &longs;ed potiùs in maiori Ration&etail;. Æream Æo
lipilam propemodùm candentem omnique
humore de&longs;titutam expendi primùm bilan
ce iu&longs;ti&longs;simâ; deinde eiu&longs;dem refrigeratæ &
ad naturalem temperiem re&longs;titutæ pondus
examinaui, & animaduerti illam facta&mtail;
fui&longs;&longs;e quatuor vt minimum grauis grauio
rem: Hinc intuli aërem, qui rarefaction&etail;
exierat, & naturali conden&longs;atione fuerat ite
rùm in Æolipilam admi&longs;&longs;us, habere in gra
uitate quatuor grana. Iterum Æolipilam,
vt priùs, calefeci, quæ eiu&longs;dem quoquè pon
deris, vt priùs, inuenta e&longs;t: & illicò eius ro
&longs;trum in aquam immer&longs;i, vt aquam &longs;ugeret:
&longs;uxit autem aquæ vncias 9, drachmas 3, gra
na 25; quæ &longs;unt in vniuer&longs;um (&longs;i fingulis
drachmis grana 72 tribuantur, & vnciis gra
na 576) grana 5425; quod e&longs;t pondus aquæ
occupantis idem &longs;patium, quod occupabat
aër vi rarefactionis exclu&longs;us. E&longs;tigitur gra
uitas aquæ granorum 5425 ad grauitate&mtail;
aëris &longs;ecundùm molem æqualis gran 4, hoc
e&longs;t vt 1356 ad 1. Quarè &longs;atis liberaliter age
re mihi videor, &longs;i dixero aërem ad aquam in
leuitate e&longs;&longs;e vt 1200 ad 1.
Quàm vellem hæc omninò veri
tati congruere! Sed veniam dabis non qui
dem improbanti conatum, &longs;ed pen&longs;icula-
tiùs examinanti, vtrùm omni pror&longs;us labe
careat tua hæc argumentatio. Sit vas cu
preum AR molis vnâ cum inclu&longs;o aëre pe
dalis, & va&longs;is pondus lib.
80, quod e&longs;t pon
dus pedis cubiciaquæ. E&longs;t igitur vas eiu&longs;
dem &longs;pecificæ grauitatis cum aquâ; atque
adeò cum grauitas cupri ad grauitatem aquæ
&longs;it vt 71. ad 8, moles cupri ad molem aëris
inclu&longs;i e&longs;t vt 8 ad 63. Quarè vas aëre plenum
nihil in aquâ grauitat, aut leuitat; &longs;ed &longs;i mer
gatur intrà aquam, quæ aëre exclu&longs;o impleat
va&longs;is capacitatem, iam vas intrà aquam pon
dus habebit lib.
80. minùs pondere, quod
habet aqua &longs;ecundùm molem cupro æqualis.
Quia igitur moles cupri e&longs;t (8/71) pedis cubici,
fiat vt 71 ad 8, ita lib.
80 pondus pedis cu
bici aquæ, ad lib.
9. (1/71) pondus aquæ æqualis
cupro va&longs;is. Erit itaque va&longs;is pleni aquâ in
trà aquam grauitas lib. (70 70/71): quod quidem
pondus tribuendum e&longs;t cupro, non autem
aquæ vas implenti, quæ intrà aquam non
grauitat: cum maximè idem e&longs;&longs;et in aquâ
eiu&longs;dem cupri pondus, etiam&longs;i in ma&longs;&longs;am
conflatum nihil aquæ contineret. Quan
quam non nego grauitatem illam cupri i&ntail;
aquâ e&longs;&longs;e æqualem grauitati molis aqueæ vas
implentis, &longs;i illa extra aquam in aere
transferatur; e&longs;t enim moles aquæ vas im-
Sivas in li
quore
retur
plenum aë
re, nunc ple
num liquo
re illo, diffe
rentia pon
derum est
pondus li
quoris im
plentis capa
citatem va
&longs;is.
At manente eâdem va&longs;is mole intelliga
tur aucta materia; & diminuta capacitas, ita
vt &longs;it va&longs;is pondus in aëre lib.
100 &longs;e&longs;qui
quartum prioris: erit moles cupri (10/71) pedis
cubici, & capacitas (61/71). Quarè aqua æqualis
cupro habet pondus lib. (11 19/71): igitur vas intrà
aquam plenum aquâ habet pondus lib. (88 52/71)
at plenum aëre intrà aquam amittet pondus
integri pedis cubici, & &longs;olùm erit lib.
20.
Horum igitur ponderum differentia (68 52/71) æ
qualis e&longs;t ponderi aquæ implentis capacita
tem va&longs;is, quæ ex con&longs;titutâ hypothe&longs;i e&longs;t (61/71)
pedis cubici.
Similiter &longs;i intrà idem vas admittantur
aquæ (10/71), reliquam capacitatem (51/71) impleat
aër: moles cupri & aëris occupat in aquâ (61/71)
pedis cubici: igitur ex lib.
100 auferuntur
lib. (68 52/71), & pondus va&longs;is in aquâ e&longs;t lib. (31 19/71).
At quando totum replebatur aquâ, pondus
erat lib. (88 52/71), igitur differentia lib. (57 33/71) e&longs;t
grauitas æqualis grauitati (51/71) pedis cubiciaquæ,
quæ impleret va&longs;is &longs;patium ab aëre occupa
tum, &longs;i aqua illa in aëre extra aquam ex
penderetut.
Quemadmodum igitur ex eiu&longs;dem va&longs;is
ponderibus intrà
ac quando e&longs;t plenum aëre, rectè infertur ho.
deri aquæ implentis locum
deretur in aëre: &longs;ic ex differentiâ ponderum
Æolipilæ intrà aërem communem con&longs;titu
tæ, quando plena e&longs;t aëre communi, ac
quando plena e&longs;t aëre ignito, ex hac, inquam,
differentiâ rectè illata videtur grauitas aëris
communis Æolipilam implentis, &longs;i aër i&longs;te
ponderaretur in aëre ignito tanquam in me
dio. Cum autem differentia huiu&longs;modi &longs;it
granorum quatuor, pondus aëris communis
æolipilam implentis e&longs;t gran. 4. non quidem
ab&longs;olutè, &longs;ed tantùm in medio leuitatis eiu&longs;
dem ac raritatis, quam habebat aër Æolipi
plens, de quo dicebam, non habet graui
tatem lib. (68 52/71) in quocunque medio, &longs;ed tan
tùm in aëre leuitatis ac raritatis eiu&longs;dem cum
aëre, qui va&longs;is capacitatem implebat. Qua
re non ego facilè dixerim quatuor illa grana
fu i&longs;&longs;e pondus aëris primùm eiecti, qu, iterùm
receptus fuit; ille enim aër nullum habebat
pondus in aëre communi; & &longs;i pondus ha
beat in aëre ignito, tota illa grauitas non
pertinet ad &longs;olum aërem, qui recipitur, &longs;ed
ad illum, qui totam Æolipilam implet. Si-
cundùm aliquam partem in aërem, & reli
qua eiiceretur, iterum autem ad naturalem
den&longs;itatem rediens eiectam aquam ex&longs;uge
ret, differentia ponderis va&longs;is non &longs;oli aquæ
eiectæ tribuenda e&longs;&longs;et, &longs;ed toti aquæ vas im
plenti. Hinc e&longs;t quod, cum aqua in Æoli
pilam admi&longs;&longs;a non &longs;it &longs;ecundùm mole&mtail;
æqualis toti aëri, qui eam implebat, ac pro
inde non &longs;it æqualis moli, quæ habet pon
dus gran. 4, non habetur præcisè Ratio gra
uitatum &longs;ecundùm &longs;peciem.
Experimen
tum Aeoli
pilæ ad &longs;um
mum osten
dit grauita
tem aëris
munis
aëre ignito,
non autem
ab&longs;olutè: nec
ex eo habe
tur quæ&longs;ita
proportio
grauitatum
aquæ & aë
ris.
Id meæ &longs;ententiæ vlteriùs fauet: &longs;i
enim aquæ moles, quam &longs;uxit Æolipila,
minor e&longs;t mole aëris, qui numerat in pon
dere grana 4, plus aliquid aquæ addendum
erit, vt moles æquales &longs;int; atque adeò mul
tò maior erit Ratio grauitatis aquæ ad gra
uitatem aëris, quam &longs;it deprehen&longs;a Ratio
1356 ad 1.
Non hæc eo con&longs;ilio af&longs;erebam, vt
irem contrà; &longs;ed tantùm vt rem paulò ob
&longs;curiorem mihi ip&longs;e enucleatiùs explicarem,
& te audirem, &longs;i quid &longs;ortè à verita
tuâ mente alienum intelligerem. Cæterùm
non nego maiorem e&longs;&longs;e Rationem grauitatis
aquæ æolipilam implentis ad grauitatem aë
ris eam pariter implentis, ac &longs;it Ratio gra-
tatem; modò hæc aquæ grauitas intelligatur
non in quocunque medio, &longs;ed in eodem, in
quo aër communis grauis e&longs;t, in aëre vide
licet ignito & rari&longs;&longs;imo. Verùm cum re
rum grauitates definiamus habitâ ration&etail;
medij omnium rari&longs;&longs;imi atque leui&longs;&longs;imi ex
ijs, quibus communiter vtimur, hoc autem
medium aër &longs;it, non video, cur aëris &
aquæ grauitates inuicem conferre oporteat
ratione medij, cuius nullus futurus e&longs;t v&longs;us.
Quod &longs;i metalla aquæ immer&longs;a dicuntur in
aquâ minùs grauitare pro ratione di&longs;crimi
nis, quod inter&longs;pecificas grauitates interce
dit; cum aquam inter & aquam, aut aërem
inter &
differentia, nulla pariter aquæ in aquâ aut aë
ris in aëre e&longs;t grauitatio: ac proinde cum
aër in aëre con&longs;tituatur, nulla e&longs;t eius graui
tas, quam cum aquâ comparare oporteat.
Sed & vnum præterea addo, &longs;i placet.
Si ferri grauitatem cum aquæ grauitate con
ferre voluero, ferrum primùm in aëre ex
pendo, deinde in aquâ; & quæ fuerit pon
derum differentia, eam tribuo grauitati aquæ
&longs;ecundùm molem æqualis ferro; & vt totum
ferri pondus ad hanc differentiam, ita graui
tatem ferri ad aquæ grauitatem &longs;ecundùm
in aëre fuerit; & &longs;it vt 42 ad 5 1/3. Nam &longs;i
tam ferrum quàm aqua intra oleum (cuius
grauitas &longs;pecifica in aëre e&longs;t 4 3/4) collocentur,
grauitas ferri in oleo erit partium 37 1/4, qua
rum grauitas aquæ erit &longs;olùm (7/12); quæ
longè maior e&longs;t eâ Ratione, quam habent
grauitates in aëre.
Di&longs;criminis huius ratio e&longs;t, quia vt inue
nirem Rationem grauitatum aquæ & ferri in
oleo, &longs;acoma &longs;eu æquipondium debui&longs;&longs;et pa
riter e&longs;&longs;e in oleo; quia autem &longs;acoma &longs;emper
fuit in aëre. & examinatum e&longs;t ferri pondus
in aëre, grauitas quoque ferri & aquæ com
paratur &longs;olum in aëre. Quod &longs;i in libræ ex
tremitate appenderetur globus ferreus vn
ciarum 42 in aëre, qui in oleo immergere
tur, re&longs;ponderet &longs;acomatiferreo in aëre exi
&longs;tenti vnciarum 37 1/4: at &longs;i &longs;acoma &longs;it pariter
in oleo, quia ferreum e&longs;t, & æqualiter à cen
tro libræ di&longs;tat, erit æquale ponderi, quod
examinatur, hoc e&longs;t vnc.42. Globus autem
ferreus &longs;i in aquâ immergeretur, re&longs;pon
deret &longs;acomati ferreo in aëre exi&longs;tenti vnc.
36 2/3: atqui &longs;i vnciæ 37 1/4 in aëre æquiponde
rant vncijs 42 in oleo, vnciæ 36 1/3 in aër&etail;
æquiponderant vnciis (41 51/149) in oleo; igitur
derat &longs;acoma ferreum in oleo vnc. (41 51/149). Sa
comatum igitur 42 & (41 51/149) differentia (98/149) da
bit grauitatem aquæ in oleo comparata&mtail;
cum grauitate ferri; ita vt ferri grauitas i&ntail;
oleo ad aquæ grauitatem in oleo &longs;it vt 42
ad (98/149), quæ e&longs;t planè eadem Ratio &longs;uperiùs in
dicata 37 1/4 ad (7/12), hoc e&longs;t 63 6/7 ad 1.
Ratio duo
rum graui
um in vno
medio, vt ba
beatur, de
bet æquipon
dium e&longs;&longs;e in
eodem me
dio.
Ex his, quæ carere videntur omni diffi
cultate, & &longs;atis manife&longs;ta &longs;unt, infeto 4 il
la grana, quæ tribuis ponderi aëris commu
nis Æolipilam implentis, nullam aëris gra
uitatem indicare per &longs;e, & immediatè; non
bam, non grauitat; non aëris communis in
aëre ignito, vt paulò ante tibi permittebam,
quia &longs;acoma &longs;eu æquipondium, quo v&longs;us es
ad grauitatem explorandam, non erat pari
ter in aëre ignito: id quod fieri oportui&longs;&longs;et.
Nam &longs;i vas aliquod aquâ plenum intrà a
quam ponderes, & &longs;acoma &longs;it pariter i&ntail;
aquâ; iterum autem &longs;acomate intrà aquam
exi&longs;tente idem vas aëre plenum in aquâ ex
pendas, vtique ponderum differentia no&ntail;
dabit aquæ pondus in aëre. Ergo pariter
cum vas aëre communi plenum expenderis
in aëre communi, & iterum in eodem
pondus aëris communis æolipilam implen
tis.
Exemplo rem declaro: & accipio illud
idem vas cupreum AR, de quo antea dixi
mus, quod cum in aëre &longs;it lib.
100, intrà
aquam plenum aquâ e&longs;&longs;et lib (88 52/71), &longs;ed intrà
aquam plenum aëre &longs;olùm e&longs;&longs;et lib.
20; vn
de jntulimus aquæ vas implentis pondus i&ntail;
aëre e&longs;&longs;e lib. (68 52/71). Ponamus &longs;acoma e&longs;&longs;&etail;
ferreum, & illud pariter intrà aquam exi&longs;te
re. Quoniam igitur ferrum 36 2/3 in aër&etail;,
æquiponderat ferro 42 in aquâ, ferrum (88 52/71)
in aëre, æquiponderabit ferro (101 499/781) in aquâ;
& hoc faciet æquilibrium cum va&longs;e cupreo
pleno aquâ. Item quia ferrum 36 2/3 in aëre
æquiponderat ferro 42 in aquâ, ferrum 20
in aëre æquiponderabit ferro (22 10/11) in aquâ; &
hoc con&longs;tituet æquilibrium cum va&longs;e cupreo
pleno aëre intrà aquam. Iam &longs;umo diffe
rentiam inter duo hæc &longs;acomata, quæ &longs;unt
vt (101 499/781) ad (22 10/11), & e&longs;t differentia lib. (78 580/781);
quod &longs;anè non e&longs;t pondus aquæ va&longs;is capa
citatem implentis, &longs;i illa in aëre ponderetur,
&longs;ëd e&longs;t multò maius. Pro diuersâ aute&mtail;
materiâ &longs;iue plumbeâ, &longs;iue æreâ, &longs;iue argen-
rent, alia atquè alia oriretur differentia, vt
patet. Id quod non in librâ tantùm, &longs;ed in
&longs;taterâ quoquè contingeret.
Aëris com
munis & a
quæ graui
tas in aëre
ignito minor
est, quà&mtail;
fuerit depre
ben&longs;a.
Quapropter aërem communem in aër&etail;
ignito con&longs;titutum leuiorem e&longs;&longs;e 4 granis,
quæ in medio eodem e&longs;&longs;ent, apertè con
&longs;tat; illis &longs;iquidem æquiponderat in medio
cra&longs;&longs;iore: atque adeò aqua deprehen&longs;a gra
norum 5425 in aëre communi, &longs;i in aër&etail;
ignito con&longs;titueretur, multò paucioribus
granis æquiponderaret in eodem medio exi
&longs;tentibus; & quidem pro ratione materiæ
&longs;acomatis; &longs;i enim e&longs;&longs;ent grana metallica,
multò pauciora requirerentur ad faciendum
æquilibrium cum aquâ, quàm &longs;i e&longs;&longs;ent gra
na hordei. Similiter &longs;i particulæ ex medul
lâ &longs;ambuci, aut materiâ leuiore quàm aqua,
fungerentur munere &longs;acomatis, tunc librâ in
aërem ignitum translatâ, minuendum e&longs;&longs;et
&longs;acomatis pondus; quemadmodum 100 vn
ciæ auri, & vnciæ (108 24/55) ferri in aere no&ntail;
æquiponderant, quæ tamen in aquâ con&longs;ti
tuerent æquilibrium, quandoquidem tam
100 vnciæ auri quàm (108 24/55) ferri in aquâ po
&longs;itæ æquiponderant &longs;acomari vnc. 94 2/3 in aë
re. Quarè nihil hìc certi mihi videor inue
nire, in quo pedem &longs;igam.
