GVIDIVBALDI
E MARCHIONIBVS
MONTIS
IN DVOS ARCHIMEDIS
ÆQVEPONDERANTIVM
LIBROS
PARAPHRASIS
Scholijs illu&longs;trata.
PISAVRI
Apud Hieronymum Concordiam;
M D LXXXVIII.
SERENISSIMO
FRANC.^{CO} MARIAE
II. VRBINI DVCI.
GVIDVSVBALDVS
E' MARCHIONIBVS MONTIS S.
I am decemnium elap&longs;um e&longs;t, DVX Sere
ni&longs;&longs;ime, ex quo de rebus machanicis volu
men, veras (ni fallor) mirabilium mechani
corum effectuum cau&longs;as manife&longs;tans, in lu
cem dedi; vbi non nulla an tiquiora,
puaque
cita ad &longs;u&longs;ceptum negotium pertinentia,
tanquam rect&ecedil; rationi magis con&longs;entanea amplexatus &longs;um.
quibus&longs;anè, tanquam &longs;olidi&longs;&longs;imis innixa fundamentis, theo
remata multa, ac varia con&longs;truxi. quippe quæ, licet non inua
lidis quoque demon&longs;trationum præ&longs;idijs à me ip&longs;o munita
fuerint; pleri&longs;què tamen, qui non admodum forta&longs;&longs;e in huiu&longs;
modi rerum cau&longs;is inue&longs;tigan disver&longs;ati exi&longs;tunt, noua pror
&longs;us (vt accepi) ac ferme inaudita, nec &longs;atis (vt opinor) apud eos
firma, atqueideo illis non omnino &longs;atisfeci&longs;&longs;e, vi&longs;a &longs;unt. Quo
circa cogitanti mihi, qua ratione fieri po&longs;&longs;et, vtopusillud à
me editum, quàm plurimorum &longs;ibi gratiam in dies magis con
ciliaret, in men tem venit, non aliunde id mihi oportun iùs
tingere potui&longs;&longs;e, quàm &longs;i pri&longs;cosip&longs;os, & graui&longs;&longs;imos alioqui
authores de hac re elegan ti&longs;&longs;imè di&longs;&longs;erentes illis offerrem. ra
tus, vt&longs;olidi&longs;&longs;imâ eorum doctrinâ, quæ à me propo&longs;ita, & ex
&longs;im ulquè alio
rum ambiguitati, ne dicam imbecillitau &longs;uccurreretur. vel&longs;al
tem ip&longs;i graui&longs;&longs;ima eorum authoritate non nullorum captiua
rentintellectum, in ob&longs;equium meliùs, rectiù&longs;què
atque intelligentium. Nihil enim tam, auta con&longs;uetudine, aut
ab opinione remotum e&longs;&longs;e &longs;olet, quod &longs;ola authoritate proba
ri non po&longs;&longs;it. Verùm ne huiu&longs;modi negotium-in recen&longs;endis
multorum ad propo&longs;itam veritatem confirmandam te&longs;timo
nijs la tiùs, quàm par e&longs;&longs;et, protraheretur; mihi con&longs;titui, ex mul
tis vnicum tantùm, eumquè reliquorum omnium hac in par
te facilè principem deligere: qui, & meam cau&longs;am tueretur: &
illis, &longs;i fieri po&longs;&longs;et, &longs;atisfaceret: vtquè coràm illis ip&longs;e &longs;e offerens,
tanquam meo quoque nomine mi&longs;&longs;us in telligeretur; quibu&longs;
dam meis notis non in &longs;ignitum certè, &longs;ed a&longs;&longs;ociatum cundem
prodire volui. E&longs;t autem graui&longs;&longs;imus hic author Syracu&longs;iusille
Archimedes de mechanicis elementis con&longs;ulu&longs;&longs;imè di&longs;&longs;erens.
cuius nimirum dignitati, atque authoritati, vtomnes probè à
me con&longs;ultum in telligerent; decreui, vt quemadmodum inter
alios illius ordinis viros primatum obtinet, ita nulli alij, quàm
amplitudini tu&ecedil; DVX Sereni&longs;&longs;ime, hac no&longs;tra &ecedil;tate, doctrina,
rerumquè omnium cognitione &longs;ingulari, citra controuer&longs;iam
Principi &longs;upremo, &longs;uum in primis hoc tempore præ&longs;taret ob&longs;e
quium. quod incredibili &longs;anè animi mei iucunditate conti
gi&longs;&longs;e fateor; non &longs;olùm, vt rur&longs;um aliquam &longs;ingularis meæ er
ga amplitudinem tuam ob&longs;eruailtiæ, ac venerationis, tot, tan
ti&longs;què nominibus iam pridem debit&ecedil; te&longs;tificationem ederem;
verùm etiam, vt munu&longs;culo illi meo tanto Principi audentiùs
forta&longs;&longs;e an tea oblato, ne pror&longs;us pr&ecedil; &longs;ua tenuitate de&longs;piceretur,
opem ferret. quanquam neque id quidem, pro eximia animi
tam excel&longs;i magnitudine, &longs;u&longs;picandum fuit. Per huncergo
celebrem authorem ad te Princeps optime, ac pr&ecedil;&longs;tanti&longs;&longs;ime
lætabundus accedo. Is enim mihi, quemadmodum & ego ip&longs;i,
ad te aditum patefeci&longs;&longs;e videtur; & &longs;icuteundem tibiloge gra
ti&longs;&longs;imum futurum confido; ita me tui amanti&longs;&longs;imum, & ob&longs;er
uanti&longs;&longs;imum, vt eâdem, qua con&longs;ueui&longs;ti, benignitate pro&longs;e
quaris, oro &longs;uplex, & ob&longs;ecro. Aueto dulce præ&longs;idium, ac &ecedil;tatis
no&longs;træ &longs;plendidum decus; & e&longs;to perpetuò f&ecedil;lix.
GVIDIVBALDI
E MARCHIONIBVS
MONTIS.
PRAEFATIO:
Mechanica facultas
verùm etiam ab eruditis admirabilis &longs;em
per habita fuit; eorum enim, qu&ecedil; in admi
rationem homines trahunt, duo e&longs;&longs;e gene
ra Ari&longs;toteles in principio
Meehanicarum a&longs;&longs;eruit; quorum &longs;anè alte
rum ad ea pertinet, quæ natura quidem,
proximis tamen ip&longs;orum cau&longs;is latentibus in lucem
alterum verò &longs;pectatad ea, qu&ecedil; preter naturam, & arte fiunt;
quibus natura &longs;uperari videtur (quamquam & ip&longs;a plurimùm
momenti ad &longs;e ip&longs;am euincendam tune quoque afferat) &
quod natur&ecedil; uiribus in lucem prodire nequit, id arte fieri con
tingat, obidquè maiorem adhuc admirationem excitat, quòd
ars natur&ecedil; çmula, qua&longs;i aduer&longs;us
ret, & tan
cau&longs;a quoque cognita admirationem parit; cùm exigua admo
dum ad tanti operisproductionem appareat. ad mirabilo e&longs;t &longs;a
nèip&longs;ius artis magi&longs;terium, cùm adeò potens &longs;it, vt effectus na
tur&ecedil; repugnantes producere tentet. quippè quibus, ni&longs;i ita &longs;en
&longs;ibus &longs;ub
vix fides adhibeatur; idquè
tum, acper&longs;ua&longs;um nobis e&longs;&longs;e po&longs;&longs;it. huiu&longs;modi autem mira
bilium operum opifex e&longs;t ip&longs;a mechanica di&longs;ciplina, tam na
tur&ecedil; emula, quàm oppugnatrix valida. H&ecedil;c enim grauia pro
prio fermè nutu &longs;ur&longs;um attolli, magnaquè pondera ab exigua
ctanda proponit. vt tum imperitis exip&longs;orummet effectuum
intuitu, tum eruditis in cau&longs;arum varia contemplatione ad
mirationem pariat. veluti &longs;i ea &longs;pectemus, qu&ecedil; neruis, vel ali
quo mouétur in&longs;trumento; vel qu&ecedil; &longs;piritibus
fiunt; de quibus Heron, & alij pertractarunt; vel denique alijs
modis. quamquam nosinijs, quæ dicenda &longs;unt, de ea mecha
nicæ facultatis parte, quæ ad pódera,
&longs;tétes pertinet,
ba faciemus. quæ quidé pars totius mechanic&ecedil; facultatis prin
cepsexi&longs;tit. ea enim e&longs;t, in qua artem &longs;uperare naturam aper
tiùs
num euadet.
Ars quippe ex Ari&longs;totele phi&longs;icorum &longs;ecundo, & ex proæ
mio quæ&longs;tionum mechanicarum triplici modo in &longs;uis opi&longs;i
cijs &longs;e&longs;e habere videtur. Nam vel immitatur naturam; vel ea
perficit, quæ natura perficere non pote&longs;t; vel denique ea, quæ
pr&ecedil;ter naturam fiunt, operatur; in quibus tamen omnibus o
perandi rationibus, &longs;i diligentereas con&longs;ideremus, artem &longs;em
perimmitari naturam per&longs;piciemus. Primùm quidem multas
artes naturam immitari aperte videmus, vt &longs;culpturam, & hu
iu&longs;modi alias. Quando autem arsea perficit, quæ&longs;ola natu
ra per&longs;icere non pote&longs;t, vtin arte medica euenire &longs;olet;
ip&longs;am pariter emulatur, & naturæ a&longs;&longs;ociata, velut in&longs;trumen
tum eius, naturalem effectum perficere dicitur: tuncquè eodé
modo operatur, ac &longs;i natura rem ip&longs;am ab&longs;que artis ope perfice
repo&longs;&longs;et, quod planè artis præ&longs;tantiam manife&longs;tat: quippè
cùm ni&longs;i ars ip&longs;inaturæ
effectus perficere ex &longs;e&longs;e minimè po&longs;&longs;it. At verò &longs;i ars
immitando ip&longs;am &longs;uperauerit; vtea, quæ ab arte fiunt, præter
naturam eueniant, longè adhuc præ&longs;tantiùs artis ingenium
apparebit. &longs;iquidem immitando naturam (paradoxum id for
tè videbitur, cùm tamen veri&longs;&longs;imum &longs;it) præter naturæ ordi
nem operari dicatur. Ars.
tura &longs;uperat; ita nimirum res di&longs;ponendo, vtip&longs;a efficeret na
tura, &longs;i eiu&longs;modi &longs;ibi producendos &longs;tatueret effectus. quod qui
dem &longs;ubiecto exemplo magis per&longs;picuum fiet.
Sint enim duo pondera
AB in aliquo vecte, A ma
ius, B minus; quorum &longs;i
mulita in vecte di&longs;po&longs;ito.
rum &longs;it centrum grauitatis
C. &longs;it autem &longs;ub vecte in
ter CA fulcimentum in D.
& quoniam pondera AB penes C grauitatis centrum inclinan
tur? tunc C deor&longs;um naturaliter mouebitur; ac per con&longs;equés Sed &longs;i B deor&longs;um mouetur,
A certè &longs;ur&longs;um eleuabitur. quippe quod,
atque &longs;olutum ab&longs;que connexione ponderis B deor&longs;um tende
ret; attamen vt adnexum ponderi B, intercedente vecte AB,
&longs;ur&longs;um mouebitur: & (vt ita dicam) pondus A contra pro
priam naturam naturaliter a&longs;cendet. Vndè
motus effectus e&longs;&longs;e naturales. Quid igitur efficit ars ip&longs;a? nil
fanè aliud, quàm quòd resita di&longs;ponit, & accomodat; vt &longs;imi
les effectus inde prodeant atque &longs;i naturales omnino exi&longs;tant,
quare opus erit, ut Ars naturam immitetur, &longs;iquidem effectus
naturales prouenire debent. propterea vectem, fulcimentum
què eodem modo di&longs;ponit; & loco ponderis B aliquam con
ti ip&longs;ius B; atque tunc ip&longs;a potentia mouens, qu&ecedil; minore&longs;t gra
uitate ponderis A, ip&longs;um A grauius nihilominus attollet.
quod quamuis propriæ ip&longs;ius naturæ repugnet, naturaliter ta
mé ab ip&longs;a potentia in B exi&longs;tente
&longs;po&longs;itæ talem habent naturam, vt A quidem &longs;ur&longs;um, B vero
deor&longs;um moueri debeant. qu&ecedil; &longs;anè ex no&longs;tro Mechanicorum
libro, & exijs, quæ in hoc pertractantur; comperti&longs;&longs;imè red
dentur, & quod diximus devecte, de alijs quoque in &longs;trumen
tis mechanicis in telligendum e&longs;t. quorum quidem apparatus
&longs;unt artis opera, effectus autem ip&longs;ius penè naturæ: cùm eius
momenta, inclinationes què lequantur, veluti præcipuas eiu&longs;
modioperum effectrices cau&longs;as: quippè quæ &longs;unt omninoad
mirabiles, acpr&ecedil;&longs;tanti&longs;&longs;ime; quemadmodum ex ip&longs;arum con
téplatione patere pote&longs;t. cuius rei
e&longs;to, Ari&longs;toteles.
rum
aperuit; qué &longs;ecutus Archimedes in his libris mechanica prin
cipia explicatiùs patefecit, eaquè planiora reddidit. Nec propte
rea Ari&longs;toteles diminutus extitit: etenim
po&longs;ita, & explicata fuere, problematum cau&longs;as egregiè patefe
cit. &longs;ed quoniam Archimedi &longs;copus fuit mechanic&ecedil; di&longs;ciplin&ecedil;
rudimenta explanare; propterea ad magis particularia
da de&longs;cendere voluit. Ari&longs;toteles.
vecte magna mouemus pondera? cau&longs;am e&longs;&longs;e ait
vectis maiorem ad partem potentiæ: & rectè quidem; cùm ex
principio ab ip&longs;o con&longs;tituto manife&longs;tum &longs;it, ea, qu&ecedil; &longs;unt in
longiori à centro Ar
chimedes verò vlteriùs adhuc progredi voluit, hoc admi&longs;&longs;o,
pè quod e&longs;t in longiori di&longs;tantia maiorem uim habere, quàm
id, quod e&longs;t in breuiori, inquirere etiam voluit, quanta &longs;it vis
eius, quod e&longs;t in longiori di&longs;tantia ad id, quod e&longs;t in breuiori;
ita vt inter h&ecedil;c nota reddatur qualis, & qu&ecedil; &longs;iteorum propor
tio determinata. atque ideo
pr&ecedil;&longs;tan ti&longs;&longs;imum manife&longs;tauit; videlicet ita &longs;e&longs;e habere pon
dus ad pondus, vt di&longs;tantia ad in&longs;tantiam, vnde pondera &longs;u
&longs;penduntur, &longs;e&longs;e permutatim habet. quo ignoto, res mechani
c&ecedil; nullo modo pertractari po&longs;&longs;e videntur. quandoquidem
huic tota mechanica facultas tanquam vnico, pr&ecedil;cipuoque Quare Archimedes
tur; quod non &longs;olùm patet exijs, quæ dicta &longs;unt; verùm etiam
&longs;i Archimedis po&longs;tulata
ea, quæ de principijs mechanicis Ari&longs;toteles patefecit, Archi
medé &longs;upponere
fiet. In ratione pr&ecedil;terea, ac modo
ma ambo affinitate coniuncti in cedere vidétur. Ari&longs;toteles.
cerea&longs;&longs;e
Mathematicè &longs;unt con&longs;ideranda, geometricè demon&longs;trauit,
vt &longs;unt di&longs;tantiæ, proportiones, & alia huiu&longs;modi: quæ verò
funtnaturalia, naturaliter
uitatis centrum &longs;pectant, & quæ &longs;ur&longs;um, & qu&ecedil; deor&longs;um moue
Ex quibus
inter tantos viros in his pertractandis con&longs;en&longs;um. Ambiget
forta&longs;&longs;e qui&longs;piam, nunquid h&ecedil;c principia rectè abillis fuerint
pertractata? &longs;ed &longs;tatim omnis ce&longs;&longs;at dubitandi occa&longs;io, &longs;i tan
torum virorum pr&ecedil;&longs;tantia ad memoriam reuocetur; quibus,
citra controuer&longs;iam in di&longs;ciplinis ab ip&longs;is traditis, omnes eru
diti vtquemadmodum
atque doctore, nemo ad rectè
ad Mathematicam,
apud peritiores authoritas meritò ob id &longs;uprema extat; quòd
abip&longs;is reseo meliori,
quo ip&longs;arum rerum natura, atque doctrin&ecedil; ratio po&longs;tulabat.s&s
qui &longs;cientiarum cupidi &longs;unt, illos &longs;equi, eorum què &longs;cripta &longs;&ecedil;pè
&longs;&ecedil;plus attentè perlegere debent. Pr&ecedil;terea philo&longs;ophi&ecedil;, ac Ma
thematic&ecedil; profe&longs;&longs;ores in hoc conueniunt; quòd cùm aliqua ad
philo&longs;ophiam &longs;pectantia tractant; mirum in modum Ari&longs;to
telem laudibus extollunt. qui verò Mathematicas pertractare
&longs;tudét, &longs;tatim ad Archimedis laudes pariter &longs;e
circa ea, qu&ecedil; nó &longs;unt Archimedis ver&longs;entur; vt
re, quod quidé optimo factum e&longs;t con&longs;ilio. etenim &longs;i ea, quæ
mathematica ope indigent, laudare volunt, ad Archimedem
confugiendum e&longs;t; vt&longs;i inuentionem, &longs;ubtili&longs;&longs;imum Archi
medis inuentum afferant, quo modum adinuenit cogno&longs;cen
d&ecedil; quantitatis argenti, quod eratin corona Regisaurea, vt Vi
truuius te&longs;tatur; & alia huiu&longs;modi; &longs;i admirabilia, &longs;tatim affe
rant Archimedis&longs;ph&ecedil;ram in globo vitreo elaboratam, in qua
omnes c&ecedil;le&longs;tis &longs;phæræ motus relucebant; ita utnatura potiùs
Archimedem immitata, quàm Archimedes naturam illu&longs;i&longs;&longs;e
videatur; nauim præterea graui pondere oneratam è mari in
littus ab Archimedeeductam; aliaquèid genus plurima. De
nique &longs;i res Mathematicas ciuitatibus e&longs;&longs;e vtiles o&longs;tendere vo
lunt, ea, quæ ab Archimede contra Marcellum in defen&longs;io
nepatriæ facta fuere, in medium afferant, quo tempore bellica
opera adeo mirabilia effecit, vt &longs;olus Archimedes contra bel
lico&longs;i&longs;&longs;imos Romanos pugnare &longs;ufficiens videretur. quæ qui
dem omnia Mechanica di&longs;ciplina Quid igitur
è qua tot, tantaquè ad
humani generis vtilitatem conferentia prodeunt? eximia cer
tè, & præclara admodum hæc Archimedisge&longs;ta fuerc; quæ ta
men, &longs;iad alia quamplurima, quæ deip&longs;o dici, ac afferri po&longs;
&longs;unt, conferantur; exigua &longs;anè mihi videntur. Nam quæ ha
ctenus commemorata &longs;unt, (quamquam forta&longs;&longs;e
multa tamen, huiu&longs;modiquè &longs;imilia alij quoque effecerunt,
& adhuc extant forta&longs;&longs;e viri co ingenij acumine pr&ecedil;diti, qui
talia aggredi non vererentur: &longs;ed
Archimedis opera, quorum &longs;imilia, nec antea, nec po&longs;t ipsú
facta fuere, neque in futurum facienda fore à nemine &longs;int ex
pectanda. omnium enim admirabili&longs;&longs;ima, præ&longs;tanti&longs;&longs;ima
què &longs;unteius &longs;eripta, in quibus, & ingenij acumen, inuentio
nes &longs;ubtili&longs;&longs;imæ, perfectaquè doctrina planè con&longs;picitur. adeo
enim his omnibus Archimedis &longs;cripta aliorum &longs;cripta mathe
maticorum excellunt, &longs;uperantquè; vt quæ aliorum, facilè
quidem inter &longs;e&longs;e comparari, cum ijsverò, qu&ecedil; ab Archimede
nobis relicta fucrunt; nullo modo po&longs;&longs;int. ut aperti&longs;simè
(alijs interim omi&longs;sis) con&longs;picuum redditur ex ijs, quæ de
&longs;ph&ecedil;ra & cylindro, & exijs, qu&ecedil; de æqueponderantibus &longs;cri
pta reliquit: quippè qu&ecedil; ob eorum
meritò literis aureis e&longs;&longs;ent imprimenda. liber enim de &longs;ph&ecedil;ra,
& cylindro inter Archimedis &longs;cripta
vt ad eius &longs;epulcrú appofita fuerit &longs;ph&ecedil;ra, & cylindr^{9}: quib^{9} a
Cicerone con&longs;pectis; &longs;tatim illud Archimedis &longs;epulcrú e&longs;&longs;e in
tellexit: de ouius inuentione ob uiri excellen
riatur: Deindè qua ratione ip&longs;um à temerario van&ecedil; orationis
proferendæ au&longs;u, (dum &longs;icloquitur, da mihi vbi &longs;i&longs;tam, ter
ramquè mouebo) vindicare po&longs;&longs;emus; ni&longs;ihec, quæ de æque
ponde
notitia proportionis ponderum, & di&longs;tantiarum, &longs;it manife
ftum non e&longs;&longs;e à ratione, nequè à natura pror&longs;us alienum, po&longs;&longs;e
terram moueri, &longs;i daretur con&longs;iftendi locus. quod etiam ex
no&longs;tro volumine Mechanico annis ab hinc aliquot elap&longs;is e
dito varijs quoquè in&longs;trumentis parere pote&longs;t.
multis modis, datum pondus à data potentia moueri, ibi
&longs;ume&longs;t. vbi demon&longs;trationes à nobis con&longs;titut&ecedil; ijs, quæ apud
ri voluntacceptam. Etne quidpiam, quod &longs;tudio&longs;is mecha
nicæ facultatis prode&longs;&longs;e po&longs;&longs;it, pr&ecedil;termitteretur, ad horum
Archimedis librorum interprætationem aliquid operis con
tuli&longs;&longs;e placuit; &longs;atisquè nobis feci&longs;&longs;e videbimur; &longs;i &longs;altem &longs;tu
dio&longs;inos Archimedis ve&longs;tigia &longs;ecutos fui&longs;&longs;e cognouerint.
Et quamuis opus hoc fuerit ab Eutocio A&longs;calonita nonnullis
commentarijs illu&longs;tratum, quia tamen propter Archimedis
&longs;criptorú ob&longs;curitaté multa adhuc remanét ab&longs;tru&longs;a, nec pror
&longs;us omnibus peruia; pr&ecedil;&longs;ertim gr&ecedil;carum literarum experti
bus; cùm liber hic in latinum ver&longs;us multis in locis ob&longs;curus,
alijsquè plerisque quodammodo mancus meritò &longs;u&longs;picetur;
ita vt adhucin tenebris iacere videatur; gr&ecedil;cusquè præterea
codex impre&longs;&longs;us, quem &longs;ecuti &longs;umus, multisin locis aliqua
correctione egere videatur; idcirco ab huiu&longs;modi munere
pr&ecedil;&longs;tando de&longs;i&longs;tere noluimus: quin &longs;imul hos libros in latinú
&longs;ermonem verteremus; commentarijsquè illu&longs;tratos redde
remus. Cùm præ&longs;ertim hinc tutus ad mechanicam
pateat aditus. Quare vtmens huius pr&ecedil;clari&longs;&longs;imi Math ema
tici magis, atque magis, quàm fieri po&longs;sit, pro virili no&longs;tra
per&longs;picua reddatur; & huius &longs;cientiæ cupidi in adipi&longs;cendis
pulcherrimis hi&longs;ce theorematibus minùs laborent; à commu
ni genere interpr&ecedil;tandi aliquamnulum in præ&longs;entia di&longs;cedere
nobis vi&longs;um e&longs;t oportunum. Nam qui res mathematicas in
terprætati &longs;unt, &longs;uos commentarios &longs;eor&longs;um à demon&longs;tratio
nibus collocauere: nos verò, qu&ecedil; no&longs;tra &longs;unt, verbis ip&longs;ius
Archimedis in&longs;eruimus, & hoc tantùm in ip&longs;is demon&longs;tra
tionibus, non in propo&longs;itionibus, & huiu&longs;modi alijs, hac
planèhabita di&longs;tinctione, vt quæ &longs;unt Archimedis (his, vel
chimedis e&longs;&longs;e intelligantur. Qu&ecedil; verò alterius &longs;unt cha
racteris, utqu&ecedil; huius exi&longs;tent formæ, no&longs;tra e&longs;&longs;e &longs;emper
&longs;int exi&longs;timanda. & quoad fieri potuit, verba omnia, qu&ecedil;
nobis declaratione aliqua, nec non correctione indigere vi&longs;a
&longs;unt (ijs tamen omi&longs;&longs;is, qu&ecedil; parui, imò nullius &longs;untmomenti,
vt e&longs;t literarum immutatio, & huiu&longs;modi alia) dilucidè expli
care, atque emendare &longs;tuduimus. quibus etiam hanc adhibui
&longs;int Archimedis in&longs;erta; &longs;iquis tamen verba tantùm Archi
medislegere maluerit, rectè id a&longs;&longs;equi poterit; &longs;iquidem ne
verbum quidem Archimedis omi&longs;im us: quinnimo ea ita di
&longs;po&longs;uimus, vt&longs;uum pror&longs;us retineant &longs;en&longs;um, po&longs;&longs;intquè
tinuatè legi; ac &longs;i nihil inter ip&longs;a in&longs;ertum fuerit. quod qui
dem &longs;tudio&longs;is non inutile fore iudicauimus; qui ab&longs;que no
&longs;tris additionibus
verò additionibus Archimedis demon&longs;trationes continua
tas, & explicatas habebunt. Huberionis autem doctrinæ gra
tia permulta adiunximus &longs;cholia, in quibus pa&longs;&longs;im ordinem,
Authori&longs;què artificium patefecimus; nec non multa lemma
ta ad Archimedis demon&longs;trationes nece&longs;&longs;aria
mus
reriam valde vtilia adiecimus. Vt etiam Archimedis dicta
magis eluce&longs;cant, antequam ad explicationem verborum
ip&longs;iusaccedamus, nonnulla prius declarare oportunum no
bis vi&longs;um e&longs;t ad ea, quæ in his libris Archimedis &longs;upponit
tanquam cognita. Deinde con&longs;iderand us proponitur &longs;copus,
atque iiitentio Archimedis; diui&longs;io item librorum; huiu&longs;
modiquè alia, quæ &longs;ummam afferent facilitatem ad intel
ligendam: mentem Archimedis.
que&longs;t. Me
chan.
huius para
phra&longs;is.
Cùm itaquè &longs;upponat &longs;unos exqui&longs;itam habere no titiam
centri grauitatis; illius definitionem afferre libuit: pro cuius
to men faciliori notitia illud quoque in primis admonen
delicet vniuer&longs;i, cen trum magnitudinis, centrum figuræ, &
centrum grauitatis, quod quidem grauitatis centrum rectè
definitur à Pappo Alexandrino in octauo libro mathemati
carum collectio num hocpacto.