Si aliquid intelligo, haud procul
abes ab eorum &longs;ententiâ, qui omn em aëri
grauitatem adimunt: idcirco enim rem hanc
ad viuum re&longs;ecas, vt illa pror&longs;us euane&longs;cat.
Quo iure omnium &longs;ententia ferro
& plumbo leuitatem negaret, quia vt pluri
mum grauitant in medijs communibus,
quamuis in hydrargyro a&longs;cendant & inna
tent; ita aërem pariter grauem apud nos e&longs;&longs;e
in&longs;icior, quamuis &longs;i in purum æthera tran&longs;
ferretur, ibi grauitaret, quemadmodum &
aër præter naturam conden&longs;atus in vtribus,
in aëre libero grauitat: ibi &longs;iquidem grauita
re poterit, vbi medium leuius, in quo &longs;it,
inueniet. Immò memini me olim vento
rum de&longs;cendentium cau&longs;am reieci&longs;&longs;e in aë
ris &longs;eu halitûs grauitatem; &longs;icut enim gra
uia de&longs;cendentia, ita & leuia a&longs;cendenti&atail;
impetum in motu concipiunt, & augent; ex
quo &longs;it, vt, quemadmodum lignum in aquam
cadens pro&longs;undiùs mergitur, quàm par &longs;it,
vnde po&longs;teà emergit, ita pariter halitus in
&longs;imo hoc aëre leuiores a&longs;cendentes conci
piant impetum, quo deferantur vltra termi
nos &longs;uæ quieti debitos in &longs;upremum aerëm
ip&longs;is halitibus leuiorem; in quo cum iam
graues &longs;int, de&longs;cendant, & obuios halitus
a&longs;cendentes reflectant. Si igitur aerem hunc
puri&longs;&longs;imum æthera, vel &longs;altem in alti&longs;&longs;imi
montis, puta Cauca&longs;i, verticem, ibique la
genam expenderemus; deinde eiu&longs;dem aquâ
plenæ pondus inue&longs;tigaremus, tunc innote
&longs;ceret Ratio grauitatum aquæ & aëris in me
dio illo leuiore. Cæterum in&longs;imæ huius re
gionis incolæ de aëris grauitate di&longs;putantes,
non video, quid certi de&longs;inire po&longs;&longs;int, neque
quos &longs;tatuant Rationum terminos.
Venti de
&longs;cendentis
cau&longs;a obiter
indicatur.
Abiret igitur in immen&longs;um hæc di
&longs;putatio, & certius nihil, vt opinor, depre
henderemus, quod in rem no&longs;tram faceret.
Quapropter præ&longs;taret ijs, quæ veritati pro
xima videntur, a&longs;&longs;umptis, progredi in eâ,
quam con&longs;titueramus, quæ&longs;tione.
Rectè mones, Galilæe.
Sed præ
&longs;tabit forta&longs;sè hæc in aliud colloquium reij
cere; neque enim mihi per tempus licet apud
vos diutiùs e&longs;&longs;e, ni&longs;i po&longs;thabito negotiolo,
quod me aliquantulum vrget, nec omnino
perire vellem.
Commodis tuis nos &longs;eruire oportot:
præterquam quod non hodie diem &longs;ine line â
duximus. Quare bonis auibus perge, quò te
negotia vocant.
QVINTA
tem in aqua ex
plicat.
POSTREMAM he&longs;terni &longs;er
monis no&longs;tri clau&longs;ulam
recolenti in memoriam
venit, ne&longs;cio quid d&etail;
aëris grauitate &longs;criptum
literis Herbipoli haud
ita pridem datis ad ami-
Excutiens itaque &longs;crinium, quod mihi hoc
epi&longs;tolarum genus &longs;eruat, demum, quod
quærebam; inueni; immo & mecum attuli,
ne, &longs;i fortè vobis aliqua &longs;uble&longs;tæ &longs;idei &longs;u&longs;pi
cio de me &longs;uboriretur, tabulæ de e&longs;&longs;ent, quas
proferrem.
Nemo no&longs;trûm e&longs;t, qui fidem in
&longs;irmare audeat Germano candori. neque ra
bulis eges, neque te&longs;tibus, me quidem iu
dice: &longs;atis e&longs;t te narra&longs;&longs;e, vt &longs;idem adhi
beam.
An aliquid, quod vel tuæ, Mer&longs;en
ne, vel meæ aduer&longs;etur &longs;ententiæ, prolatu
rus Guldinus, idquè nobis minùs placitu
rum, &longs;u&longs;picans, ideo literarum exemplar at
tulit, vt omnem à &longs;e inuidiam auertens i&ntail;
alium declinet? Quæcunque tandem ill&atail;
&longs;int, in medium fidenti animo profer, Gul
dine. Non hìc partium, &longs;ed vno veritatis
&longs;tudio tenemur. Nulla ne aëri tribuend&atail;
grauitas?
Immo verò non modica; &longs;i eum,
à quo datæ &longs;unt literæ, audiamus; vocat &longs;i
quidem luculenti&longs;simum grauitatis aeris ar
gumentum, id, quod ab Eruditi&longs;&longs;imis Vi
ris ob&longs;eruatum e&longs;t ex occa&longs;ione experimen
ti, quod ad Vacuum inue&longs;tigandum in&longs;ti-Paratum e&longs;t vas vitreum eximiæ
&longs;pi&longs;&longs;itudinis, & capacitatis tantæ, vt men&longs;u
ras 32, hoc e&longs;t aquæ vncias, vt minimum,
circiter mille, contineret, in Recipientis
Chymici formam elaboratum. Huius col
lo adglutinatum e&longs;t &longs;ingulari arte ac firmi
tudine epi&longs;tomium ex orichalco tran&longs;uer&longs;am
habens clauem ver&longs;atilem apti&longs;&longs;imè &longs;uo lo
culamento congruentem, ne quid ex va&longs;&etail;
po&longs;&longs;it effluere, aut in illud &longs;e in&longs;iuare ni&longs;i eâ
conuersâ. Ex orichalco pariter con&longs;tructus
e&longs;t firmi&longs;&longs;imus tubus hians vtrinque, &longs;ed in
angulum inflexus, vt parti breuiori immitti
po&longs;&longs;it epi&longs;tomium Recipientis (&longs;ic liceat ap
pellare) pars verò longior fungi po&longs;&longs;it mu
nere antliæ duobus a&longs;&longs;ariis ritè in&longs;tructæ, vt
per alterum quidem Recipienti proximum
reducto embolo attrahi po&longs;&longs;it corpus, quo
illud impletur, per alterum verò a&longs;&longs;arium
congruo loco in antliæ dor&longs;o con&longs;titutum
corpus illud atractum exprimi po&longs;&longs;it, cum
impellitur adduciturque tru&longs;illum: cuius
capiti tran&longs;uer&longs;um manubrium adnectitur,
vt duo &longs;imul po&longs;&longs;int antliam agitare: immò
quia, licèt initio facilis &longs;it antliæ agitatio,
paulatim tamen adeò cre&longs;cit difficultas em
bolum reducendi ab a&longs;&longs;ario Recipienti proxi
mo, vt demum duo robu&longs;ti&longs;&longs;imi iuuenes &longs;u-
illum ampliùs extrahere po&longs;&longs;int, aut prohi
bere, ne &longs;ponte relabatur intus, & cum im
petu ac &longs;onitu illidatur ad partes tubi prexi
mas Recipienti; ideò extremo manubrio fu
niculi duo adnectuntur, vt plures opem fer
re po&longs;&longs;int. Ne verò, dum antlia agitatur,
po&longs;sit aër per tenui&longs;simas rimulas (&longs;i fortè
Recipientis epi&longs;tomium & extremus antliæ
tubus non &longs;ibi exqui&longs;itâ collabellatione con
gruerent) &longs;e latenter in&longs;inuare; antlia ligneæ
cupæ firmi&longs;simè affigiturita, vt immi&longs;&longs;a in
cupam aqua Recipientis epi&longs;tomium, eius
que clauem ver&longs;atilem contegat, & &longs;olum
illud antliæ o&longs;culum ex aquâ emineat, cu
pæ &longs;cilicet labro incumbens, per quod em
bolus agitatur.
Experimen
tum, quo ten
tatum est
Vacuum, &
quæ&longs;ita aë
ris grauitas.
His omnibus ritè paratis, ac reuolutâ ver
&longs;atili claue, vt pateret exitus aëri Recipien
tis (quod antequam imponeretur antliæ, fuit
&longs;taterâ examinatum) agitata e&longs;t aliquot ho
ras antlia; ac demum vbi præ nimiâ diffi
cultate extrahendi pi&longs;tillum, ce&longs;&longs;atum e&longs;t,
clau&longs;um e&longs;t vas clauis conuolutione, ac refi
xum ab antliâ iterum expen&longs;um e&longs;t, & anti
quo ponderi deerant lotones 2 3/5, hoc e&longs;t vn
cia (1 3/10).
Et dubitabis adhuc, an aëri vas
re extracto defuit? tibi certè non ad&longs;tipu
larentur, quicunque liquorem, puta oleum
aut mel, ad libram vendunt; quæ enim in
ter vas plenum ac vacuum differentia pon
derum intercedit, eam liquori tribuendam
nemo negat.
Sed quid, &longs;i vas in aquâ expende
rent nunc quidem melle, nunc verò aër&etail;
plenum? an grauitatum differentiam it&atail;
melli tribueres, vt velles pro eâ pretium &longs;ol
uere?
Ex differen
tia
va&longs;is pleni,
& vacui,
quæ&longs;ita aë
ris grauitas
Minimè omnium: quia præter
mellis pondus &longs;ublatum, etiam de va&longs;is gra
uitate non parùm demeret inclu&longs;i aëris le
uitas.
Quid ni igitur va&longs;is grauitati de
tractum pariter a&longs;&longs;eras ab inclu&longs;o aëre ma
ximè raro, ac proinde longè leuiore, quàm
aër i&longs;te communis? Ex quo illud vnum con
&longs;icitur, quod vltro do, aërem &longs;cilicet no&longs;trum
futurum grauem, &longs;i phialæ inclu&longs;us tran&longs;
ferretur in aërem rati&longs;simum Recipientis, &
in eodem medio e&longs;&longs;et æquipondium, quo
aëris communis grauitas exploraretur.
Ex Ari&longs;totelis Vacuum pro&longs;cribentis
officinâ i&longs;thæc depromere oportet; & quan
doquidem negari non pote&longs;t aërem vi antliæ
corpus extrin&longs;ecùs aduocatum &longs;ucceder&etail;
queat, mauultis aliquid aëris contumaci&longs;si
mi, qui modò minorem modò maiorem oc
cupet locum, violentâ ratiocinatione in va
&longs;e concludere, quàm Vacuum, quod &longs;ponte
menti occurrit, admittere, vobisque inqui
rendæ veritatis viam ob&longs;truitis.
Non opus e&longs;t corpora amouere,
vt progrediatur Philo&longs;ophia in Vacuo. Mi
hi &longs;anè nullus relinquitur ambigendi locus,
an aere per vim rari&longs;simo Recipiens implea
tur; aperto enim re&longs;ixi ab antliâ va&longs;is epi&longs;to
mio, tantâ vi externus aër irrumpebat i&ntail;
vas, vt illud hominem è regione &longs;tantem ad
&longs;e raperet: Vacuo autem, quod, præter cor
pus aptum continere, nil addit ni&longs;i corporis
contenti negationem, nullam attrahendi fa
cultatem phy&longs;icam concedo. Sed nequ&etail;
aërem externum &longs;ponte irrupi&longs;&longs;e cen&longs;eo, vt
&longs;uppleret Vacuum; palàm enim aërem at
trahi deprehendit non-nemo, qui apertum
os &longs;uum&longs;atis eminus admouit, & confe&longs;tim
de&longs;iciente &longs;piritu manibus &longs;ignum; vt &longs;&etail;
amouerent, dedit, ip&longs;e ad di&longs;cedendum
impotens factus. Nec diffirebitur alius, cui
digitum admouenti cutem cum carne pænè
ab&longs;traxit, non negatio corporis, &longs;ed vis aëris
&longs;e&longs;e re&longs;tituentis. Ne quod autem &longs;upere&longs;&longs;e
de veritate dubium po&longs;&longs;it, ip&longs;e &longs;e aër prodi
dit, vbi re&longs;ixum ab antliâ vas optimè clau
&longs;um immer&longs;um fuit collo tenus in aquam
puri&longs;simam, & conuolutâ denuo claue ver
&longs;atili re&longs;eratum; magno &longs;iquidem impetu
atque tumultu, ebullientis aquæ in&longs;tar, vel
potiùs fontis copio&longs;i&longs;&longs;imi, irrumpebat aqua
in vas per collum (non tamen &longs;inè bullis at
què copiosâ &longs;pumâ) illudque paulatim ad
&longs;ummum v&longs;que repleuit. Nunquam tamen
id a&longs;&longs;equi potuerunt, vt omnem pror&longs;us aë
rem exclu&longs;um viderent; nam cum in expe
rimentum adhibitum primùm fui&longs;&longs;et vas
27 men&longs;urarum capax, non &longs;uxit aquæ men
&longs;uras ni&longs;i 26 3/4: cumque iterum minori va&longs;e
20 circiter men&longs;urarum capace idem tenta&longs;
&longs;ent, adeò repletum fuit, vt vix &longs;patium,
quod auellana nux repleret, reman&longs;erit aquâ
vacuum, quod &longs;patium aër replebat in &longs;phæ
rulam conglobatus, qui & ad aquæ motum
hùc illùc manife&longs;tè di&longs;currebat. Et quamuis
hinc &longs;pes &longs;acta e&longs;&longs;et, adhibito minore Reci
piente, omnem prorsus aërem extrahendi,
rem tamen eò deducere nunquam potue
runt.
Experimen
tum probat
non dari Va
cuum.
Nullus dubitabam, quin contin-
geret in minori Recipiente minus quoque
ticula certos habeat raritatis terminos, quos
nequit tran&longs;ilire, patet paucioribus particulis
ad eam raritatem adductis impleri &longs;patium
minus, pluribus verò &longs;patium maius. Hinc
e&longs;t longiori tempore opus e&longs;&longs;e ad extrahen
dum aërem ex maiore va&longs;e quàm ex minore,
non &longs;ecundùm Rationem capacitatis eorum,
qua&longs;i &longs;emper æquales particulas antlia extra
heret, &longs;ed &longs;pectatâ raritate minore, quam, cæ
teris paribus, in va&longs;e maiore &longs;u&longs;cipiunt. Po
namus enim, exempli gratiâ,
&longs;tractione extendi po&longs;&longs;e ad &longs;patium centu
plò maius, ac naturaliter occupet, & antliam
primo ductu extrahere quatuor digitos cu
bicos aëris: hi autem &longs;int pars mille&longs;ima ca
pacitatis va&longs;is: igitur po&longs;t primum antliæ du
ctum illa pars mille&longs;ima &longs;patij di&longs;tribuitur
inter re&longs;iduas aëris partes 999: po&longs;t &longs;ecun
dum ductum duæ &longs;patij partes mille&longs;imæ
inter seliquas 998 vnâ cum aliquâ aëris ap
pendice; quia &longs;ecundo ductu non extrahitur
vna mille&longs;ima integra totius primi aëris, &longs;ed
aliquid minus, &longs;cilicet (999/1000) vnius mille&longs;imæ;
& &longs;ic deinceps, donec demùm decem partes
aëris ab initio in va&longs;e exi&longs;tentis, a&longs;&longs;umptâ
centuplâ raritate, impleant totum &longs;patium.
rali&longs;tatu aër recedat, nil mirum &longs;i &longs;emper
extrahendi difficultas augeatur; quia æquale
incrementum raritatis &longs;emper paucioribus
partibus communicatum maiorem &longs;ingulis
infert violentiam. Ex quo patet ad eam ex
trahendi difficultatem citiùs perueniri in va
&longs;e minori, quia &longs;ingulæ aëris particulæ &longs;in
gulis antliæ ductibus plus di&longs;trahuntur, quàm
in va&longs;e maiori: nam &longs;i quatuor digiti &longs;int
&longs;olùm pars va&longs;is cente&longs;ima, primo ductu
partes 99 aëris debent inter &longs;e di&longs;tribuer&etail;
quatuor digitos &longs;patij, quod &longs;patium in va&longs;e
ampliore di&longs;tribuebatur inter partes 999.
Hæc autem maior violentia, quam patitut
aër maximè rarus, in cau&longs;â e&longs;t, cur primùm
tanto impetu irrumpat aqua, & po&longs;tmodum
paulatim a&longs;cendat; quò enim magis à natu
rali &longs;tatu remouetur aër, &longs;icut vim vlterio
rem inferenti contumaciùs re&longs;i&longs;tit, ita maio
re impetu re&longs;tituit &longs;e&longs;e, & aquæ re&longs;i&longs;tentiam
faciliùs vincit, cum maximè aqua ad mino
rem altitudinem euehatur; at vbi aqua adhuc
altiùs eleuanda e&longs;t, atque aër à minori rari
tate violentâ recedit, & hic &longs;egniùs &longs;e&longs;e con
trahit, & illa validiùs re&longs;i&longs;tit, ac proinde len
tior quoque e&longs;t motus aquæ paulatim a&longs;cen
dentis, & vas replentis.
Indicatur
cau&longs;a eo
rum, quæ in
allato expe
rimento con
tingunt.
Aquæ tu
multuantis
cau&longs;a expli
catur.
Sed quid illud e&longs;t, quod tantum
a&longs;cendentis aquæ tumultum, bullas, atque
&longs;pumam excitat? An quia colli angu&longs;tias
prætergre&longs;&longs;a aqua à plurimis aëris particulis
totius va&longs;is latitudinem occupantibus qua&longs;i
in fru&longs;ta di&longs;cerpitur, dum &longs;ingulæ aliquid
liquoris ad &longs;e rapere conantur? An verò quia
tanto impetu &longs;e&longs;e contrahit aër, vt. &longs;piritus
aquæ admi&longs;ti cogantur ita rare&longs;cere, vt in
termi&longs;ti aqueis particulis & bullas & &longs;pumam
creent? An veròquia aër ip&longs;e non planè ho
mogeneum corpus e&longs;t, &longs;ed alias habet par
tes alijs ad raritatem proniores, atque adeò
inæquali impetu attractæ aqueæ particulæ
di&longs;trahuntur, atque tumultum excitare vi
dentur? Quàm varias autem corporum om
nium expirationes aër excipiat, atque inui
cem permi&longs;ceat, nemo e&longs;t Philo&longs;ophus, qui
ignoret; & argumento e&longs;t ip&longs;a naturalis re
&longs;pirandi facultas, quæ licet aërem promi&longs;cuè
attrahat in pulmones, minùs tamen pro&longs;i
cuum expiratione reijcit, retentis vtiliori
bus particulis, quas cum &longs;anguine commi
&longs;ceat, vt per arteriam veno&longs;am in &longs;ini&longs;trum
cordis ventriculum de&longs;cendentes elaboren
tur in &longs;piritus vitales primùm, ac deinde ani
males. Quod &longs;i quis in cubiculo eodem diu
tiùs
imbutum excipere, &longs;atis experitur, quantum
aërem inter & aërem inter&longs;it. Hinc vix po&longs;
fum aliquando non &longs;ubridere, cum diuer&longs;a
rum gentium mores aut corporis habitum
viribus è cælo dimi&longs;&longs;is tribui audio ab ali
quibus, qui vbi caloris aut frigoris mentio
nem fecerint, illicò propo&longs;itæ quæ&longs;tioni &longs;e
feci&longs;&longs;e &longs;atis exi&longs;timant. Cau&longs;a tamen i&ntail;
promptu e&longs;t, cum ex vitalium atque anima
lium &longs;pirituum diuer&longs;itate alia atque alia cor
poris habitudo, ingenium, mores pro&longs;ici
&longs;cantur; &longs;piritibus autem fabricandis no&ntail;
vltimo loco in&longs;eruit aër, quem in&longs;piramus;
hic verò pro regionum varietate alias atque
alias recipit ex tellure expirationes. Quar&etail;
non facilè mihi per&longs;uadere po&longs;&longs;um vas ali
quod omninò homogeneis aëris particulis
impleri; atque adeò, &longs;i per vim rare&longs;cant,
inæqualiter etiam eas rare&longs;cere, atque inæ
qualem &longs;ubire violentiam oporter: & quod
hinc &longs;equitur, inæquali impetu &longs;e&longs;e po&longs;tmo
dum contrahant, nece&longs;&longs;e e&longs;t.
Eiu&longs;de&mtail;
experimenti
exten&longs;io: &
eorum, quæ
accidunt,
cau&longs;æ indi
cantur bre
uiter.
Huic inæquali particularum rari
tati ego pariter plurimum tribuendum cen
&longs;eo; cum enim vas Recipiens primùm aquâ
repletum fuerit, & antliæ impo&longs;itum, faci
opus fuerit ce&longs;&longs;are, quamuis non omnis a qua
exhau&longs;ta fui&longs;&longs;et; id &longs;cilicet nunquam potuit
obtineri. Refixo autem ab antliâ va&longs;e, & re
&longs;erato epi&longs;tomio in aquâ mundi&longs;&longs;imâ, hæc
vt prius a&longs;cendebat, &longs;ed excitatâ copio&longs;iore
&longs;pumâ & pluribus bullis, ac quando &longs;olo aë
re vas fuerat repletum: id quod contigi&longs;&longs;e
exi&longs;timo, quia maior e&longs;t inæqualitas raritatis
in partibus illis partim aqueis, partim ex aquâ
eductis. Nam cum primùm aqua extrahi
tur, &longs;eparantur ab illâ &longs;piritus aliqui & parti
culæ, quæ facilè expirarent, & &longs;uprema va
&longs;is &longs;patia occupant; deinde etiam partes
aqueæ minùs contumaces rare&longs;cunt, inter
quas aliqua datur inæqualitas: &longs;icut enim
aqua igni appo&longs;ita non tota &longs;tatim in vapo
res &longs;oluitur, &longs;ed aliæ præ alijs particulæ &longs;aci
liùs &longs;oluuntur, ita vbi per di&longs;tractionem &longs;ol
ui debet carum compages, aliæ &longs;e exhibent
præ alijs contumaces ad rare&longs;cendum. Fieti
autem eam, quam dixi &longs;pirituum &longs;epara
tionem ex eo con&longs;tat, quod po&longs;tmodum vas
totum aquâ repleri nequit, partes enim va
&longs;is &longs;upremas &longs;piritus illi occupant. Immo
cùm per apertum epi&longs;tomium immi&longs;&longs;us
fui&longs;&longs;et copio&longs;us fumus, hic quidem per a
quam a&longs;cendebat, &longs;ed eius &longs;uperficiei in-
ni&longs;i cum demum calido linteo obuolutum
fui&longs;&longs;et vas: tunc enim vi caloris faciliùs rare
&longs;cebat fumus, quàm &longs;piritus illi per vim ad
huc rari; ac proinde cum fumus amplrot&atail;
&longs;patia exigens comprimeret vicinum &longs;piri
tum, qui propterea à violento &longs;tatu recede
bat, tunc poterat fumus in &longs;uperiora &longs;pati&atail;
leuior factus a&longs;cendere.
Sed & illud hanc &longs;pirituum &longs;eu halituum
&longs;eparationem o&longs;tendit; quod non nemo ex
cogitauit, vt citiùs, quando opus e&longs;&longs;et, ex
perimentum in&longs;titueret, & vitreum vas Re
cipiens exhauriret. Ingens vas æneum plu
rium vrnarum capax con&longs;tructum fuit opti
mè clau&longs;um, præterquam in imo, vbi epi
&longs;tomium cum claue ver&longs;atili habebat, vt po&longs;
&longs;et antliæ imponi, & in &longs;ummo pariter tu
bus extabat claue ver&longs;atili tran&longs;uersâ accu
rati&longs;simè clau&longs;us, ac &longs;ummo tubo imponi
poterat Recipiens vitreum. Et vt omnis &longs;u
&longs;picio aëris &longs;e furtim in&longs;inuantis auerteretur,
tubum circumplectebatur va&longs;culum aquâ
repletum, ne aut per clauis ver&longs;atilis rimulas,
aut per tubi o&longs;culum &longs;e aër in&longs;inuaret. Ænei
va&longs;is aquâ pleni epi&longs;tomio infimo applicita
e&longs;t antlia, & aliquot horas agitata, ita vt
cen&longs;eretur aquâ pænè vacuum; tum clau&longs;o
perimenti in&longs;tituendi. Vbi igitur rem per
ficere oportuit, impo&longs;itum e&longs;t &longs;uperiori tubo
vas vitreum Recipiens aquâ plenum, & re
uolutâ claue ver&longs;atili, quæ tubum va&longs;is ænei
claudebat, apertoque Recipientis epi&longs;tomio,
cæpit illicò aqua Recipientis delabi in vas
æneum, & &longs;imul ex va&longs;e æneo a&longs;cendebant
bullæ infinitæ ac radii quidam vnionum in
&longs;tar lucidi&longs;simi, & frequenti&longs;simi: nec ce&longs;&longs;a
uit hic aquæ & bullarum radiorumque con
flictus, quamdiu durauit aquæ in &longs;ubiectum
vas æneum vel lap&longs;us, vel attractio. Po&longs;t
horam integram, cum nec dum tota exhau
&longs;ta fui&longs;&longs;et aqua, remotum fuit vas vitreum
Recipiens, eius epi&longs;tomio priùs clau&longs;o, &
aquæ puri&longs;simæ, vt aliàs, impo&longs;itum; con
uolutâque denuò claue ver&longs;atili irrumpebat,
vt aliàs, aqua cum tumultu, & &longs;pumâ, non
tamen eâ copiâ, qua cùm per antliam extra
cta fuerat aqua ex Recipiente. Ex quibus
apertè con&longs;tat in va&longs;e æneo &longs;patium fui&longs;&longs;&etail;
occupatum à &longs;piritibus ex aquâ eductis, qui
proinde patente exitu in vas vitreum Reci
piens erumpebant, cùm aqua ex Recipiente
in &longs;ubiectum vas æneum tum attraheretur,
rum etiam fortè dilaberetur: quis enim dicat
id, quod a&longs;cendebat fui&longs;&longs;e Vacuum, nimi-Quoniam verò Reci
pientis &longs;patium ab aquâ de&longs;cendente reli
ctum non &longs;olis &longs;piritibus eductis ex aquâ Re
cipientis replebatur, &longs;ed alijs præterea, qui ex
&longs;ubiecto va&longs;e a&longs;cenderant, quid mirum, &longs;i
&longs;inguli minorem violentiam in raritate pa&longs;
&longs;i, minore quoquè impetu &longs;e contrahentes
minorem aquæ copiam demùm attrahe
rent?
Quare nihil e&longs;t in toto hoc experimento;
quod vel leuem inferat Vacui &longs;u&longs;picionem;
immò in oculos &longs;emper incurrit corpus, te
nue illud quidem ac rarum, à quo va&longs;is &longs;pa
tia occupantur. Hoc itaque experimento
non &longs;atis probari no&longs;tri aëris grauitatem ab
&longs;olutè, mihi certi&longs;simum e&longs;t; &longs;ed illud vnum
ex inæquali Recipientis pondere antè ac po&longs;t
aëris extractionem vi antliæ, confici pote&longs;t,
quod non inficior, &longs;cilicet communem hunc
no&longs;trum aërem in aëre alio magis raro gra
uitare po&longs;&longs;e: Id autem nil facit ad quæ&longs;tio
nem, quam heri examinandam &longs;u&longs;cepimus,
aquis in partem vnam delap&longs;is. Cum eni&mtail;
aquæ & cæterorum corporum grauitates
nobis tantùm innote&longs;cant cum hoc commu
ni aëre comparatæ, quid confugiendum e&longs;t ad
aërem ne&longs;cio quem rariorem, quo caremu
Quam igitur excogita&longs;ti viam,
qua ad propo&longs;itum quæ&longs;tionis terminum
euadamus?
Ex ijs, quæ minimum habent du
bitationis, ad ignota progrediendum cen&longs;eo.
Illud autem carere videtur omni dubitatio
ne, quod &longs;i aqua in aëre grauis e&longs;t vt 5 1/3, aër
vici&longs;sim in aquâ leuis e&longs;t vt 5 1/3. Nam, vt
herì ratiocinabamur, vas cupreum pedal&etail;
lib.
80. nihil grauitat in aquâ, &longs;i aëre implea
tur, quia &longs;cilicet aëris inclu&longs;i leuitas elidit
eam cupri grauitatem, quam aqua non tol
lit: Cum enim cuprum amittat &longs;olùm lib.
(9 1/71) ratione aquæ &longs;ecundùm molem æqualis,
reliquum pondus decedit ratione leuitatis aë
ris: illud autem e&longs;t pondus aquæ &longs;ecundùm
molem aëri æqualis. Quarè &longs;i æolipila can
dens ami&longs;it 4 grana ponderis, licebit argue
re aërem ignitum in aëre communi leuitare
vt 4. Item quia in eodem aëre communi
aqua æolipilam implens grauior e&longs;t granis
5425, &longs;acomate &longs;emper in eodem medio exi
&longs;tente, colligere po&longs;&longs;umus Rationem ali
quam inter aquam & aërem ignitum, mu
tuantes ab Algebrâ numeros defectiuos.
Quomodo
inueniatur
quanta &longs;it
aëris igniti
leuitas i&ntail;
aqua.
E&longs;t &longs;iquidem in aëre communi æqualium
molium grauitas &longs;imili granorum men&longs;urâ
deprehen&longs;a, Aquæ quidem &longs;altem gran&atail;
Aëris igniti grauitas grana o——4, quoniam
aër ignitus non grauitat &longs;ed leuitat, & com
munis non grauitat aut leuitat. In no&longs;tro
igitur aëre communi aqua ad aërem ignitum
e&longs;t in pondere vt 5425 ado——4. Et quo
niam aquæ grauitatem in aëre communi po
nimus 5 1/3, fiat vt 5425 ad o——4, ita 5 1/3 ad
o——(64/16275). E&longs;t ergo aëris igniti grauitas in
aëre communi o——(64/16275). Atqui aëris com
munis grauitas in aquâ e&longs;t o——5 1/3, igitur &longs;i
iungantur hæ differentiæ grauitatum, erit
o——(5 5489/16275) grauitas aëris igniti in aquâ, hoc
e&longs;t leuitas.
Quod verò hac methodo ritè inue&longs;tigata
&longs;it leuitas aeris igniti in aquâ, con&longs;tabit hoc
exemplo. Grauitas Mercurii e&longs;t 71 1/2, aquæ
5 1/3, olei 4 3/4: igitur grauitas olei in aquâ e&longs;t
o——(7/12), & grauitas aquæ in Mercurio e&longs;t
o——(66 2/12). & &longs;i differentiæ i&longs;tæ iungantur,
erit grauitas olei in mercurio o——(66 9/12); quæ
eadem e&longs;t cum illâ, qua immediatè oleum
cum mercurio comparatur, & e&longs;t 4 3/4 ——
71 1/2. Si igitur aeris igniti grauitas in aër&etail;
communi, iungatur aëris communis graui
tati inaquâ, Hanc autem grauitatem defecti
uam vocemus leuitatem, nec ab hac voc&etail;
abhorreamus.
Et &longs;anè v&longs;us i&longs;te numerorum defectiuo
rum
tur apti&longs;&longs;imus, ita vt leui&longs;&longs;imo negotio con
&longs;
&longs;it, an grauior quàm aqua. Sienim datâ Ra
tione molis corporum totam molem com
ponentium, & datâ Ratione grauitatis &longs;in
gulorum aut leuitatis &longs;pecificæ intrà aquam,
ducatur numerus partium in numerum, quo
leuitas, aut grauitas de&longs;ignatur, horum &longs;um
ma &longs;i defectiua &longs;it, molem illam aquâ leuio
rem indicabit. Quæramus ex.gr. an dolium
ferreis circulis firmatum, ac vino plenum,
&longs;it leuius aquâ, nec ne. Et primùm quarum
partium ferrum e&longs;t 3, earum &longs;it lignum 400,
& vinum 100000. Deinde &longs;pecifica in aëre
grauitas ferri &longs;it 42, ligni 4. vini 5 1/4 ad a
quam 5 1/3. E&longs;t igitur grauitas &longs;pecifica ferri
in aquâ + 36 2/3, quæ per 3 numerum partium
ducta dat totam ferri grauitatem + 110. Gra
uitas &longs;pecifica ligni in aquâ, &longs;eu potius leui
tas e&longs;t——1 1/3, quæ per 400 numerum par
tium ducta dat totam ligni leuitatem——
533 1/3. Demum &longs;pecifica leuitas vini in aquâ
100000 ducta dat totam vini leuitatem——
8333 1/3. Si igitur inuentæ &longs;ingularum par
tium grauitates &longs;eu leuitates in &longs;ummam re
digantur, erit &longs;umma——8756 2/3 defectiu&atail;
indicans totius dolii leuitatem in aquâ, &longs;eu
differentiam, qua aquæ grauitas excedit dolii
grauitatem in aëre. Nam &longs;i aquæ grauitas
5 1/3 ducatur per 3 fit 16, ducta per 400 dat
2133 1/3, ducta per 100000 dat 533333
in &longs;ummam redactæ dant totam aquæ graui
tatem 535482 2/3, à qua deficit dolii grauitas
particulis 8756 2/3. Id quod con&longs;tabit, &longs;i par
tium dolii grauitatem in aëre examinemus;
ligni enim partes 400 per &longs;pecificam graui
tatem 4 ductæ dant ab&longs;olutam ligni grauita
tem 1600; iterum partes 100000 vini per gra
uitatem &longs;pecificam 5 1/4 ductæ dant ab&longs;olutam
vini grauitatem 525000: demum partes 3
ferri per 42 ductæ dant totam grauitatem
ferri 126: quæ &longs;i in &longs;ummam redigantur, erit
tota dolii grauitas in aere 526726. Hæc au
tem grau
qualis relinquit
Inuenire an
moles com
po&longs;ita po&longs;sit
innatare a
quæ nec ne,
ab&longs;que cal
culo aquæ
æqualis.