DEFINITIO CENTRI GRAVITATIS
Centrum grauitaris vniu&longs;cuiu&longs;que corporis e&longs;t pun ctum
quoddam intra po&longs;itum, à quo &longs;
conçipiatur, dum fertur, quie&longs;cit
principio habebat po&longs;itionem, nequeinip&longs;a latione circum-
EIVSDEM ALIA DEFINITIO.
Centrum grauitatis vniu&longs;cuiu&longs;que &longs;olidæ figuræ e&longs;t
illud in tra po&longs;itum, circa quod vndique partes &ecedil;qualium mo
mentorum con&longs;i&longs;tunt. &longs;i.
guram quomodo cunque &longs;ecans, &longs;emper in partes æqueponde
rantes ip&longs;am diuidet.
Hanc po&longs;tremam definitionem, &longs;eu potiùs de&longs;eriptionem
tradidit Federicus Commandinus in libro de centro grauita
tis &longs;olidorum. ex quipus &longs;anè definitionibus eluce&longs;cit natura,
atque facultas
vt &longs;i punctum A fuerit
grauitatis corporis BC, tunc
ex Pappi&longs;ententia, &longs;i BC
datur ex A, magnitudo BC
eadem, qua reperitur, di&longs;po
&longs;itionelocata manebit; neque
partes ullasip&longs;ius corporis, vt qu&ecedil; &longs;untad
BC, circumuerti, neque omnino &longs;uum
mutare &longs;itum depræhendetur. &longs;i verò vt
grauitatis magnitudinis BCD, eadem
què per punctum A vtcunque &longs;ecúdùm
rectitudinem diuidatur, veluti per EAF.
tunc pars EBF ip&longs;i ECDF æqueponde
rabit, quamuis EBF, & ED &longs;int magni
tudines inæquales. &longs;æpenumero enim e
uenire &longs;olet, vt in diui&longs;ione figuræ per eius centrum graui
tatis ip&longs;a aliquando in partes diuidatur æquales, ali
quando in partes inæquales: vt &longs;uo loco o&longs;tendemus:
&longs;emper tamen in partes diuiditur hinc inde æquepon
derantes; non tamen &longs;eor&longs;um con&longs;titutas, ab inuicen
què &longs;eiunctas, & veluti ad æquilibrium examinatas; vt pu
ta &longs;i EBF decem pondo ponderet; ED quoque totidem
pependiffe oporteat. res quippe non &longs;ic &longs;e habet, &longs;ed cas e&longs;&longs;e
in eo &longs;itu æqueponderantes, in quo reperiun tur; vt neutra
ex quibus colligipote&longs;t, &longs;i graue quidpiam
in centro mundi collo catum fuerit, oportere centrum graui
tatis illius in centro mundi con&longs;titutum e&longs;&longs;e: &longs;iquidem vt
graue illud tunc quie&longs;cat, partes vn dique ip&longs;um ambientes ç
qualium momentorum exi&longs;tere, atque manere oporteat.
Quare dum a&longs;&longs;eritur, graue quod cumque naturali propen
fione &longs;edem in mundi centro appetere, nil aliud &longs;ignifica
tur, quàm quòd eiu&longs;modi graue proprium centrum grauitatis
cum centro vniuer&longs;i coaptare expetit, vt optimè quie&longs;cere va
leat. Ex quo &longs;equitur motum deor&longs;um alicuius grauis fieri
per rectam lineam, quæ centrum grauitatis ip&longs;ius grauis, cen
trumquè mundi connectit. quandoquidem grauia deor&longs;um
rectà feruntur. Vnde manife&longs;tum e&longs;t, Grauia &longs;ecundum gra
uitatis centrum deor&longs;um tendere. quod nos in no&longs;tro Mecha
nicorum libro &longs;uppo&longs;uimus.
mi buius.
Ex ijs omnibus, quæ hactenus de centro grauitatis dicta
&longs;unt, per&longs;picuum e&longs;t, vnumquodque graue in eius centro
grauitatis propriè grauitare, veluti nomen ip&longs;um centri gra
uitatis idip&longs;um manife&longs;tè præ&longs;eferre videtur. ita vt tota vis,
grauita&longs;què ponderisin ip&longs;o grauitatis centro coaceruata, col
lectaquè e&longs;&longs;e, ac tanquam in ip&longs;um vndiquè fluere videatur.
Nam ob
uenire cupit; centrum verò graui tatis (exdictis) e&longs;t id, quod
propriè in centrum mundi tendit. in centro igitur grauitatis
pondus propriè grauitat. Præterea quando aliquod pondus
ab aliqua potentia in centro grauitatis &longs;u&longs;tinetur; tunc pon
dus &longs;tatim manet, to taquè ip&longs;ius ponderis grauitas &longs;en&longs;u per
cipitur. quod etiam contingit, &longs;i &longs;u&longs;teneatur pondus in ali
quo puncto, à quo per centrum grauitatis ducta recta linea
in centrum mundi tendat. hoc namque modo idem e&longs;t, ac
Quod
quidem non contingit, &longs;i &longs;u&longs;tineatur pondus in alio pun
cto. neque enim pondus manet, quin potiùs
grauitas percipi po&longs;&longs;it, vertitur vtique pondus, donec &longs;imi
liter à &longs;u&longs;pen&longs;ionis puncto ad centrum grauitatis ducta re
cta linea in vniuer&longs;i centrum recto tramite feratur.
quæ quidem ex prima no&longs;trorum Mechanicorum pro-
zonti erecta, tuncidem pror&longs;us e&longs;t (vt mox diximus) perinde
ac &longs;i pondus in centro grauitatis ad vnguem &longs;u&longs;tineretur.
Quocirca &longs;i pònderis grauitas minimè percipi pote&longs;t, ni&longs;i in
Centrum figuræ apud Mathematicos e&longs;t punctum, à quo
&longs;emidiametri exeunt; vel per quod
li centrum, & ellip&longs;is, necnon oppo&longs;itarum &longs;ectionum.
Centrum verò magnitudinis e&longs;t id, quod medium figuræ
obtinet; vel quod &ecedil;qualiter ab exteriori &longs;uperficie di&longs;tat. vt
&longs;phær&ecedil; centrum.
Centrum denique mundi e&longs;t punctum in medio vniuer&longs;i
&longs;itum, omniumquè rerum infimum.
Cæterùm ad meliorem horum notitiam ob&longs;eruandum e&longs;t,
h&ecedil;c centra aliquando &longs;imul omnia inter &longs;e conuenire,
do nonnulla; aliquando autem minimè. &longs;imul verò omnia
conueniunt. vt centrum vniuer&longs;i, centrum magnitudinis ter
ræ (&longs;ph&ecedil;ræ &longs;cilicet ex aqua, terraquè compo&longs;it&ecedil;, quam nos bre
uitatis &longs;tudio terram tantùm nuncupabimus) centrum figu
r&ecedil; terr&ecedil;; ac centrum grauitatis terr&ecedil;. Cùm enim terra &longs;it &longs;phæ
rica (vt omnes fatentur.) eius medium erit centrum figur&ecedil;, à
quo &longs;emidiam etri exeunt. idip&longs;um què erit centrum magnitu
dinis, &longs;iquidem ip&longs;ius figur&ecedil; medium obtinet. Pr&ecedil;terea idem
punctum e&longs;t centrum grauitatis terr&ecedil;. & quoniam terra in me
dio
collocatum. & hoc dun taxat modo centra omnia in
uenire po&longs;&longs;unt. quamquam verò &longs;ph&ecedil;ra, qu&ecedil; continet
aqu&acedil;, compo&longs;ita e&longs;t ex corporibus diuer&longs;&ecedil; &longs;peciei,
grauitatis, nimirum ex terra, & aqua; non
Ari&longs;to telis &longs;ententia terra circa mundi centrum vn dique
&longs;tit; & Archimedes affirmat,
cum&longs;i ita que terra, & aqua ma
rumatque adeo quatuorpr&ecedil;dicta
centra in Quod
mul centra in vnum co
naturæ intuenti; &longs;iquidem eius medium erit centrum magni
tudinis, & centrum &longs;iguræ; idemquè punctum crit ip&longs;ius cen
& quoniam hæc &longs;phæra non e&longs;t in centro mundi; propterea
tria tantùm centra &longs;imul conuenient. &longs;i verò &longs;phçra non &longs;imi
laris, &longs;ed di&longs;&longs;imilaris fuerit, veluti altera ip&longs;ius meditate plum
bea, altera verò medietate lignea exi&longs;tente, tunc eius medium
erit quippe centrum magnitudinis, & figur&ecedil;, grauitatis verò
centrum nequaquam. Nam partes vndique circa medium æ
queponderare non po&longs;&longs;ent; &longs;ed grauitatis centrum ad grauio
rem partem, nimirum plumbeam declinabit. & hoc modo
duo tantùm centra inter &longs;e conuenient. vt etiam (modo ta
men diuer&longs;o) accidit ellip&longs;i; cuius centrum e&longs;t centrum figu
r&ecedil;, &longs;iquidem per ip&longs;um tran&longs;eunt diametri; idemquè
quod cùm non &longs;it propriè me
dium figuræ, non erit quoque centrum magnitudinis.
enim figuræ propriè circulo, ac &longs;phæræ tantùm competit.
Quare duo centra hoc quoque modo &longs;imul tantùm conue
nient. In figura paraboles recta linea terminat&ecedil; centrum gra
figuræ, neque centrum magnitudinis e&longs;&longs;e pote&longs;t. etenim in
hac figura non pote&longs;t dari medium, vnde neque centrum ma
gnitudinis dabitur, & quoniam in parabole diametri &longs;unt in
ter&longs;e &ecedil;quidi&longs;tantes, vt ex primo libro conicorum Apollonij
pergei con&longs;tat; neque etiam centrum figuræ dabitur. &longs;ic igi
tur centra nullo modo conuenient.
de cælo
de iis
qu&ecedil; uehun
tur in aqua
ci
centro gra
uitatis &longs;oli
dorum.
com
man. de cen
tro graui
tatis &longs;olido
rum.
libro huius
Noui&longs;&longs;e quoque oportet centrum grauitatis communius
e&longs;&longs;e, in pluribu&longs;què reperiri, quàm centra magnitudinis, & fi
guræ: centrum verò figuræ communius e&longs;&longs;e centro magnitu
dinis. In trin&longs;ecùs vt
figuræ, vel alicuius figuræ vt A; cuius centrum grauitatis &longs;it
in ambitu figuræ, vt in puncto B; extrin &longs;ecùs verò vt figura
C, cuius centrum grauitatis extrin&longs;ecus &longs;it, vt in D; quod
e&longs;t in telligendum, &longs;i graue C in centrum mundi ten deret,
ueniret; &longs;iguraquè C quie&longs;ceret circa cen
trum vniuer&longs;i, veluti &longs;e habetcirca
D. partes enim figuræ talem po&longs;&longs;untha
bere &longs;itum, vt inter &longs;e &ecedil;queponderare po&longs;
&longs;int. vt ex &longs;ubiectis figuris per&longs;picuum e&longs;t.
& ad huc clariùs, &longs;i in telligatur figura, vt
E circulo tum exteriori, tum interiori ter
minata, cuius centrum grauitatis extra fi
guram eritin F. quod quidem cum cir
culorum centro conueniet. circa quod
(exi&longs;tente centro F in centro mundi)
partes vndique &ecedil;queponderabunt: cùm
omnes &ecedil;qualiter à centro grauitatis
præterea in hac figura E centrum graui
tatis (quamuis &longs;it extra &longs;iguram) cum cen
tro figuræ,
figuræ conuenire, forta&longs;&longs;e non eritincon
ueniens a&longs;&longs;erere. At verò figuræ AC nul
lo pacto figuræ, magnitudinisquè
habebunt. & quamuis dictum &longs;it
grauitatis corporum regularium e&longs;&longs;e me
dium ip&longs;orum, non tamen propterea dicen dum e&longs;t, idem e&longs;&longs;e
centrum magnitudinis, atque figuræ, ni&longs;i impropriè;
enim his impropriè attribuitur, &longs;icuti etiam centrum figuræ;
cùm lineæ ex ip&longs;o prodeuntes non &longs;int ip&longs;orum corporum
(quatenus regularia &longs;unt) &longs;emidiametri. quare centrum gra
uitatis reperiri pote&longs;t ab&longs;que alijs centris; at non è conuer&longs;o.
Rur&longs;us commune magis e&longs;t
dinis; quia præter circulum, & &longs;phæram, quæ tam figuræ,
magnitudinis centrum habent, nonnullæ figuræ &longs;uum ha
bent figuræ centrum in ip&longs;is, & extra ip&longs;as; in ip&longs;is, vt ellip&longs;is,
cuius centrum in tùs habetur; &longs;emicirculus etiam, dimidia què
&longs;phæra centrum habent in limbo. extra figuram verò veluti
hyperbolæ centrum, quod extra figuram exi&longs;tit; vbi nempè
diametri concurrunt. Quæ quidem omnia &longs;unt figuræ cen
tra; magnitudinis verò minimè. verùm obijciet hoc loco for
tiones allatas, diminutas e&longs;&longs;e; vel ijs, quæ modò à nobis de
tro grauitatis dicta &longs;unt, repugnare; cùm o&longs;tenderimus cen
trum grauitatis aliquando e&longs;&longs;e, vel in ambitu figuræ, vel extra
figuram; definitiones verò allat&ecedil; &longs;emper &longs;upponunt illud e&longs;&longs;e
in ip&longs;is intra po&longs;it
dem, neque huiu&longs;modi centrum extra figuram con&longs;titutum,
fui&longs;&longs;e Archimedi pror&longs;usignotum, exi&longs;timare debemus; vt
colligere licet ex nono po&longs;tulato huius libri; cùm inquit.
grauitatis intra ip&longs;am e&longs;&longs;e oportet.
metrum non ad eandem partem concauum habenti, extra
ip&longs;am grauitatis centrum obtinere. Cui obiectioni in hunc
modum occurri poterit, &longs;i dixerimus, quòd quamuis exempli
gratia in figura C dictum &longs;it centrum grauitatis D extra fi
guram exi&longs;tere, id ip&longs;um etiam intra figuram e&longs;&longs;e affirmati
poterit. &longs;iquidem ambitus figur&ecedil; C centrum D intra &longs;e
tinct; ita vt re&longs;pectu tötius &longs;it intra. idemquè dicen dum e&longs;t de
altera figura A. hoc autem euidenti&longs;&longs;imum e&longs;t in figura E.
& hic e&longs;t &longs;en&longs;us definitionum centri grauitatis. His itaque pri
mùm cognitis con&longs;ideranda e&longs;t intentio Archimedis in his li
bris, quç quidem vt plurimum à librorum in&longs;criptionibus e
luce&longs;cere &longs;olet.
DE SCOPO HORVM LIBR ORVM
Si Archimedis propo&longs;itum in his libris ex ip&longs;a operis in
&longs;criptione, vt in alijs quoque aliorum authorum volumini
bus fieri vt plurimùm &longs;olet, inue&longs;tigandum erit, partim &longs;anè
con&longs;picuum illud e&longs;&longs;e videbitur, partim verò ignotum adeò,
vt potiùs nullius fermè rei &longs;e habiturum e&longs;&longs;e &longs;ermonem profi
teatur Archimedes. quid enim (ob&longs;ecro) verbis illis &longs;ignificari
potuit, que primilibri initio ita &longs;e
<10>o w_ixw_n, h_ ke(nt<10>a <32>a/<10>wn e)w_iw_e\dwn.
derantium, pel centra grauitatum planornm.
tur Archimedes rem pror&longs;us
gnantem &longs;ibi contemplandam proponere. dùm enim polli-
ue de centris grauitatum planorum; cùm ea, quæ æqueponde
rare debent, ponderare quoque oporteat; &longs;i plana æqueponde
rare quod
valdè à planorum natura abhorret, cùm grauitas, nonn&longs;ii cor
poribus, neque tamen omnibus comperat. ip&longs;e tamen, dum
plana æqueponderantia, vel centra grauitatum planorum &longs;e
explicaturum pollicetur, apertè &longs;upponit plana, ac &longs;uperficies
graues exi&longs;tere, rem &longs;anè immaginariam pror&longs;us, ip&longs;iusquè rei
naturæ nuiiatenus re&longs;pondentem. ita vt Archimedes circa ea,
quæ omnino rei naturæ aduer&longs;antur, negotium &longs;ump&longs;i&longs;&longs;evi
deatur. Verùm enimuero &longs;i Authoris
mur, rem planè egregiam, naturæquè rei apprimè con&longs;enta
neam ip&longs;um pertra ctandam &longs;ump&longs;i&longs;&longs;e depræhendemus. Nam
quamuis plana, quatenus plana &longs;unt, nuiiam habeant graui
tatem, non e&longs;t tamen à rei natura, neque à ratione alienum,
quin po&longs;&longs;imus planorum, &longs;uperficierum què centra grauitatis
depræhendere, ex quibus&longs;i &longs;u&longs;pendantur, planorum partes
vndiquè &ecedil;qualium momentorum confi&longs;ten tes maneant.
doquidem centrum grauitatis talis e&longs;t naturæ, vt &longs;i mente
cipiamus
&longs;e, eo pror&longs;us modo, quo reperitur, quie&longs;cat, & maneat. vt
antea declarauimus. & quamuis re ip&longs;a, actùque plana
à corporibus reperiri nequeant; in ip&longs;is tamen hæcip&longs;orum
circa centra grauitatis æqueponderatio ad actum facilè redigi
poterit. Vt &longs;it &longs;olidum AB pri&longs;
ma, cui^{9} latera AE CF DB &longs;int
horizonti erecta, &longs;upetiorquè ba
&longs;is ACD, quem ad modum & in
ferior EFB &longs;it horizonti æquidi
&longs;tans; &longs;it autem plani ACD cen
trum grauitatis G, exquo G &longs;i
&longs;u&longs;pendatur totum AB patet
planum ACD horizonti æqui
di&longs;tans permanere, ac plopterea
circa
ponderare. quod quidem, quamuis egeat demon&longs;tratione,
&longs;at
autem nobis nunc &longs;it o&longs;tendi&longs;&longs;e, hæc ad praxim reduci, ma
nibu&longs;què (vt dicitur.) contrectari po&longs;&longs;e. Quòd &longs;i hæc ita &longs;e ha
bent, huiu&longs;modi con&longs;ideratio non erit vana, neque vt inuti
lis reijcienda. Sed vlteriùs adhuc progrediamur, dicamu&longs;
què, quoniam planum ACD, quatenuse&longs;t corpori coniun
ctum, horizonti æquidi&longs;tans permanere debet; &longs;i &longs;eor&longs;um à
corpore illud in telligamus, vt &longs;i ADC ex eius centro graui
tatis G &longs;u&longs;pendatur, tunc quocunque modo reperiatur, hoc
e&longs;t &longs;iue horizonti &ecedil;quidi&longs;tans, &longs;iuè
minùs, idip&longs;um perman&longs;urum ni
hilominus in telligere po&longs;&longs;umus,
parte&longs;què vndique æqualium mo
men torum con&longs;i&longs;tentes. Neque
enim Ari&longs;to teles grauibus dunta
xat, &longs;ed etiam leuibus momenta
tribuit, idip&longs;um què (vt Eutocius
in horum librorum comentarijs
refert) Ptolæmeo quoque placuit, vt habetur in líbro (à nobis
ramen de &longs;iderato) quem de momen tis &longs;crip&longs;it. Pr&ecedil;terea alij
quoque Philo&longs;ophi id ip&longs;um &longs;en&longs;i&longs;&longs;evidentur. quod e&longs;t qui
dem rationi con&longs;en taneum, &longs;uperuolant enim, quæ leuia &longs;unt,
& &longs;i mente concipiatur
&longs;i detineatur in G, partes vndique &ecedil;qualium
con&longs;i&longs;tent, e&longs;&longs;etquè G (vtita dicam) centrum leuitatis. Quo
niam autem circa centrum grauitatis &ecedil;queponderationem
con&longs;ideramus, id circo plana, tanquam no bis apparentia gra
uitatem habere, mente concipimus. Non e&longs;t igitur à ratio
ne alienum, æqueponderantiam in planis, vt grauibus con&longs;i
deratis intelligere, conciperequè. Nec quicquam nobis offi
cit, quòd definitiones centri grauitatis priùs allatæ non pla
norum, &longs;ed corporum centra explicarunt, ita vtgrauitatis
trum
men propterea impropriè re&longs;picit plana, &longs;ed quia primò re&longs;pi
cit corpora; in quib^{9} actu ine&longs;&longs;e
met
milibri.
DEFINITIO CENTRI GRAVITATIS PLANORVM.
Cen trum grauitatis vniu&longs;cuiu&longs;que plani e&longs;t punctum quod
dam intra po&longs;itum, à quo &longs;i planum appen&longs;um mente con
cipiatur, dum fertur, quie&longs;cit; & &longs;eruat eam, quam in princi
pio habcbat po&longs;itionem, neque in ip&longs;a latione
EIVSDEM ALIA DEFINITIO.
Centrum grauitatis vniu&longs;cuiu&longs;que plani e&longs;t punctum il
lud intra po&longs;itum, circa quod vndique partes æqualium mo
mentorum con&longs;i&longs;tunt. &longs;i enim per tale centrum recta du
catur linea figuram quomodocunque &longs;ecans, &longs;emper in par
tes
Vtitaque in planis quoque centrum grauitatis con&longs;ide
ratur, ita etiam plana grauitate prædita con&longs;iderare, non e
rit ab&longs;urdum. &longs;i enim impo&longs;&longs;ibile e&longs;&longs;et con&longs;iderare plana gra
uitate prædita, centrum quoque grauitatis in ip&longs;is nullo mo
do concipi po&longs;&longs;et; atque per&longs;picuum e&longs;t, centrum grauita tis in
ip&longs;is admitti, ac de&longs;ignari po&longs;&longs;e, igitur & plana grauitate in&longs;i
gnita. Et &longs;i mathematicus con&longs;i derat corpora &longs;eclu&longs;a interim
ip&longs;orum grauitate, & leuitate: & A&longs;tronomus corpora con&longs;i
derans cæle&longs;tia, quæ neque grauia, neque leuia &longs;unt, non pro
pterea
que leuia e&longs;&longs;e; etenim quamuis grauia, vel leuia e&longs;&longs;ent, nihilo
minus neque grauia, neque leuia e&longs;&longs;e ea con&longs;ideraret. quòd &longs;i
Mathematicus hoc pacto huiu&longs;modi corpora intelligere po
te&longs;t; quid prohibet rur&longs;um
nequeleuia &longs;int; vel grauia, vel leuia e&longs;&longs;e concipere?
modum
plo res magis eluce&longs;cet
veluti &longs;i intelligamus ex
AC appen&longs;a e&longs;&longs;e plana
DE, quæ &longs;int æqualia; &longs;u
&longs;pendaturquè AC in me
dio pror&longs;us in B; cur mente intelligere non po&longs;&longs;umus, quan
lia?
num D deor&longs;um tendere concipiemus. & hoc nulla alia de
cau&longs;a, quàm quòd cùm D maius &longs;it, quàm E, &longs;tatim
D, quàm E grauius quoque e&longs;&longs;e concrpimus. Con&longs;iderare
igitur plana cum grauitate non e&longs;t omnino à ratione
Quare vtrum que titulum, nempe planorum æqueponderan
tium, vel centra grauita tis
Verùm quoniam Archimedes &longs;ecundum librum &longs;implicivo
cabulo, nimirum (qua&longs;i &longs;imul omnia complectens)
derantium
brum (æqueponderantium) in&longs;cribendum exi&longs;timamus. eo
quèlibentiùs; quoniam ip&longs;emet Eutocius horum quoque li
brorum explanator ho&longs;ce libros hoc tantùm nomine æque
ponderantium nuncupauit: alijquè omnes, qui hos Archime
dis libros nominant; hoc titulo de æqueponderantibus nun
cupant. Præterea titulus hic magis operi congruere mihi vide
tur; quoniam nonnulla Archimedes in principio pertractat,
quæ tam &longs;olidis, quàm planis communia exi&longs;tunt; quamuis
cætera ad plana &longs;int tantu
rumquod facilè con&longs;tat inpri
mis ip&longs;iu&longs;met Archimedis
in libro de quadratura paraboles
&longs;tituta, ip&longs;ius paraboles
Deinceps ex cognitione
cognitionem centrorum grauitatum &longs;olidorum deducimur.
Denique adeo pro&longs;icua e&longs;t hæc doctrina, quam nobis in his
libris Archimedes præ&longs;tat; vt affirmare non verear, nullum
e&longs;&longs;e Theorema, nullum què problema ad rem mechanicam
pertinens, quod in &longs;ui &longs;peculatione peculiare quem
admodum (cæteris interim omi&longs;&longs;is) patet ex vulgata illa pro
po&longs;itione enunciante, ita &longs;e habere pondusad pondus, vt di
&longs;tantia ad di&longs;tantiam permutatim &longs;e habet, ex quibus &longs;u&longs;pen
duntur. quæ præclari&longs;&longs;imè ab ip&longs;o in primo libro demon&longs;tra
tur. Et quamuis Iordanus Nemorarius (quem &longs;ecutus e&longs;t
dem
bationi demon&longs;trationis nomen conuenire pote&longs;t. cùm vix
ex probabilibus, & ijs, quæ nullo modo nece&longs;&longs;itatem
& forra&longs;&longs;e neque ex probabilibus &longs;uas componat rationes.
Cùm in mathematicis demon&longs;trationes requirantur exqui&longs;i
ti&longs;&longs;imæ. acpropterea neque inter Mechanicos videtur mihi
Iordanus ille e&longs;&longs;e recen&longs;endus. Quapropter ad Archimedem
confugien dum e&longs;t, &longs;i fundamenta mechanica, veraquè huius
&longs;cientiæ principia perdi&longs;cere cupimus: qui (meoiudicio) ad
hoc poti&longs;&longs;imùm re&longs;pexit; vt elementa mechanica traderet. vt
ctiam Pappus in octauo Mathematicarum collectionum li
bro &longs;entit; quod quidem ex diui&longs;ione, ac progre&longs;&longs;u horum li
brorum facilè digno&longs;cetur.
DE DIVISIONE HORVM LIBRORVM.