TABELLE WAR HIER
Porrò &longs;pecificam dolii grauitatem in aëre
habemus, &longs;i ab&longs;olutam grauitatem 526726
diuidamus per 100403 numerum partium,
& erit grauitas &longs;pecifica molis compo&longs;itæ
diuidatur leuitas ab&longs;oluta——8756 2/3, erit &longs;pe
cifica molis compo&longs;itæ leuitas in aquâ(27/1000).
Quarè &longs;i inter nos conueniat, quota portio
terrenihuius globi inclu&longs;is ignibus tribuen
da &longs;it, quota aëri cæterisque &longs;ialitibus aquâ le
uioribus, facilè innote&longs;cet, vtrùm leuitas vin
cat grauicatem, ducto partium numero per
leuitatis aut grauitatis denominatorem.
Quod quidem &longs;pectat ad ingen
tem illam ignium cauernam, qui terræ cen
trum circumob&longs;ident, memini me ab acu
ti&longs;simo æquè ac diligenti&longs;simo &longs;ydereorum
arcanorum &longs;crutatore Gottifredo Wendeli
no aliquid planè &longs;ingulare accepi&longs;&longs;e, quod
&longs;um moperè placuit. Hic terræ &longs;emidiame
trum in partes 100 tribuens, cauernæ æ&longs;tuan
tis &longs;emidiametro partes 63 concedebat, reli
quas 37 den&longs;itati cru&longs;tæ huius extimæ; vnde
fit ignem illum e&longs;&longs;e quartam totius globi
partem (cubus enim ex 63 e&longs;t &longs;ubquadru
plus cubi ex 100) cum &longs;it globus ille igneus
ad totum globum in triplicatâ ratione dia
metrorum. Ne verò id ab ip&longs;o temerè ex
cogitatum videretur, addebat, præter pro
babiles coniecturas & rationes non paucas,
ne id &longs;ibi innotui&longs;&longs;e, cum videret Lunam
ad intimam, hoc e&longs;t concauam, huius cru
&longs;tæ &longs;uperficiem ita motus &longs;uos temperate, vt
crect admirabilem illam librationem, quæ
torquet A&longs;tronomico-Phy&longs;icorum ingenia:
cuius motûs hypothe&longs;es nondum, quod
&longs;ciam, publici iuris fecit.
Ignis &longs;ub
terraneus
est quarta
pars glohi
terreni ex
&longs;ententi&atail;
VVendeli
ni
Tent
innestigatio
leuitatis i
gnis.
Nollem tamen, Guldine, eam tantù&mtail;
igni leuitatem tribueres, quam aëri ignito
intrà Æolipilam candentem conuenire ob
&longs;eruaui: longè enim maiore leuitate prædi
tam ignis &longs;ub&longs;tantiam &longs;tatuere æquum vide
tur, quàm &longs;it leuitas aëris per vim à calor&etail;
extrin&longs;ecùs adueniente rarefacti. Nam cum
aquæ intrà Æolipilam ex&longs;uctæ pondus fue
rit granorum 5425, facilè reperiemus illius
&longs;oliditatem; in pede &longs;iquidem cubico &longs;unt
digiti &longs;olidi 4096: Et quia pes cubicus aquæ
habet in pondere lib. Rom. 80, quarum &longs;in
gulæ &longs;unt vnc. 12, hoc e&longs;t gran. 6912, to
tius pedis cubici aquæ pondus e&longs;t gran.
552960. Si igitur grana 552960 dant digi
tos &longs;olidos 4096, grana 5425 dabunt di
gitos &longs;olidos (40 5/27). Quis autem &longs;ub&longs;tantiæ
ignis habenti in mole 40 & eo amplius digi
tos &longs;olidos, non facilè tribuat longi&longs;simè
vt ob&longs;i&longs;tat quatuor granorum grauitati? Ignis
enim naturam in motum maximè pronam
&longs;i attentiùs con&longs;ideremus, nihilquè igne le
uius nobis innotui&longs;&longs;e anim aduertamus, cum
in tertiâ poti&longs;simum & &longs;upremâ aëris regio
ne ignitæ impre&longs;siones Meteorologicæ ac
cendantur, nemo facilè negauerit ignis leui
tatem &longs;altem tantam e&longs;&longs;e, quæ vincat me
dullæ &longs;ambuceæ grauitatem, datâ molis æ
qualitate; nihil quippe medullâ &longs;ambuceâ
leuius, hoc e&longs;t minùs graue, potui adhuc
deprehendere. Medullam enim &longs;ambuci
vtcunque viridis cylindricam expendi, cuius
altitudo erat digitorum 2 2/3 & ba&longs;is diameter
digiti 1/4, fuitque pondus gran. 3: æqualis au
tem cylindri cerei pondus fuit gran. 72. At
qui &longs;i ignis tanta &longs;it leuitas &longs;ur&longs;um, quant&atail;
e&longs;t medullæ &longs;ambuceæ grauitas deor&longs;u&mtail;,
multò maior e&longs;t ignis leuitas quàm aeris igni
ti in Æolipilâ. Nam quia ba&longs;is diameter
e&longs;t digiti 1/4, & altitudo dig. 2 2/3, &longs;i diametri
quadratum 1/16 ducatur in altitudinem
betur parallelepipedum digiti &longs;olidi 1/6: hoc
autem parallelepipedum e&longs;t ad cyliudrum
in Ratione ba&longs;ium, quadratum autem dia
metri ad circulum habet maiorem rationem
ad 223: igitur vt 14 ad 11 ita 1/6 ad (11/14) maio
rem vero; & vt 284 ad 223, ita 1/6 ad (223/1704) mi
norem vero. Inuentæ quantitates (11/14) & (223/1704) re
ducantur ad eandem denominationem, &
&longs;unt (78744.18732/143136), differentia autem e&longs;t 12 parti
cularum; & &longs;umpto medio Arithmetico &longs;o
liditas cylindruli dati e&longs;t (18738/143136), hoc e&longs;t (3123/23856). Si
igitur medulla &longs;ambucea, cuius &longs;oliditas e&longs;t
dig. (3123/
cuius &longs;oliditas e&longs;&longs;et dig. (40 5/27), haberet pondus
gran. 920. Ignis itaque leuitas in aëre &longs;i tan
ta fuerit, quanta e&longs;t medullæ &longs;ambuceæ gra
uitas, erit gran.——920, quorum 5425 e&longs;t
grauitas aquæ &longs;ecundum molem æqualis.
Ergo &longs;i aquæ grauitas in aëre 5425 dat ignis
in aëre leuitatem——920, aquæ grauitas 5 1/3
dat ignis leuitatem——(4906/5425); quæ &longs;i addatur
leuitati aëris in aqua——5 1/
in aquâ &longs;altem——(6 25
dine, tuà methodo: quanquam ægetrimè
mihi per&longs;uadeo tantulam leuitatem e&longs;&longs;e igni
tribuendam.
Vtinam non adeò vorax ignis e&longs;&longs;et,
& pateretur &longs;e va&longs;culo concludi: ve&longs;tra&mtail;
ha
aliquando excogitaui ad examinandam cor-
porum leuitatem in mercurio. Tran&longs;uer&longs;a
ria duo EF, CD planè æqualia, bifariam di
ui&longs;a in A & I, in
&longs;erui tigillo AB ita,
vt circà axiculos A
& I facilè ver&longs;atilia
e&longs;&longs;ent, additâ lin
gulà, quæ æquili
brium indicaret.
Extremitati C &longs;ty
lum deor&longs;um ver
gentem adieci, qui
&longs;ubiectum corpus tangeret; & ex &longs;uperioris
iugi EF extremitate E, lanx H pendebat:
reliquæ extremitates FD filo iungebantur æ
quali interuallo AI, ne alterum iugum &longs;inè
altero moueretur; additumque e&longs;t plumbi
momentum brachijs. AF & ID, vt fieret
æquilibrium cum lance H & itylo addito in
C; adeò vt iuga EF, CD &longs;emper inuicem &
horizonti parallela e&longs;&longs;ent, ni&longs;i accepto ex
trin&longs;ecùs impetu remouerentur à paralle
li&longs;mo.
Instrumen
tum, quo ex
aminari po
&longs;unt leuita&longs;
tes. corpora
in medio
grauiore, &
media ip&longs;a
comparari
&longs;ecundù&mtail;
grauitatem.
Tum va&longs;culum KL ritè collocatum mer
curio implebam, & &longs;olidum, cuius exami
nanda erat leuitas, mercurio impo&longs;itum ita
&longs;ubijciebam &longs;tylo C, vt pondere in lance H
deor&longs;um vrgente attolleretur F & D, ac pro-
dum O intrà mercurium, cuius partes æqua
les moli immer&longs;æ effluebant: atquè ita pon
dus in H temperabam, vt iuga EF, CD, ho
rizonti parallela con&longs;i&longs;terent, quando &longs;uper
&longs;icies &longs;olidi immer&longs;i vnicam cum circumfu
&longs;o mercurio &longs;uperficiem con&longs;tituebant. Qua
re pondus in H grauitans æqualiter re&longs;i&longs;te
bat leuitati &longs;olidi &longs;ursùm conantis in mercu
rio: cumque ex Archimede lib.
de in&longs;id.
hum. prop.
6. con&longs;tet &longs;olidum leuius humi
do immer&longs;um tantâ vi &longs;ur&longs;um ferri, quantâ
humidum molem &longs;olido æqualem habens
grauius e&longs;t ip&longs;o &longs;olido, colligebam, quantò
grauior &longs;ecundùm &longs;peciem e&longs;&longs;et mercurius,
quàm immer&longs;um &longs;olidum. Deinde ide&mtail;
&longs;olidum in aquâ expendebam vel eadem li
brâ, &longs;i in aquâ pariter leuitabat, vel librâ
communi, &longs;i grauius erat quàm aquâ. Col
lectis demùm in &longs;ummam huiu&longs;modi pon
deribus &longs;eu differentijs grauitatum, &longs;i i&ntail;
mercurio leuitans &longs;olidum grauitabat in aquâ,
&longs;iue &longs;ubtracto minori pondere ex maiore, &longs;i
in vtroque leuitabat; &longs;umma vel re&longs;iduum
dabat mihi exce&longs;&longs;um grauitatis mercurij &longs;u
pra grauitatem aquæ: alterutrius aute&mtail;
pondere cognito, reliqui pondus innote&longs;ce
bat.
Sic &longs;i ferri leuitatem in mercurio ex.
gr.
deprehendebam æqualem drachmis 29 1/2 e
iu&longs;dem autem ferri grauitas in aquâ erat dra
chm. 36 2/3, arguebam ferrum medium e&longs;&longs;e
inter mercurium & aquam; adeòque addi
tis hi&longs;ce differentijs, nota erat differentia in
ter mercurium & aquam 66 1/6: quarè ponde
rato ferro in aëre & inuento drachm. 42, a
qua reperitur 5 1/3, atque adeò mercurius 71 1/2.
Quod &longs;i ferri loco &longs;olidum ligneum v&longs;urpa
bam, cuius leuitas in mercurio e&longs;&longs;et 67 1/2, le
uitas verò in aquâ 1 1/3; quia in vtroque leui
tabat, intelligebam aquæ grauitatem me
diam e&longs;&longs;e inter lignum & mercurium: qua
propter harum leuitatum differentia 66 1/6, vt
priùs, e&longs;t differentia inter aquam & mercu
rium.
Hac, inquam, librâ, quam tunc eum in
v&longs;um excogitaui, &longs;i va&longs;culum igne plenum
& clau&longs;um, ne auolet, dederitis, eius leuita
tem intrà aquam in va&longs;e KL explorabimus,
additâ, prout opus fuerit, vel demptâ ip&longs;ius
va&longs;culi intrà aquam grauitate vel leuitate.
Nolo me duriorem præbere i&ntail;
extenuandâ ignis leuitate, quamuis aëris in
æolipilâ candenti leuitatem non adeò mulVnam aliquam &longs;tatue
&longs;imè ab&longs;it à tuâ illâ plu&longs;quam millecuplâ Ra
tione, quam inter aeris & aquæ grauitates in
tercedere opinaris; non enim facilè acquie
&longs;cerem.
Galilæo &longs;altem poteris acquie&longs;ce
re aërem quadringenties tantùm aquâ leuio
rem &longs;tatuenti; ignis autem adhuc aëre le
uior e&longs;t. Quid caput abnuentis in morem
nutat? Hoc certè, quod addo, nemo æquus
iudex reijciat; quandoquidem probabiles
coniecturas per&longs;equi nece&longs;&longs;e e&longs;t, vbi ip&longs;am
veritatem certò a&longs;&longs;equi non datur. Inter
grauia nihil adhuc nobis inno
uius, inter leuia nihil igne leuius: aqua &
aër medio loco con&longs;i&longs;tunt. Quamobrem
&longs;icut grauium grauitates in aere tanqua&mtail;
communi medio inuicem comparamus, ita
leuium leuitates in aquâ tanquam communi
medio conferri inuicem po&longs;&longs;e videntur. Hin
&longs;i ponatur leui&longs;fimum corpus ita &longs;e habere
ad medium, vnde leuitas de&longs;umitur, vt cor
pus graui&longs;&longs;imum &longs;e habet ad medium, ex
quo grauitas incipit denominari, nihil ab
&longs;urdum &longs;tatuitur, & corporum motrices fa
cultates &longs;ibi proportione re&longs;pondent. Sicu
igitur in aere grauitas aquæ e&longs;t partium 5 1/3,
quarum grauitas auri e&longs;t 100, quidni parites
aëris in aquâ, earum——100 &longs;tatuas ignis
leuitatem in aquâ? Vide, quò demum ve
nerim, Guldine, vt tibi morem geram.
Alia bypo
th
gnis leuit
Iam planè videris paulò mitiùs
atque veri&longs;imiliùs opinari. Et vt meam fa
cilitatem tibi probem, &longs;iquidem placuit
VVendelini &longs;ententia quartam orbis huius
partem tribuens æ&longs;tuanti cauernæ ignium
plenæ, eam vltrò admitto, & ignis leuita
tem ad aquam a&longs;&longs;umo partium——100, qua
rum aëris leuitas e&longs;t——5 1/3. Reliquis verò
tribus orbis huius quadrantibus quam conce
dis grauitatem?
Si perpendantur omnia, forta&longs;sè
non deberent cen&longs;eri grauiores argillâ. Pri
mùm quia aqua, quæ terræ faciem alluic
aëri finitima, &longs;erè tota demenda e&longs;t ex pon
dere, cùm illa, ex hypothe&longs;i initio con&longs;titutâ,
habeat rationem medij, in quo motus per
ficitur, & ad vnam partem confluxerit. De
inde quia aqua terræ venas permeans benè
multa e&longs;t, & intrà aquam con&longs;tituta nihil
addit ponderis. Præterea reliqua moles, quæ
demptâ aquâ &longs;upere&longs;t, tam multum habet
aeris halituumque intrà aquam leuitantium,
vt plurimum elidant grauitatis metallorum Quarè &longs;i tribus orbis qua
drantibus argillæ grauitas tribuatur, maxi
mè probabili coniecturâ vti videatur, qui &longs;i
opinetur.
Globus ter
renus in ea
bypothe &longs;i in
nataturus
aquæ vide
tur.
Quoniam verò aquæ grauitas ad argillæ
grauitatem in aëre, vt olim me dicere me
mini, e&longs;t vt 16 ad 27, po&longs;itâ aquæ grauita
te 5 1/3, argillæ grauitas in aere e&longs;t 9: igitur
argillæ grauitas in aquâ e&longs;t 3 (
partes orbis &longs;i per 3 2/3 ducantur, erit grau itas
11, & vna pars ignis e&longs;t leuitatem habens
——100: &longs;i addantur, erit globi huius con
flati ex hac variâ corporum mi&longs;cellâ leuitas
——89 intrà aquam: ac proinde &longs;i aqua v
nam in partem &longs;ecederet, tellure
globus hic innataret, cum leuior &longs;it aquâ.
Nam &longs;i comparetur grauitas ab&longs;oluta aquæ
&longs;ecundùm molem æqualis toti globo, cum
grauitate ab&longs;olutâ ip&longs;ius globi, reperietur di&longs;
ferentia 89. Quia enim ignis leuitas ad aquam
e&longs;t——100, aquæ grauitas ad ignem e&longs;t +
100: quia verò argillæ grauitas ad aquam e&longs;t
3 2/3, argillæ grauitas ad ignem e&longs;t 103
qui tres globi partes ex hypothe&longs;i &longs;unt argil
la, reliqua pars e&longs;t ignis, igitur &longs;i argillæ gra
uitas 103 2/3 ducatur per 3, erit globi grauitas
311: &longs;i verò aquæ grauitas ad ignem 100
æqualis 400. Cum itaque grauitas aquæ ad
grauitatem molis confla
ignis, &longs;it vt 400 ad 311, con&longs;tat mole&mtail;
conflatam innataturam.
Quin aquæ grauitatem in
tiùs, quàm in igne, con&longs;ideras? An timui
&longs;ti, ne globus hic ex aëre communi in purio
rem æthera auolaret? Quando quidem &longs;i le
uitas aeris ad aquam e&longs;t——5
ignis ad aquam e&longs;t——100, leuitas ignis ad
aërem e&longs;t——94 2/3: e&longs;t autem argillæ grauitas
in aëre 9. Igitur tres orbis quadrantes ha
berent grauitatem vt 27, & reliquus qua
drans leuitatem vt——94 2/3, atquè adeo totus
orbis leuitatem haberet in aere vt—— 67 2/3;
hoc e&longs;t &longs;i per 4 numerum partium diuida
tur, leuitas &longs;pecifica totius globi in aere e&longs;
&longs;et——(16 11/
Imò leuior
e&longs;&longs;et &longs;ecun
dùm
aëre.
Lynceus es, Galilæe, nihil no&ntail;
vides. Mirabar pariter, cur omi&longs;sâ aquæ &
argillæ grauitate in aëre, confugerit Mer&longs;en
nus ad earum grauitatem in igne; cum ta
men ex aquâ emergens globus in aërem ve
niat; qui e&longs;t aquæ circum&longs;u&longs;us, non verò in
ignem. Quemadmodum enim &longs;i mercurio
aëre ferrum (cuius grauitas 42) innatatio
ferri debet con&longs;iderari iuxta differentiam gra
uitatum in aëre, & leuitatio ferri ex mercurio
in aërem e&longs;t——29 1/2. At &longs;i mercurio &longs;uper
fu&longs;a &longs;it aqua, leuitas ferri de&longs;umenda e&longs;t ex
differentiâ inter grauitatem mercurij & gra
uitatem ferri, quod intrà aquam non am
pliùs graue e&longs;t vt 42, &longs;ed vt 36 2/3: quare leui
tas ferri tunc e&longs;t maior, videlicet vt——34 5/6.
Ita &longs;imiliter cùm globus terrenus ex aquâ in
aerem, non in ignem veniret, eius grauitas
cum aëre debuit, non cum igne, compa
rari.
Solida plus
leuitant e
mercurio in
aquam
in aërem.
Qui aquam mercurio &longs;uper&longs;udi&longs;ti,
cur pariter non circumfudi&longs;ti? tunc enim
mercurij in aquâ exi&longs;tentis grauitas non e&longs;
&longs;et 71 1/6 &longs;ed 66 1/2, & in illo leuitas ferri (cuius
grauitas in aquâ 36 2/3) e&longs;&longs;et pariter——29 1/2.
Duo &longs;iquidem æqualia corpora inæqualiter
grauia &longs;ecundùm &longs;peciem &longs;i in eodem me
dio con&longs;tituantur, quodcunque illud &longs;it, ean
dem &longs;emper &longs;eruant differentiam, quia vtri
que æqualis fit grauitatis dece&longs;sio aut acce&longs;sio
pro maiori aut minori grauitate medij. Igi
run &longs;i terreni globi'grauicas &longs;pecifica compa
rata cum &longs;pecificâ grauitate aquæ in vno me-
dixi exce&longs;&longs;um grauitatis aquæ &longs;upra grauita
tem globi terreni ex igne & argillâ conflati
e&longs;&longs;e 89; in quocunque medio con&longs;tituantur,
eundem habebit exce&longs;&longs;um. Sic in aëre gra
uitas 3/4 argillæ & 1/4 ignis e&longs;&longs;et defectiua, vt re
ctè Galilæus ratiocinabatur, videlicet——
67 2/3, & grauitas æqualis molis aquæ in aëre
e&longs;&longs;et po&longs;itiua, nimirùm 21 1/3 (ducto 4 per
5 1/3) differentia autem, qua maior numerus
21 1/3 excedit minorem —— 67 2/3, e&longs;t 89 planè
eadem ac prius. Satis igitur fuit in vno ali
quo medio differentiam reperire, cùm illa
&longs;emper eadem maneat. Opportunius verò
accidit in medio omnium leui&longs;simo, in quo
ignis ip&longs;e globi molem componens nihil
grauitat, examen illud in&longs;tituere, quia om
nes partium grauitates po&longs;itiuæ &longs;unt, nulla
defectiua.
Ratio
uitatum
rum corpo
rum in vno
medio, est
eadem i&ntail;
omni medio.
Cæterùm non timui, ne terrenus globus
auolaret, quemadmodum ouorum putami
na, vt aiunt, matutino rore impleta & meri
diano Soli expo&longs;ita. Qui enim tres orbis qua,
drantes intrà aquam exi&longs;timat grauiores non
e&longs;&longs;e, quàm &longs;i ex merâ argilla con&longs;tarent,
quia aqua illis permi&longs;ta nihil grauitat, & aër
valde leuitat, forta&longs;sè non item a&longs;&longs;erat tres
&longs;ecundùm &longs;peciem grauitate cum argillâ pr
ditos e&longs;&longs;e: quia iam aër in aëre nihil leuitat,
& hal
tis grauitat, & cætera omnia corpora tan
tum addunt ponderis, quantum illis detrahe
bat aqua, intrà quam exi&longs;tebant, &longs;cilicet iux
tà molis æqualitatem. Id quod pariter di
ctum velim, &longs;i globus i&longs;te intrà leui&longs;&longs;imum
ignem con&longs;titutus intel igatur, neque enim
ibi tres orbis quadrantes æqualiter cum ar
gillâ grauitarent, &longs;ed longè validiùs. Quan
do autem paulò antè grauitatum rationes
iniens tres orbis quadrante, qua&longs;i ex merà ar
gillâ in igne con&longs;titutâ a&longs;&longs;ump&longs;i, id fuit &longs;o
lùm ad explicandam hypothe&longs;im, qua tri
buebatur grauitas &longs;pecifica æqualis grauitati
argillaceæ. Quarè &longs;i orbis in aëre con&longs;titua
tur, non ea tantùm e&longs;t illius grauitas, quæ
tribus quadrantibus ex argillâ conueniat, &longs;ed
perinde &longs;e habet, ac &longs;i totus ex argillâ con
admittebamus; adeòque nullum &longs;ube&longs;&longs;et pe
riculum, ne auolaret. Nam &longs;i pofito ign&etail;
omnium leui&longs;simo aëris grauitas e&longs;t + 94
argillæ grauitas e&longs;t + 103
orbis totus ex argillâ con&longs;titutus ponatur.
grauior e&longs;t aëre; cum tamen intrà aqua&mtail;
tantùm quadrantes argillacei e&longs;&longs;ent, & reli
quus igneus.
Vereor plurimùm, ne i&longs;ta, quæ
po&longs;tremo loco attuli&longs;ti, plus habeant &longs;peciei
quàm veritatis. & facilè &longs;u&longs;picor non omni
nò &longs;eriò à te prolata; &longs;ed quia vidi&longs;ti me i&ntail;
tabellam illam oculos curio&longs;iùs coniicien
tem,, volui&longs;ti o&longs;citantiam tentare, & ad at
tentionem reuocare. Attenti&longs;simo tamen
animo excepi omnia; &longs;olùm enim vt phan
ta&longs;iæ contentioni parcerem, labrum illud,
in quo moriens Seneca &longs;anguinem cum vitâ
effundit, re&longs;piciebam; illudque mihi nunc
aquâ, nunc &longs;tanno liquente plenum finge
bam perpendens, vtrùm fieri po&longs;sit corpus
aliquod, cui in vno medio conuenit grauitas
&longs;pecifica argillæ, in alio medio maiorem aut
minorem grauitatem
&longs;atis poteram percipere, quî fieri po&longs;&longs;et, vt
totus globus haberet in aëre grauitatem ar
gillæ. non autem &longs;imilem argillæ grauitatem
obtinere po&longs;&longs;et in aquâ.
Eadem mo
les compo &longs;i
ta quam &longs;pe.
cificam gra
uitatem ba
bet in vno
medio, babet
in quocun
cunque me
dio.
Fingamus enim corpus, cuius duæ partes
&longs;int ferrum, vna matmor, duæ cera, vna
aër, & vna lapis: e&longs;t autem in aere &longs;pecifica
grauitas ferri 42, marmoris 21, ceræ 5, aë
ris 0, lapidis 14. igitur &longs;ingularum partium
tas tota in aëre 129; quæ &longs;i per 7 diuidatur,
quia &longs;unt &longs;eptem æquales dati corporis par
tes, erit grauitas &longs;pecifica huiu&longs;modi corporis
in aere 18 3/7. Iam corpus hoc, cuius grauitas
&longs;pecifica 18 3/7, comparemus cum &longs;tanno
TABELLE WAR HIER
communi, cuius grauitas &longs;pecifica 39; erit
dati corporis leuitas in &longs;tanno liquente——
20 4/7; quæ &longs;i per 7 numerum partium duca
tur, erit tota leuitas ——144. Eadem autem
leuitas habetur, &longs;i &longs;ingulæ partes cum &longs;tan
no conferantur; e&longs;t enim duarum partium
ferri in &longs;tanno grauitas + 6, vnius partis
marmoris leuitas——18, duarum partium
—— 39, & vnius partis lapidis leuitas——25:
quæ &longs;i in &longs;ummam referantur, erit corporis
dati leuitas in &longs;tanno liquente ——144. Qua
propter tam in &longs;tanno quàm in aëre perinde
&longs;e habet, ac &longs;i corpus homogeneum e&longs;&longs;et,
cuius grauitas &longs;pecifica e&longs;&longs;et 18 3/7.
Sed in aquâ etiam non dubito, quin pari
ter &longs;e habeat vt corpus, cuius grauitas &longs;it
18 3/7 ad aquam, cuius grauitas 5 1/3: erit enim
corporis huius &longs;pecifica grauitas in aquâ +
(13 2/21), quæ per 7 ducta dat + ab&longs;olutam corpo
ris in aquâ grauitatem + 91 2/3. Hæc verò e&longs;t
planè eadem, ac &longs;i in &longs;ummam conferantur
grauitas duarum partium ferri + 73 1/3, graui
tas vnius partis marmoris + 15 2/3, leuitas du&atail;
rum partium ceræ——
tis aëris —— 5 1/3, & grauitas vnius partis lapi
dis + 8 2/3: ex his enim pariter habetur graui
tas in aquá + 91 2/3. In quocunque igitur me
dio con&longs;tituatur, perinde &longs;e habet, atque &longs;i
grauitas &longs;pecifica in medio leui&longs;simo, i&ntail;
quo nulla po&longs;iti corporis pars leuis e&longs;t, hoe
e&longs;t in aëre, e&longs;&longs;et + 18 3/7.
Ex his infero terrenum globum eandem
&longs;emper habere &longs;pecificam grauitatem in quo
cunque medio; ac proinde &longs;i in aquâ habet
&longs;tinctus ponitur (cum corpori leui&longs;&longs;imo mo
lem componenti 1/4 tribuatur) diuidatur 89
per 4, & erit leuitas &longs;pecifica in aquâ——
22 1/4: Si igitur grauitas aquæ in igne + 100 illi
addatur, erit grauitas &longs;pecifica terreni globi
in igne + 77 3/4. In quocunque igitur medio
terrenus globus &longs;emper haberet &longs;pecificam
grauitatem vt + 77 3/4 in medio leui&longs;simo, hoc
e&longs;t in igne, &longs;i cæterorum corporum grauitas
ad idem medium, hoc e&longs;t ignem, compare
tur. Quia verò aëris communis in igne gra
uitas e&longs;t + 94 2/3, grauitas autem &longs;pecifica glo
bi terreni e&longs;t + 77 3/4, &longs;equitur terreni globi &longs;pe
cificam leuitatem in aëre communi e&longs;&longs;e——
(16 11/12), quæ &longs;i per 4 numerum partium duca
tur, dabit ab&longs;olutam globi leuitatem in aëre
——67 2/3, vt Galilæus argumentabatur: tellus
igitur aere leuior e&longs;&longs;et.
Qui inter &longs;alebras ambulat, non
&longs;emper ad numerum gre&longs;&longs;us ponit. Id
quoquè contigi&longs;&longs;e videtur, qui non &longs;atis di
lucidè verbis &longs;um complexus, quod vole
bam. Sed mihi nunc per ve&longs;tram humani
tatem licebit meam mentem interpretari.
Toti huic globo, quem incolimus, aquam
terramque complectenti, arg
&longs;tit, nec vllum &longs;ube&longs;t auolandi in æther&atail;
periculum. At &longs;ublatis marium atquè flu
minum aquis, quas in latus &longs;ece&longs;&longs;i&longs;&longs;e poni
mus, reliquæ molis grauitas minor e&longs;t; quia,
&longs;ublatâ marium grauitate, ad &longs;olius terræ &
metallorum grauitatem leuitas ignis inclu&longs;i
& halituum habet maiorem Rationem, quàm
haberet ad grauitatem terræ &longs;imul & aquæ; ac
proinde cum maneant eadem corpora leuia,
& minuantur grauia, minor quoquè e&longs;&longs;e vi
detur &longs;pecifica grauitas totius globi. Quem
admodum &longs;i ex æreo va&longs;e aëris pleno aufe
ram partem metalli, manente eodem aëre,
fit minor &longs;pecifica va&longs;is grauitas. Quamuis
autem &longs;ublato vno globi quadrante, qui igni
tribuitur, & &longs;ublatis marium, lacuumque,
& fluminum aquis, quæ &longs;upere&longs;t moles &longs;it
minor tribus totius orbis quadrantibus, ac
proinde minor tribus quadrantibus molis,
quæ re&longs;tat &longs;ublatâ tantùm aquâ, mihi tamen
minutas quæ&longs;tiunculas non con&longs;ectanti &longs;atis
videbatur tribus quadrantibus molis reliquæ
argillaceam grauitatem tribuere; vt &longs;i fortè
grauitas i&longs;ta iu&longs;to minor e&longs;&longs;et, compen&longs;aretur
diminutâ ignis mole, quæ &longs;tatuebatur &longs;olùm
quadrans reliquæ molis, cum tamen e&longs;&longs;et 1/4
totius globi.
Moli
&longs;i
add
d
ius grauit is
&longs;pe
tatur.
Sed &longs;i placeat grauitatem illam augere,
&longs;tatuamus tres illos quadrantes non argillæ,
&longs;ed Magnetis grauitatem habere; e&longs;t autem
magnetis grauitas 26 ad aquæ grauitatem
5 1/3 in aëre: quare magnetis grauitas in aquâ
e&longs;t 20 2/3, & grauitas in igne 120 2/3; ideòque
grauitas trium quadrantum in aquâ, cu&mtail;
ex magnete con&longs;tent, e&longs;t 62, & leuitas
conueniens vni quadranti ignis in aquâ e&longs;t
——100: igitur leuitas globi in aquâ e&longs;t
—— 38. Quando verò totus globus ex aquâ
pariter & terrâ conflatus accipitur, iam au
ctâ grauitate, alia e&longs;t &longs;pecifica totius globi
grauitas, qualis e&longs;&longs;et ex. gr. grauitas mar
moris aut alia, cum non videatur a&longs;&longs;umi
po&longs;&longs;e
tes ex magnete &longs;tatuantur.
At &longs;i globo non re&longs;tituatur aqua,
quæ in latus &longs;ece&longs;&longs;it, augebiturnè grauitas
&longs;pecifica molis, quæ ex tribus quadrantibus
magnetis, & vno quadrante ignis conflatur?
Ni&longs;i quid addatur aut dematur,
fieri non pote&longs;t vt grauitas &longs;pecifica variatio
nem &longs;ubeat,
Nobis igitur conuenit.
Quapropter
&longs;i globi leuitas in aquâ e&longs;t——38, per 4 nu
merum partium diuidatur, & erit leuitas
&longs;pecifica terreni globi in aquâ——9 1/2; cui &longs;i
cifica globi leuitas in aere ——4 1/6. Quà&mtail;
bellè itaque res Archimedi &longs;ucce&longs;&longs;i&longs;&longs;et
doquidem vbi cò terram eleua&longs;&longs;et, vt aqua
in latus &longs;ece&longs;&longs;i&longs;&longs;et, telluris globus non &longs;olùm
ex aquâ emergeret, &longs;ed relictâ in vniuer&longs;i
centro aquâ auolaret &longs;upra aërem, donec of
fenderet medium æqualis leuitatis &longs;upra aë
rem communem. Quarè machinationibus
opus e&longs;&longs;et non ad mouendam, &longs;ed ad reti
nendam tellurem.