Diuiditur enim in primis hic tractatus in duos libros diui
&longs;us, in po&longs;tulata, & theoremata: theoremata verò &longs;ubdiui
duntur in duas &longs;ectiones, quarum prima continet priora o
cto theoremata; ad alteram verò reliqua theoremata
quæ quidem adhuc in alias duas partes diuidi pote&longs;t; nempè
in theoremata primo libro examina ta, & in ea, quæ &longs;ecun
dus liber contemplatur. Hanc autem horum librorum con
&longs;tituimus diui&longs;ionem, quoniam imprimis Archimedes, (o
mi&longs;&longs;is po&longs;tulatis, quæ primum locum obtinere debent) quæ
dam tractauit communia in pricribus octo theorema tibus;
quorum &longs;copus e&longs;t inuenire fundamentum illud
mechanicum, quòd &longs;cilicet ita &longs;e habet grauitas ad grauita
tem, vt di&longs;tan tia ad di&longs;tantiam permutatim. ad quod demo
&longs;trandum quinque præmittit theoremata, quæ paulatim
deducunt nos in cognitionem demon&longs;tra tionis præfati fun
damenti. quo loco illud &longs;ummoperè notandum e&longs;t, nimi
rum &longs;undamentum illud, nec non octo priora theorema
ta communia e&longs;&longs;e tam planis, quàm &longs;olidis; atque promi&longs;
cuè de vtri&longs;que quòd &longs;i quis aliter
&longs;timauerit, vel de &longs;olidis, non autem
rectilineis, vel de homogeneis tantùm, & de ijs, quæ inter &longs;e
&longs;unteiu&longs;dem &longs;peciei, longè aberrat à &longs;copo, & mente Archi
medis. etenim in his &longs;emper loquitur. vel de grauibus &longs;impli
citer, veluti in primis tribus theorematibus; vel de magnitu
dinibus, vt in reliquis quinque quod quidem nomen tam
planis, quàm &longs;olidis quibu&longs;cunque e&longs;t
qui parùm in Mathematicis ver&longs;ati &longs;unt, &longs;atis norunt. ficu
ti etiam Euclides, dum quinti libri propo&longs;itiones pertracta
uit, quantitatem continuam &longs;ub nomine magnitudinis
prehendit. quòd
per &longs;e con&longs;tat. Per&longs;picuum e&longs;t igitur priora hæc octo Theo
remata communia e&longs;&longs;e, tam planis, quàm &longs;olidis. ac non &longs;o
lùm &longs;olidis eiu&longs;dem &longs;peciei, & homogeneis, verùm etiam &longs;oli
dis diuer&longs;æ &longs;peciei, & hçterogeneis, vt&longs;uo loco manife&longs;tum
fiet. Iactoquè hoc fundamento, quod Archimedes in duob^{9}
propo&longs;itionibus, &longs;exta nempè, & &longs;eptima demon&longs;trauit; in o
ctaua tanquam corrollarium colligit. Deinceps peculiariter
pertractat de centro grauitatis planorum, nec amplius plana
nominat magnitudinis nomine, &longs;ed proprijs cuiu&longs;cun que
nominibus; vt parallelogrammi, trianguli, & aliorum huiu&longs;
modi. & in hac parte de&longs;cendit ad particularia. quippè cùm
& &longs;i non actu forta&longs;&longs;e, virture tamen cuiu&longs;libet particularis
plani centrum grauitatis nos doceat. in primo enim libro
&longs;at &longs;i bi vi&longs;um e&longs;t o&longs;tendi&longs;&longs;e centra grauitatum trianguloru,
ac parallelogrammorum, ex quibus cæterarum &longs;igurarum,
veluti pen tagoni, hexagoni, & aliorum &longs;imilium centra gra
uita tis inue&longs;tigare non admodum erit difficile. &longs;iquidem hu
iu&longs;modi plana in triangula diuiduntur. vt in &longs;ine primi li
bri attingemus. In &longs;ecundo autem libro altiùs &longs;e extollit, &
moro &longs;uo circa &longs;ubtili&longs;&longs;ima theoremata ver&longs;atur; nompè cir
ca centrum grauitatis conice &longs;ectionis, quæ parabole nun
cupatur. nonnullaquè præmittit theorema ta, quæ &longs;unt tan
quam præuie di&longs;po&longs;itiones ad inue&longs;tigandam demon&longs;tra
tionem centri grauitatis in parabole. Itaque per&longs;picuum e&longs;t,
Archimedem propriè elementa mechanica tradere. quando-
&longs;cientiæ. fundamentum nempè illud præ&longs;tanti&longs;&longs;imum iam
to ties præfatum, deinde centra grauitatis planorum o&longs;tendit.
& quamuis hi duo Archimedis libelli pauca continerevidean
tur, non tamen pauca docui&longs;&longs;e Archimedem exi&longs;timandum
e&longs;t. multa enim &longs;unt mole exigua, quæ tamen virtute maxima
habentur. quod planè Archimedis &longs;criptis accidit; hi&longs;què pr&ecedil;
&longs;ertim, ex quibus patet aditus ad multa, ac penè in&longs;inita theo
remata, problemataquè mechanica. nihil enim in hoc gene
re demon&longs;trari pote&longs;t, quod his non indigeat &longs;criptis. &
quod admirabilius e&longs;t, nos non &longs;olùm pro fundamento &longs;u
&longs;cipere po&longs;&longs;e ad aliquod demon&longs;trandum theoremata in his
libris demon&longs;trata, verùm etiam ab his demon&longs;trationibus
perdi&longs;cerere ip&longs;um modum argumentandi, & demon&longs;trandi;
vt&longs;uis locis o&longs;tendemus. ita vt verè concludendum &longs;it, nemi
nem pror&longs;us inter mechanicos connumerandum fore, qui
hæc Archimedis &longs;cripta ignorat. ignoratis enim principijs
nulla e&longs;t &longs;cientia, vt apud omnes &longs;apientes per&longs;picuum e&longs;t.
Ip&longs;um igitur Archimedem audiamus, eiu&longs;què &longs;cripta diligen
ti&longs;&longs;imè perpendamus.
GVIDIVBALDI
EMARCHIONIBVS
MONTIS.
IN PRIMVM ARCHIMEDIS
AEQVEPONDERANTIVM
LIBRVM
PARAPHRASIS
SCHOLIIS ILLVSTRATA.
Archimedis tamen huius primi libri
titulus &longs;ic &longs;e habet.
VEL CENTRA GRAVIT ATVM PLANORVM.
AR CHIMEDIS POSTVLATA.
I.
Grauia æqualia ex æqualibus di&longs;tantijs æque
ponderare.
SCHOLIVM.
Dvobvs modis grauia in di&longs;tantijs
collocata in telligi po&longs;&longs;unt. quod &
in cæteris po&longs;tulatis, & in propo&longs;i
tionibus intelligendum e&longs;t. etenim
vel grauia
gura æqualia grauia AB &longs;unt in CD
appen&longs;a; ita vt di&longs;tantia EC &longs;it di
&longs;tatiæ ED æqualis. intelligaturquè
CD tanquam libra, quæ &longs;u&longs;pendatur
in E. vel vtin &longs;ecunda figura grauia AB habent ip&longs;orum
centra grauitatis, quæ &longs;int CD, in ip&longs;a DC linea, in pun-
con&longs;tituta. li
braquè &longs;imili
ter ex puncto
E &longs;u&longs;pendatur;
&longs;itquè di&longs;tátia
EC di&longs;tantiæ
ED æqualis.
vtraque figura
pondera AB
in di&longs;tantijs &ecedil;
qualibus con
&longs;tituta. ac pro
pterea æquepondera bunt, atque manebunt. nulla enim ratio
afferri pote&longs;t, cur ex parte A, vel ex parte B deor&longs;um, vel &longs;ur
&longs;um fieri debeat motus; cùm omnia &longs;int paria. ea verò æque
ponderare debere, aliqua ratione manife&longs;tari pote&longs;t ex eo,
quod o&longs;ten&longs;um e&longs;t à nobis in no&longs;tro mechanicorum libro,
tractatu de libra: quod quidem ab Ari&longs;to tele quoque in prin
cipio quæ&longs;tionum mechanicarum elici pote&longs;t: idem &longs;cilicet
pondus longius a centro grauius e&longs;&longs;e eodem pondere ip&longs;i cen
tro propinquiori. Vnde &longs;i duo e&longs;&longs;ent pondera æqualia alte
rum altero propinquius centro, quod remotius e&longs;t, grauius al
tero appareret. &longs;i igitur grauia æqualia à centro æqualiter di
&longs;tabunt, æque grauia erunt. ac propterea æqueponderabunt.
quod quidem &longs;upponit Archimedes. Punctum autem illud,
quod Archimedes accipit, vnde &longs;umuntur di&longs;tantiæ, ex qui
bus grauia &longs;u&longs;penduntur, veluti punctum E, Ari&longs;toteles cen
rum appellat. & hæc quidem æqueponderatio tam ponderi
bus in libra appen&longs;is, quàm in ip&longs;a (vt dictum e&longs;t) con&longs;titutis
competit: dummodo ea, quibus appenduntur pondera, libe
re &longs;emper in centrum mundi tendere po&longs;&longs;int. vtroque enim
modo in punctis CD grauitant, vt diximus etiam in eodem
uactatu de libra. Noui&longs;&longs;e tamen oportet Archimedem in his
libris potiùs in tellexi&longs;&longs;e pondera e&longs;&longs;e in di&longs;tantijs collocata, vt
in &longs;ecunda &longs;igura, quàm appen&longs;a; vt ex quarta, & quinta
demon&longs;trationes enim cla
riores redduntur.
Porrò non ignoran
dum hoc Archimedis
po&longs;tulatum verificari
deponderibus quocun
que &longs;itu di&longs;po&longs;itis, &longs;iuc
CED fuerit horizonti
vtin hac prima figura,
codem modo femper
verum e&longs;&longs;e pondera æ
qualia CD ex &ecedil;quali
bus di&longs;tantijs EC ED
æqueponderare, vt in
fra (po&longs;t &longs;cilicet
huius propo&longs;itionem)
per&longs;picuum erit. Qua
re cùm Archimedes
in hoc po&longs;tulato,
in &longs;equentibus, &longs;uppo
nit pondera in di&longs;tan
tijs e&longs;&longs;e collocata, intel
ligendum e&longs;t
ex vtraque parte in ea
dem recta linea exi&longs;te
re. Nam &longs;i (vtin &longs;ecun
da figura)
fuerit &ecedil;qualis di&longs;tanti&ecedil; BC, quæ non indirectum iaceant,
&longs;ed angulum con&longs;tituant; tunc pondera AB, quamuis &longs;int
&ecedil;qualia, non &ecedil;queponderabunt. ni&longs;i quando (vt in tertia fi
gura) iuncta AC, bifariamquè diui&longs;a in D, ductaquè BD,
fuerit h&ecedil;c horizonti perpendicularis, vt in eodem tractatu
no&longs;tro expo&longs;uimus. Di&longs;tantias igitur in eadem recta linea
&longs;emper exi&longs;tere intelligendum e&longs;t. vt ex demon&longs;trationibus
Archimedis per&longs;picuum e&longs;t.
II.
Aequalia verò grauia ex inæqualibus di&longs;tátijs
non æqu eponderare, &longs;ed præponderare ad gra
ue ex maiori di&longs;tantia.
SCHOLIVM.
Si enim
tia EC maior
fuerit di&longs;tantia
ED, grauibus
AB &longs;imiliter æ
qualibus
tibus, & in CD po&longs;itis, tunc concedendum videtur graue A
præponderareip&longs;i B, quandoquidem EC longior e&longs;t, quàm
ED. &longs;upponit autem Archimedes hoc po&longs;tulatum reipiciens
forta&longs;&longs;e ad ea, quæ Ari&longs;toteles in principio quæ&longs;tionum me
chanicarum o&longs;tendit, vbi colligit Ari&longs;toteles idem pondus ce
leriùs ferri, quò magis à centro di&longs;tat, vel quod idem e&longs;t, duo
pondera æqualia inæqualiter à centro di&longs;tantia, quod magis
di&longs;tat, celeriùs ferri. quod autem æqualium ponderum cele
riùs fertur, grauius exi&longs;tit; erit igitur A grauius, quàm B.
quia EC longior e&longs;t, quàm ED. Nos quoque (vt diximus)
in libro no&longs;trorum Mechanicorum tractatu de libra, alijs
quoque rationibus o&longs;tendimus, quo pondus e&longs;t in longiori
di&longs;tantia grauius e&longs;&longs;e. ex quibus &longs;equitur propter longiorem
di&longs;tantiam EC pondus A præponderare ponderi B. acpro
pterea deor&longs;um ferri.
III.
Grauibus ex aliquibus di&longs;tantijs æqueponderá
tibus, &longs;ialteri grauium aliquid adijciatur, non æ
queponderare; &longs;ed ad graue, cui adiectum fuit,
deor&longs;um ferri.
SCHOLIVM
Grauia enim
AB &longs;iuè æqua
lia, &longs;iue in &ecedil;qua
lia æqueponde
rent ex di&longs;tan
tijs AC CB, al
teri verò gra
uium, putà B,
adijciatur pon
dus D. per&longs;picuum e&longs;t pondera BD &longs;imul magis ponderare,
quàm A. &longs;i enim B &ecedil;queponderat ip&longs;i A; erit pondus B in
hoc &longs;itu æquegraue, vt A: pondera igitur BD in hoc &longs;itu
erunt æquegrauia, vt pondus A. &longs;ed grauiora exi&longs;tent, quàm
A. quare BD deor&longs;um tendent.
IIII.
Similiter autem, &longs;i ab altero grauium auferatur
aliquid, non æqueponderare; verùm ad graue, à
quo nil ablatum e&longs;t, deor&longs;um tendere.
SCHOLIVM.
Aequeponderent grauia BD &longs;imul, & A
&longs;tantias CB CA; vtin eadem figura, & ab altero eorum, putà
BD, auferatur D, remanenbunt grauia BA; eritquè A gra
uius ip&longs;o B. Nam &longs;i BD &longs;imul æqueponderant ip&longs;i A, B
tantùm eidem A non æqueponderabit, &longs;edleuius erit. vnde
&longs;equitur ex parte A motum fieri deor&longs;um.
ra.
V
Aequalibus, &longs;imilibu&longs;què figuris planis inter &longs;e
coaptatis, centra quoque grauitatum inter &longs;e coa
ptati oportet.
SCHOLIVM.
Aequales,
figuræ ABC DEF, qua
rum centra grauitatis &longs;int
GH; &longs;i ABC &longs;uperpona
tur ip&longs;i DEF, & hoc
dùm laterum
hoc e&longs;t &longs;i latus AB fuerit
æquale lateri DE, tunc
ponatur AB &longs;uper DE; &longs;imiliter AC &longs;uper DF, & BC &longs;uper
EF; tunc manife&longs;tum e&longs;t centrum grauitatis G &longs;uper centro
grauitatis H ad unguem conuenire; ita vt &longs;int vnum tan
punctum. Plana enim quæ &longs;e inuicem contingunt, non ef
ficiunt, ni&longs;i vnum tantùm planum. Solius autem figuræ ex
planis ABC DEF inuicen coaptatis, vnum tantùm erit cen
trum grauitatis, vt nos in no&longs;tro mechanicorum libro &longs;up
po&longs;uimus; centra igitur grauitatis inter &longs;e&longs;e conuenire nece&longs;
&longs;e e&longs;t. &longs;i enim centra grauitatis inter &longs;e non conuenirent, v
na tantùm figura duo po&longs;&longs;et centra grauitatis habere. quod
e&longs;&longs;et omnino
re has figuras e&longs;&longs;e &longs;imiles, & æquales, nam figuræ æquales,
&longs;ed non &longs;imiles, item &longs;imiles, & qua
re, vt inter &longs;e&longs;e coaptari po&longs;&longs;int, & &longs;imiles, & æquales e&longs;&longs;e ne
ce&longs;&longs;e e&longs;t.
VI
Inæ qualium autem, &longs;ed &longs;imilium centra graui
tatum e&longs;&longs;e &longs;imiliter po&longs;ita.
SCHOLIVM.
Inæquales &longs;int figuræ, &longs;i
miles verò ABCD EFGH,
quarum cétra grauitatis &longs;int
KL. &longs;upponit Archimedes
h&ecedil;e grauitatis centra KL e&longs;
&longs;e in figuris ABCD EFGH
&longs;imiliter po&longs;ita.
&longs;imilium figurarum, & late
ra, & &longs;pacia &longs;int &longs;imilia, nece&longs;&longs;e e&longs;t in ip&longs;is &longs;imili quo que mo
do centra grauitatis e&longs;&longs;e po&longs;ita. vt in &longs;equenti clariùs apparebit.
quomodo autem Archimedes intelligathanc po&longs;itionis &longs;imi
litudinem, hoc modo definit.
VII.
Dicimus quidem puncta in &longs;imilibus figuris e&longs;
&longs;e &longs;imiliter po&longs;ita, à quibus ad æquales angulos
ductæ rectæ lineæ cum homologis lateribus angu
los æquales efficiunt.
SCHOLIVM.
In &longs;imilibus figuris ABCD EFGH &longs;int homologa latera
AB EF, BCFG, CD GH, AD EH. anguli verò æquales, qui
ad AE, BF, CG, DH, primum quidem o&longs;tendendum e&longs;t fie
ri po&longs;&longs;e, ut à duobus punctis intra figuras con&longs;titutis, duci
po&longs;&longs;int rect&ecedil; line&ecedil; ad angulos æquales, qu&ecedil; cum lateribus an
gulos &ecedil;quales efficiant. Qua&longs;i dicat Archimedes, quoniam
&longs;upponere po&longs;&longs;umus puncta in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter
po&longs;ita, ideo &longs;upponere quoque po&longs;&longs;umus centra grauiratisin
ip&longs;is e&longs;&longs;e &longs;imiliter po&longs;ita. Itaque &longs;int figuræ ABCD EFGH &longs;i
miles, vt dictum e&longs;t, &longs;umaturquè in ABCD vteumque pun
ctum K à quo ducatur KA KB KC KD. deinde fiatan
gulus FEL angulo BAK æqualis; & EFL ip&longs;i ABK. Iun
ganturquè GL LH. Dico L e&longs;&longs;e &longs;imiliter po&longs;itum, vt K.
Quoniam enim anguli BAK ABK &longs;unt angulis FEL EFL
æquales, erit reliquus BKA ip&longs;i FLE æqualis, eritquè ob &longs;i
verò AB ad AD, vt EF ad EH propter &longs;imilitudinem fi
& quoniam angulus BAD angulo FEH e&longs;t æqualis, & BAK
ip&longs;i FEL æqualis; erit & reliquus angulus KAD angulo
Quare triangulum KAD triangulo LEH &longs;i
mile exi&longs;tit, eodemquè modo o&longs;tendetur BKG &longs;imile e&longs;&longs;e
FLG, & KCD ip&longs;i LGH. ex quibus con&longs;tat angulos KBC
LFG, KCB LGF, & huiu&longs;modi reliquos reliquis æquales e&longs;&longs;e.
& ob id puncta KL in figuris ABCD EFGH e&longs;&longs;e &longs;imili
ter po&longs;ita.
Itaque demon&longs;trato dari po&longs;&longs;e puncta in figuris fimiliter
po&longs;ita, potuit &longs;anè Archimedes antecedens po&longs;tulatum &longs;up
ponere, nempè inæqualium, &longs;ed &longs;imilium figurarum centra
grauitatis e&longs;&longs;e &longs;imiliter po&longs;ita. quod quidem po&longs;tulatum e&longs;t
rationivalde con&longs;entaneum. ex dictis enim (&longs;uppo&longs;itis KL
centris grauitatum) triangulum ABK triangulo EFL &longs;imi
Quarevt
AK ad KB, &longs;ic EL ad LF, ac permutando vt AK ad EL,
ita BK ad FL. &longs;imiliter o&longs;tendetur ita e&longs;&longs;e BK ad FL, vt
KC ad LG, & KD ad LH. quare centra grauitatis KL
16
KM KN KO KP, LQ LR LS LT. & quoniam anguli
KMA LQE &longs;unt recti, ac propterea æquales, & KAM LEQ
&longs;unt æquales, ut o&longs;ten&longs;um e&longs;t; erit reliquus MKA reliquo
QLE &ecedil;qualis, triangulumquè AKM triangulo ELQ &longs;imile.
vtigitur AK ad KM; &longs;ic EL ad
ad EL, vt KM ad
lum BKM triangulo FLQ &longs;imile exi&longs;tere; e&longs;&longs;equè BK ad
FL, vt KM ad
detur, ita e&longs;&longs;e Bk ad FL, vt KN ad LR; & Ck ad GL e&longs;&longs;e, vt
kO ad LS; atque kD ad LH, vt kP ad LT. quia verò AK
EL, Bk FL, Ck GL, Dk HL in eadem &longs;untproportione, vt
proximè demon&longs;tratum fuit; in eadem quoque proportione
erit kM ad LQ, & KN ad LR; & KO ad LS, atque kP ad
LT. ex quibus &longs;equitur centra grauitatis KL, non &longs;olùm ab
angulis in cadem proportione di&longs;tare; verùm etiam à lateri
ribus in eadem quoque proportione di&longs;tare. Itaque cognito,
quomodo intelligar Archimedes centra grauitatis in &longs;imili
bus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; nunc con&longs;iderandum e&longs;t præ
cedens po&longs;tulatum, quatenus nimirum oporteat grauitatis
tra in &longs;imilibus figuris &longs;imiliter e&longs;&longs;e con&longs;tituta. Nam inti
miùs con&longs;iderando hanc &longs;imilem horum grauitatis
po&longs;itionem, congruum, & nece&longs;&longs;arium videtur, &longs;imiles &longs;igu
ras &longs;ecundùm eandem proportionem e&longs;&longs;e æquepon
eademquè ratione (ob earum &longs;imilitudinem) circa grauita
tis centra æqueponderare, veluti &longs;i figuræ: AC EG (quarum
centra grauitatis &longs;int KL) à rectis lineis PN TR vrcumquè
diuidantur, quæ percentra KL tran&longs;eant; dummodo in figu
ris &longs;int &longs;imiliter ductæ; hoc e&longs;t, vellatera, vel angulos in
proportione di&longs;pe&longs;cant: vt &longs;it AP ad PD, vt ET ad TH. æ
queponderabunt vtique partes PABN PNCD, veluti partes
TEFR TRGH. & hæc non e&longs;t &longs;implex æqueponderatio; ve
rùm etiam (vtita dicam) &longs;imilis, & æqualis æqueponderatio.
cùm &longs;it &longs;ecundùm eandem proportionem, quandoquidem
e&longs;t PB ip&longs;i TF &longs;imilis, cùm triangula AKB ELF, AKP ELT,
BKN FLR, &longs;int inter &longs;e &longs;imilia, quæ quidem efficiunt, figuras
ob eademquè cau&longs;am e&longs;t PC &longs;i
milis TG. quod quidem ex dem on&longs;tratis etiam facilè con
&longs;tat. cùm anguli &longs;int &ecedil;quales, & latera proportionalia. Vtau
tem clariùs intelligatur hæc &longs;imilis, & æqualis æquepondera
rio, adducerelibuit nonnulla ex ijs, quæ po&longs;teriùs tractanda
&longs;umentur. Itaque intelligatur punctum V centrum e&longs;&longs;e gra
uitatis figuræ PB, X verò centrum grauitatis figure TF. &longs;i
militer punctum Y centrum e&longs;&longs;e grauitatis figuræ PC, Z
verò figur&ecedil; TG. Iunganturquè VY XZ. quæ quidem per
centra grauitatis KL tran&longs;ibunt. quòd ex ijs, qu&ecedil; dicenda
&longs;unt, manife&longs;tum erit, percipuèque ex octaua proportione
primi huius. quod tamen interim &longs;upponatur. At verò quo
niam PB PC &ecedil;queponderant &longs;ecundùm proportionem,
quam habet YK ad KV; TF verò & TG &ecedil;queponderant
&longs;ecundùm proportionem, quam habet ZL ad LX. e&longs;t.
ac &longs;i AN e&longs;&longs;et appen&longs;a in V, & PC in Y; ER in X, &
TG in Z. vt in &longs;equentibus manife&longs;ta erunt. Atverò quo
proportionem eius, quam habet latus PN ad TR. pariquè
ratione quoniam PC &longs;imilis e&longs;t TG, habebit PC ad TG
duplam proportionem eius, quam habet idem latus PN ad
Y K ad KV, & vt ER ad TG. &longs;ic ZL ad LX. eandem igitur
AN PC, & ER TG &longs;ecundùm eandem proportionem æ
queponderabunt. quod quidem contingit ex &longs;imilitudine fi
gurarum, & ex centris grauitatum KL &longs;imiliter po&longs;itis, qu&ecedil;
quidem magnitudines, &longs;i non e&longs;&longs;ent &longs;imiles, diui&longs;&ecedil; quide per
centrum grauitatis, partes vtique &ecedil;queponderarent; non ta
men &longs;emper &longs;ecundùm eandem proportionem. quod tamen
&longs;emper figuris &longs;imilibus (cùm in ip&longs;is grauitatis centra &longs;int &longs;i
militer po&longs;ita) contingit; dummodo (vt dictum e&longs;t) diui
dantur. Vnde con&longs;tat, quam &longs;it conueniens grauitatis centra
in figuris hac ratione e&longs;&longs;e con&longs;tituta. ex quibusomnibus per
&longs;picuum e&longs;t, centra grauitatis debere in figuris &longs;imilibus e&longs;&longs;e &longs;i
militer po&longs;ita. vt Archimedes in pr&ecedil;cedeti po&longs;tulato pr&ecedil;mi&longs;it.
16
VIII.
Simagnitudines ex æqualibus di&longs;tantijs æque
ponderant, & ip&longs;is æquales ex ij&longs;dem di&longs;tantijs æ
queponderabunt.
SCHOLIVM.
Hoc e&longs;t per&longs;picuum,
&longs;i magnitudines AB ex di
&longs;tantijs CA CB &ecedil;quepon
derant: &longs;itautem D ip&longs;i A
&ecedil;qualis, & E ip&longs;i B.
turquè magnitudines AB à
linea AB, ip&longs;arumquè loco ponatur D in A, & E in B, ma
gnitudines DE fimiliter
magnitudines AB inter &longs;e&longs;e &ecedil;queponderare dicuntur; eadem
pror&longs;us, & magnitudines DE ex ij&longs;dem di&longs;tantijs &ecedil;quepon
derabunt. quandoquidem omnia data &longs;unt paria. illud ta
men non e&longs;t pretereundum, nimirum non oportere DE ip&longs;is
AB &ecedil;quales e&longs;&longs;e in magnitudine, &longs;ed in grauitate. pote&longs;t enim
ob naturæ diuer&longs;itatem, ac propterea cùm inquit Archimedes
telligendum e&longs;t e&longs;&longs;e omnino æquales in grauitate. grauitas.
cau&longs;a e&longs;t, vt magnitudines æqueponderare debeant.
VIIII,
Omnis figuræ, cuiusperimeter &longs;it ad
tem concauus, centrum grauitatis intra figuram
e&longs;&longs;e oportet.
SCHOLIVM.