Quod tellus, &longs;ublatâ aquâ, leuior
&longs;it quàm aer i&longs;te communis, quem &longs;piritu
ducimus, vix adduci po&longs;&longs;um, vt credam: id
autem cùm ex hypothe&longs;i à vobis con&longs;titutâ
&longs;equatur, hypothe&longs;i ip&longs;i probabilitatem de
mit. Quapropter aut nimiam igni leuita
tem, aut nimiam molem tribui&longs;tis; atqu&etail;
adeò vtramque aut alterutram temperar&etail;
oportet. Equidem exi&longs;timo in leuitate po
tiùs peccatum fui&longs;&longs;e quàm in mole: mihi &longs;i
quidem facilè per&longs;uadeo ingentem ignium
vim intimis terræ vi&longs;ceribus concludi, quo
rum poti&longs;&longs;imùm admini&longs;tratione natur&atail;
perficit pretio&longs;am illam metallorum &longs;uppel
lectilem, quibus &longs;odinæ ditantur: certum
e&longs;t autem calote humidum attenuante com
mi&longs;ceri &longs;piritus metallicos partibus fixis, at-
&longs;olis calor pertingere nequit: quamobrem
cum in tantá à centro di&longs;tantiâ producantur
metalla, totque milliarium cra&longs;&longs;itudine&mtail;
permeet calor, haud ægrè venio in docti&longs;&longs;i
mi VV endelini &longs;ententiam, &longs;i maximè cum
Lunæ libratione cohæreat. Sed quoniam
ignis ille acerrimus e&longs;t (quò autem tenuior
ignis, eò languidior e&longs;t) non adeò multam
forta&longs;sè teuitatem obtinet, vt eius leuitas ad
aquam &longs;it vt auri grauitas ad aerem, quem
admodum a&longs;&longs;um p&longs;i&longs;tis. Animum autem
aduertite, vtrùm eam potiùs Rationem ha
beat ignis leuitas ad aëris leuitatem in aquâ,
quam habet terræ &longs;eu argillæ grauitas ad
aquæ grauitatem in aere.
Si me audieritis, nullas hìc conie
cturas con&longs;ectabimur, præter eam, quam
olim à nobis con&longs;titutam nemo facilè infi
cietur, videlicet telluris globum vniuer&longs;um
in aere grauitate argillaceâ præditum e&longs;&longs;&etail;.
Globi &longs;oliditatem inue&longs;tigemus, eamqu&etail;
per &longs;pecificam argillæ grauitatem ducamus;
ex quo innote&longs;cet ab&longs;oluta grauitas globi:
Hinc demamus ab&longs;olutam aquæ grauitatem,
quæ obtinetur ductâ aquæ &longs;pecificâ grauitate
in eius &longs;oliditatem. Re&longs;iduum grauitatis ab
&longs;olutæ globi diui&longs;um per re&longs;iduum &longs;oliditatis
grauitatem globi demptâ aquâ.
Terreni glo. &longs;bi grauitas
peci fic&atail;
maior est
grauitat&etail;
argillæ, &longs;i
aqua dema
tur ex gio
bo.
Primùm globi &longs;oliditatem habemus, &longs;i
inuentæ &longs;ub initium no&longs;tri he&longs;terni &longs;ermonis
&longs;uperficiei &longs;phæricæ mill. quad. 214.201996.
pa&longs;&longs;. 716000, accipiamus trientem mill. quad.
71.400665. pa&longs;&longs;. 572000, & ducamus per
telluris &longs;emidiametrum mill. 4128. pa&longs;&longs;.
638: fiet enim totius globi &longs;oliditas mill. cu
bicorum 294787.501105. pa&longs;&longs;. 850936000.
Ex hac globi &longs;oliditate &longs;i inuenta &longs;uperiùs
aquæ &longs;oliditas mill. cub. 80.325748. pa&longs;&longs;.
768500000. auferatur, relinquitur mill. cub
294707.175357. pa&longs;&longs;. 82.436000. &longs;olidita
reliqui globi demptâ aquâ. s
Deinde globi &longs;oliditas pa&longs;&longs; cub.
294.
787501.105850.936000 ducatur per 9 &longs;pe
cificam grauitatem argi læ in aere, & e&longs;t ab
&longs;oluta globi grauitas 2653.087509.952658.
424000. Item aquæ &longs;oliditas pa&longs;&longs;.cub 80325.
748768.500000. ducatur per 5 1/3 &longs;pecificam
aquæ grauitatem, & fit ab&longs;oluta aquæ graui
tas 428403.993432.000000. Dematur hæc
aquæ grauitas ex terreni globi grauitate, &
remanet 2652.659105.959226.424000.
grauitas re&longs;idui globi demptâ aquâ.
Demùm re&longs;idua hæc grauitas 2652.
659105.959226.424000. diuidatur per re-
numerum pa&longs;&longs;. cub.
294.707175.357082.
436000. Et qui prodibic Quotiens (9 9994/10.000000)
proximè dabit re&longs;idui globi &longs;pecificam gra
uitatem pauló maiorem grauitare argillaceâ.
Quia nimirum aquæ demptæ grauitas &longs;peci
fica minor e&longs;t grauitate argillæ. Quod &longs;i
pars dempta fui&longs;&longs;et argilla, aut argillâ gra
uior, diminuta fui&longs;&longs;et grauitas &longs;pecifica; vt
quando ex va&longs;e metallico aëris pleno pars ali
qua metalli aufertur: Et hoc &longs;olùm in ca&longs;u
admittenda &longs;unt, quæ paulò antè Mer&longs;ennus
dicebat; quia tunc re&longs;iduum pondus ad pon
dus ablatum habet minorem Ratione&mtail;,
quàm re&longs;idua moles ad molem ablatam; hìc
autem contrà, quia grauitas aquæ minor e&longs;t
grauitate argillæ, re&longs;iduum pondus ad pon
dus ablatum habet maiorem Rationem,
quàm re&longs;idua moles ad molem aquæ abla
tam.
Quarè &longs;i totius globi terraquei grauitas
&longs;pecifica non fuerit in aere minor grauitate
&longs;pecificâ aquæ, fieri non pote&longs;t, vt aquis in
vnam partem &longs;ecedentibus terra reliqua in
natet, &longs;emper enim grauitas &longs;pecifica terræ
maior erit &longs;pecificâ grauitate aquæ.
Si telluri
tribuatur
grauit as ar
gillacea, a
cuæ in latus
&longs;ecedenti
innataret.
Si
grauitas ad aquæ grauitatem &longs;pecificam vt
rerraquei comparatur cum æquali mole a
quæ, erit pariter grauitas ab&longs;olu
ta globi terraquei ad grauitatem
ab&longs;olutam æqualis globi tantum
modò aquei vt RT ad SV. Dein
de ex globo terraqueo auferatur
aqua, cuius grauitas TO; æqua
lis aqua ex globo aqueo dempt&atail;
æqualem habet grauitatem VN:
& &longs;unt re&longs;iduæ grauitates RO &
SN inæquales, quia ex inæqualibus RT, SV,
ablatæ &longs;unt æquales grauitates OT, VN.
Cum itaque maior &longs;it Ratio RT maioris ad
OT, quàm SV minoris ad VN, erit per
conuer&longs;ionem Rationis minor Ratio RT ad
RO, quàm SV ad SN; igitur & vici&longs;&longs;i&mtail;
minor erit Ratio RT ad SV, quàm RO ad
SN. Igitur re&longs;idua globi terreni grauitas
RO ad re&longs;iduam globi aquei &longs;ecundùm mo
lem æqualis grauitatem SN habet maiorem
Rationem, quàm totius globi terraquei gra
uitas RT ad totius globi aquei æqualis gra
uitatem SV, hoc e&longs;t, quàm argillæ grauitas
ad grauitatem aquæ. Fieri itaque non pote&longs;t,
vt terra aquis innatet.
Valeat igitur Thales Mile&longs;ius cum
&longs;uo illo nauigio. F
enim in eam &longs;ententiam delabi quis po&longs;&longs;et,
vt &longs;ibi per&longs;uaderet ab inclu&longs;is terræ halitibus
atque ignibus tantum demi grauitatis, vt illa
intrà aquam leuior e&longs;&longs;et &longs;ecundùm &longs;peciem:
quæ quidem vnica e&longs;t innatationis cau&longs;&atail;.
Ni&longs;i quis fortè ex eorum numero, qui la
minam metallicam ratione figuræ in mul
tam latitudinem explicatæ, ideòque diffi
ciliùs, (vt ip&longs;i quidem loquuntur) &longs;ubie
ctam aquam diuidentis, innatare cen&longs;ent,
ambigeret pariter, an idem terræ quoqu&etail;
contingere po&longs;&longs;et.
Errorem hunc iamdudum aureo
illo tuo de Innatantibus libello profliga&longs;ti;
Galilæe: vixque puto aliquem &longs;upere&longs;&longs;e, qui
cramben hanc recoquat, præter eos, quos
iuuat ex antiquioribus tantùm codicibus ru
dioris &longs;æculi puluerem colligere. Nihil Gul
dino, nihil mihi e&longs;t cum huiu&longs;modi homi
num genere commercij: quapropter mi&longs;&longs;os
illos prorsùs faciamus: quamuis enim motui
tarditatem aliquam inferre po&longs;&longs;it figura, im
pedire tamen omninò non pote&longs;t, &longs;i illa qui
dem per &longs;e re&longs;piciatur; quod &longs;i ea fuerit cor
poris grauis figura, quæ leuioris aeris partem
buenda e&longs;t innatatio, quatenus ex aere, &
pondus minus e&longs;t pondere æqualis molis
aquæ. Cum autem nihil &longs;imile in terrâ con
tingere po&longs;&longs;it, quandoquidem aër ille, qui
valles ex aquâ extantes impleret, nihil con
ferret leuitatis (quemadmodum & in naui,
qui &longs;upra aquæ &longs;uperficiem aer, nauis pon
dus non minuit, nam intrà aerem non leui
tat) nulla &longs;pes nobis reliqua e&longs;t innatationis
terræ: &longs;ed quicquid in tellure machinationi
bus mouendâ compendij haberet Archime
des ex aquis, totum illud e&longs;&longs;et ex eâ ponde
ris diminutione, quam &longs;ubeunt &longs;olida gra
uiora aquis immer&longs;a.
Ita planè, &longs;i terræ tantùm ratio ha
beatur: nam compendium aliud, nec illud
contemnendum, antequam heri aduenires
Mer&longs;enne, ob&longs;eruabamus ex ip&longs;ius aquæ &longs;e
iunctione, cuius grauitatem ad calculos re
uocantes deprehendebamus non minorem
libris Rom. 803.257487.685000.000000.
Quæ &longs;i ex totius globi grauitate argilla
ceâ dematur, relinquet reliquæ terræ pon
dus.
Terraintra
aguam mi
nùs ponde
raret.
Vt autem totius globi grauitatem ad li
bras reuocem, accipio vnius milliaris cubici
ex argillâ grauitatem: cum verò aquæ gra
uitas ad argillæ grauitatem &longs;it vt 5
inuenerim lib.
10.000000.000000. erit quar
tus analogiæ terminus lib.
16.875000.000000
pondus vnius milliaris cubici ex argillâ. Per
hanc igitur grauitatem duco inuentam paulò
antè globi &longs;oliditatem mill. 294787.501105.
pa&longs;&longs;. 850936000, & prouenit demùm to
tius globi terraquei grauitas, &longs;i mera argilla
effet, lib.
4.974539.081161.234545.
000000. quæ forta&longs;sè paulò maior erit eâ gra
uitate, quam Mer&longs;enne, &longs;tatuebas, quia in
terræ diametro inueniendâ minùs accuratis
rationibus vtebaris, &longs;i &longs;atis memini. Ex in
uentâ itaque globi grauitate &longs;i dematur con
&longs;tituta aquarum grauitas, remanet &longs;olius ter
ræ pondus lib.
4.973735.823673.549545.
000000.
Terræ gra
uitas libra
rum nume
ro
explicata.
Cum verò grauitas hæc intrà aquam non
tota percipiatur, &longs;ed &longs;olùm iuxta exce&longs;&longs;um
grauitatis &longs;pecificæ argillæ &longs;upra grauitatem
aquæ &longs;pecificam, &longs;atis con&longs;tat, quantùm
minueretur terræ pondus, & quantò faciliùs
moueretur.
Eam tamen hic intelligis, puto, ad
hibendam circum&longs;criptionem, vt non toti
globo, &longs;ed illi tantùm parti, quæ aquis cir
cum&longs;underetur, grauitatem adimerent aquæ,
parem ponderi molis aqueæ æqualis; neque
delite&longs;ceret. Prætereà quamuis totius terreæ
portionis in aquâ exi&longs;tentis grauitas minuere
tur, &longs;egmenti tamen vltra Vniuer&longs;i cen
trum po&longs;iti grauitas imminuta vel auct&atail;
nihil iuuaret, cùm motui illa non repugnet,
dum de&longs;cendit: ideò &longs;olùm &longs;egmenti &longs;upe
rioris pars aquis immer&longs;a attendenda e&longs;&longs;et:
atque adeò quò magis terræ centrum ab vni
uer&longs;i centro remoueretur, eò augeretur ma
gis pondus, quia plus terrenæ molis ex aquis
extaret. Neque enim aliud terræ conringe
ret, quâm &longs;olido cuilibet corpori, quod ex
humore in va&longs;e extrahitur, cuius maior mo
les emergit ex aquâ, quàm &longs;it moles aquæ
accurrentis ad replendum &longs;patium à corpo
re &longs;olido relictum. Quia &longs;cilicet &longs;i aqua ma
neret in eâdem &longs;uperficie, nec deprimere
tur, &longs;olidi tanta moles emergeret, quanta e&longs;t
moles, quæ relinquit &longs;patium intra aquam:
&longs;ed quoniam aqua infra illam &longs;uperficie&mtail;
deprimitur, quam priùs con&longs;tituebat, & re
linquit aliam prætereà &longs;olidi partem ab aquâ
immunem, ideò moles, quæ ex aquâ emer
git, maior e&longs;t mole aquæ accurrentis ad re
plendum &longs;patium relictum à corpore eleua
to. Sic in va&longs;e HX, corpus AB totum in
trà aquam, cuius &longs;uperficies &longs;it CD, &longs;i ex-
trahatur ex A in H, pars CH, quæ &longs;upr&atail;
&longs;uperficiem aquæ CD eleuatur, æqualis e&longs;t
parti, quæ replebat &longs;patium GB: hoc autem
&longs;patium relictum implet aqua de&longs;cendens ex
C in E, ac proinde totum corpus HE extr&atail;
aquam manet; quæ moles maior e&longs;t mol&etail;
HC, hoc e&longs;t mole aquæ de&longs;cendentis DE ad
replendum &longs;patium &longs;ibi æquale GB. Idem
igitur terræ contingeret, quæ & eleuaretur
&longs;upra locum, vbi erat &longs;uperficies aquæ, &
infra illum locum &longs;uperficies aquæ deprime.
retur; ex quo fieret maiorem terræ partem
ex aquis emergers.
Solidi moles
ex bumido
emerg&ebreve;s, ma
ior est mole
humidi ac
currentis a
replendum
&longs;patium.
At aquæ illæ nullo va&longs;e contine
rentur, aut alueo.
Perinde e&longs;t &longs;i aqua va&longs;is lateribus
circum&longs;cripta certam figuram induat, ac &longs;i
&longs;uis &longs;e nutibus ip&longs;a in &longs;phæram di&longs;ponat. Il
lud quidem contingeret, quod cùm aqua ex
maiori eleuatione terræ in minorem &longs;emper
tionem haberet va&longs;is minoris & minoris;
atque adeò aqua magis & magis deprimere
tur, etiam datâ æquali terræ eleuation&etail;.
Sint enim duo va&longs;a &longs;imilia &longs;ed inæqualia, in
quibus &longs;int duo corpora &longs;imilia & æquali&atail;
AB in minori, & KL in maiori &longs;imiliter po
&longs;ita: & &longs;it continua &longs;olidorum immer&longs;orum
& aquæ circumfu&longs;æ &longs;uperficies. Extrahatur
ex aquâ vtrumque &longs;olidum pari velocitat&etail;;
vtique citiùs emerget omninò illud, quod
e&longs;t in minori va&longs;e, quàm quod e&longs;t in maiori;
& multò plus aquæ de&longs;cendere debet in ma
iori quàm in minori, ac proinde plus eleuari
debet &longs;olidum in maiori va&longs;e, quàm in mi
nori, vt æqualis moles emineat &longs;upra aquæ
de&longs;cendentis &longs;uperficiem.
Idem
vt extraba
tur ex bu
mido, plus
debet eleua
ri in va&longs;&etail;
maiori
in minori,
& plus a
quæ de&longs;ien
dit in va&longs;e
maiori,
in minori.
Hæc veritas
infertur ex
contradicto
ria hypothe
&longs;i.
Sit enim primò aquæ in va&longs;e minori &longs;u
perficies CD, in va&longs;e autem maiori OM:
deinde ita extrahatur vtrumque &longs;olidum, vt
æquales partes HE & VS emineant &longs;upra &longs;u
perficiem, quam denuò acqui&longs;iuit aqua de
&longs;cendens in locum à &longs;olido eleuato relictum.
Dico maiorem e&longs;&longs;e eleuationem VK &longs;upr&atail;
primam &longs;uperficiem OM, in va&longs;e maiori,
quàm &longs;it eleuatio AH &longs;upra primam &longs;uper
ficiem CD in va&longs;e minori; ac proinde &longs;pa
tium relictum NL maius e&longs;&longs;e &longs;patio relicto
maiorem aquâ CDFE de&longs;cendente.
Nam &longs;i KV non e&longs;t maior quàm AH,
ergo aut æqualis, aut minor. Sit æqualis: er
go quia HG & VN æquales &longs;unt ex hypothe
&longs;i, & VK ip&longs;i HA æqualis dicitur, etiam
reliquæ KN & AG, hoc e&longs;t OS & CE, æqua
les &longs;unt: ergo aquæ OMRS & CDFE æqua
lem habentes altitudinem &longs;unt inter &longs;e vt
ba&longs;es, hoc e&longs;t vt &longs;uperficies OM & CD. At
qui &longs;uperficies OM maior e&longs;t &longs;uperficie CD,
ergo aqua OMRS maior e&longs;t quàm aqu&atail;
CDFE: &longs;ed aqua de&longs;cendens e&longs;t æqualis mo
li corporis NL, quæ replebat &longs;patium reli
ctum; igitur maior e&longs;t moles NL quàm GB:
e&longs;t autem NL æqualis parti eleuatæ VO, &
GB æqualis e&longs;t parti eleuatæ HC, ergo VO
maior e&longs;t moles
VO & HC &longs;unt vt altitudines, quia ex hypo
the&longs;i data &longs;olida &longs;unt æqualia, &longs;imilia, & &longs;i
militer po&longs;ita; ergo maior e&longs;t altitudo KV
quam altitudo AH.
Quod &longs;i KV dicatur minor quàm AH
ergo ex æqualibus VN, HG, demptis inæ
qualibus, remanet KN maior quam AG,
hoc e&longs;t OS maior quam CE: aqua igitur
OMRS maiorem habens ba&longs;im ac maiorem
altitudinem, quam aqua CDFE, maior quo-
què erit; ac proinde & moles NL, hoc e&longs;t VO,
maior mole GB, hoc e&longs;t HC; & vt priùs ele
uatio KV maior eleuatione AH. Quare vide
tis hoc adeò certum e&longs;&longs;e, vt
eleuationem in maiori va&longs;e maiorem e&longs;&longs;&etail;
eleuatione in minori va&longs;e, veritatis huius con
fe&longs;&longs;io extorqueatur, ijs admi&longs;&longs;is, quæ con
tradicens ponit.
Priu&longs;quàm vlteriùs progrediatis,
vnum vellem ex te quærere; an &longs;cilicet aliqua
&longs;it inter eleuationes &longs;olidorum & depre&longs;&longs;io
nes aquatum in va&longs;is inæqualibus analogi&atail;
&longs;altem reciproca, ita vt quæ Ratio e&longs;t eleua
tionis KV in va&longs;e maiori ad eleuationem AH
in minori, eadem Ratio &longs;it depre&longs;&longs;ionis aquæ
CE, hoc e&longs;t AG, in minori ad depre&longs;&longs;ionem
aquæ OS, hoc e&longs;t KN in maiori.
Nulla e&longs;t &longs;iue directa, &longs;iue recipro-
quæ analogia, præterquàm in vno ca&longs;u. Non
quidem directa, quia, vt dicebam, KV ma
ior e&longs;t quàm AH, ergo maior e&longs;t Ratio KV
ad AG, quàm HA ad AG; atqui AG maior
e&longs;t quàm KN, ergo maior e&longs;t Ratio KV ad
KN quàm KV ad AG; ergò multò maior e&longs;t
Ratio KV ad KN quàm HA ad AG. No&ntail;
e&longs;&longs;e autem reciprocam analogiam &longs;ic o&longs;tedo.
Eleuationes
&longs;olidi, &
depre&longs;&longs;iones
bumidi i&ntail;
v
qualibus
&longs;unt propor
tionales, ni &longs;i
in vno c
Aqua OMRS circumfu&longs;a e&longs;t æqualis moli
NL, hoc e&longs;t VO; addatur vtrique commu
nis moles NO, erittota moles duobus pla
nis KOM & NSR parallelis contenta æqualis
moli corporis KL &longs;eu VS. Item aqua CDFE
circumfu&longs;a æqualis e&longs;t moli GB, hoc e&longs;t HC;
& additâ communi mole AE, erit tota mo
les planis ACD & GEF parallelis content&atail;
æqualis moli corporis AB &longs;eu HE. Atqui HE
& VS æquales &longs;unt moles ex hypothe&longs;i; igi
tur & moles GD æqualis e&longs;t moli NM: ergo
per 34. lib.
11. ba&longs;es cum altitudinibus re
ciprocantur, & vt AG ad KN, ita &longs;uperficies
va&longs;is maioris ad &longs;uperficiem va&longs;is minoris.
Prætereà aqua circumfu&longs;a OMRS æqua
lis e&longs;t moli VO, ergo ad molem KS habet
eandem Rationem quam VO ad KS, hoc e&longs;t
quam altitudines KV ad KN. At aqua cir
cum&longs;u&longs;a ad mo'em KS, quia in eâdem &longs;unt
&longs;uperficie &longs;olidi VY ad ip&longs;am &longs;uperficiem &longs;o
lidi; igitur VK eleuatio &longs;olidi ad KN depre&longs;
&longs;ionem aquæ e&longs;t vt &longs;uperficies va&longs;is maioris
minùs &longs;uperficie &longs;olidi ad ip&longs;am &longs;uperficiem
&longs;olidi VY: & componendo vt VN corporis
emer&longs;io ad KN aquæ depre&longs;&longs;ionem, ita &longs;u
perficies va&longs;is maioris ad &longs;uperficiem &longs;olidi
VY. Eâdem methodo o&longs;tenditur HA eleua
tionem &longs;olidi ad AG depre&longs;&longs;ionem aquæ e&longs;&longs;e,
vt e&longs;t &longs;uperficies va&longs;is minoris minùs &longs;uper
ficie &longs;olidi HT ad ip&longs;am &longs;uperficiem &longs;olidi: &
componendo vt HG ad GA, ita &longs;uperficies
va&longs;is minoris ad &longs;uperficiem &longs;olidi HT.
Quoniam igitur vt &longs;uperficies va&longs;is maio.
ris ad &longs;uperficiem minoris, ita GA ad NK;
&longs;i e&longs;&longs;et VK ad HA vt GA ad NK, iam e&longs;&longs;et vt
VK ad HA ita &longs;uperficies va&longs;is maioris ad &longs;u-
moles VO ad molem HC, hoc e&longs;t aqu&atail;
OMRS ad aquam CDFE; ergo vt &longs;uperficies
va&longs;is maioris ad &longs;uperficiem minoris, hoc e&longs;t
vt GA ad NK, ita aqua OMRS ad aqua&mtail;
CDFE. Atquiaquæ i&longs;tæ circumfu&longs;æ habent
Rationem compo&longs;itam ex Rationibus altitu
dinum & ba&longs;ium; ergo Ratio GA ad NK
æqualis e&longs;t
bus altitudinum GA ad N
ad OM. Cum verò &longs;ieri non po&longs;&longs;it vt (quan
do Ratio non componitur ex duabus Ratio
nibus, quarum altera &longs;it alterius &longs;ubduplica
ta, vt Ratio compo&longs;ita ex Rationibus 4 ad 2
& 2 ad 8, e&longs;t eadem cum Ratione 2 ad 4
con
modum hic non contingere &longs;uppono) Ratio
aliqua compo&longs;ita eadem &longs;it directè cum vnâ
ex Rationibus componentibus, ni&longs;i alter&atail;
Ratio componens &longs;it Ratio æqualitatis (&longs;ic
Ratio compo&longs;ita ex Rationibus 4 ad 4 & 4 ad
3 e&longs;t 16 ad 12 eadem cum Ratione 4 ad 3,
quia Ratio æqualitatis aliam Rationem mul
tiplicans eam non mutat) con&longs;tet autem ex
demon&longs;tratis AG maiorem e&longs;&longs;e quàm KN,
&longs;
Rationem æqualitatis. At ba&longs;es i&longs;tæ &longs;unt &longs;u
perficies va&longs;orum minùs &longs;uperficie &longs;olidi im-
ip&longs;æ æquales &longs;uperficies &longs;olidi, erunt &longs;uperfi
cies va&longs;orum æquales: id quod e&longs;t contra hy
pothe&longs;im. Non igitur e&longs;t vt VK ad HA ita
AG ad KN.
HA, vt AG ad KN: VK prima vel e&longs;t maior
quàm AG tertia, vel minor, vel æqualis. Si
maior, ergo per 14. 5. etiam HA &longs;ecund&atail;
maior e&longs;t quàm KN quarta; e&longs;t autem VK
ex dictis etiam maior quàm HA; igitur VK
e&longs;t maxima & KN minima; igitur per 25. 5.
VK &longs;imul cum KN maior e&longs;t quàm HA &
AG &longs;imul, quod e&longs;t contra hypothe&longs;im, iux
ta quam VN & HG æquales &longs;unt. Si VK mi
nor e&longs;t quàm AG, etiam HA minor e&longs;t quàm
KN; &longs;ed HA minor e&longs;t quàm VK ex dictis;
ergo HA e&longs;t omnium minima & AG omniu
maxima; ergo per 25.5. HG maior e&longs;t quàm
VN, contra hypothe&longs;im.
At verò &longs;i demùm VK prima æqualis &longs;it
AG tertiæ, etiam HA &longs;ecunda æqualis e&longs;t
KN quartæ: ergo per 7.5. vt VK ad KN ita
GA ad AH. At ex demon&longs;tratis vt VK ad KN,
ita &longs;uperficies va&longs;is maioris minùs &longs;uperficie
&longs;olidi ad &longs;uperficiem &longs;olidi VY, & vt GA ad
AH, ita &longs;uperficies &longs;olidi HT, hoc e&longs;t VY,
ad &longs;uperficiem va&longs;is minoris minùs &longs;uperfi-Igitur &longs;uperficies &longs;olidi e&longs;t medio
loco proportionalis inter differentias, quibus
&longs;uperficies &longs;olidi exceditur à &longs;uperficiebus va
&longs;orum: ergo componendo & permutando vt
&longs;uperficies va&longs;is maioris ad &longs;uperficiem mi
noris, ita &longs;uperficies &longs;olidi HT ad &longs;uperficiem
va&longs;is minoris minùs &longs;uperficie &longs;olidi, hoc e&longs;t
&longs;uperficiem aquæ CD. Sed vt &longs;uperficies &longs;o
lidi HT ad &longs;uperficiem CD, ita moles AE
ad aquam CDFE eiu&longs;dem altitudinis: & quia
AG ex hypothe&longs;i e&longs;t æqualis ip&longs;i VK, moles
AE e&longs;t æqualis ip&longs;i moli VO, hoc e&longs;t aquæ
OMRS. ergo vt &longs;uperficies HT ad &longs;uperfi
ciem CD, ita aqua OMRS ad aquam CDFE;
ergo per 11. 5. vt &longs;uperficies va&longs;is maioris ad
&longs;uperficiem minoris, hoc e&longs;t vt aquarum al
titudines AG ad KN, ita aqua OMRS ad a
quam CDFE. Sunt igitur aquæ inter &longs;e re
ciprocè vt earum altitudines: Ratio aute&mtail;
molium ex Rationibus altitudinum &
componitur, ba&longs;es verò non habent rationem
æqualitatis; ergo aquarum &longs;uperficies OM
ad CD &longs;unt in duplicatâ Ratione altitudinum
reciprocè &longs;umptarum, hoc e&longs;t vt quadratu&mtail;
GA ad quadratum KN. Id quod &longs;ic breui
ter demon&longs;tro Algebricis notis.
Sit GA altitudo R, & KN altitudo &longs;it S:
&longs;uperficies CD &longs;it D planum, & &longs;uperficies lgitur aqua CDEF e&longs;t
D plan. in R; & aqua OMRS e&longs;t Z plan; i&ntail;
S. Quare cum &longs;it Z plan. in S ad D plan. in R,
vt R ad S. erit per 16. 6. vel 19. 7. Z pl. in S.
quadr. æquale D plano in R quadratum: ‘er
go Z planum ad D planum, hoc e&longs;t &longs;uperfi
cies OM ad &longs;uperficiem CD, e&longs;t vt R qua
dratum ad Squadratum: &longs;unt igitur &longs;uperfi
cies aquarum in duplicatâ Ratione altitudi
num AG ad KN. At in Ratione AG ad KN
e&longs;t &longs;uperficies va&longs;is maioris ad &longs;uperfieie&mtail;
minoris, ergo &longs;uperficies OM ad &longs;uperficiem
CD e&longs;t in duplicatâ Ratione &longs;uperficiei va&longs;is
maioris ad &longs;uperficiem minoris.
Datis itaque va&longs;is &longs;imilibus inæqualibus,
& datâ Ratione &longs;uperficierum huiu&longs;modi va
&longs;orum, poterimus reperire &longs;uperficiem &longs;oli
di VY aut HT; quæ ex va&longs;orum &longs;uperficie
bus dempta reliquam aquæ &longs;uperficiem relin
quat in duplicatâ Ratione &longs;uperficierum va
&longs;orum. Cum enim &longs;olidorum &longs;uperficies HT,
VY habeant Rationem æqualitatis, maior e&longs;t
Ratio totius &longs;uperficiei va&longs;is maioris ad totam
&longs;uperficiem minoris, quàm ablatæ VY ad
ablatam HT, ergo per 33 5. apud Clau. ma
ior e&longs;t Ratio reliquæ OM ad reliquam CD,
quàm
haberi Ratio duplicata Rationis totius ad to-Sic autem inue&longs;ligo &longs;uperficiem VY,
quæ &longs;it media proportionalis inter differen
tias illius & &longs;uperficiei va&longs;orum, hoc e&longs;t in
ter OM & CD.
Ex datis
duabus &longs;i
perficiebus
auferre ean
dem &longs;uper fi
cie
linquat re
sidua in Ra
tione dupli
cata data
rum.
Ratio &longs;uperficierum va&longs;orum &longs;it data 8
ad 4; Ratio duplicata e&longs;t 8. ad 2. Pono &longs;u
perficiem &longs;olidi VY Algebricè 1℞. Quare &longs;u
perficies aquæ OM e&longs;t 8——1℞, & &longs;uperfi
cies CD e&longs;t 4——1℞: quæ &longs;unt inter &longs;e in du
plicatâ Ratione &longs;uperficiei va&longs;orum: igitur
8——1℞ ad 4——1℞. e&longs;t vt 8 ad 2 ergo per
19. 7. 16——2℞ æquatur 32——8℞. Et fa
ctâ Antithe&longs;i æquatio demum e&longs;t inter 16 &
6℞. In &longs;titutâ it aque diui&longs;ione 1℞ e&longs;t 2 2/3 &longs;uper
ficies VY; quæ ablata ex 8 &longs;uperficie va&longs;is
maioris relinquit &longs;uperficiem OM 5 1/3, &
ablata ex &longs;uperficie va&longs;is minoris, hoc e&longs;t ex
4, relinquit &longs;uperficiem CD 1 1/3: e&longs;t autem
2 2/3 medio loco proportionalis inter 5 1/3 & 1 1/3,
quæ præterea &longs;uperficies 5 1/3, & 1/3 &longs;unt in Ra
tione duplicatâ Rationis 8 ad 4, hoc e&longs;t in
Ratione 8 ad 2. Iam verò &longs;i ba&longs;es aquarum
OM ad CD &longs;int vt 8 ad 2, altitudines ve
rò KN & GA reciprocè vt &longs;uperficies va
&longs;orum, hoc e&longs;t KN 4, & GA 8, erit aqu&atail;
OMRS 32, & aqua CDFE 16, planè i&ntail;
Ratione, quam habent &longs;uperficies va&longs;orum,
in quibus ip&longs;æ aquæ exi&longs;tunt.
Quare hoc vno in ca&longs;u quando &longs;uperficies
&longs;olidi immi&longs;&longs;i e&longs;t media proportionalis inter
exce&longs;&longs;us, quos relinquit in &longs;uperficiebus va
&longs;orum, pote&longs;t contingere eleuationes &longs;olidi
reciprocari cum depre&longs;&longs;ionibus aquæ; quan
do &longs;cilicet eleuatio &longs;olidi in maiori va&longs;e e&longs;t
æqualis depre&longs;&longs;ioni aquæ in minori, & con
trà eleuatio &longs;olidi in minori æqualis e&longs;t de
pre&longs;&longs;ioni aquæ in maiori:
Operæ profectò pretium fuit hac
&longs;uper re te interrogare, de qua neminem di
&longs;putantem audi&longs;&longs;e me memini aut legi&longs;&longs;e.
Sed vt ad terram aquis delap&longs;is circumfu&longs;am
reuertamur, eadem-nè erit depre&longs;&longs;ionis aquæ
Ratio, quæ in va&longs;is, de quibus hactenus fuit
&longs;ermo?
Eadem e&longs;&longs;e ratio omnino non po
te&longs;t; quia aquæ de&longs;cen&longs;us non ex &longs;olo &longs;patio,
quod ab eleuatâ terrâ relinqueretur, pende
ret, quemadmodum &longs;i globus ex aquâ i&ntail;
va&longs;e eximeretur, &longs;ed etiam ex diuersâ ip&longs;ius
aquæ in &longs;phæram conformatione. Cum
verò aliam &longs;emper & aliam diuei&longs;aru&mtail;
&longs;phærarum portionem con&longs;titueret, &longs;eu po
tius Meni&longs;corum &longs;olidorum, donec demùm
in &longs;phæram integram aqueam di&longs;poneretur,
nulla certa & con&longs;tans Ratio afferri pote&longs;t;
&longs;ed dato certo &longs;patio, quod à centro terræ
centro di&longs;tantiâ, inue&longs;tigare oporteret, cuius
in &longs;phæræ &longs;uperficiem &longs;e data aquæ moles
di&longs;poneret in vtroque motûs termino, vt in
de colligeretur, quantum terræ ex aquis i&ntail;
motu illo emer&longs;i&longs;&longs;et.