Quid intelligat Ar
chimedes per has figu
ras ad eandem partem
concauas, apertiùs &longs;i
gnificauit initio libro
rum de&longs;ph&ecedil;ra, & cylin
dro. vbi primùm vult
has figuras e&longs;&longs;e termina
tas; quod non &longs;olùm in
telligendum e&longs;t decur
uilineis, verùm etiam
de rectilineis, & de mi
xtis. rectiline&ecedil; quidem
erunt trium, quattuor,
quinque & plurium la
terum; quamuis latera
non &longs;int æqualia, ne
que anguli &ecedil;quales, vt
partem. & hocmodo perimeter huius figuræ erit ad eandom
partem con cauus. vnde excludun tur figuræ, exempli gratia
FGHKL; cùm angulus K non &longs;it &longs;inuo&longs;us, & con oauus ad
eandem partem, vt reliquidnguli; qui &longs;unt &longs;in uo&longs;
riorem pamem &longs;igur&ecedil; K vero bd exterioitem. &longs;imili modo
intelligen dum e&longs;t ded
generis&longs;igræ, vt &longs;unt MN, quæ &longs;uam habent conqau tatem
adiean dem partem: &longs;ed curuline¸ OP ilnon &longs;unt ad ea n dem
partem concau&ecedil;. Mixtæ quoque figuræ, ut&longs;unt portiones eil
culi, hyperbab&ecedil; ac para bod&ecedil; rectis linen
rius gen erisfigur&ecedil;, vt &longs;pnt QR. h&ecedil; quidemom nes&longs;unt ad ea
dem partem concauç Mixcæ verò ST minimè Regulgm au
tem qua
elicere po&longs;&longs;unus, vtoog nofcere valeam us, an figu
eandem partem concauæ, vel minùs vt fcilicet inboblata figu
ra vbicum que duo &longs;umi po&longs;&longs;int puncta, quæ &longs;i rectal
nectantur, tota recta li
nea, velip&longs;ius pars ali
qua extra figuram non
cadat. vt in figuris A,
quæ &longs;unt ad
tem concauæ, vtcum
que duo &longs;umantur
cta
ctantur, tota utique re
cta linea inter puncta
BC exi&longs;tens, extra figu
ram non cadet. Quòd
&longs;i hæclinea cum termino, hoc e&longs;t eum latere figur&ecedil; conueni
ret, vt &longs;i &longs;iguræ latus fueritrectum, in quo duo &longs;umantur pun
cta, nihilominus recta linea inter hæc puncta extra figuram
non cadei: quandoquidem figuræ terminus extra figuram mi
nimè roperitur atque hac ratione quomodocunque, & vbicú
que in his figuris duo &longs;um a ntur puncta, idem &longs;emper con tin
get. Quod tamen figuris D &longs;emper euenite non pote&longs;t in qui
bus (cùm non &longs;int ad eandem partem concau&ecedil;) duo &longs;umero
&longs;iguram cadet. vel fumerepo&longs;&longs;umus puncta FG, ita vt rect&ecedil;
line&ecedil; FG pars EG extra figuram cadat. figur&ecedil; igitur, quæ
ad ean dem partem &longs;unt concauæ, ill&ecedil; &longs;unt, qu&ecedil; &longs;inuo&longs;itatem,
concauitatemquè &longs;uam habent &longs;emper interiorem ip&longs;ius fi
gur&ecedil; partem re&longs;picientem. Harum què rectè &longs;upponit Archi
medes centrum grauitatis &longs;emperle&longs;&longs;e intra ip&longs;am figuram.
ita vt neque centrum e&longs;&longs;e po&longs;&longs;icin ambitu ip&longs;ius figur&ecedil; ete
nim &longs;i extra figuram, &longs;iue in ambitu ip&longs;ius e&longs;&longs;e po&longs;&longs;et, num
quam circa centrum grauitatis partes figur&ecedil; vndiquè
vbicumque, & in omni &longs;itu maneret. quod ramen ex ratione
centri grauitatis efficere deberet. to ta nimirum figura ex vna
e&longs;&longs;et parte, & ex altera nihil e&longs;&longs;et, quod ip&longs;i figur&ecedil; &ecedil;queponde
rare po&longs;&longs;et. Nece&longs;&longs;e e&longs;t igitur centrum grauitatis cuiu&longs;libet fi
gur&ecedil; ad ean dem partem concau&ecedil; e&longs;&longs;ein &longs;pacio à figur&ecedil; ambi
tu contento. vt figur&ecedil; AB
centrum grauitatis erit in
tra ip&longs;am, putà in C. quod
quidem non euenit &longs;emper
in alijs figuris, qu&ecedil; &longs;uum
cauitatis ambitum interio
rem figur&ecedil; partem
cientem habent. cùm varijs
modis po&longs;&longs;itcentrum graui
tatisin figuris e&longs;&longs;e
vt &longs;uperius quoque diximus.
Nam &longs;igur&ecedil; D
uitatis erit extra ambitum fi
gur&ecedil;, vt in E. figura verò F
ita &longs;e habere poterit, vt cen
trum grauitatis &longs;it in perime
tro, vt in G.
grauitatis L intra ip&longs;am figuram reperiatur; quamuis conca
uitates la torum interiorem partem minimè
po&longs;&longs;unt e&longs;&longs;e, & non e&longs;&longs;e, vt in figura M, cuius centrum extra
e&longs;&longs;e pote&longs;t in N. quamuis (vt an tea diximus) centrum graui-
te&longs;t.
Refort Eutocius hocloco, Geminum rectè dicere, dum a&longs;&longs;e
rit Archimedem dignitates peritiones apellare. æqualia enim
grauia ex di&longs;tantijs æqualibus æqueponderare, dignitas eft; &
quæ deinceps.
ctèperpendamus, omnia dignitates e&longs;&longs;e minimè reperiemus.
nam &longs;eptimum po&longs;tulatum e&longs;t definitio, non dignitas. veluti
alia forta&longs;&longs;e nonnulla non &longs;unt dignitates, vt &longs;ecundum; quod
aliquo modo probari pote&longs;t, vt diximus. &longs;extum quoque po
tiùs e&longs;t &longs;uppo&longs;ito, quàm dignitas. Quoniam autem vt clarè
con&longs;picitur Archimedes &longs;ub vno tantùm titulo pauca hæc
principia complecti voluit; quippe quod in&longs;titutum quàm plu
rimis mathematicis &longs;olemne fuit, qui principia vnico tantum
nomine nuncuparunt, modò vno, modò altero; nimirum,
vel petitionis, vel dignitatis, vt refert Proclus &longs;ecundo libro, &
tertio &longs;uorum commentariorum in primum elementorum. Eu
clidis; qui de Archimede peculiariter mentionem faciens, in
quit illum in his libris ptincipia vnico tantùm nomine (peti
tionis &longs;cilicet) nuncupa&longs;&longs;e. Hæc tamen potiùs petitionum,
quàm definitionum, vel dignitatum nomine nuncupare vo
luit; nam &longs;i dignitares appella&longs;&longs;et; ea principia, quæ non &longs;unt
dignitates, inter dignitates malè collo ca&longs;&longs;ct. nulla quippè defi
nitio dignitas dici debet; quandoquidem definitio terminos
declarat, atque con&longs;tituit. dignitas verò notos terminos copu
lat. Pariquè ratione &longs;i de&longs;initionis nomine hæc principia nun
cupa&longs;&longs;et. dignitates malè &longs;ub hoc nomine complexus fui&longs;&longs;et,
quæ nullo modo rem definiunt, &longs;ed cùm &longs;int communes no
tiones, &longs;tatim cùm eas in tellectus apprehendit, quie&longs;cit. Qua
reomnia &longs;ub petitionum nomine recte collocauit, non e&longs;t.
ab&longs;urdum dignitates, definitione&longs;què po&longs;&longs;e apellari petitio
nes. etenim petimus, quæ &longs;unt concedenda, atque dignitates
&longs;unt concedend&ecedil;, ergo eas petere quoque po&longs;&longs;umus. Definitio
nibus verò rectè quoque hoc nomen conuenire pote&longs;t. Nam
dùm definitio terminos con&longs;tituat, atque declaret, cur non pe
tere po&longs;&longs;umus, terminos &longs;ic &longs;e habere, vel &longs;ice&longs;&longs;e rectè definitos?
vt exempli gratia, petit Archimedes puncta in figuris fimilitel
nita puncta, quæ &longs;unt in figuris &longs;imilibus po&longs;ita. Quapropter
hæc principia, quoniam pauca &longs;unt, &longs;ub petitionum nomine
Archimedes rectè collocauit. quòd &longs;i multa extiti&longs;&longs;ent, ea for
ta&longs;&longs;e di&longs;tinxi&longs;&longs;et.
mata &longs;e conuertit, & inquit,
po&longs;uimus, &longs;ufficiuntad o&longs;ten denda theoremata, veluti.
PROPOSITIO. I.
Grauia, quæ ex æqualibus di&longs;tantijs æquepon
derant, æqualia &longs;unt.
Sint AD, & B grauia,
quæ ex æqualibus di&longs;tantijs
CA CB æqueponderent. di
co grauia AD, & B inter
&longs;e&longs;e æqualia e&longs;&longs;e.
ri pote&longs;t)
AD e&longs;&longs;et grauius, quàm B,
&longs;it D exce&longs;&longs;us, quo AD grauius e&longs;t, quàm B.
queponderare deberent; tamen cùm
po&longs;itum &longs;it AD B &ecedil;queponderare, &
libus di&longs;tantijs CA CB non &ecedil;queponderabunt quod fieri
non pote&longs;t; &longs;iquidem AB inter &longs;e &longs;unt &ecedil;qualia.
quæ ex æqualibus
mon&longs;trare oportebat.
tum huius
mum post
huius.
SCHOLIVM.
Cùm &longs;it &longs;copus Archimedis (vt diximus) in primis octo
theorematibus, fun damentum tradere in hac &longs;cientia præci-
re, vt di&longs;tantiæ permutatim ex quibus &longs;u&longs;penduntur &longs;e
primùm incipit o&longs;tendere, quomodo &longs;e habeant grauia in di
&longs;tantijs &ecedil;qua ibuspo&longs;ita; primùmquè in hac prima propo&longs;itio
ne o&longs;tendit, &longs;i grauia &ecedil;queponderant ex di&longs;tantijs &ecedil;qualibus,
&ecedil;qualia e&longs;&longs;e. in &longs;equenti verò, &longs;i grauia &longs;unt in&ecedil;qualia, ex di
&longs;tantijs &ecedil;qualibus nullo modo æqueponderare o&longs;tendet; &longs;ed
præponderare ad maius.
PROPOSITIO. II.
Inæqualia grauia ex æqualibus di&longs;tantijs non
æqueponderabunt, &longs;ed præponderabit ad maius.
Sint gra
uia in&ecedil;qua
lia AB C in
di&longs;tantijs &ecedil;
qualib^{9} DA
DC. &longs;itquè
grauius AB,
quàm C. di
co grauia AB C non &ecedil;queponderare, &longs;ed maius AB
ferri. &longs;it B exce&longs;&longs;us, quo AB &longs;uperat C.
iori AB
DA DC cùm æqualia grauia ex distantiis æquali-
igitur
&longs;um tendet.
fuit
tijs &ecedil;qualibus po&longs;itis. quod demon&longs;trare oportebat.
ius.
ius.
SCHOLIVM.
Hæc duo theoremata in gr&ecedil;co exemplari impre&longs;&longs;o &longs;equun
tur
beant propo&longs;itiones, &longs;uafquè &longs;eor&longs;um habeant demon&longs;tratio
nes, ideo inter propo&longs;itiones ip&longs;a collocare nobis vi&longs;um e&longs;t.
cùm pr&ecedil;&longs;ertim nonnulla ex &longs;equentibus theorematibus, po
ti&longs;&longs;i mùm verò proximum eiu&longs;dem cum his duobus ordinis,
& naturæ &longs;int. Neque enim propterea peruertitur ordo; non
enim h&ecedil; propo&longs;itiones in alium transcerun tur locum. &longs;ed
ùexi&longs;tim andum enim e&longs;t,
Archimedem propo&longs;itiones in &longs;erie propo&longs;itionum colloca&longs;
&longs;e. hanc verò exiguam muta tionem accidi&longs;&longs;e
temporis; cuius proprium e&longs;t, res potiùs de&longs;truere, quàm ac
comodare. Hocautem nobis hanc præbebit commoditatem,
vt, quando libuerit, has propo&longs;itiones numeris nominare
po&longs;&longs;imus. idip&longs;umquè numeri po&longs;tulata di&longs;tinguentes præ
&longs;tant, quamuis in Gr&ecedil;co codice po&longs;tulata (Gr&ecedil;corum more)
numeris adnotata non &longs;int.
PROPOSITIO. III.
Inæqualia grauia ex di&longs;tantijs inæqualibus æ
rò, quo AD &longs;uperat B, &longs;it
D.
di&longs;tantiis AC C B. o&longs;tendendum
e&longs;t, minorem e&longs;&longs;e
ip&longs;a CB. Non &longs;it quidem, &longs;i fie
ripotest
vel maior. Quòd &longs;i AC fuerit &ecedil;qualis ip&longs;i CB,
exce&longs;&longs;u
li CB
ce&longs;&longs;us D,
ponderant, quod e&longs;t ab&longs;urdum. di&longs;tantia igitur AC ip&longs;i CB
æqualis e&longs;&longs;e non pote&longs;t.
militer exce&longs;&longs;u D, nihilominus &ecedil;qualia grauia AB non &ecedil;que
ponderabunt, &longs;ed
distantiampo&longs;ita enim &longs;unt æqueponde
rare. Quare AC maior e&longs;&longs;e non pote&longs;t, quàm CB. &longs;ed o&longs;ten&longs;a
e&longs;t, neque ip&longs;i CB æqualis e&longs;&longs;e:
CB. Mani&longs;estum e&longs;t itaque grauia ex distantiis inæqualibus æquepon
derantia, inæqualia e&longs;&longs;e; maiu&longs;què in minori
oportebat demon&longs;trare.
ius.
ius.
ius.
SCHOLIVM.
In propo&longs;itione verba illa,
tur integra in codice græco, qui &longs;ic habet,
vbi de&longs;iderari viderur
a)w_o\ tou_ e)la/ssonos.
plendum e&longs;t,
tuenda, quia in vltima demon&longs;trationis conclu&longs;ione inquit
Archimedes,
æqueponderantia inæqualia e&longs;&longs;e; maiu&longs;què in minori existere.
&longs;e
po&longs;tulato
qualibus
rat
&longs;tratquè graue maius in breuiori
rò graue in
ducit nos in
ue ad graue e&longs;t, vt Ex hoc.
mùm cogno&longs;cimus grauius in minori, leuius
di&longs;tantia e&longs;&longs;e debere, &longs;i &ecedil;queponderare debent.
PROPOSITIO. IIII.
Si due magnitudines æquales non idem
grauitatis habuerint, magnitudinis ex vtri&longs;que
magnitudinibus compo&longs;itæ centrum grauitatis
er it medium rectæ lineæ grauitatis centra magni
tudinum coniungentis.
tis magnitudi
nis A. B uerò
&longs;it
uitatis
tudinis B iun
staquè AB bifariam diuidatur in C. dico magnitudinis ex utri&longs;què ma
gnitudinibus compo&longs;itæ centrum
utrarumquè magnitudinum AB centrum grauitatis D, &longs;i
autem &longs;it in linea AB, præo&longs;ten&longs;um est. Quoniam igitur punstum D
tudines AB magnitudines igitur AB
ponderant ex di&longs;tantiis AD DB
æqualia.
rant
re manifestum est punstum C
compo&longs;itæ.
centri
grauit.
contra 2.
post huins
ius.
SCHOLIVM.
Po&longs;&longs;unt magnitudines &ecedil;quales
grauitatis habere, vt duo
qualia ad rectos &longs;ibi
tia:
ter&longs;e æqualia.
dros, & nuiu&longs;modi alias magnitudines &ecedil;qua
les
mus. propterea in propo&longs;itione cùm inquit Archimedes
nere magnitudines ita e&longs;&longs;e con&longs;titutas, vt à centro ad centrum
duci po&longs;&longs;it recta linea. quod idem ob&longs;eruandum e&longs;t in prima
propo&longs;itione &longs;ecundi libri huius.
Súmoperè
Archimedes in hac propo&longs;itione, cùm &longs;int communi&longs;&longs;ima,
& maximè vtilia in hac &longs;cientia. ac primùm quidem con&longs;ide
randum occurrit, quid &longs;ibi vult Archimedes per magnitudi
nem ex vtri&longs;que magnitudinibus AB compo&longs;itam. Nam ma
gnitudines AB &longs;unt inuicem &longs;eparat&ecedil;, & &longs;unt du&ecedil;, ip&longs;e autem
vtram quovnam tantùm con&longs;iderat. quod quidem ita in
gendum
Archimedes con&longs;iderat vnam tantùm e&longs;&longs;e
con&longs;tat ex ip&longs;is AB, & efficitur vna magnitudo àlinea AB.
cuius munus e&longs;t non &longs;olùm connectere magnitudines AB,
ita vtneque ad &longs;e ampliùs accedere, neque recedere inuicem
po&longs;&longs;int; &longs;intquè ab hac linea qua&longs;i compul&longs;&ecedil; eundem &longs;emper
in ter&longs;e &longs;eruare &longs;i tum: verum etiam &longs;i &longs;u&longs;pendantur ex C, in
tellig endum e&longs;t linea AB in rectitudin em iacere, in&longs;uperquè
&longs;u&longs;tinere magnitudines AB. Neque magis vna e&longs;t magnitudo
quadrilaterum,
&longs;it magnitudo, quæ componitur ex magnitudinibus AB v
nà cum linea AB. quòd &longs;i e&longs;t vna tantùm magnitudo, ergo
vnum habet Archimed esigitur qu&ecedil;rit cen
trum grauitatis huiu&longs;ce magnitudinis; demon&longs;tratquè cen
trum e&longs;&longs;e in puncto C. quod e&longs;t medium lineæ AB. notan
dum e&longs;t autem Archimedem non con&longs;iderare grauitatem li
ne&ecedil; AB. vt potè, qu&ecedil; longitudo tantùm exi&longs;tat. Quòd &longs;i quis
etiam mente concipere vellet lineam AB grauitate
e&longs;&longs;e; nihilominus centrum grauitatis line&ecedil; AB &longs;imiliter e&longs;&longs;et
in eius medio C. nam longitudo AC longitudini CB e&longs;t
æqualis; ac propterea h&ecedil; quidem longitudines e&longs;&longs;ent inter &longs;e&longs;e
&ecedil;queponderantes. Quare, &longs;iue
&longs;iue minùs, centrum grauitatis magnitudinis ex AB compo
&longs;it&ecedil; e&longs;t mediu rect&ecedil; line&ecedil;, quæ centra grauitatis
coniungit. Et hoc modo &longs;i plures etiam e&longs;&longs;ent magnitudines
à recta linea coniunct&ecedil;, eodem modo eas pro vna tan tùm ma
terimus, veluti Archimedes in &longs;equenti bus accipiet.
Argumen tandi modus in e&longs;t in hac demon&longs;tratione maxi
ma con&longs;ideratione dignus, & huius &longs;cientiæ maximè pro
prius. cùm enim dixi&longs;&longs;et Archimedes po&longs;ito centro grauitatis
magnitudinis ex AB compo&longs;itæ in puncto D, &longs;tatim infert.
AB compo&longs;ita, &longs;u&longs;pen&longs;o puncto D, magnitudines AB æquepondera
bunt.
D, manebit, vt reperitur; nec amplius in alteram partem in cli
nabit. quod euenit ob naturam centri grauitatis, quod talis
e&longs;t naturæ (&longs;icuti initio explicauimus) ut &longs;i graue in eius cen
tro grauitatis &longs;u&longs;tineatur, eo modo manet, quo reperitur,
&longs;u&longs;penditur; parte&longs;què undiquè æqueponderant. & ob id &longs;i
magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur in eius centro gra
uitatis, manet; parte&longs;què AB æqueponderant. ac propterea
quando in &longs;equentibus quærit Archimedes, quoniam grauia
æqueponderare debent, tunc tan tùm quærit ip&longs;orum
grauitatis, utin &longs;exta, &longs;eptimaquè propo&longs;itione in quit Archi
medes magnitudines &ecedil;queponderare ex di&longs;tantijs, quç permu
tatim proportionem habent, utip&longs;arum grauitates, in
&longs;tratione tamen quærit, vbi nam e&longs;t
tudinis ex vtrisquè compo&longs;it&ecedil;. quo inuento, &longs;tarim nece&longs;&longs;ariò
&longs;equitur, magnitudines, &longs;i ex ip&longs;o centro &longs;u&longs;pendantur, æque
ponderare.
Hinc colligere po&longs;&longs;umus alterum argumentandi modum,
conuer&longs;o nempè modo, veluti in eadem &longs;igura, &longs;i dicamus
grauia AB &longs;u&longs;pen&longs;a ex C æqueponderant, &longs;tatim inferre
po&longs;&longs;umus, punctum C ip&longs;orum &longs;imul grauium, hoc e&longs;t ma
gnitudinis ex ip&longs;is AB compo&longs;it&ecedil; centrum e&longs;&longs;e grauitatis.
Quare ad &longs;e inuicem conuertuntur, hoc punctum e&longs;t horum
grauium cen trum grauitatis; ergo h&ecedil;c grauia ex hoc puncto
æqùeponderant; & è conuer&longs;o, nempè hæc grauia ex hoc pun
cto æqueponderant, ergo idem punctum e&longs;t ip&longs;orum
grauitatis. &longs;ed ad uertendum hanc &longs;equi
do
tum ponderum coniungit; deinde quando h&ecedil;c linea non e&longs;t
&longs;ecus aurem minimè. Nam &longs;i pon
dera AB &longs;int in libra ADB, qu&ecedil; &longs;itarcuata, vel angulum
&longs;tituat
perpendicularis CD. vt in tractatu de libra no&longs;trorum Me
chanicorum diximus. &longs;u&longs;pendantur autem pondera AB ex
D, & æqueponderent;
&longs;equitur tamen, ergo D
gnitudinis ex AB com
po&longs;it&ecedil;. centrum enim gra
uita tis in linea exi&longs;tit AB
quæ centra grauitatis ma
gnitudinum AB coniun
git, nempein C. Verùm coniungat recta linea AB centra
grauita tis æqualium ponderum AB, lineaquè
AB, cuius medium &longs;it C, in centrum mundi
dat
cunque &longs;u&longs;pendatur in linea AB, vtin E; ma
nebuntvtique pondera AB ex E &longs;u&longs;pen&longs;a, vt in
prima propo&longs;itione de libra no&longs;trorum Mecha
nicorum o&longs;ten dimus. cùm C &longs;it ip&longs;orum
grauita tis, & EC &longs;it horizonti erecta. Et quam
uis magnitudo ex ip&longs;is AB compo&longs;ita ex E &longs;u
&longs;pen&longs;a maneat; non propterea &longs;equitur ergo E
centrum e&longs;t grauitatis magnitudinis ex ip&longs;is AB
compo&longs;it&ecedil;. ni&longs;i fortè accidat &longs;u&longs;pen&longs;io ex puncto
C. Præterea verò aduertendum e&longs;t in hoc ca&longs;u
dera AB, dici quidem po&longs;&longs;e, manere, non autem
æqueponderare. omnia nimirum, qu&ecedil; æqueponderant, ma
nent; &longs;ed non è conuer&longs;o, quæ manent, æqueponderant. Nam
&longs;i pondus A maius fuerit pondere B; &longs;iue B maius, quàm
A, vbicunque fiat &longs;u&longs;pen&longs;io in linea AB, &longs;emper ob
cau&longs;am, quomodocun que &longs;int pondera, manebunt; non ta
men æqueponderabunt. Vt enim pondera æqueponderent,
requiritur, vt pars parti, virtu&longs;què vnius virtuti alterius hinc
inde re&longs;i&longs;tere, & æquipollere po&longs;&longs;it; vt propriè dici po&longs;&longs;int
dera æqueponderare. & vt hoc euenire po&longs;&longs;it, oportet, vt par
grauita tes; &longs;i ex dato puncto æqueponderare debent. Quòd
&longs;i in hoc ca&longs;u datum fuerit punctum C, ex quo pondera AB
ex æqualibus di&longs;tantijs CA CB &ecedil;quepo nderare debeant: o
porteret, vt pondera AB (ex demon&longs;tratis) &longs;emper e&longs;&longs;ent æ
qualia.
ue pondus A maius, &longs;iue minus fuerit, quàm B, manent, &longs;i
igitur dixerimus, ergo pondus A ponderi B &ecedil;queponderat;
e&longs;&longs;et o mnino inconueniens. cùm ex ijsdem di&longs;tantijs
deri pondus quandoquè maius, quandoquè minus &ecedil;quepon
derare non po&longs;&longs;it; vt in hoc ca&longs;u accidere pote&longs;t. Quocirca
nec propriè dici po&longs;&longs;unt pondera, &longs;iue in libra AB, &longs;iue ex
di&longs;tantijs CA CB con&longs;tituta e&longs;&longs;e. Vndè neque Archimedis
propo&longs;itiones in hoc ca&longs;u &longs;unt in telligend&ecedil; quandoquidem
in his propriè quærit ponderum, magnitudinumquè æque
ponderationes. neque enim in hac quarra demon&longs;tratione in
hoc ca&longs;u potui&longs;&longs;et Archimedes ab&longs;urdum o&longs;tendere, &longs;i C
e&longs;t grauitatis centrum magnitudinis ex AB compo&longs;itæ, &longs;it
E. facta igitur ex E &longs;u&longs;pen&longs;ione, magnitudines æquales AB
ex in æquali bus di&longs;tantijs EA EB &ecedil;queponderabunt. quod
&longs;ieri non pote&longs;t. non enim hoc e&longs;t ab&longs;urdum; cùm pondera
ex E &longs;u&longs;pen&longs;a
ti erecta; propriè ad rem no&longs;tram minimè pertinet. Ex dictis
igitur &longs;emper valet con&longs;equentia, hoc punctum horum pon
derum centrum e&longs;t grauitatis, ergo &longs;i ex hoc &longs;u&longs;pendantur,
dera &ecedil;queponderant. non autem è conuer&longs;o. ni&longs;i quando ar
gumentatio &longs;umitur &longs;emper ex recta linea, quæ centra graui
tatis magnitudinum coniungit, & quando h&ecedil;c linea non e&longs;t
horizonti erecta. hac enim
ratione quocunque modo
recta linea &longs;e habeat, &longs;em
per &longs;equitur idem. Vt &longs;i li
nea AB fuerit, &longs;iue
rit horizonti æquidi&longs;tans,
ip&longs;ius medium C centrum
erit grauitatis magnitudi
nis ex magnitudinibus AB æqualibus compo&longs;it&ecedil;. vnde &longs;equi
&
è conuer&longs;o, &longs;i AB pondera ex C æqueponderant, ergo C
centrum grauitatis exi&longs;tit. ex quibus &longs;equitur lineam AB,
deraquè manere eo modo, quo reperiuntur. vt in no&longs;tro me
chanicorum libro in codem tractatu de libra demon&longs;traui
mus, & aduer&longs;us illos, qui aliter &longs;entiunt, abundè &longs;atis
tauimus.
tam propo
&longs;itionem.
*
In demon&longs;tratione autem huius quartæ propo&longs;itionis in
quit Archimedes.
&longs;i dicat Archimedes, &longs;e priùs o&longs;ten di&longs;&longs;e centrum grauitatis ma
gnitudinis ex AB compo&longs;itæ e&longs;&longs;ein linea AB; quod tamen
in ijs, quæ dicta &longs;unt, non videtur expre&longs;&longs;um. virtute tamen &longs;i
con&longs;ideremus ea, qu&ecedil; in prima, tertiaquè propo&longs;itione dicta
&longs;unt, facilè ex his concludi pote&longs;t, centrum grauitatis magni
tudinis ex duabus magnitudinibus compo&longs;itæ e&longs;&longs;e in recta li
nea, quæ ip&longs;arum centra grauitatis coniungit. Quare memi
ni&longs;&longs;e oportet eorum, qu&ecedil; a nobis in expo&longs;itione primi po&longs;tu
lati huius dicta fuere, nempè Archimedem &longs;upponere, di&longs;tan
tias e&longs;&longs;e in vna, eademquè recta linea con&longs;titutas. ideoquè in
prima propo&longs;itionec inquit, Grauia, qu&ecedil; ex
bus
&longs;trat, quòd quando æqueponderant, &longs;unt æqualia: ex dictis
&longs;equitur, &longs;i æqueponderant, ergo centrum grauitatis magni
tudinis ex ip&longs;is compo&longs;it&ecedil; erit in eo puncto, vbi æqueponde
rant; hoc e&longs;t in medio di&longs;tantiarum, line&ecedil; &longs;cilicet, qu&ecedil;
centra grauitatis coniungit. quod idem e&longs;t, ac &longs;i Archimedes
dixi&longs;&longs;et. Grauia, qu&ecedil; habent centrum grauitatis in medio li
ne&ecedil;, qu&ecedil; magnitudinum centra grauitatis coniungit, &ecedil;qua
lia &longs;unt inter &longs;e. cuius quidem h&ecedil;c quarta propo&longs;itio videtur
e&longs;&longs;e conuer&longs;a. quamuis Archimedes loco grauium nominet
magnitudines. Pr&ecedil;terea in tertia propo&longs;itione, quoniam
dit
in&ecedil;qualibus, ita vt grauius &longs;it in minori di&longs;tantia, &longs;equitur er
go centrum grauitatis e&longs;t in eo puncto, vbi æqueponderant;
& idem e&longs;t, ac &longs;i dixi&longs;&longs;et, in æqualium grauium centrum gra
uitatis e&longs;t in recta linea, quæ ip&longs;orum centra grauitatis con
iungit; ita vt &longs;it propinquius grauiori, remotius uerò leuiori.
que e&longs;&longs;e po&longs;&longs;e in recta linea, qu&ecedil; ipiorum centra grauitatis
iungit. Ex quibus concludi potelt,
tudinis ex duabus magnitudinibus compo&longs;it&ecedil; e&longs;&longs;e in recta li
nea, quæ ip&longs;orum centra grauitatis connectit.
Po&longs;tremò notandum e&longs;t, Archimedem ea, quæ in &longs;uperio
ribus propo&longs;itionibus nuncupauit grauia, in hac quarta pro
po&longs;itione, veluti etiam in &longs;equentibus, non ampliùs grauia,
&longs;ed (vti diximus) magnitudines nominare. quod quidem his
de cau&longs;is id ab ip&longs;o factum exi&longs;timo. primùm enim, quia in
his expre&longs;se quærit centrum grauitatis; quod quidem
quamuis &longs;it centrum grauitatis, potiùs re&longs;picit
quàm graue aliquod. Nam cùm dicim us centrum grauitatis,
&longs;tatim innuim us &longs;i tum, &longs;itum inquàm determinatum &longs;igu
ræ, in qua e&longs;t; &longs;iquidem centrum grauitatis e&longs;t punctum, &
(vtita dicam) punctum grauitatis eius, in quo e&longs;t. & ideo,
quoniam magnitudo formam habet dete mina tam,
grauitatis rectè pote&longs;t re&longs;picere &longs;itum re&longs;pectu magnitudinis,
in qua e&longs;t; quod tamen efficere non pote&longs;t re&longs;pectu grauis.
etenim graue, ut graue e&longs;t, non habet formam determina
cùm eadem grauitas e&longs;&longs;e po&longs;&longs;itin cubo, in piramide, alii&longs;què
corporibus quibu&longs;cunque, modò minoribus, modò maiori
bus, prout &longs;unt diuer&longs;arum &longs;pecierum. quare centrum grauita
tis non pote&longs;t re&longs;picere &longs;itum in grauibus, quatenus grauia
&longs;iderantur; &longs;ed quatenus magnitudines exi&longs;tunt. Præterea Ar
chimedes loco grauium magnitudines nominat, quia eas di
ui&longs;ibiles con&longs;iderat, quod e&longs;t proprium magnitudinis; vt in &longs;e
xta, &longs;eptima, & octaua propo&longs;itione. & quamuis, dum
tur magnitudines, grauia quoque diui&longs;a proueniant; non ta
men propterea grauia diuiduntur, ut grauia. hoc ip&longs;is
competit, vt grauibus; &longs;ed vt magnitudinibus, quæ &longs;unt por
&longs;e diui&longs;ibiles. Archimedes igitur his de cau&longs;is nomen
in magnitudines mutauit. in &longs;uperioribus enim theoremati
bus pertractauit, quomodo res æqueponderant ex di&longs;tantijs
modò æqualibus, modò in æqualibus. & quoniam res
derant, prout &longs;unt magis grauia, & minùs grauia; non ut
maiores, vel minores magnitudines, &longs;iquidem talis naturæ
rius nature grauior exi&longs;tat; proindé Archimedesin &longs;uperiori
busrectè grauia nuncupauit; optimèquè in his magnitudines
vocat. Atverò aduertendum e&longs;t, quòd quamuis Archimedes
in his magnitudines nominet, non propterea exi&longs;tim andum
e&longs;t, eum intelligere magnitudines tantùm; &longs;ed magnitudines
grauitate prçditas, ita utin ip&longs;is omnino grauitatem re&longs;piciat.
Etenim pluribus modis in telligere po&longs;&longs;umus magnitudines,
vel enim ut &longs;int inter &longs;e eiu&longs;dem &longs;peciei, vel diuer&longs;æ; nec
in&longs;uper homogeneæ, vel heterogeneæ. vt in hac propo&longs;itione
intelligere po&longs;&longs;umus eas e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas;
quæ, cùm &longs;int æquales, erit & grauitas vnius grauita ti alterius
æqualis. &longs;i verò con&longs;ideremus eas e&longs;&longs;e diuer&longs;æ &longs;peciei, & e
tiam heterogeneas; tunc quando Archimedes proponit has
magnitudines æ quales; intelligendum e&longs;t, eas e&longs;&longs;e æ quales in
grauita te; quæ quidem efficit, vt demon&longs;tratio, quod propo
&longs;itum e&longs;t, concludat. vtex eius demon&longs;tratione patet. Et his
quoque modis intelligere po&longs;&longs;umus magnitudines in &longs;equen
tibus v&longs;que ad nonam propo&longs;itionem in quibus &longs;cilicet intel
ligere po&longs;&longs;umus magnitudines e&longs;&longs;e non &longs;olùm eiu&longs;dem &longs;pe
ciei, vel diuer&longs;æ, verùm etiam & homogeneas. & heteroge
neas. ut po&longs;t &longs;eptimam clariùs o&longs;tendemus. Verùm de
mon&longs;trationes clariores red duntur, &longs;i intelligamus magnitu
dines e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas, in quibus graui
tas magnitudini re&longs;pondet, vt &longs;i ip&longs;arum altera fuerit alte
rius dupla, & grauitas vnius grauitatis alterius dupla exi&longs;tat.
Quòd &longs;i magnitudo fuerit alterius tripla, vel quadrupla, &c.
erit & grauitas grauitatis tripla, vel quadrupla, & &longs;ic dein
ceps. deinde &longs;i magnitudo bifariam diui&longs;a fuerit, & ip&longs;ius gra
uitas in duas &ecedil;quas partes &longs;it quoque diui&longs;a. quòd &longs;i magnitu
do in plures diuidatur partes, & grauitas quoque in totidem
eiu&longs;dem proportionis diui&longs;a proueniat.
PROPOSITIO. V.
Si trium magnitudinum centra grauitatis in re
cta linea fuerint po&longs;ita, & magnitudines æ qualem
habuerint grauitatem, acrectæ lineæ inter centra
fuerint æ quales, magnitudinis ex omnibus magni
tudinibus compo&longs;itæ centrum grauitatis erit
ctum, quod & ip&longs;arum mediæ centrum grauitatis
exi&longs;tit.
puncta ACB in resta linea&longs;int verò magnitudines ACB
æquales; rectæquè lineæ AC CBDi
co magnitudims ex omnibus
uitatis e&longs;&longs;e punetum C.
gnitudinis.
tatis erit punctum C: cùm &longs;int AC CB æquales.
punctum C medium rectæ line&ecedil; AB.
trum grauitatis est
gnitudinum ABC centrum quoque grauitatis erit.
tet magnitudinis ex omnibus magnitudinibus
grauitatis e&longs;&longs;e punctum, quod &
tatis existit.
COROLLARIVM. I.
Ex hoc autem manife&longs;tum e&longs;t, &longs;i quotcunquè
magnitudinum, & numero imparium, centra
uitatis in re cta linea con&longs;tituta fuerint; & magni
tudines æ qualem habuerint grauitatem; rectæquè
lineæ inter ip&longs;arum centra fuerint æ quales, ma
gnitudinis ex omnibus magnitudinibus compo&longs;i
tæ centrum grauitatis e&longs;&longs;e punctum, quod & ip&longs;a
rum mediæ centrum grauitatis exi&longs;tit.
SCHOLIVM.
Ex demon&longs;tratione colligit Archimedes &longs;i plures fuerint
magnitudines,
ABCDE; quarum centra grauitatis ABCDE reperiantur in li
nea recta AE. fuerint autem h&ecedil; magnitudines æ quales in gra
uitate. in&longs;uper rect&ecedil; line&ecedil; AB BC CD DE, qu&ecedil; &longs;unt in ter
tra
gnitudinibus ABCDE compo&longs;itæ centrum grauita tis e&longs;&longs;e
punctum C. quod e&longs;t centrum grauitatis magnitudinis
mediæ.
Eodem enim modo, ac primùm quidem ex demon&longs;tratio
ne patet
BCD, & quoniam AB BC &longs;unt æquales ip&longs;is CD DE,
cùm què &longs;it grauitas magnitudinis
tudinum AE centrum grauitatis. ergo punctum C magni
tudinis ex omnibus magnitudinibus ABCDE compo&longs;itæ
centrum grauitatis exi&longs;tit.
Quòd &longs;i fuerint ad huc plures magnitudines, impares verò
extiterint; quæ ita &longs;e habeant, vt expo&longs;itum e&longs;t; &longs;imiliter
detur, centrum grauitatis mediæ magnitudinis centrum e&longs;&longs;e
grauitatis magnitudinis ex omnibus magnitudinibus com
po&longs;itæ.
rint grauitatem
ta a)w_o\ tou= me\sou mege\qeos i)\sonba/<10>os e)/xwnt
nea nobis vi&longs;a &longs;unt; loco quorum (vt arbitror) rectè Nam &longs;i ordinis atque
tet vt magnitudines &ecedil;qualem habeant grauitatem; Nam &
Archimed es in &longs;equentibus demon&longs;trationibus ijs vtitur, ut
&longs;unt æquegraues. Adhuc tamen veritatem habebit &longs;i cæteris
conditionibus illud quoque addere voluerimus, nempe &longs;i
gnitudines à media magnitudine æqualiter di&longs;tantes æqualem habuerint
grauitatem
magnitudinis ex omnibus ABCDE compo&longs;it&ecedil;, Nam &longs;i ma
gnitudines à media magnitudine &longs;unt &ecedil;quegraues; &ecedil;qualem
quoque habebunt grauitatem magnitudines AE; veluti ma
gnitudines BD, quæ æ qualiter à media magnitudine C di
&longs;tant. & quam uis non &longs;int omnes æ quegraues, &longs;ufficit, vt AE
quæ &ecedil;qualiter à media magnitudine di&longs;tant, &longs;int &ecedil;quegraues.
&longs;imiliter BD &ecedil;quegraues. Eadem enim ratione, quoniam
BD &longs;untæ quegraues, & di&longs;tantiæ BC CD &ecedil;quales; erit C ip&longs;a-
pari què ratione C erit centrum
grauitatis magnitudinum AE &ecedil;quegrauium. cum &longs;int AC
CE &ecedil;quales, & idem C e&longs;t grauitatis centrum magnitudinis
C. ergo punctum C magnitudinis ex omnibus magnitudini
bus ABCDE compo&longs;it&ecedil; centrum grauitatis exi&longs;tit.
COROLL ARIVM. II.
Si verò magnitudines fuerint numero pares;
& ip&longs;arum centra grauitatis in recta linea extite
rint, magnitudine&longs;què æ qualem habuerint graui
tatem, rectæ què lineæ inter centra fuerint æ qua
les: magnltudinis ex omnibus magnitudinibus
po&longs;itæ centrum grauitatis erit medium rectæ li
neæ, quæ magnitudinum centra grauitatis
git
SCHOLIVM.
Colligit præterea Archimedes &longs;i magnitudines ABCDEF
fuerint numero pares, quarum centra grauitatis ABCDEF in
recta linea AF &longs;int con&longs;tituta; magnitudine&longs;què &longs;int æquales
in grauitate; &longs;intquè inter centra line&ecedil; AB BC CD DE EF
æ quales. diuidatur autem AF bifariam in G. erit punctum
G centrum grauita tis magnitudinis ex omnibus compo&longs;i
tæ quod quidem, figura tantùm in&longs;pecta, per&longs;picuum e&longs;t.
Cùm enim magnitudines AF &longs;int æquegraues, & AG GF
compo&longs;itæ. quia verò AB e&longs;t ip&longs;i EF æqualis, reliqua BG
ip&longs;i GE æqualis exi&longs;ter. & &longs;unt magnitudines BE çquegra
ues, eritidem G centrum grauitatis
ter cùm &longs;it BC æ qualis DE, relin quetur CG ip&longs;i GD &ecedil;qua
lis; magnitudinesquè CD &longs;unt &ecedil;quegraues. ergo
trum e&longs;t quoque magnitu dinum CD. Vnde &longs;equitur,
G magnitudinis ex omnibus magnitudinibus ABCDEF
po&longs;itæ
grauitatem.
ba/<10>os e)/xwnti
In præcedenti propo&longs;itione o&longs;tendir Archimedes, quomo
do &longs;e habet centrum gra uitatis magnitudinis ex duabus ma
gnitudinibus equalibus compo&longs;itæ. In hac autem
vbi &longs;imiliter grauitatis cen trum reperitur inter plures magni
tudines æquegraues, & inter &longs;e &ecedil;qualiter di&longs;tantes. ex quibus
randem colliget fun damentum &longs;æpiùs dictum. nempè &longs;i ma
gnitudines &ecedil;queponderare debent; ita &longs;e habebit magnitudi
num grauitas ad grauitatem, ut &longs;e habent di&longs;tantiæ permuta
tim, ex quibus &longs;u&longs;pen duntur. & hoc demon&longs;trat Archimedes
in duabus &longs;equen tibus propo&longs;itionibus. nam magnitudines,
vel &longs;unt commen&longs;urabiles in ter&longs;e&longs;e, vel incommen&longs;urabiles.
de commen&longs;urabilibusaget in &longs;equenti: de incommen&longs;urabi
libusverò in &longs;eptima propo&longs;itione. & Archimedes duas
tesNam in &longs;exta
inquit in &longs;
quit,
tùm &longs;it propo&longs;itio in duas partes diui&longs;a. ita ut neque numeris
e&longs;&longs;ent di&longs;ting uende, &longs;ed pro vna tantùm propo&longs;iuone &longs;um n
dæ, ob&longs;equen tis autem demon&longs;trationis faciliorem
tiam hecpriùs præmittimus.
LEMMA.
Si du&ecedil; fuerint magnitudines in æquales, quarum maior &longs;it
alterius dupla, tertia verò qu&ecedil;dam magnitudo minorem me-
maiorem quoque in partes numero pares metietur.
Sint du&ecedil; in &ecedil;quales magni
tudines AB, &longs;itquè A ip&longs;ius
B duplex. magnitudo
C
tur. Dico C
A metiri, men&longs;urationesquè numero pares e&longs;&longs;e. Quoniam
enim C metitur B, eodem numero C metietur medietates
ip&longs;ius A, quæ &longs;untip&longs;i B æquales. ergo duplo plures erunt nu
mero men&longs;urationes ip&longs;ius A, quàm ip&longs;ius B. quare men&longs;u
rationes ip&longs;ius A &longs;unt numero pares. duplum enim &longs;emper
paritatem &longs;ecum affert. quod demon&longs;trare oportebat.
Porrò maxima in his duabus &longs;equentibus propo&longs;itionibus
adhibenda e&longs;t diligentia; quibus tota rerum Mechanicarum
ratio in nititur. Quocirca vt harum propo&longs;itionum demon
&longs;trationes perfectè intelligere po&longs;&longs;imus; præter eos argumen
tandi modos, quorum ante quintam huius propo&longs;itionem
meminimus; alterum quoque modum, quo Archimedes in
hac&longs;exta propo&longs;itione vtitur, noui&longs;&longs;e oportet. vt &longs;cilicet, &longs;i ma
gnitudo A æqueponderatip&longs;is BC facta &longs;u&longs;pen&longs;ione ex
cto
ex omnibus ABC magnitudinibus compo&longs;itæ; ip&longs;arum verò
tæ centrum grauitatis &longs;it punctum E; auferantur verò BC
à linea EA, & ip&longs;arum loco ponatur in E magnitudo;
quæ &longs;it vtri&longs;que &longs;imul BC &ecedil;qualis, vtin &longs;ecunda figura. Dico
eodem modo pondera ABC &ecedil;queponderare in prima figu
ra, veluti grauia AE in &longs;ecunda.
Primum autem, vthoc recte per
pendamus, intelligantur pondera
BC (vt in tertia figura) &longs;eor&longs;um
à linea CA, & penes di&longs;tantias EC
EB con&longs;tituta. quorum quidem
derum
compo&longs;itum: pondera utique manebunt. quòd &longs;i ambo pe
penderint, vt quinquaginta, potentia in E tantùm quinqua
ginta &longs;u&longs;tinebit. quoniam totum &longs;u&longs;tinebit pondus ex ip&longs;is
compo&longs;itum, auferantur verò pondera BC à &longs;itu BC, intelli
ganturquè pondera e&longs;&longs;e in E con&longs;tituta; hoc e&longs;t vnum &longs;it
pondus ex ip&longs;is &longs;imul iun ctis compo&longs;itum, cuius
uitatis &longs;itin E con&longs;titutum; tunc eadem potentia in E eo
dem modo hoc pondus &longs;u&longs;tinebit; propterea quod
do quinquagin ta tantùm &longs;u&longs;tinebit. Quare pondera BC
ex di&longs;tan tijs EC EB grauitant, quàm &longs;i vtraque in E con
&longs;tituta fuerint; vel quod idem e&longs;t, quàm pondus ip&longs;is BC &longs;i
mul æ quale in E po&longs;itum. Ex quo patetid, quod initio pr&ecedil;
fati &longs;um us, nempe, vnumquodquè graue in eius centro gra
uitatis propriè grauitare. Quocum que enim modo
uia &longs;e&longs;e habent, eodem &longs;emper modo in eius grauitatis
grauitant.
cent. grau.
Quibus cognitis, intelligantur nunc grauia BC in linea
CA po&longs;ita e&longs;&longs;e; ut in &longs;uperiori figura: & ut quod propo&longs;itum
fuit, o&longs;tendatur; hoc modo argumentari licebit. Quoniam
enim magnitudines BC &longs;uam habent grauitatem in E, &longs;iqui
dem pro vna tantùm in telliguntur magnitudine ex BC com
po&longs;ita, cuius punctum E centrum grauitatis exi&longs;tit. in
da verò figura magnitudo E &longs;imiliter &longs;uam habet
in puncto E; quod e&longs;t eius atque magnitu
di&longs;tanti&ecedil; verò AD
DE &longs;unt æquales, cum &longs;int &ecedil;edem; erit vtique punctum D in
&longs;ecunda figura centrum grauitatis magnitudinis ex AE com
po&longs;itæ, veluti D in prima figura ip&longs;arum ABC centrum gra
uitatis exi&longs;tit. ac propterea in vtraque figura pondera æque
ponderabunt:
Cæterum hoc quoque o&longs;tendemus hoc pacto.
Ii&longs;dem namque po&longs;itis; æqueponderarent &longs;cilicet grauia
ABC facta ex D &longs;u&longs;pen&longs;ione. &longs;itquè punctum E
centrum grauitatis ponderum CB. quæ quidem pondera
CB grauitatis centrum habeantin linea CB. Dico pondus
A ponderi ip&longs;is CB &longs;imul &longs;umptis æquali in E con&longs;ti
tuto æqueponderare. Mente concipiamus di&longs;tantias EC
EB, manente centro E, circa ip&longs;um circumuerti po&longs;&longs;e;
vt modò &longs;int in FEG, modòin HEK. &longs;imiliter in
telligantur pondera CB, modò in FG, modò in HK
exi&longs;tere. Quoniam igitur punctum E. centrum e&longs;t
grauitatis ponderum CB; erit idem E (cùm &longs;itum
nonmutet) centrum grauitatis ponderum in &longs;itu FG, ac
ponderum in HK exi&longs;tentium. Quiaverò vnumquod
que pondus (ex dictis) propiè in eius centro grauitatis graui
tat; pondera &longs;imul CB &longs;iue &longs;int in FG, &longs;iuein HK, proprie
in puncto E grauitabunt. At verò quoniam idem
ex CB compo&longs;itum æquegraue, tam in &longs;itu CB, quàm in
FG, & in &longs;itu HK. con&longs;iderando nempe pondera CB (ut
revera &longs;unt) nilaliud e&longs;&longs;e ni&longs;i vnum tantùm pondus ex CB
compo&longs;itum. Ex quibus per&longs;picuum e&longs;t, punctum E eodem
&longs;emper modo grauitare. Quare quoniam pondera CB in &longs;i
tu CB ip&longs;i A &ecedil;queponderant, &longs;uamquè habent grauitatem
in puncto E; eadem pondera CB &longs;iue &longs;int in FG, &longs;iue in
HK, eidem ponderi A æqueponderabunt. &longs;iquidem propriè
&longs;emper grauitantin E, & eandem &longs;emper habent
tem
runtvtique vtraque pondera HK, tanquam in puncto E
&longs;tituta, vt ex prima propo&longs;itione no&longs;trorum Mechanicorum
elici pote&longs;t, quamuis per&longs;e notum &longs;it. &longs;iquidem &longs;eor&longs;um pon
dus H &longs;ecundùm eius centrum grauitatis propriè grauitat &longs;u
per puncto E; pondus verò K e&longs;t, tanquam ex E appen&longs;um;
vndè & in eodem puncto E quoque grauitat. Itaque
ambo propriè grauitant in E, erunt pondera HK perinde,
ac&longs;i vnum e&longs;&longs;et pondusip&longs;is HK, hoc e&longs;tip&longs;is CB æquale, cu
ius centrum grauitatis &longs;itin E con&longs;titutum. atverò pondus
A ip&longs;is CB in &longs;itu HK exi&longs;tentibus æqueponderat. ergo
pondus A ip&longs;is CB in E con&longs;titutis, hoc e&longs;t ponderi ip&longs;is CB
&longs;imul &longs;umptis &ecedil;quali in E po&longs;ito æqueponderabit. quod de
mon&longs;trare oportebat.
Quod idem quoque, &longs;i plura e&longs;&longs;ent pondera, &longs;imiliter o
&longs;tendetur.
Valetitaque con&longs;equentia, punctum D centrum e&longs;tgra
uitatis magnitudinis ex ponderibus ABC compo&longs;it&ecedil;; ergoi
dem punctum D centrum e&longs;t grauitatis ponderis in A, &
derisip&longs;is BC &longs;imul &ecedil;qualis in E con&longs;tituti. ex quo con&longs;equi
tur, quòd &longs;i magnitudines ABC ex D æqueponderant, ergo
ex eodem D magnitudo ip&longs;is BC &longs;imul æqualis in E po&longs;ita,
& magnitudo A æqueponderabunt. quòd &longs;i rectè perpenda
mus, nil aliud &longs;unt pondera in BC, ni&longs;i magnitudo in E con
&longs;tituta. &longs;iquidem punctum E ip&longs;ius centrum grauitatis
exi&longs;tit
In no&longs;tro autem Mechanicorum libro in quinta propo&longs;i-
tes
vtraque ex puncto E &longs;u&longs;pendantur. At verò quo niam demon
&longs;trationes ibi allatæ ijs indigent, qu&ecedil; Archimedes in &longs;equen
ti &longs;exta propo&longs;itione demon&longs;trauit, idcirco demon&longs;trationes
illæ huic loco non &longs;unt oportunæ; vt ex ip&longs;is&longs;umi po&longs;&longs;it tan
quam demon&longs;tratum pondera CB, tam in punctis CB pon
derare, quàm &longs;i vtraque ex E &longs;u&longs;pendantur. Quare hoc loco h&ecedil;
tantùm &longs;ufficiant rationes, quæ dictæ &longs;unt. Ex quibus pote&longs;t
Archime des di&longs;tam con&longs;equentiam colligere; nempè magni
tudines ABC ex D æqueponderant, auferantur autem BC,
& loco ip&longs;arum vtri&longs;que &longs;imul &ecedil;quegrauis ponatur magnitu
do in E; &longs;imiliter h&ecedil;c magnitudo ip&longs;i A æqueponderabit. Po
&longs;tea verò ex ijs, quæ Archimedes demon&longs;trauit, fieri pote&longs;t re
gre&longs;&longs;us; v
dera BC ita ponderare, ac &longs;i vtraque ex puncto E &longs;u&longs;pen
dantur.
C&ecedil;terum hoc loco Archimedes non &longs;olùm de duobus,
etiam de pluribus ponderibus idip&longs;um
vt &longs;i magnitudines STVXZM æqueponderent facta
ne ex puncto C. &longs;itquè magnitudinum MZ
D; ip&longs;arum verò STVX &longs;it centrum grauitatis E. &longs;i itaque ma
gnitudines STVX, & ZM ex C æqueponderant; auferantur
STVX, quarum loco ponatur in E magnitudo ip&longs;is STVX &longs;i
mul &longs;umptis &ecedil;qualis: auferanturquè ZM, atque
natur in D magnitudo ip&longs;is ZM &longs;imul &ecedil;qualis; tunclicetinfer
re, ergo hæ magnitudines in ED po&longs;itæ &ecedil;quepondera
bunt. Quod quidem ijsdem pror&longs;us modis o&longs;tendentur.
præ&longs;ertim &longs;i mente concipiamus di&longs;tantias ES EX,
grauitatis E circumuerti po&longs;&longs;e; veluti di&longs;tantias DZ DM, ma
gnitudine&longs;què ZM circacentrum D. moueantur autem
SEX, & ZDM, donec in centrum mundi vergant. &longs;imiliter
o&longs;tendetur magnitudines STVX e&longs;&longs;e, ac &longs;i in E e&longs;&longs;ent appen
&longs;&ecedil;, &longs;iue con&longs;titut&ecedil;; magnitudines verò ZM ac &longs;i in D po&longs;i
tæ fuerint. &c. Ex quibus &longs;equitur, &longs;i punctum C centrum
e&longs;t grauitatis magnitudinum STVXZM. ponatur magnitu
do ip&longs;is STVX &longs;imul &longs;umptis &ecedil;qualis in E; magnitudo au
tem ip&longs;is ZM &longs;imul æqualis in D; punctum C &longs;imiliter
ip&longs;arum quoque centrum grauitatis exi&longs;tet. vnde vtroque mo
do æqueponderabunt. & ita in alijs, &longs;i plures fuerint magni
tudines.
PROPOSITIO. VI.