Ab&longs;tineamus nunc, &longs;i placet, ab
hoc labore: quamuis datâ &longs;ectione lunulari,
& inuentis dimidiæ &longs;ectionis centro grauita
tis, ac viâ rotationis, po&longs;&longs;emus &longs;oliditatem
Meni&longs;coidis &longs;phærici inuenire; &longs;atis nunc
nobis e&longs;&longs;e puto inquirere; quantam in alti
tudinem &longs;u&longs;tolli terram oporteret, vt nul
lum amplius &longs;ub&longs;idium afferret aqua circum
fu&longs;a minuens terræ grauitatem.
Res e&longs;t non adeò difficilis aut ope
ro&longs;a. Inuenire &longs;cilicet oportet &longs;emidiame
& huic addere &longs;emidiametrum terræ; hæc
enim e&longs;&longs;et ea centri terræ atque centri vni
uer&longs;i di&longs;tantia, in qua nihil terræ intrà aquam
e&longs;&longs;et. Vt autem &longs;phæræ, quam aqua effi
ceret, &longs;emidiametrum habeamus; cum data
&longs;it aquæ &longs;oliditas, quam &longs;uperius po&longs;uimus,
fiat vt 11 ad 21, ita data &longs;phæræ aqueæ &longs;o
liditas pa&longs;&longs;. cub.
80325. 748768. 500000,
ad aliud, & prouenit 153349. 156739.
863636. cubus diametri minoris verâ. Item
tas 80325. 748768. 500000. ad 153447.
394508. 434977. cubum diametri maioris
verâ. Si igitur horum numerorum radix cu
bica extrahatur, habebimus &longs;phæræ diame
trum tùm minorem, tùm maiorem verâ.
Quantu&mtail;
elenanda e&longs;
&longs;et tellus, vt
omnino ab
aqua &longs;eiun
geretur.
tantùm vt propo&longs;itæ quæ&longs;tionis metam ali
quam attingamus, &longs;ed etiam vt methodum
ob&longs;eruem, qua cubicam radicem eruere &longs;o
les; alijs enim alia e&longs;t methodus, & nimis
attentum animum exigunt, dum &longs;eor&longs;im in
&longs;tituendæ &longs;unt multiplicationes, quæ&longs;ub da
tum numerum transferantur: & periculu&mtail;
&longs;æpius &longs;ube&longs;t, ne per imprudentiam aliu&mtail;
pro alio numerum &longs;upponas.
Vtrum ea, quæ mihi familiaris e&longs;t
methodus, omnium facillima &longs;it, ignoro:
mihi tamen arridet magis, cum in pote&longs;tate
meâ &longs;emper &longs;it operationem totam ex ordi
ne recogno&longs;cere. Cæterùm fieri non pote&longs;t,
quin plures requirantur operationes, cu&mtail;
præter cubum primi lateris oporteat inueni
re &longs;olidum ex triplo quadrato primi lateris in
latus &longs;ecundum, & &longs;olidum ex triplo later&etail;
primo in quadratum lateris &longs;ecundi, & de
mum ip&longs;ius lateris &longs;ecundi cubum. Quar&etail;
hic in &longs;chedula, quoniam &longs;ic placet, dato-
no&longs;træ quæ&longs;tionis. Hæc autem mihi e&longs;t re
gul&atail;.
Po&longs;t quartam quamque figuram, vt mo
ris e&longs;t, puncto notatam, primi puncti latus
de&longs;cribo & eius cubum extraho, id quod om
nibus methodis commune e&longs;t: deinde late
ris primi inuenti triplum &longs;cribo &longs;ub penulti
má figurâ &longs;ecundi puncti, ip&longs;um verò latus
primum &longs;ub antepenultimâ figurâ: & hos
duos numeros inuicem duco, &longs;cilicet triplum
lateris primi in ip&longs;um: & qui producitur nu
merus (&longs;ub antepenultimâ pariter figurâ col
locatus) e&longs;t Diui&longs;or; Quotiens verò e&longs;t latus
&longs;ecundum.
Radicis cu
bicæ facilis
extractio.
Tum latus &longs;ecundum duco in triplum la
teris primi, & productum &longs;cribo &longs;ub penul
timâ figurâ; atquè hunc addo priori produ
cto. Summam multiplico per latus &longs;ecun
dum, & producto addo Cubum &longs;ecundi la
teris collocatum &longs;ub figurâ puncto notatâ.
Summam demùm ex propo&longs;ito numero &longs;ub
duco, & &longs;ic deinceps.
Quoniam it aque numeri hìc propo&longs;iti &longs;unt
18 ciphrarum, & &longs;unt &longs;ex puncta, vt tempo
ri parcam & labori, prima quatuor punct&atail;
accipto pro primo puncto; & ex Tabulà, in
qua habeo myriadem cuborum, inuenio ma-
ho C ex D, & remanet E, cui addo tres fi
guras ad &longs;equens punctum pertinentes. De
inde triplico A, & e&longs;t F, quem &longs;ub penulti
mâ figurâ colloco, cui &longs;ub&longs;cribo A &longs;ub &longs;igu
râ antepenultimâ. Duco A in F, & produ
citur G: & a&longs;&longs;umpto G tanquam diui&longs;ore nu
meri E, habeo Quotientem B, &longs;cilicet latus
&longs;ecundum: Tum B in F duco, & 'produci
tur H &longs;ub penultimâ figurâ. Additis G & H,
fit &longs;umma I: quæ per latus &longs;ecundum duct&atail;
producit K; cui &longs;ub puncto dati numeri &longs;ub
&longs;cribitur lateris &longs;ecundi B cubus L; atque ex
K & L fit &longs;umma M auferenda ex E: & qui re
linquitur numerus N, pertinet ad &longs;equens
punctum.
Quod &longs;pectat ad numerorum collocatio
nem &longs;ub penultimâ vel antepenultimá figu
râ, &longs;atis videtis id factum e&longs;&longs;e propter com
pendium, quo omittuntur ciphræ nullita
tis, vltimo loco addendæ; e&longs;t &longs;iquidem la
teris primi numerus decadicus, &longs;i cum late
re &longs;ecundo comparetur: quare dum triplica
tur, & fit F, vna ciphra nullitatis e&longs;&longs;et &longs;ub
puncto: & dum hoc triplum per ip&longs;um la
tus ducitur, in producto e&longs;&longs;ent vltimo loco
duæ ciphræ ideò &longs;ub antepenultimâ figurâ
collocatur numerus productus, vt relinqua-
Huius me
th
ostenditur.
TABELLE WAR HIER
colloco A &longs;ub F ita, vt A &longs;it &longs;ub antepenulti
mâ figurâ. Quia verò quando B in F duci
tur, vnica tantùm e&longs;&longs;et ciphra nullitatis ad
ip&longs;um F pertinens, ideò productus H &longs;cribi
tur &longs;ub penultimâ figurâ. Cubus demùm L
&longs;ub puncto collocatur, quia nullam habet
ciphram nullitatis, quæ omittatur.
Quod autem pertinet ad ip&longs;am metho
dum, res clara e&longs;t. Dum enim triplum la
teris primi ducitur in latus ip&longs;um, hoc e&longs;t F
in A, producitur G, quod e&longs;t triplum qua
drati ip&longs;ius A. Dum verò B in F ducitur, &
fit H, planum, quod fit, e&longs;t ex latere &longs;ecun
do in triplum lateris primi. Additis G & H
fit &longs;umma I, hoc e&longs;t 3 A Quad. + 3 A in B.
Hæc &longs;umma multiplicatur per B, & fit K,
hoc e&longs;t 3 A Quadr. in B + 3 A in B Quadr.
Additur demum L cubus ip&longs;ius B, vt ip&longs;ius
lateris A + B cubus &longs;it A cub. + 3 A Quadr.
in B + 3 A in B Quadr. + B cub.
Præ&longs;agiebat animus me aliquid ex
te auditurum, quod operationem hanc fa
cilem redderet atque expeditam, vixque pu
to aliquid addi po&longs;&longs;e facilitatis atque per&longs;pi
cuitatis; cum nullâ hìc opus &longs;it numerorum
translatione, & triplicatio lateris primi A,
aut multiplicatio lateris &longs;ecundi B in F tri-
que eo, quod opus &longs;it multiplicatorem &longs;ub
multiplicando de&longs;cribere. E&longs;t itaque &longs;phæ
ræ aqueæ diameter minor verâ pa&longs;&longs;. 535254.
maior autem verâ pa&longs;&longs;. 535368. Quare &longs;ta
tui pote&longs;t diameter vera pa&longs;&longs;. 535300 rotun
dè, & &longs;emidiameter mill. 267. pa&longs;&longs;: 650:
quæ &longs;i addatur terreni globi &longs;emidiametro à
nobis &longs;uperiùs a&longs;&longs;umptæ mill. 4128 pa&longs;&longs; 638,
dabit mill. 4356 pa&longs;&longs;. 288. di&longs;tantiam cen
triterræ ab vniuer&longs;i centro, quando iam aqua
in &longs;phæram conglobata nihil iuuaret terræ
motionem. Nunquam autem, puto, ab Ar
chimede aliquis exigat machinationum &longs;pe
cimen exhiberi tanto motu. Ex quo fit nun
quam totius telluris pondus debui&longs;&longs;e ab Ar
chimede &longs;u&longs;tineri, atque adeò faciliùs perfi
ci potui&longs;&longs;e illam motionem, ac vulgus exi
&longs;timet, modò locus &longs;uppeteret, in quo ma
chinæ firmarentur. Aqua enim dum deor
&longs;um niteretur, quamuis ob minorem in &longs;pe
cie grauitatem non po&longs;&longs;et terram &longs;u&longs;tiner&etail;,
aliquantulum tamen repugnaret de&longs;cenden
ti, minueretque &longs;u&longs;tinentis laborem.
Archimedi
tellurem mo
uenti per ali
quot millia
ria, no&ntail;
fui&longs;&longs;et opus
totum pon
dus &longs;ustine
re.
Rectè intuli&longs;ti, quod voleba&mtail;;
aquâ videlicet infimam terræ portione&mtail;
&longs;ubeunte ita futurum vt minueretur terræ
pondus, vt hinc aliqua mouendi aut &longs;u&longs;tinen-Quamuis verò facilè per
mittam aquam repugnare terræ de&longs;cenden
ti, hæc enim intrà aquam de&longs;cendere no&ntail;
pote&longs;t, quin aquæ &longs;uperficies in maiorem
&longs;phæram conglobata magis ab vniuer&longs;i cen
tro recedat; quia tamen dubitare quis po&longs;
&longs;et, an corpora grauia motui &longs;ur&longs;um relu
ctentur, quando ita mouerentur, vt nihil
leuius infrà &longs;e, ac centro vicinius haberent;
aqua autem tunc ita moueretur, & in orbem
di&longs;poneretur, vt nihil e&longs;&longs;et centro vicinius,
infra quod con&longs;i&longs;tere expeteret, nihil enim
circumfiui aëris propiùs abe&longs;&longs;et à centro;
ideò exipsâ minori terræ grauitatione potiùs
quàm ex aquæ deor&longs;um nitentis re&longs;i&longs;tentiâ
rem explicandam cen&longs;erem. Grauitas &longs;i
quidem e&longs;t vis di&longs;ponendi &longs;e in vniuer&longs;o i&ntail;
loco &longs;ibi debito infra alia corpora: quò au
rem magis di&longs;&longs;imilia &longs;unt corpora &longs;ecundùm
locum, quem exigunt, & quò plures corpo
rum &longs;pecies inter illa deberent intercipi, &longs;i
iuxta naturæ propen&longs;ionem &longs;ingulæ di&longs;po
nerentur, eò etiam grauiora &longs;ecundum &longs;pe
ciem &longs;unt ea, quæ centro viciniora e&longs;&longs;e exi
gunt. Quarè tota grauitationis ratio & ni&longs;us,
quo vnum corpus infra aliud de&longs;cendere co
natur, in quo exi&longs;tit tanquam in medio, ori
tur ex di&longs;&longs;imilitudine &longs;ecundùm grauitatem.
eò pariter maior e&longs;t grauitatio, & conatus
deor&longs;um validior. Atqui quodcunque pen
dus grauius e&longs;t aquâ, e&longs;t multo grauius aëre;
igitur magis ab aëre differt quàm ab aquâ,
magi&longs;què in aëre grauitat quàm in aquá: cum
autem grauitet propter di&longs;&longs;imilitudinem, to
ta grauitatio petenda e&longs;t iuxta exce&longs;&longs;um, quo
&longs;uperat aquæ grauitatem.
Eâdem ratione ea quæ leuiora &longs;unt aquâ,
intrà aquam leuitatem habent iuxta differen
tiam, qua vincuntur à grauitate aquæ. At
que hinc facilè definitur cuiu&longs;cunque &longs;olidi
innatantis quota portio emergat ex aquâ aut
alio humido: ibi enim &longs;olidum intrans hu
morem con&longs;i&longs;tit, vbi grauitas partis in aëre
extantis æqualis e&longs;t leuitati portionis in hu
mido immer&longs;æ: id autem fit, quando pars
demer&longs;a ad extantem e&longs;t vt grauitas &longs;pecifi
ca &longs;olidi innatantis ad differentiam grauita
tum &longs;olidi & humidi. Si enim quæratur
quota portio plumbi extaret ex mercurio (e&longs;t
autem plumbum ad mercurium vt 11 ad
13) &longs;tatim dicetur molis plumbeæ (2/13) extare,
& (11/13) immergi; quia nimirum &longs;ingulæ partes
immer&longs;æ in mercurio leuitant vt 2, &longs;ingulæ
autem in aëre extantes grauitant vt 11. Si
igitur vt grauitas in aëre ad leuitatem in mer-
tem, paria &longs;unt grauitatis & leuitatis mo
menta: nam partes 11 leuitantes vt 2 ha
bent momentum leuitatis 22; & duæ partes
in aëre grauitantes vt 11 habent momentum
grauitatis vt 22; igitur &longs;equitur con&longs;i&longs;tentia.
Quod &longs;i vlteriùs deprimatur plumbum in
trà mercurium, augentur leuitatis momen
ta, & minuitur grauitas, ideò &longs;ibi relictum
a&longs;cendit; & &longs;i plures plumbi partes quàm
(2/13) extrahantur ex mercurio, de&longs;cendet, quia
grauitatis momenta augentur &longs;upra momen
ta leuitatis; & ibi demum quie&longs;cet, vbi fit
momentorum æqualitas.
Quota pars
&longs;olidi inna
tantis emer
gat ex hu
Habetis &longs;uper quo ad rauim v&longs;que
di&longs;putetis, &longs;i hanc per&longs;equi quæ&longs;tionem vo
lueritis; & illud forta&longs;&longs;e demum conficietis,
incertum e&longs;&longs;e, vtrùm grauia in aquâ minùs
conentur deor&longs;um
uis æquè conentur, minùs tamen proficiant
propter aquæ in oppo&longs;itum conantis re&longs;i&longs;ten
tiam: cum vtroque &longs;cilicet experimenta co
hærent.
Haud tanti e&longs;t: &longs;ed i&longs;tis dimi&longs;&longs;is,
ad alia, &longs;i lubet, tran&longs;eamus: &longs;atis enim pro
Archimede di&longs;&longs;eruimus.
paginam indicat.
n ari, num.
tympanis res perficitur, n.
marit, n.
&longs;it, num.
num.
stantia, n.
num.
num.
quàm fuerit deprehen&longs;a, n.
pleni et vacui, n.
culo aquæ æqualis, n.
&longs;et opus totum pondus &longs;u&longs;tinere, n.
parantur, n.
Fabula de altitudine Phari Alexandrinæ reijcitur, n.
tur, n.
dem in omni medio, n.
tia VV endelini, n.
vel &longs;ubten&longs;am, n.
mento, n.
n
&longs;ecundum magnitudinem, n.
m
num.
in vno meaio habet in quocunque meaio n.
&longs;pecifica
plus po&longs;&longs;unt, quam duæ trochleæ ex multis millibus
orb culo
aquæ &longs;ece&longs;&longs;ione, n.
num.
in liquore ponderetur.
terminum rationis &longs;im
æquipondium e&longs;&longs;e in eodem medio, n.
Stadij Græci quantitas, n.
gare inuentis aliquibus, n.
num.
ac urrentis ad replendum &longs;patium, n.
in va&longs;e maiori quàm in minori, & plus aquæ de&longs;cendit
in va&longs;e maiori quàm in minori, n.
libus, non &longs;unt proportionales, ni &longs;i in vno ca&longs;u, n.
quæ relinquat re&longs;idua in ratione duplicata dataru&mtail;,
num.
tate mouetur, n.
res habeat, n.
ro
P
quam plures in maioribus, n.
queat, n.
inuestigare, n.
num.
cum Trigonometria, & Algebra, n.
vix errari po&longs;&longs;it à Geometra, n.
mero
num.
læ, &longs;i aqua dematur ex globo, n.
cedenti non innataret, n.
mero
geret
ri, & &longs;piritum vini, n.
uitas, n.