Magnitudines commen&longs;urabiles ex di&longs;tantijs
eandem permutatim proportionem habentibus,
vt grauitates, æqueponderant.
tis
gnitudinis
DC ad distantiam CE.
rint in punctis ED con&longs;tituta, hoc e&longs;t A in E, & B in D;
grauitatis e&longs;&longs;e punctum C. Quoniam enim ita est
magnitudinem
recta linea rectæ lineæ
CD communis reperitur men&longs;ura. quæ quidem &longs;it N. deinde ponatur
ip&longs;i EC æqualis vtraque DG DK; ip&longs;i verò DC æqualis EL. &
quoniam æqualis est DG ip&longs;i CE
ip&longs;i EG æqualis
lis ip&longs;i EG.
facta e&longs;t ip&longs;i CE, erit
N
metiatur.
enim vtriu&longs;que duplex exi&longs;tit
& GK itidem ip&longs;ius CE duplex)
tudinem
magnitudinem A, vt KG ad GL.
N, totuplex &longs;it
magnitudo
magnitudinem
psam
quemultiplex e&longs;t
magnitudo A ad ip&longs;am F, vt LG ad N, quæ quidem LG mul
tiplex e&longs;t ip&longs;ius N.
&longs;ura. Jtaque diui&longs;a LG in partes
cadent vtique diui&longs;iones in punctis EC, quoniam
metitur, nec non ip&longs;am quoque LE metitur; cùm &longs;it LE ip&longs;i
CD æqualis. eruntquè diui&longs;iones LH, HE, EC, CG, numero
pares; cùm N dimidiam ip&longs;ius LG, hoc e&longs;t CD metiatur.
nes
les, erunt numero æquales &longs;ectionibus
existentibus ip&longs;i F æqualibus.
CG bifariam in punctis STVX.
ne ip&longs;ius LG apponatur magnitudo æqualis ip&longs;i F, quæ centrum gra
uitatis babeat in medio &longs;ectionis
S, in HE magnitudo T, in EC magnitudo V, & in
CG magnitudo X; ip&longs;arum què vna quæque STVX &longs;it ip&longs;i
F æqualis: habeat verò magnitudo S &longs;uum grauitatis
quod &longs;it punctum S, in medio &longs;ectionis LH, nempè in
cto
grauitatis; quæ &longs;int puncta TVX, in medio &longs;ectionum HE,
EC, CG, in punctis nempè TVX, erunt centra grauitatisma
gnitudinum STVX in recta linea con&longs;tituta, & quoma
SH dimidia e&longs;t ip&longs;ius LH, veluti HT ip&longs;ius HE, erit ST,
ip&longs;ius LE dimidia, vnaquæque verò LH HE dimidia
quoque e&longs;t ip&longs;ius LE, &longs;iquidem LH, HE inter &longs;e &longs;unt &ecedil;qua
les; eritigitur ST vnicuique LH, & HE æqualis. eodem què
pror&longs;us modo o&longs;tendeturi TV &ecedil;qualem e&longs;&longs;e vnicuique HE
EC. & VX æqualem EC. & CG. & quoniam omnes
ter&longs;e æquales. quare lineæ inter centra grauitatis magnitudi
num STVX exi&longs;tentes &longs;unt inter &longs;e &ecedil;quales.
tudines
OPQR, & numero, & magnitudine &longs;unt &ecedil;quales; ergo
tudinis ex omnibus
uitatis erit punstum E. cùm omnes
mero pares.
mero paribus. &
EG æqualis, demptis æqualibus LS GX æqualibus, &longs;iquidem
&longs;unt dimidiæ &longs;ectionum LH CG æqualium: erunt SE EX
ter&longs;e æquales, vnde ex præcedenti colligitur, punctum E cen
trum e&longs;&longs;e grauitatis magnitudinum STVX.
detur, quòd &longs;i
cadetvtique diui&longs;ionum aliqua in
GD DK metitur; cùm vtraque &longs;it æqualisip&longs;i EC. diui&longs;ione&longs;
què GD DK numero pares erunt; cùm N dimidiam ip&longs;ius
GK, ip&longs;am &longs;cilicet EC metiatur. &longs;i itaque diuidatur GD DK
bifariam in punctis ZM. deinde diuidatur magnitudo B
in partes ip&longs;i F æquales; &longs;ectiones GD DH in GK exi&longs;tentes
ip&longs;i N æquales, erunt numero æquales &longs;ectionibus in ma
gnitudine B exi&longs;tentibus ip&longs;i F æqualibus. quare
partium ip&longs;ius GK apponatur magnitudo æqualis ip&longs;i F; centrum gra
uitatis habens in medio &longs;ectionis
&longs;ectionibus GD DK, ita vt magnitudinum centra grauita
tis, quæ &longs;int ZM, in medio &longs;ectionum GD DK, in punctis
nempè ZM &longs;int con&longs;tituta,
mul
ZM
&ecedil;qualis DM.
æquales, & ZM ip&longs;i B ergo
ad E, ip&longs;a verò B ad D.
gnitudo A impo&longs;ita ad E, vt &longs;e habent magnitudines STVX;
ip&longs;a verò B &longs;e habebit ad D, vt magnitudines ZM.
tem magnitudines
ip&longs;i F &ecedil;qualis: &longs;untquè omnes, (hoc e&longs;t ip&longs;arum centra graui
tatis)
&longs;e æquales e&longs;&longs;e. Eodemquè modo o&longs;tendetur XZ ZM cæteris
æquales e&longs;&longs;e.
cùm &longs;ectiones totius LK, (in quibus in&longs;unt) ip&longs;i N æquales
&longs;int inter &longs;e &ecedil;quales, & numero pares. cùm o&longs;ten&longs;um &longs;it &longs;ectio
tudinis ex omnibus
dinum habentur. Itaque cùm LE &longs;it æqualis C D, EC verò ip&longs;i D
qui dem &longs;unt eidem N æquales, & harum medietates, hoc e&longs;t
LS ip&longs;i MK &ecedil;qualis erit. & ob id SC ip&longs;i CM e&longs;t æqualis.
at verò linea SM magnitudinum centra grauitatis
tæcentrum grauitatis est punctum C. Quare
STVX,
ZM po&longs;ito
gnitudinis ex vtri&longs;que magnitudinibus AB compo&longs;itæ. ac
prop terea
ex di&longs;tantijs DC CE, qu&ecedil; permutatim eandem habent pro.
portionem, vt grauitates, &ecedil;queponderant. quod demon&longs;trare
oportebat.
cimi.
cor.
ti.
quin
tæ huius.
SCHOLIVM.
qua&longs;i dicat, centrum grauitatis magnitudinis ex omnibus
magnitudinibus STVXZM compo&longs;it&ecedil; medietatem e&longs;&longs;e rect&ecedil;
line&ecedil; VX, qu&ecedil; centra mediarum magnitudinum VX coniun
git; quòd cùm &longs;int omnes magnitudines numero pares;
e&longs;&longs;et punctum C, & quamuis hoc &longs;it verum, non tamen ad hoc
re&longs;pexit Archimedes duabus de cau&longs;is.
pr&ecedil;cedentis o&longs;tendit centrum grauitatis omnium magnitu
dinum e&longs;&longs;e medietatem rect&ecedil; line&ecedil;, qu&ecedil; grauitatis centra om
nia coniungit. Dein de concludere volens punctum C
e&longs;&longs;e grauito tis omnium magnitudinum, &longs;tatim inquit hoc &longs;e
qui, quia LC e&longs;t ip&longs;i CK &ecedil;qualis, qu&ecedil; &longs;unt medietates totius
Quare codicem græcum ita re&longs;tituendum cen&longs;eo.
tou= ba\<10>eos megeqw=n
Ob &longs;equentis verò demon&longs;trationis cognitionem, hoc pro
blema priùs o&longs;tendemus.
PROBLEMA.
Duarum expo&longs;itarum magnitudinum incommen&longs;urabi
lium altera vtcumque &longs;ecetur; magnitudinem tota &longs;ecta ma
gnitudine minorem, & altero &longs;egmentomaiorem, alteri ve
rò expo&longs;itæ magnitudini commen&longs;urabilem inuenire.
Sint duæ magnitudi
nes incommen&longs;urabiles
AE BC. &longs;eceturquè ip&longs;a
rum altera, putà BC, vt
cumque in D. oportet
magnitudinem inuenire
minorem quidem BC,
maiorem verò BD, quæ &longs;itip&longs;i AE commen&longs;urabilis. Au
feratur ab AE pars dimidia, rur&longs;us dimidiæ partis ip&longs;ius AE
dimidia auferatur; & eius, quæ remanet, adhuc dimidia; idquè
&longs;emper fiat, donec relinquatur magnitudo minor, quàm DE.
quod quidem per&longs;picuum e&longs;t po&longs;&longs;e fieri ex prima decimi Eu
clidis propo&longs;itione. &longs;ititaque AF, quæ minor exi&longs;tat, quàm
DC. quippe qu&ecedil; AF, cùm &longs;it abla ta ex AE &longs;emper per dimi
diam partem, metietur vtique AF ip&longs;am AE. Deinde mul
tiplicetur AF &longs;uper BD, tum demum multiplicatio vltima,
vel in puncto D cadet, vel minus. &longs;i cadet; &longs;eceturex DE
magnitudo DG &ecedil;qualis AF. quod quidem fiet,
minor e&longs;t DC. Quoniam igitur AF metitur BD, & DG;
metietur AF totam BG. Sed & ip&longs;am AE metitur; etgo
AF ip&longs;arum BG AE communis exi&longs;tit men&longs;ura, ac propte
rea BG ip&longs;i AE commen&longs;urabilis exi&longs;tir; quæ quidem BG
minor e&longs;t BC, maior verò BD. Si verò vltima
plicatio ip&longs;ius AF &longs;uper BD non cadet in D. &longs;ed in H,
erit vtique HD minor AF. nam &longs;i HD ip&longs;i AF e&longs;&longs;et &ecedil;qualis,
quàm AF tunc non e&longs;&longs;et vltima multiplicatio. quare cùm &longs;it
DC maior AF; erit & HC ip&longs;a FA maior. &longs;i itaque fiat HK
æqualis AF; erit punctum K inter puncta DC. BK igitur
minor erit, quàm BC, & maior BD; eodemquè modo o
&longs;tendetur AF ip&longs;arum Bk AE communem e&longs;&longs;e men&longs;u
ram. & obid BK ip&longs;i AF commen&longs;urabilem exi&longs;tere. quod
facere oportebat.
mi.
Cùm autem verba &longs;equentis demon&longs;trationis aliquantu
lum &longs;int ob&longs;cura, vt vim demon&longs;trationis rectè petcipiamus,
hoc quoque theorema ex ijs, quæ ab Archimede hactenus de
mon&longs;trata &longs;unt, o&longs;tendemus. ad quod demon&longs;trandum com
muni notione indigemus, quam nos in no&longs;tro Mechanico
rum libro po&longs;uimus. Nempè.
Quæ eidem æquepondeiant, inter &longs;e æquè &longs;unt grauia.
PROPOSITIO.
Si commen&longs;urabiles magnitudines minorem habuerint
proportionem, quàm di&longs;tanti&ecedil; permutatim habent; vt &ecedil;que
ponderent, maiori opus erit magnitudine, quàm &longs;it ea, qu&ecedil;
ad alteram magnitudinem minorem proportionem habet.
Sint magnitudines AC commen&longs;urabiles, di&longs;tanti&ecedil; ve
rò &longs;int ED EF. minorem autem habeat pro-
dines ex di&longs;tantijs ED EF æqueponderent, maiori o
pus e&longs;&longs;e magnitudine in F, quàm &longs;it magnitudo A;
ita vt ip&longs;i C in D æqueponderare po&longs;&longs;it. fiat ED
ad EG, vt magnitudo A ad magnitudinem C.
Deindefiat EK æqualis EG. exponaturquè altera ma
gnitudo L ip&longs;i A &ecedil;qualis. Quoniam igitur minorem
habet proportionem A ad C, quàm ED ad EF, &
vt A ad C, ita ED ad EG; habebit ED ad
EG minorem proportionem, quàm ad EF. ac propterea
EF minor e&longs;t, quàm EG. quoniam ausem A ad C
e&longs;t, vt ED ad EG, commen&longs;urabiles magnitudines
AC ex di&longs;tantijs ED EG æqueponderabunt. Cùm
verò EK &longs;it æqualis EG, magnitudines AL æ
quales ex di&longs;tantis æqualibus EK EG &longs;imiliter æque
ponderabunt. At verò quoniam C in D æque
ponderat ip&longs;i A in G, &longs;imiliter L in K eidem A in
G &ecedil;queponderat; &ecedil;qualem habebit grauitatem C in D, vt
L in K. Itaque quoniam di&longs;tantia EG æqualis e&longs;t di&longs;tan
tiæ Ek, longitudo EK maior erit longitudine EF. ergo
magnitudines AL &ecedil;quales ex inæqualibus di&longs;tantijs EK
EF non &ecedil;queponderabunt. &longs;ed magnitudo L deor&longs;um ver
get. &longs;i igitur in F collocanda &longs;it magnitudo, quæ æquepon
deret ip&longs;i L in K, proculdubiò h&ecedil;c magnitudine A ma
ior exi&longs;tet. Inæqualia enim grauia, nempè L, & magnitu
do maior, quàm A, exinæqualibus di&longs;tantijs EK EF æ
queponderant, dummodo maius, hoc e&longs;t magnitudo maior,
quàm A, &longs;it in di&longs;tantia minori EF. minusverò, hoc e&longs;t ma
gnitudo L, &longs;it in minori EK. Quoniam itaque magnitudo
C in D e&longs;t &ecedil;quegrauis, vt L in K, magnitudo, quæ in F
ip&longs;i L in K æqueponderat, eadem quoque in F ip&longs;i C in D
æqueponderabit maior verò magnitudo, quàm &longs;it A, in F ip&longs;i
L in K æqueponderat, ergo maior magnitudo, quàm A in
F, ip&longs;i C in D æqueponderabit. quod demon&longs;trare opor
tebat.
ius.
His cognitis po&longs;&longs;umus ad Archimedis demon&longs;trationem
accedere.
PROPOSITIO. VII.
Si autem magnitudines fuerint incommen&longs;ura
biles, &longs;imiliter æqueponderabunt ex di&longs;tantijs per
mutatim eandem, atque magnitudines, propor
tionem habentibus.
DE EF. Habeat autem AB ad C proportionem eandem, quam di
stantia ED ad ip&longs;am EF. Dico,
rò ad D,
uitatis e&longs;&longs;e punctum E. &longs;i enim non æqueponderabit
vtSit maior
exce&longs;&longs;us HL; ita vt KH ad F, & C ad D &ecedil;queponderent.
HL,
e&longs;t.
Et quoniam minor e&longs;t kN quàm KM, minorem quoque
C. tota verò KM ad C e&longs;t, vt DE ad EF; ergo KN ad
C minorem habet proportionem; quàm DE ad EF.
niam igitur magnitudines AC,
les, & minorem habet proportionem A,
ad EF; non æqueponderabunt A C,
vt æqueponderent, oporter, vt in F maior &longs;it magnitudo,
quàm KN; ita vt ip&longs;i C in D æqueponderate po&longs;&longs;it. Ac
propterea cùm &longs;it kH adhuc minor, quàm KN, &longs;i igitur
KH ponatur ad F, & C ad D, nullo modo æqueponde
rabunt. quod tamen fieri non pote&longs;t. &longs;upponebatur enim eas
æqueponderare. Non igitur magnitudo minor, quàm tota
KM in F magnitudini C in D æqueponderat.
tem ratione, neque &longs;i C maior fuerit, quàm vt æqueponderet ip&longs;i A
hoc e&longs;t ip&longs;i KM. etenim grauiore
ad F. primùm auferatur ex C exce&longs;&longs;us, quo C grauior e&longs;t,
quàm KM, ita vt æqueponderet ip&longs;i KM. Deinde rur&longs;us
auferatur quædam magnitudo minor exce&longs;&longs;u, quo grauior
e&longs;t C, quàm kM, ita vt æqueponderent; re&longs;iduum verò &longs;it
ip&longs;i KM commen&longs;urabile, & c. &longs;imiliter o&longs;tendetur
magnitudinem ip&longs;a C minorem po&longs;itam ad D vllo modo
æqueponderare ip&longs;i KM ad F po&longs;itæ. Quare magnitudo
C ad D, kM verò ad F &ecedil;queponderant. Vnde &longs;equitur ma
gnitudinis ex vtri&longs;que magnitudinibus compo&longs;itæ centrum
grauitatis e&longs;&longs;e punctum E. ac propterea incommen&longs;urabiles
magnitudines AB C ex di&longs;tantiijs ED EF, quæ permutatim
eandem habent proportionem, vt magnitudines, æquepon
derare. quod demon&longs;trare oportebat.
mo proble
mate.
8.
denti.
ex prima
propo&longs;itio
ne.
SCHOLIVM.
In demon&longs;tratione occurrit ob&longs;eruandum, quòd &longs;i exce&longs;
&longs;us HL ita diuideret magnitudinem KM, vt re&longs;iduum KH
fuerit commen&longs;urabile ip&longs;i C; tunc ab&longs;que alia con&longs;tructio
ne, magnitudines commen&longs;urabiles KH C ex di&longs;tantijs DE
EF æqueponderarent; quod fieri non pote&longs;t. cùm minorem
&longs;upponitur KM ad C ita e&longs;&longs;e, vt ED ad EF. Archimed es ve
iò, vt demon&longs;tratio ab&longs;que di&longs;tinctione &longs;it vniuer&longs;alis, pr&ecedil;
cipit (exi&longs;tente KH ip&longs;i C commen&longs;urabili, &longs;iu e incommen
&longs;urabili) vt auferatur pars aliqua minor exce&longs;&longs;u HL, ut AL,
ita tamen, vt reliqua KN &longs;it commen&longs;urabilis ip&longs;i C. quod qui
dem fieri po&longs;&longs;e o&longs;ten&longs;um e&longs;t in proximo problemate. ex tota
enim magnitudine KM partem ab&longs;cindere po&longs;&longs;umus, vt KN
minorem quidem tota KM, maiorem verò KH, quæ ip&longs;i
C commen&longs;urabilis exi&longs;tat.
Cognita Archimedis demon&longs;tratione de incommen&longs;ura
bilibus magnitudinibus, idem alio quoque modo o&longs;tendere
po&longs;&longs;umus, applicando nempè diui&longs;ibilitatem, & commen&longs;ura
bilitatem non magnitudinibus, verùm di&longs;tantijs. hac autem
priùs demon&longs;trata propo&longs;itione.
PROPOSITIO.
Si commen&longs;urabiles di&longs;tanti&ecedil; maiorem habuerint pro
portionem, quàm magnitudines permutatim habent; vt
&ecedil;queponderent, maiori opus erit longitudine, quàm &longs;it
ea, ad quam altera longitudo maiorem habet proportio
nem.
Sint di&longs;tantiæ DE EH commen&longs;urabiles, magnitudines
verò &longs;int A C. habeatquè ED ad EH maiorem proportio
nem, quàm A ad C. Dico vt AC &ecedil;queponderent, maiori opus
do G, quæ ad C eandem habeat proportionem, quàm habet
DE ad EH. erunt vtique magnitudines GC inter &longs;e
&longs;urabiles. Deinde fiat EK æqualis EH, exponaturquè ma
gnitudo L ip&longs;i G æqualis. Quoniam igitur G ad C e&longs;t,
vt DE ad EH, ob commen&longs;urabilitatem æquepondera bunt
G in H, & C in D. &longs;imiliter æquepondera bunt magnitudi
nes æquales GL ex æqualibus di&longs;tantijs EK EH. Cùm igitur
C in D ip&longs;i G in H æqueponderet; L verò in K ip&longs;i quo
que G in H æqueponderet; eandem habebit grauitatem C
in D, ut L in K. Quoniam autem maiorem habet propor
tionem DE ad EH, quàm A ad C, & vt DE ad EH, ita e&longs;t
G ad C; maiorem habebit proportionem G ad C, quàm A
ad C. ergo maior e&longs;t G, quàm A. ac propterea magnitudo A
minor e&longs;t magnitudine L. po&longs;ita igitur magnitudine L in K,
& A in H, non æquepondera bunt; & vt &ecedil;queponderent, o
portet, vt A in longiori &longs;it di&longs;tantia, quàm &longs;it EH: In&ecedil;qualia
enim grauia LA ex in&ecedil;qualibus di&longs;tantijs &ecedil;queponderant,
maius quidem L in minori di&longs;tantia EK, minus verò graue
A in maiori, quàm &longs;it EK, hoc e&longs;t in maiori, quàm &longs;it EH.
Itaque cùm &longs;it C in D æquegrauis, vt L in k; longitudo,
quæ efficit, vt A æqueponderetip&longs;i L in K; eadem pror&longs;us
efficiet, vt A ip&longs;i C in D &ecedil;queponderare po&longs;&longs;it. A verò in
maiori di&longs;tantia, quàm EH, ip&longs;i L in K &ecedil;queponderat; ergo
in maiori di&longs;tantia, quàm EH, magnitudo A ip&longs;i C in D
&ecedil;queponderabit. quod demon&longs;trare oportebat.
tio &longs;upradi
cta.
Hoc demon&longs;trato Archimedis propo&longs;itionem de incom
men&longs;urabilibus magnitudinibus aliter o&longs;tendemus hoc
pacto.
ALITER.
Incommen&longs;urabiles magnitudines ex di&longs;tantijs permuta
tim eandem, atque magnitudines, proportionem habenti
bus; &ecedil;queponderant.
Sint incom
gnitudines AC,
di&longs;tantiæ verò
DE EF. &longs;itquè vt
A ad C, ita DE
ad EF. Dico A
in F, C verò in
D æqueponde
rare. Si autem (&longs;i fieri pote&longs;t) non æquepondera bunt;
tiæ DE EF aliter &longs;e&longs;e habere debebunt, vt magnitudines AC
&ecedil;queponderent. Quocirca vel longior e&longs;t EF, quàm opus
&longs;it, vel longior e&longs;t ED. &longs;it EF longior. &longs;itquè exce&longs;&longs;us GF, ita
vt po&longs;ita magnitudine A in G ip&longs;i C in D æqueponde
Fiat EH maior EG, minor verò EF. &longs;it autem EH
ip&longs;i ED commen&longs;urabilis. Quoniam igitur DE ad EH
maiorem habet proportionem, quàm ad EF; & vt DE ad
EF, ita e&longs;t A ad C; maiorem habebit proportionem DE
ad EH, quàm A ad C. &longs;untquè longitudines ED EH in
ter&longs;e commen&longs;urabiles; ergo magnitudo A in H ip&longs;i C in
e&longs;t longitudine, quàm &longs;it EH; ita vt A ip&longs;i C in D æque
ponderare po&longs;&longs;it. atque adeò cùm adhuc minor &longs;it EG, quàm
EH; magnitudo A in G magnitudini C in D nullo modo
æqueponderabit. quod fieri non pote&longs;t. &longs;upponebatur enim
A in G, & C in D &ecedil;queponderare. eademquè pror&longs;us ra
tione, &longs;i ED longior fuerit, quàm opus &longs;it, ita vt magnitu
dines æqueponderent, o&longs;tendetur
cto æqueponderare po&longs;&longs;e ip&longs;i A in F in minori di&longs;tantia,
quàm DE. Quare magnitudines in commen&longs;urabiles AC ex
di&longs;tantijs ED EF, quæ eandem permutatim habent propor
tionem, vt magnitudines, æqueponderant. quod demon&longs;tra
re oportebat.
ante
ius
ppo&longs;itione
In prioribus &longs;ermonibus ante quintam propo&longs;itionem ha
bitis, diximus propo&longs;itionum præcedentium demon&longs;tratio
nes planiores euadere, &longs;i intelligamus magnitudines eiu&longs;dem
e&longs;&longs;e &longs;peciei, & homogeneas. Quòd quidem &longs;i Archimedem
nulli
chimedis demon&longs;trationibus non &longs;it adhuc vniuer&longs;aliter de
mon&longs;tratum hoc pr&ecedil;cipuum fun damentum; nempè magni
tudines ex di&longs;tantijs permutatim
ip&longs;arum grauitates, &ecedil;queponderare; in hoc certè rationes ab
Archimede allatas, ip&longs;arum què demon&longs;trationum vim mini
mè percipiemus. Quapropter ea, quæ demon&longs;trauit, omni
bus magnitudinibus vniuer&longs;aliter competere ip&longs;um volui&longs;&longs;e
nullatenus e&longs;t dubitandum. Neque enim, vt perfectè, & vni
uer&longs;aliter&longs;ciamus, magnitudines çqueponderare ex di&longs;tantijs
permutatim proportionem habentibus, vt ip&longs;arum grauita
tes, alijs, quàm pr&ecedil;cedentibus propo&longs;itionibus indigemus.
In hoc enim fundamento demon&longs;trando minimè diminu
tus extitit Archimede. Nam &longs;i ad propo&longs;itiones ab ip&longs;o alla
tas, pr&ecedil;cipuèquè ad vim demon&longs;trationum re&longs;piciamus, &longs;iuè
magnitudines intelligantur eiuldem &longs;peciei, &longs;iue diuer&longs;&ecedil;, &longs;i
ue homogene&ecedil;, &longs;iue heterogene&ecedil;, &longs;iue plan&ecedil;, &longs;iue &longs;olid&ecedil;, &
h&ecedil; quidem, &longs;iue rectiline&ecedil;, &longs;iue quom odocunque mixt&ecedil;; ni
hilominus demon&longs;trationes idem pror&longs;us concludent, ita vt
Archimedes non de aliquibus magnitudimbus tantùm de
mon&longs;trationes attulerit; &longs;ed de omnibus pror&longs;us demon&longs;tra
uerit. In his enim Archimedes non ad magnitudines tantùm,
verùm ad magnitudinum grauitates poti&longs;&longs;imùm re&longs;pexit.
quandoquidem loco grauium magnitudines nominat; vt
po&longs;t quartam huius propo&longs;itionem adnotauimus. quod qui
dem facilè ex verbis ip&longs;ius rectè in tellectis apparere pote&longs;t.
in quærta propo&longs;itione cùm inquit,
æquales
e&longs;&longs;e grauitate. quod non &longs;olùm ex eius demon&longs;trationeli
quet, verùm etiam ex modo loquendi, quo v&longs;us e&longs;t Archime
des in alijs propo&longs;itionibus. In quinta enim propo&longs;itione,
qu&ecedil; eiu&longs;dem e&longs;t cum quarta ordinis, & natur&ecedil;, in quit;
ta, & magnitudines æqualem habuerint grauitatem.
ter po&longs;t quintam demon&longs;trationem bis quoquè eodem v
titur loquendi modo, nempè cùm adhuc proponit
grauitatem.
grauitates omnino re&longs;pexi&longs;&longs;e. ita vt quando Archimedes in
quit,
dines æqualem habuerint grauitatem.
ne inquit magnitudines &ecedil;queponderare ex di&longs;tantijs permu
tàtim proportionem habentibus, vt grauitates. ita ut cau&longs;a
huius æqueponderationis &longs;it (vt reuera e&longs;t) magnitudinum
grauitas. &
gnitudines æqueponderare ex di&longs;tantijs permutatim propor
tionem habentibus, vt magnitudines, & non dixit, vt grauita
tes; intelligendum tamen e&longs;t, ac &longs;i dixi&longs;&longs;et, eas &ecedil;quepondera
re, vt magnitudinum grauitates. h&ecedil;c enim &longs;eptima propo&longs;i
tio e&longs;t pars &longs;extæ propo&longs;itionis, vt iam pr&ecedil;fati fum^{9}; vnde &longs;i in
&longs;exta magnitudines &ecedil;queponderant ob earum grauitatem, ob
eandem quoque cau&longs;am & in hac &longs;eptima æqueponderare de
bent. Pr&ecedil;terea in &longs;equenti etiam propo&longs;itione dum proponit
o&longs;tendere quam proportionem habere debent &longs;ectiones line&ecedil;
intercentra grauitatum diui&longs;&ecedil; magnitudinis
autem deinceps exponens,
eam habere proportionem, quàm grauitas ad grauitatem ha
bet; &longs;ed horum loco inquit, quàm magnitudo ad magnitudi
nem. ex quibus omnibus clarè per&longs;picitur, quòd quando Ar
chimedes magnitudines nominat, omnino magnitudinum
grauitates vult intelligere.
Ad eorum autem
maquè propo&longs;itione,
e&longs;t, quòd in &longs;exta propo&longs;itione pro magnitudinibus commen
&longs;urabilibus intelligere oportet magnitudines grauitate com
men&longs;urabiles; ita nempe, vt numeris exprimi po&longs;&longs;int; quam
quam non &longs;int mole, & magnitudine commen&longs;urabiles, vt
in figura &longs;ext&ecedil; propo&longs;itionis magnitudo A ponderet exempli
gratia vt XVI. B verò vt VIII.
cuique parti OPQR, quæ quidem, & &longs;i non &longs;int magnitu
dine inter &longs;e &ecedil;quales, &longs;ufficit, vt &longs;int æquegraues: veluti magni
tudines quoque STVX inter &longs;e,
graues; ita ut vnaquæque ponderet, vt IIII. veluti etiam par
tes ip&longs;ius B, & vnaquæque ZM. hi&longs;què ita po&longs;itis
tio rectè concludet.
In hacverò &longs;eptima Archimedis propo&longs;itione &longs;imiliter
telligantur magnitudines kMC incommen&longs;urabiles graui
tate, vt in eius figura grauitas ip&longs;ius C ponderet, vt XII. gra
uitas verò ip&longs;ius KM maior &longs;it, quàm XX. ita vth&ecedil; graui
tates &longs;int in
commen&longs;urabiles. auferaturquè grauitas exce&longs;&longs;us
HL, quæ &longs;it vt IIII. ita vt quæ relinquiturgrauitas, ip&longs;ius
pè
uitati ip&longs;ius C, quæ e&longs;t XII, in D po&longs;itæ æqueponderet,
Auferatur deinde NL minor exce&longs;&longs;u HL; cuius quidem gra
uitas &longs;it maior, quàm II. ita vt grauitas re&longs;idui KN, quæ
nimirum &longs;it XVIII, &longs;it commen&longs;urabilis grauitati
XII. ip&longs;ius C. &
vel
quæ quidem omnia in &longs;equenti quoque propo&longs;itione
derandaVnde per&longs;picuum e&longs;t has Archime dis pro
po&longs;itiones, ac demon&longs;trationes vniuer&longs;ali&longs;&longs;imas e&longs;&longs;e, arque o
mnibus, & quibu&longs;cunque magnitudinibus conuenientes.
guram
mæ propo&longs;i
tionis Ar
chimedis.
Iacto hoc pr&ecedil;cipuo, ac pr&ecedil;&longs;tanti&longs;&longs;imo mechanico funda
mento; in &longs;equenti propo&longs;itione colligit ex hoc Archimedes,
quomodo &longs;e habent centra grauitatis magnitudinis diui&longs;æ.
PROPOSITIO. VIII.
Si ab aliqua magnitudine magnitudo aufera
tur; quæ non habeat idem centrum cum tota; re
liquæ magnitudinis centrum grauitatis e&longs;t in re
cta linea, quæ coniungit centra grauitatum to tius
magnitudinis, & ablatæ, ad eam partem produ
cta, vbi e&longs;t centrum to tius magnitudinis, ita vt a&longs;
&longs;umpta aliqua ex producta, quæ coniungit
prædicta eandem habeat proportionem ad eam,
quæ e&longs;t inter centra, quam habet grauitas magni
tudinis ablatæ ad grauitatem re&longs;iduæ, centrum e
rit terminus a&longs;&longs;umptæ.
què ex AB magnitudo AD; cuius centrum grauitatis &longs;it E. coniuncta
verò EC, &
dem habeat proportionem, quam habet magnitudo AD ad DG. osten
dendum est, magnitudinis DG centrumgrauitatis e&longs;&longs;e punctum F.
&longs;it autem; &longs;ed, &longs;i fieri potest, &longs;it punctum H. Quoniam igitur magnitudi
nis AD centrum grauitatis est punctum E; magnitudinis verò DG
e&longs;t punctum H; magnitudinis ex vtri&longs;que magnitudinibus AD DG,
permutatim eandem Quare non
etæ.
ad DG; ita
cundùm proportionem ip&longs;ius AD ad DG; non terminabit
diui&longs;io ad punctum C. cùm &longs;it impo&longs;&longs;ibile eandem habere
proportionem FC ad CE, quam. HC ad eandem CE. di
ui&longs;io igitur ad aliud terminabitur punctum, vt K; ita vt HK
ad KE &longs;it, vt AD ad DG. vnde &longs;equitur punctum K cen
trum e&longs;&longs;e grauitatis magnitudinis ex AD DG compo&longs;itæ.
&longs;itæ; hoc est ip&longs;ius AB. e&longs;t autem; &longs;uppo&longs;itum e&longs;t enim
go neque punctum H centrum est grauitatis magnitudinis DG.
igitur punctum F; quod quidem e&longs;t terminus product&ecedil; line&ecedil;
CF; quæ eandam habet proportionem ad lineam CE inter
centra exi&longs;tentem; quam habet grauitas magnitudinis AD
ad grauitatem ip&longs;ius DG. quod demon&longs;trare oportebat.
dentibus.
dentibus.
SCHOLIVM.
In hac demon&longs;tratione intelligendum e&longs;t etiam punctum
H e&longs;&longs;e po&longs;&longs;e extra lineam EF, ita vt EFH non &longs;itirecta linea.
quòd &longs;i H non e&longs;&longs;et in linea EF, idem &longs;equi ab&longs;urdum adeò
per&longs;picuum e&longs;t; vt nec demon&longs;tratione egeat. Quoniam &longs;i in
telligatur H extra lineam EF; iuncta EH, & ita diui&longs;a intel
ligatur, vt ip&longs;ius partes permutatim grauitatibus magnitudi
num AD DG re&longs;pondeant; e&longs;&longs;et vtique hoc punctum
tum&longs;iquidem e&longs;t punctum C, vt
&longs;uppo&longs;itum fuit. Vnde neque illud punctum H ip&longs;ius DG
trum grauitatis exi&longs;teret.
Hic e&longs;t terminus prim&ecedil; partis principalis, in qua Archime
des (vt initio dixim^{9}) de magnitudinib^{9}, & degrauibus in
communi pertractauit; quandoquidem propo&longs;itiones, ac de
mon&longs;trationes tam planis, quàm &longs;olidis quibu&longs;cunque &longs;unt
accomodatæ; vt manife&longs;tum fecimus.
Nunc ita que &longs;e conuertit Archimedes ad
tra grauitatis planorum. primùm què perquirit centrum gra
uitatis parallelogrammorum; o&longs;ten detquè centrum grauitatis
cuiu&longs;libet parallelogrammi e&longs;&longs;e in recta linea, quæ coniungit
oppo&longs;ita latera bifariam diui&longs;a. ob cuius intelligentiam hæc
priùs lemmata in vnum collecta noui&longs;&longs;e erit valdè vtile.
LEMMA.
Sit parallelogrammum ABCD, cuius oppo&longs;ita latera AB
CD &longs;int bifariam diui&longs;a in EF. connectaturquè EF, quæ ni
mirum æquidi&longs;tans eritip&longs;is AC BD. Deinde diuidatur v
naquæque AE EB in partes numero pares, & inuicem &ecedil;qua
les; vt in AG GE; & EH HB.
EF &ecedil;quidi&longs;tantes. &longs;it verò centrum grauitatis ip&longs;ius AK pun
ctum M. ipfius verò GF punctum N, & ip&longs;ius EL pun
ctum O deniquè ip&longs;ius HD punctum P. Dico primùm
cta MNOP e&longs;&longs;e in linea recta. deinde lineas MN NO OP
inter centra exi&longs;tentes inter &longs;e æquales e&longs;&longs;e. Denique centrum
grauitatis parallelogrammi AD e&longs;&longs;e in linea NO, qu&ecedil; con
iungit centra grauitatis &longs;patiorum mediorum; parallelogram
morum &longs;cilicet GF EL.
SNT. erunt vtique AQRG, & GSTE parallelogramma.
Quoniam igitur parallelogramma AK GF in æqualibus
&longs;untba&longs;ibus AG GE, & in ij&longs;dem parallelis; erunt AK GF
inter &longs;e &ecedil;qualia. & quoniam AC GK EF &longs;unt
erit angulus CAG ip&longs;i KGE &ecedil;qualis, & KGA ip&longs;i FEG
æqualis; & horum oppo&longs;iti inter &longs;e &longs;unt &ecedil;quales; ergo
logrammum GF ip&longs;i AK &ecedil;quale, & &longs;imile exi&longs;tit. Itaque
&longs;i GF collocetur&longs;uper AK, rectè congruet: eruntquè paral
lelogramma inuicen coaptata. line&ecedil;què GE AG, GK AC, &
reliquæ coaptatæ erunt. quare eorum centra grauitatis
cem coaptata erunt. hoc e&longs;t N erit in puncto M. Quoniam
autem à punctis MN (quod nunc intelligitur vnum tantum
e&longs;&longs;e punctum) ductæ fuerunt ST QR ip&longs;i AGE æquidi
&longs;tantes, linea ST coaptabitur cum QR, quippe cùm ambæ
hæ lineæ ab vno puncto prodeuntes ip&longs;i AG &ecedil;quidi&longs;tantes
e&longs;&longs;e debeant. punctum igitur S in Q, & T in R coaptabi
tur. eritquè QM ip&longs;i SN &ecedil;qualis, & MR ip&longs;i NT. ac pro
pterea linea GS parallelogrammi GT erit coaptata in
& ET coaptata eritin GR parallelogrammi AR. Vnde e
rit AQ &ecedil;qualis GS, cùm &longs;int coaptatæ; & GR ip&longs;i ET &ecedil;
qualis; cùm &longs;int quoque coaptat&ecedil;. Quocirca quoniam
rallelogramma AR GT &longs;unt inuicem coaptata, paral
lelogrammorumquè oppo&longs;ita latera &longs;unt inter &longs;e &ecedil;qualia,
AQ GS GR ET inter &longs;e &ecedil;qualia. Nunc autem
parallelogramma AK GF non ampliùs coaptata. &
line&ecedil; QMR, & SNT &longs;untip&longs;i AGE parallel&ecedil;; & AQ GR,
GS ET, inter &longs;e &longs;untæquales, & &ecedil;quidi&longs;tantes; puncta RS in
vnum coincident punctum. eritquè QST linea recta. ex qui
bus patet, rectam
ip&longs;i AGE æquidi&longs;tantem exi&longs;tere. eodemquè modo o&longs;tende
tur rectas lineas, quæ coniungunt grauitatis centra NO, cen
traquè OP, ip&longs;i AB Vnde &longs;equitur lineam
MNOP rectam e&longs;&longs;e. Quare primùm con&longs;tat grauitatis
in recta linea exi&longs;tere.
ius.
Quoniam autem o&longs;ten&longs;um e&longs;t QM æqualem e&longs;&longs;e ip&longs;i SN,
& MR ip&longs;i NT, eodem quoque modo o&longs;tendetur OT &ecedil;qua-
demquè TN SM æquales, erit ON ip&longs;i NM æqualis. ea
demquè ratione o&longs;tendetur OP &ecedil;qualem e&longs;&longs;e ip&longs;i ON. vn
de colligitur lineas MN NO OP inter centra exi&longs;tentes in
rer&longs;e &ecedil;quales e&longs;&longs;e.
Po&longs;tremò quoniam parallelogramma AK GF EL HD
&longs;unt inuicem æqualia, & numero paria, centraquè grauitatis
&longs;unt in recta linea po&longs;ita. line&ecedil;què MN NO OP inter cen
tra &longs;unt &ecedil;quales, magnitudinis ex omnibus AK GF EL HD
MP bifariam diui&longs;a. Et quoniam MN e&longs;t æqualis ip&longs;i OP,
punctum, quod bifariam diuidit MP cadet in linea NO.
centrum ergo grauitatis omnium magnitudinum AK GF
EL HD, hoc e&longs;t parallelogrammi AD e&longs;t in linea NO, qu&ecedil;
coniungit centra &longs;patiorum mediorum GF EL. qu&ecedil;
omnia o&longs;tendere oportebat.
quin
tæhuius.
Quoniam autem centrum grauitatis
e&longs;t in linea NO, & in linea MP bifariam diui&longs;a; non repu
gnare videtur, quin inferri po&longs;&longs;it, hoc centrum e&longs;&longs;e in puncto
T, in linea EF exi&longs;tente. Quòd tamen fal&longs;um e&longs;t. nam po&longs;
&longs;et quidem concludi centru e&longs;&longs;e in medio line&ecedil; NO (
e&longs;t in medio line&ecedil; MP, vt
&longs;tratione enim o&longs;tenditur NS æqualem e&longs;&longs;e ip&longs;i TO. at verò
NT &ecedil;qualem e&longs;&longs;e ip&longs;i TO, nullo modo demon&longs;trari pote&longs;t;
ni&longs;i &longs;upponeremus centra grauitatis MNOP in parallelogra
mis ita &longs;e habere, vt MQ MR, & MR RN, & RN NT &
NT TO, &c. inter &longs;e &ecedil;quales e&longs;&longs;ent. quod nullo modo &longs;up
poni pote&longs;t nam hoc modo centra grauitatis parallelogram
morum AK GF &c. e&longs;&longs;entin lineis, qu&ecedil; bifariam &longs;ecant op
po&longs;ita latera. e&longs;&longs;ent quippè in lineis à punctis MN OP du
ctisip&longs;is AC GK EF &c. æquidiftantibus, quæ oppo&longs;ita la
tera AG CK, GE KF, EH FL, &c. bifariam &longs;ecarent. quod
e&longs;t id, quod Archimedes demon&longs;trare in quod
quidem in cau&longs;a e&longs;t, vt demon&longs;tratione ad impo&longs;&longs;ibile id de
ducat. &longs;uppo&longs;uimus autem (vt pare&longs;t) parallelogramma cen-
rallelogramma exi&longs;tere, quoniam parallelogramma &longs;unt
guræ ad ea&longs;dem partes concauæ. quod quidem eodem modo
ab Archimede in &longs;equenti &longs;upponitur.
ius.
PROPOSITIO. IX.
Omnis parallelogrammi centrum grauitatis
e&longs;t in recta linea, quæ oppo&longs;ita latera parallelo
grammi bifariam diui&longs;a coniungit.
tera AB CD. Dico parallelogrammi ABCD centrum grauitatis e&longs;&longs;e
ad lineam EF
&longs;emper bifariam
&longs;emper fiat, tandem
ip&longs;a HI. Diuidaturquè vtraque AE EB in partes
LE GO OB
diui&longs;a e&longs;t EB in partes &longs;emper &ecedil;quales.
ctis ducantur
diui&longs;um enim erit totum parallelogrammum in parallelogramma æqualia
& &longs;imiliaip&longs;i
EL LM MN NA KG GO OB ip&longs;i KE æquales,
grammaquè in ij&longs;dem &longs;int parallelis AB CD con&longs;tituta;
erunt parallelogramma æqualia. &longs;imilia verò, quoniam
&longs;unt &ecedil;quiangula.
nient.
&ecedil;quales erunt parallelogramma in ED numero paria. ac per
con&longs;equens & qu&ecedil; &longs;unt in EC numero paria. vnde & qu&ecedil; sut
in toto AD numero paria
nes æquidi&longs;tantium laterum æquales ip&longs;i KF numero pares,
omnibus compo&longs;itæ centrum grauitatis erit in recta linea, quæ coniungit
centra grauitatis mediorum &longs;patiorum,
cet LF KF.
e&longs;&longs;e centrum grauitatis omnium magnitudinum, hoc e&longs;t pa
rallelogrammi AD,
etenim cùm &longs;it EK minor HI, linea KS ip&longs;i EF
lineam HI ip&longs;i EK æquidi&longs;tantem &longs;ecabit, quippè quæ re
linquet punctum H extra figuram KF, ac per con&longs;equens ex
tra media parallelogramma LF KF. quare punctum H non
e&longs;t centrum grauitatis parallelogrammi AD, vt &longs;upponeba
tur.
cta linea EF.
pr&ecedil;cedenti
SCHOLIVM.
linea &longs;unt constituta,
tw=n me)swn auta/ te i)/sa e)nti/
uiNam &longs;i Archimedes di
xit omnia parallelogramma e&longs;&longs;e inter &longs;e, & &ecedil;qualia, & &longs;imilia;
non opus e&longs;t addere, media LF ES e&longs;&longs;e inter &longs;e &ecedil;qualia, &
qu&ecedil; ab his &longs;unrad vtramque partem, vt MR KT, NQ GV,
AP OD, e&longs;&longs;e inter &longs;e æqualia; cum omnia (vt dictum e&longs;t) &longs;int
&ecedil;qualia. quare verba h&ecedil;c (meo quidem iudicio) delenda &longs;unt.
demon&longs;trationes enim mathematic&ecedil; nullum admittunt &longs;u
perfluum. & Archim edes non tantùm &longs;uperfluus, quin potiùs
ob cius breuitatem diminutus ferè videatur.
Ex hac nona propo&longs;itione duo corolloria elicere po&longs;&longs;um^{9};
quæ quidem tanquam valde nota fortaf&longs;e videtur omi&longs;i&longs;&longs;e Ar
chimedes. quamuis
COROLLARIVM. I.
Ex hoc per&longs;picuum e&longs;t cuiu&longs;libet parallelogrammi
grauitatis e&longs;&longs;e punctum, in quo coincidunt rectæ lineæ, quæ
oppo&longs;ita latera bifariam &longs;ecant.
Nam (vt Archimedes etiam &longs;e
quenti demon&longs;tratione inquit)
&longs;i parallelogrammi ABCD line&ecedil;
EF GH bifariam diuident late
ra oppo&longs;ita AB DC, & AD BC.
patetin EF centrum e&longs;&longs;e graui
tatis parallelogrammi AC. &longs;imi
liter con&longs;tat idem centrum e&longs;&longs;e
in linea GH, quæ oppo&longs;ita latera AD BC bifariam &longs;ecat. e
ritigitur in K, vbi EF GH &longs;einuicem &longs;ecant.
COROLLARIVM. II.
Ex hoc patet etiam, cuiu&longs;libet parallelogrammi
uitatis e&longs;&longs;e in medio rectæ line&ecedil;, quæ bifariam oppo&longs;ita latera
di&longs;pe&longs;cit.
Cùm enim o&longs;ten&longs;um &longs;it centrum grauitatis parallelogram
mi AC e&longs;&longs;e punctum K. & ob parallelogrammum EH e&longs;t
EK æqualis BH. propter parallelogrammum verò KC
linea KF e&longs;t æqualis HC. &longs;untquè BH HC æqua
les. erit EK ip&longs;i KF æqualis. punctum ergo K e&longs;tin medio
rectæ line&ecedil; EF, quæ oppo&longs;ita latera AB DC bifariam diui
dit.
GH, quæ bifariam &longs;ecat oppo&longs;ita latera AD BC.
In &longs;equenti Archimedes adhuc per&longs;i&longs;tit in inuentione cen
tri grauitatis parallelogrammorum, alia tamen methodo.
nam hoc perip&longs;orum parallelogrammorum diametros duo
bus modis a&longs;&longs;equitur.
PROPOSITIO. X.
Omnis parallelogrammi centrum grauitatis
e&longs;t punctum, in quo diametri coincidunt.
ABCD. & in ip&longs;o &longs;it li
nea EF
què &longs;it KL
bifariam. conueniant
què EF kL in H.
vtique parallelogrammi
tis in linea EF. hoc enim
o&longs;ten&longs;um e&longs;t. eadem verò de cau&longs;a
etiam in linea
trum grauitatis existit. Verùm in puncio H diametri parallelogram
mi concurrunt.
lineæ AE EB EF FD inter &longs;e &longs;unt &ecedil;quales. &longs;imiliter quoque
AK KC BL LD inter &longs;e &ecedil;quales; erit EH ip&longs;i HF &ecedil;qua
lis, cùm &longs;int ip&longs;is BL LD &ecedil;quales. duæ igitur AE EH dua
ac
propterea angulus EHA angulo FHD æqualis. cùm igitur
&longs;it EHF recta linea, eruntangnli EHA FHD adverticem,
& obid AHD recta exi&longs;tit linea. ac per con&longs;equens diame
ter parallelogrammi AD. pariquè ratione o&longs;tendetur BHC
rectam e&longs;&longs;e lineam. ex quibus patet in puncto H
metrum conuenire. centrum igitur grauitatis parallelogram
mi AD e&longs;t
stratume&longs;t, quod propo&longs;itum fuit.
ALITER.
que o&longs;tendetur. &longs;it paralle
ip&longs;ius verò diameter &longs;it
ABD BDC
ter&longs;e æqualia, & &longs;imilia.
quare triangulis inuicem
coaptatis; centra quoque
grauitatis ip&longs;orum inuicem coaptabuntur. Sit autem trianguli ABD cen
nectaturquè EH, & producatur. &longs;umaturquè FH æqualisip&longs;i HE.
Itaque coaptato triangulo ABD cumtriangulo B DC, po&longs;itoquè latere
AB in DC,
BC;
DB coaptatur, B &longs;cilicet in D, & D in B. quia verò pun
ctum H &longs;ibi ip&longs;i coaptatur, cùm fitmedium line&ecedil; BD. & an
guli EHD FHB ad verticem &longs;unt æquales; lineaquè EH e&longs;t
ip&longs;i HF &ecedil;qualis;
ctum
e&longs;t grauitatis trianguli ABD idem punctum E
tiam grauitatis trianguli B DCergo punctum F
trum
triangula non ampliùs coaptata.
tatis trianguli ABD e&longs;t punctum E, ip&longs;ius verò DBC est punctum F,
triangulaquè ABD DBC &longs;unt &ecedil;qualia,
tri&longs;que triangulis compo&longs;itQuoniam autem dia
metri cuiu&longs;libet parallelogrammi &longs;e&longs;e bifariam di&longs;pe&longs;cunt, e
rit punctum H, vbi diametri parallelogrammi ABCD con
currunt. ergo punctum H, in quo diametri coincidunt; ip&longs;ius
ABCD centrum grauitatis exi&longs;tit. quod demon&longs;trare opor
rebat.
mi.
ius.
SCHOLIVM.
Cognito centro grauitatis cuiu&longs;libet parallelogrammi,
vult Archimedes o&longs;ten dere centrum grauitatis triangulorum.
& quoniam in hac po&longs;trema demon&longs;tratione a&longs;&longs;ump&longs;it cen
trum grauitatis trianguli ABD e&longs;&longs;e punctum E, videtur or
dinem peruerti&longs;&longs;e, & per ignotiora doctrinam tradidi&longs;&longs;e; cùm
non &longs;it adhuc o&longs;ten&longs;um, in quo &longs;itu dictum centrum in
gulisquod tamen &longs;i rectè perpendamus, non ita &longs;e
habet. Nam vis demon&longs;trationis e&longs;t in hoc con&longs;tituta, vt
&longs;upponatur triangulum habere centrum grauitatis, idquè tan
te&longs;t. cùm triangulum &longs;it figura ad ea&longs;dem partes concaua. ne
que enim refert, &longs;iuè centrum &longs;it in E, &longs;iuè in alio &longs;itu, dum
modo intra triangulum exi&longs;tat. demon&longs;tratio enim
do &longs;emper concludet punctum H centrum e&longs;&longs;e grauitatis pa
rallelogrammi AC, quod idem ob&longs;eruandum e&longs;t in
alijs demon&longs;trationibus. vt in &longs;ecunda demon&longs;tratione deci
mæ tertiæ, hui^{9} & in prima &longs;ecun dilibri. Antequam
chimedes centrum grauitatis triangulorum o&longs;tendat, nonnul
las pr&ecedil;mittit propo&longs;itiones.
ius.
PROPOSITIO. XI.
Si duo triangula inter &longs;e &longs;imilia fuerint, & in i
p&longs;is &longs;int puncta ad triangula &longs;imiliter po&longs;ita & alre
rum punctum trianguli, in quo e&longs;t, centrum fue
rit grauitatis, & alterum punctum trianguli, in
quo e&longs;t, centrum grauitatis exi&longs;tet.
Dicimus quidem punctain &longs;imilibus figuris e&longs;&longs;e
&longs;imiliter po&longs;ita, è quibus ad æquales angulos du
ctæ rectæ lineæ, æqual es efficiunt angulos ad ho
mologalatera. Vt dictum fuit in &longs;eptimo po&longs;tulato.
AB ad DE, & BC ad EF. & in præfatis triangulis ABC DEF
&longs;int puncta HN &longs;imiliter po&longs;ita &longs;itquè punctum H centrum grauitatis
trianguli ABC. Dico & punctum N centrum e&longs;&longs;e grauitatis trianguli
DEF. non &longs;it quidem, &longs;ed, &longs;i fieripote&longs;t, &longs;it punctum G centrum grauita
tis trianguli DEF.
DG EG FG. Quoniamigitur &longs;imile e&longs;t triangulum ABC triangulo
DEF, &
lium autem figurarum centra grauitatum &longs;unt &longs;imiliter po&longs;ita; ita vt
ab ip&longs;is ad &ecedil;quales angulos ductæ rectæ line&ecedil;
angulos ad homologa latera, vnumquemquè vnicuiquè; erit angulus
GDE ip&longs;i HAB aqualis. at verò anguius HAB aqualis est angulo
EDN, cùm &longs;int puncta HN &longs;imiliter po&longs;ita: angulus igitur EDG
angulo EDN æqualis existit. maior minori quòd fierinon potest. Non
igitur punctum G centrum e&longs;t grauitatis trianguli DEF. Quare e&longs;t
punctum N. quod demonstrare oportebat.
huius.
SCHOLIVM.
In hac propo&longs;itione &longs;upponit Archimedes dari po&longs;&longs;e pun
cta in triangulis &longs;imilib^{9} &longs;imiliter po&longs;ita, qd
o&longs;tendimus in &longs;cholijs &longs;eptimi po&longs;tulati. Præterea idem vide
tur Archimedes in triangulis demon&longs;trare, quod in &longs;exto po
&longs;tulato vniuer&longs;aliter in figuris &longs;uppo&longs;uit. Nam &longs;i centra gra
uitatis &longs;upponuntur in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita;
& in &longs;imilibus triangulis quoque erunt &longs;imiliter po&longs;ita. In
ter h&ecedil;c tamen maxima e&longs;t differen tia, nam in po&longs;tulato inquit,
centra grauitatum in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; cu
ius quidem conuer&longs;um, nempè puncta in &longs;imilibus figuris &longs;i
militer po&longs;ita e&longs;&longs;e ip&longs;arum centra grauitatis, e&longs;t falium. quod
e&longs;t quidem manife&longs;tum ab&longs;que alio exemplo. ac propterea
Archimedes hoc in loco inquit, &longs;i duo erunt pun&longs;ta in &longs;imi
libus triangulis &longs;imiliter po&longs;ita, & alterum ip&longs;orum fuerit
trum& alterum quoque
Vnde propo&longs;itio h&ecedil;c potiùs e&longs;t conuer&longs;a po&longs;tulati, quàm
eadem.
Ob demon&longs;trationem autem noui&longs;&longs;e oportet, quòd &longs;i pun
ctum G fuerit in linea DN, tuncanguli EDG EDN e&longs;&longs;entin
ter&longs;e &ecedil;quales, ac propterea demon&longs;tratio nihil ab&longs;urdi conclu
deret. In hoc autem ca&longs;u o&longs;tendendum e&longs;&longs;et, angulum EFG
ip&longs;i EFN &ecedil;qualem e&longs;&longs;e, vel FEG ip&longs;i FEN. quæ quidem eo
dem pror&longs;us modo o&longs;tendentur. comparando nempè angu
los EFG EFN angulo BCH; angulos verò FEG FEN ip&longs;i
CBH. Quòd &longs;i G fuerit in alio &longs;itu, vtin triangulo EDN,
tuncanguli FDG FDN o&longs;tendentur &ecedil;quales. & ita in alijs
ca&longs;ibus, vbicunque &longs;cilicet fuerit punctum G, &longs;emper ali
quod inuenietur huiu&longs;modi ab&longs;urdum. quæ quidem omni
nò fieri non po&longs;&longs;unt.
PROPOSITIO. XII.
Si duo triangula &longs;imilia fuerint, alterius verò
trianguli centrum grauitatis in rectalinea fuerit,
quæ &longs;it ab aliquo angulo ad dimidiam ba&longs;im du
cta; & alrerius trianguli centrum grauitatis erit in
linea &longs;imiliter ducta.
AB ad DE, & BC ad FE. Diui&longs;aquè AC bifariam in G, iunga
tur BG. centrum verò grauitatis trianguli ABC &longs;it punctum H in li
nea BG. Dico centrum grauitatis trianguli EDF e&longs;&longs;e in recta linea &longs;i
militer ducta. &longs;ecetur DF bifariam in puncto M. & iungatur EM.
& vt BG ad BH, ita fiat ME ad EN. connectanturquè AH
HC, DN NF. Quoniam enim
ad ED, vt AG ad DM.
ABC DEF &longs;imilitudinem angulus BAC angulo EDF e&longs;t &ecedil;
qualis. & vt AB ad DE, ita AG ad DM;
AG, vt DE ad DM; erit erit
igitur angul^{9} AGB angulo
&longs;i DEM æqualis quare
ita BG ad EM; & pmu
tado AB ad BG, vt DE
ad EM.
æquales, erit triangulun. ABH triangulo DEN &longs;imile. qua
re anguli &longs;unt inter &longs;e æquales,
proportionalia &longs;i autem hoc, angulus BAH angulo EDN est æqualis.
Vnde & reliquus angulus HAC angulo NDF æquolis exi&longs;tit.
dem totius BAC ip&longs;i EDF e&longs;t æqualis.
& angulas HCG angulo NFM
æqualis, o&longs;ten&longs;um est autem angulum ABH ip&longs;i DEM aqualem e&longs;&longs;e.
ob &longs;imilitudinem autem riangulorum ABC DEF totus an
ip&longs;i NEF æqualis exi&longs;tit. Porrò ex his omnibus patet puncta HN ad
homologa latera e&longs;&longs;e &longs;imiliter po&longs;ita, &
cere. Cùm igitur puncta HN &longs;int &longs;imiliter po&longs;ita; & punctum H cen
trum e&longs;t grauitatis trianguli ABC, & puncium N trianguli DEF
trum
nea BG ab angulo ad dimidiam ba&longs;im ducta. & alterum gra
uitatis centrum N in linea EM &longs;imiliter ducta reperitur.
quod demon&longs;trare oportebat.
ius.
SCHOLIVM.
In &longs;equenti Archimedes o&longs;tendet, in qua linea reperitur
trum grauita tis cuiu&longs;libet trianguli. quod quidem duobus a&longs;
&longs;equitur medijs. Diligenter autem omnia &longs;unt con&longs;ideranda;
quoniam in hoc con&longs;i&longs;tit tota per&longs;crutatio centri grauitatis
triangulorum. Quapropter vt prior demon&longs;tratio appareat
per&longs;picua, h&ecedil;c antea dem on &longs;trabimus.
LEMMA. I.
Æquidi&longs;tantes lineæ lineas in eadem proportione di&longs;pe
&longs;cunt.
Sintline&ecedil; AB CD, quas &longs;ecent æqui
di&longs;tantes lineæ AC EF BD. Dico ita e&longs;
&longs;e BE ad EA, vt DF ad FC. primùm
quidem AB CD vel &longs;unt &ecedil;quidi&longs;tantes,
vel minùs. &longs;i &longs;unt æquidi&longs;tantes, iam habe
tur in tentum. Nam BE erit æqualis DF,
& EA ip&longs;i FC. vnde &longs;equitur ita e&longs;&longs;e BE
ad EA, vt DF ad FC.
Si verò AB CD non fuerint æquidi
&longs;tantes, concurrantin G, vt in &longs;ecunda fi
gura, & quoniam BD EF &longs;unt
&longs;tantes, erit GB ad BE, vt GD ad DF.
&
conuertendoquè BE ad EG, vt DF ad
FG, rur&longs;us quoniam EF AC &longs;unt æquidi
&longs;tantes; erit GE ad EA, vt GF ad FC, e
ritigitur ex æquali BE ad EA, vt DF ad FC.
Secent verò &longs;e&longs;e lineæ AB CD, vt in tertia figura, ob
litudinem triangulorum BGD EGF, it a erit BG ad GE, vt
DG ad GF. & componendo BE ad EG, vt DF ad FG. e&longs;t
verò GE ad EA, vt GF ad FC. ergo ex æquali BE ad EA
erit, vt DF ad FC. quod demon&longs;trare oportebat.
LEMMA. II.
Sit A ad B, vt C ad D; rur&longs;us A ad E &longs;it, vt C ad F.
Dico primùm A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF.
erit BE ad A, vt DF ad C. ac denique conuertendo A e
rit ad BE, vt C ad DF.
Si verò fuerint quattuor magnitudines; vt adhue A (in ea
dem figura) ad G &longs;it, vt C ad H. &longs;imili
ter o&longs;tendetur A ad omnes BEG &longs;imul
&longs;umptas ita e&longs;&longs;e, vt C ad omnes &longs;imul
DFH. &longs;umendo vt in &longs;ecunda figura BE
pro vna tan ùm magnitudine, & DF pro
alia; eruntque ex vtraque parte tres
magnitudines; eritquè A ad BE &longs;imul,
vt C ad DF &longs;imul, vt o&longs;ten&longs;um e&longs;t, dein
de A ad G e&longs;t, vt C ad H, erit igitur
A ad BEG &longs;imul, vt C ad DFH.
Pariquè ratione &longs;i quinque fuerint magnitudines, eodem
modo tres mediæ
tudines. & &longs;ic in infinitum. quod demon&longs;trare oportebat.
COROLLARIVM.
Ex hoc elici pote&longs;t. quòd &longs;i fuerint quotcun que magnitudi
nes proportionales; & ali&ecedil; ip&longs;is numero æquales, & in eadem
proportione, vt&longs;cilicet &longs;it (vt in prima figura) A ad B, vt C
ad D, B verò ad E, vt D ad F. deinde vt E ad G, &longs;ic F
ad H, & ita deinceps, &longs;i plures fuerint magnitudines, &longs;i
militer erit A ad omnes BEG &longs;imul &longs;umptas, vt C ad om
nes &longs;imul DFH.
Primùm quidem A e&longs;t ad B, vt C ad D. & quoniam ma
gnitudines &longs;unt proportionales, ex &ecedil;quali erit A ad E, vt C
ad F. &longs;imiliter A ad G, vt C ad H. Ex quibus &longs;equitur
A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF. A verò ad omnes
BEG &longs;imul, vt C ad omnes &longs;imul DFH. & ita &longs;i plures fue
rint mag nitudines.
LEMMA. III.
Sit triangulum ABC, cuiuslatus BC in quotcunque di
uidatur partes æquales BE ED DF FC. & a punctis EDF
ip&longs;i AB equidi&longs;tanres ducantur EG DH FK. rur&longs;us à pun
ctis GHK ip&longs;i BC &ecedil;quidi&longs;tantes ducantur GL HM KN.
Dico triangulum ABC ad omnia triangula ALG GMH
HNK KFC &longs;imul&longs;umpta eandem habere proportionem,
quam habet CA ad AG.
vt CK ad KH.
ter&longs;e &longs;unt æquales. &longs;imiliter propter lineas æquidi&longs;tantes FK
æqualis DE; eritigitur KH ip&longs;i HG æqualis. Pariquè ra
tione o&longs;tendetur ob &ecedil;quidi&longs;tantes lineas DH EG BA,
HG ip&longs;i GA æqualem e&longs;&longs;e. Ex quibus patet CK KH HG
GA inter &longs;e æquales e&longs;&longs;e. Quoniam autem trianguloru ABC
kFC angulusad C e&longs;tvtrique communis; & ABC ip&longs;i kFC,
erit triangulum ABC ip&longs;i KFC &longs;imile. & quonian NK FC,
& HN KF &longs;unt &ecedil;quidi&longs;tantes, erunt anguli KCFCkF angu
lis HkN KHN &ecedil;quales; ac propterea reliquus CFK reliquo
KNH &ecedil;qualis: latus verò CK lateri KH e&longs;t &ecedil;quale; erit igi
&longs;imi
literquè
in ter&longs;e&longs;e &longs;imilia, & æqualia e&longs;&longs;e. & obid ip&longs;i ABC &longs;imilia e&longs;&longs;e.
Fiat igit vt AC ad AG, ita AG ad alia O. &longs;imiliterv AC ad GH,
ita GH ad P. rur&longs;usvt AC ad Hk, ita HK ad
vt AC ad Ck, ita CK ad R. & quoniam AG GH HK KC
dem habebit proportionem, ergo eandem quoque habebit
propo&longs;itionem AG ad O, vt GH ad P, & HK ad Q, &
Atverò quoniam ita e&longs;t AC ad AG, vt AG ad O, & vt
AC ad GH, ita GH, hoc e&longs;t AG ip&longs;i &ecedil;qualis, ad P. rur&longs;us
vt AC ad HK, ita HK, hoc e&longs;t AG ad
AC ad KC, ita KC, hoc e&longs;t AG ip&longs;i &ecedil;qualis, ad R. erit AC
ad omnes con&longs;equentes &longs;imul &longs;umptas AG GH HK KC,
hoc e&longs;t erit AC ad eandem AC, vt AG ad omnes &longs;imul
OPQR. vnde &longs;equitur omnes &longs;imul OPQR ip&longs;i AG &ecedil;qua
les e&longs;&longs;e. Itaque quoniam &longs;imilia triangula in dupla &longs;unt
portione laterum homologorum, erit triangulum ABC ad
ALG, vt AC ad O. eodemquè modo erit triangulum ABC
ad GMH, vt AC ad P. rur&longs;us ABC ad HNK, vt AC ad
Q, & vt idem ABC ad KFC, ita AC ad R. triangulum
igitur ABC ad omnes con&longs;equentes, videlicet ad omnia
gula &longs;imul &longs;umpta ALG GMH HNK KFC, eritvt AC ad
omnes &longs;imul OPQR. hoc e&longs;t ad AG. o&longs;ten&longs;um e&longs;t igitur,
quod propo&longs;itum fuit.
ti lemmate
ti lemmate
PROPOSITIO. XIII.
Omnis trianguli centrum grauitatis e&longs;t in recta
linea ab angulo ad dimidiam ba&longs;im ducta.
diamba&longs;im BC ducta. o&longs;tendendum est, centrum grauitatis trianguli
ABC e&longs;&longs;e in linea AD. Non &longs;it quidem, &longs;ed &longs;i fieri potest &longs;it punctum
H. & ab ip&longs;o ducatur HI æquidi&longs;tansip&longs;i BC,
in I.
tur linea
in partes æquales
b
a &longs;ectionum punctis ducantur
tes ip&longs;i AD. & connectantur EF G
æquidistantes erunt.
dem OB ZC æquales; erit DO ip&longs;i DZ &ecedil;qualis. quare DO
ad OB e&longs;t, vt DZ ad ZC. Quoniam autem EO FZ &longs;unt
ZC. erit igitur AE ad EB, vt AF ad FC. quare EF ip&longs;i BC
GB, vt AK ad KC, & AL ad LB, vt AM ad MC. ex quib^{9}
&longs;equitur LM GK EF non &longs;olùm ip&longs;i BC, verùm etiam inter
&longs;e&longs;e parallelas e&longs;&longs;e. &longs;ecct EF lineas G
AD in T. lineaquè GK &longs;ecet L
linea deniquè LM ip&longs;am AD in S di&longs;pe&longs;cat. Quoniam au
tem D
nea ac propterea
punctum H centrum grauitatis trianguli ABC extra paral
At verò quoniam LD DM
&longs;unt para lelogramma, erunt LS
ter SM D&longs;untverò
SM inter &longs;e &longs;unt &ecedil;quales. eademquè rarione NY Y
&longs;e, & ip&longs;is LS SM &ecedil;quales exi&longs;tent. quarelinea SY bifariam
diuiditlatera oppo&longs;ita parallelogrammi MN. pariquè ratio
ne o&longs;tendetur lineam YT bifariam diuidere oppo&longs;ita latera
parallelogrammi KX; lineamquè TD latera oppo&longs;ita paral-
mi MN centrum grauitatis est in linea
KX grouitatis centrum est in linea T
linea TD; magnitu linis igitur ex
MN KX FO
itaque punctum R.
LNGXEOZF
&longs;ecet in P.
ip&longs;i RH occurrat in V.
la ex AM MK
gula ASM M
tionem, quam habet CA ad AM. &longs;iquidem &longs;unt AM MK
EB de&longs;cripta triangula &longs;imilia
bet proportionem, quam ‘BA ad AL
omnes con&longs;equentes, hoc e&longs;t totum triangulum ABC ad on
nia triangula &longs;imul &longs;umpta, quæ &longs;unt in AB, & in AC con&longs;ti
tuta, eandem habebit proportionem, quam habet AC AB &longs;i
mul ad AM AL &longs;imul, quia verò ob
ABC ALM CA ad AM e&longs;t, vt BA ad AL; erit CA ad AM, vt
CA BA &longs;imul ad AM AL &longs;imul.
Atque CA ad AM maiorem habet proportionem quàm VR ad RH; e
tenim proportio ip&longs;ius CA ad AM e&longs;t eadem, quæ est totius VR
R. p.
M
D
diuiduntur; erit C
ad D
quia verò VR ad RP maiorem habet proportionem, quàm
ad RH. maiorem quoque habebit proportionem CA ad
AM, quàm VR ad RH. e&longs;t autem CA ad AM, vt
ABC ad omnia triangula in lineis AC AB. (vt dictum e&longs;t)
con&longs;tituta; ergo
rem habet proportionem, quàm VR ad RH. Quare & diuidendo pa-
circumrelicta triangula
habeat proportionem ad HR, quam parallelogramma MN
kX FO ad circumrelicta triangula, maior erit, quàm VH
triangula;
magnitudo ABC, cuius centrum grauitatis est H, & ab ea magnitudo
nis ablatæ centrum grauitatis e&longs;t punctum R; magnitudinis reliquæ ex
circumrelictis triangulis compo&longs;itæ centrum grauitatis erit in recta li-
HR eam habeat proportionem, quam habet magnnudo
grammis MN KX FO con&longs;tans
qua triangula,
ex ip&longs;is circumrelictisquoa fieri non pote&longs;i aucta
enim recta linea
guli ABC,
gulorum,
grauitatis trianguli ALM, ac centrum magnitudinis ex vtri&longs;
què triangulis LGN MK
&longs;it&ecedil;, ac magnitudinis ex. EBO FZC compo&longs;&longs;tæ, e&longs;&longs;ent in par
te Q
gulis compo&longs;itæ centrum e&longs;&longs;et grauitatis. quæ
nino ab&longs;urda. Quòd &longs;i ducta linea per Q, non fuerit etiam
ip&longs;i AD &ecedil;quidi&longs;tans, eadem &longs;equentur in conuenien tia.
ni&longs;estum e&longs;t igitur; quod propo&longs;itum fuerat.
mi.
add.
SCHOLIVM.
Id ip&longs;um vult ad huc Archimedes aliter o&longs;tendere. ob
tem verò demon&longs;trationem hoc priùs cogno&longs;cere oportet.
LEMMA.
Si intra triangulum vni lateri &ecedil;quidi&longs;tans ducatur, ab op
po&longs;ito autem angulo intra triangulum quoquè recta ducatur
linea, æquidi&longs;tantes lineas in eadem proportione di&longs;pe&longs;cet.
Hoc in &longs;ecundo no&longs;trorum plani&longs;ph&ecedil;riorum libro in ea
parte o&longs;tendimus, vbi quomodo conficienda &longs;it ellip&longs;is, in&longs;tru
mento à nobis inuento demon&longs;trauimus. hoc nempè modo,
Sit triangulum ABC, ip&longs;iquè BC in
tra triangulum ducatur vtcumquè æ
quidi&longs;tans DE. à punctoquè A intra
triangulum &longs;imiliter quocumque du
catur AF; quæ lineam BC &longs;ecet in F;
lineam verò DE in G. Dico ita o&longs;&longs;e
CF ad FB, vt EG ad GD.
enim GE FC &longs;unt æquidi&longs;tantes, erit
triangulum AFC triangulo AGE æquiangulum, vt igitur
AF ad AG, ita CF ad EG. ob ean demquè cauíam ita e&longs;t FA
ad AG, vt FB ad GD. quare vt CF ad EG, ita e&longs;t FB ad GD.
ac permutando, vt CF ad FB, ita EG ad GD. quod demon
&longs;trare oportebat.
ba&longs;im
N on &longs;it autem, &longs;ed &longs;i fieri pote&longs;t; &longs;it H. iunganturquè AH HB HC, &
ED
&longs;am AD in M. &
iungaturquè
gulo DFC, cùm &longs;it BA ip&longs;i FD æquidistans
ad DB.
intra vtrumquè triangulum &longs;unt &longs;imiliter po&longs;ita. etenim ad homologa
latera angulos efficiunt æquales. hoc enim per&longs;picuum. est
&longs;int triangulorum ABC DFC homologa latera AC FC,
gulus LFC angulo HAC &ecedil;qualis. &longs;ed angulus CFD e&longs;t ip&longs;i
æqualis exi&longs;tit. & quoniam ita e&longs;t CF ad FA, vt CL ad LH,
cùm &longs;int FL AH &ecedil;quidi&longs;tantes. CF verò dimidia e&longs;t ip&longs;ius
CA, erit & CL ip&longs;ius quoque CH dimidia. at CD ip&longs;ius
CB dimidia exi&longs;tit; erit igitur DL ip&longs;i BH &ecedil;quidi&longs;tans. ac
propterea angulus LDC e&longs;t ip&longs;i HBC &ecedil;qualis, & LDF ip&longs;i
HBA &ecedil;qualis. cùm &longs;ittotus CDF toti CBA &ecedil;qualis; anguli
verò ACH & HCB tam &longs;unt trianguli ABC, quàm FDC.
lo EBD e&longs;&longs;e &longs;imiliter po&longs;itum, vt H in triangulo ABC.
centrum grauitatis e&longs;t in medietate lineæ
& in ij&longs;dem parallelis EF BC, &longs;iquidem e&longs;t AE ad EB, vt
AF ad FC. quippè cùm latera AB AC &longs;int bifariam diui
&longs;a.
&ecedil;quidi&longs;tans, & ob id &longs;it
ad EA, ita CF ad FA;
quare vt BK ad KH, ita CL ad LH. æquidi-
dium e&longs;t ip&longs;ius KL.
gulorum
vbi &longs;imiliter diametri concurrunt,
EBD FDC, &longs;imulquè parallelogrammi AEDF, hoc e&longs;t totius
KN ip&longs;i BD æquidi&longs;tans; erit BK ad KH, vt DN ad
NH: vtautem BK ad KH, ita e&longs;t BE ad EA, & vt BE ad
EA, ita e&longs;t DM ad MA, cùm &longs;it EM ip&longs;i BD æquidi&longs;tans.
erit igitur DM ad MA, vt DN ad NH. quare MN ip&longs;i AH
e&longs;t &ecedil;quidi&longs;tans; ideoquè MN numquam cùm AH conueni
re pote&longs;t.
demonitrare oportebat.
lemma.
SCHOLIVM.
ctum H. quod e&longs;&longs;e non pote&longs;t,
&longs;ed eius pars, &longs;iuead M, &longs;iue ad N producta cum H conue
nireoporteat. cùm tamen ip&longs;amet linea MN per punctum
H tran&longs;ire debeat. ita vt punctum H &longs;it inter puncta MN;
hoc e&longs;t in linea MN, & non in eius parte producta. Nam &longs;i
punctum H centrum e&longs;t grauitatis totius trianguli ABC.
punctum verò N centrum grauitatis magnitudinis ex
lis EBD FDC compo&longs;it&ecedil;; atque punctum M centrum gra
uitatis parallelogrammi AEDF; oportet vt punctum H ita li
neam diuidat MN; vt eius partes magnitudinibus permuta
tim re&longs;pondeant. vt nimirum pars ad M ad partem ad N &longs;it,
vt magnitudo ex triangulis EBD FDC con&longs;tans ad parallelo
grammum AEDF. vt ex &longs;exta, & octaua huius propo&longs;itione
per&longs;picuum e&longs;t. Quare punctum H in linea MN e&longs;&longs;e debe
ret; vt ip&longs;emet Atchimedes paulò &longs;uperiùs affirmauit; cùm in
tis e&longs;t in linea MN.Quodiv
ca vel del
additum, vel ideo tamen hoc dixi&longs;&longs;e voluit Archimedes, vt o
&longs;tenderet lineam MN nullo modo (etiam &longs;i produceretur)
H conuenire po&longs;&longs;e.
PROPOSITIO. XIIII.
Omnis trianguli centrum grauitatis e&longs;t
in quo rectæ lineæ ab angulis trianguli ad dimidia
later a ductæ concurrunt.
diam BC. BE verò
line&ecedil; AD BE &longs;einuicem &longs;ecent in
cto H.
tis trianguli ABC est in vtraque linea
AD BE; hoc enim demonstratum e&longs;t
pr&ecedil;cedenti. erit vtique centrum graui
tatis, vbilineç AD BE &longs;e
&longs;ecant verò &longs;e&longs;e in H.
H centrum e&longs;t grauitatis
quod demon&longs;trare oportebat.
SCHOLIVM.
Similiter &longs;i ducta fuerit CH, & producta, bifariam &longs;ecaret
AB. In hac enim linea e&longs;&longs;et centrum grauitatis trianguli;
trum verò e&longs;t in linea ab angulo ad dimidiam ba&longs;im ducta:
ergo hæc linea ab angulo C ad dimidiam AB ducta e&longs;&longs;et.
Præterea &longs;i linea à puncto C ad dimidiam AB ducta
&longs;iret per H; e&longs;&longs;et vtique in hac linea centrum grauitatis; &longs;ed
trum
H; vnius igitur figur&ecedil; plura darentur centra grauitatis. quod
fieri non pote&longs;t. quod quidem, cùm &longs;it in con ueniens, nos in
no&longs;tro Mechanicorum libro dari non po&longs;&longs;e &longs;uppo&longs;uimus.
Quare linea CH in directum ducta, bifariam &longs;ecaret AB.
quod quidem paulò infra aliter quoque o&longs;tendemus,
lis prius demon&longs;tratis; quæ Archimedes ob &longs;equentem
&longs;trationemVult enim Ar
chimedes, po&longs;tquam inuenit centrum grauitatis cuiu&longs;libet
trianguli, centrum quoque grauitatis quærere trapetij duo la
tera &ecedil;quidi&longs;tantia habentis. quod e&longs;t quidem pars trianguli,
& tanquam fru&longs;tum a triangulo ab&longs;ci&longs;&longs;um. &longs;upponitquè den
trum grauitatis cuiu&longs;libet trianguli e&longs;&longs;e in recta linea ba&longs;i du
cta &ecedil;quidi&longs;tante, quæ latera ita diuidat, vt partes ad uerticem
&longs;int reliquarum partium duplæ. quod quidem ortum ducit
ex cognitione alterius theorematis o&longs;tendentis centrum gra-
midiam ba&longs;im ducta (vt Archimedes demon&longs;trauit) & in&longs;u
per in eo puncto, quod dictam lineam diuidatita, vt pars ad
angulum reliqu&ecedil; ad ba&longs;im &longs;it dupla. Quare hoc prius ita
demus.
PROPOSITIO.
Omnis trianguli centrum grauitatis e&longs;t punctum in recta
linea ab angulo ad dimidiam ba&longs;im ducta exi&longs;tens, quod li
neam diuidat, ita vt poitio ad angulum reliquæ ad ba&longs;im, &longs;it
dupla.
Sit triangulum ABC, in quo ab an
gulo A ad dimidiam ba&longs;im BC re
cta ducatur linea AD. Ducaturquè
ab angulo B ad dimidiom ba&longs;im
AC linea BE, quæ&longs;ecet AD in F. Et
quoniam centrum grauitatis
e&longs;t lineam FA ip&longs;ius FD duplam e&longs;
&longs;e. iungatur FC. quoniam enim AE
e&longs;t equalis ip&longs;i EC, erit triangulum
&longs;int &longs;ub eadem altitudine. Ob eandemquè cau&longs;am
AFE triangulo EFC exi&longs;tit æquale. &longs;i igitur à triangulo ABE
auferatur triangulum AFE, & à triangulo EBC triangulum
auferatur EFC; relinquetur triangulum ABF triangulo BFC
æquale. Rur&longs;us quoniam BD e&longs;t æqualis ip&longs;i DC; erit trian
bentaltitudinem. duplum igitur e&longs;t triangulum BFC
li
quia verò triangula ABF FBD in eadem &longs;unt altitudi
ne, idcirco &longs;e&longs;e habebunt, vt ba&longs;es AF FD. atque triangulum
ABF. duplum e&longs;t ip&longs;ius FBD; ergo portio AF ip&longs;ius FD dupla
exi&longs;tit. quod demon&longs;trare oportebat.
ALITER.
Sit rur&longs;us triangulum ABC, & AD BE ab angulis ad di
midias ba&longs;es ductæ &longs;int erit vtique punctum, F (vbi &longs;e in ui
cen fecant) centrum grauita tis triangulb ABC. Drco AF a
p&longs;ius FD duplam e&longs;&longs;e. Iungatur DE. Quoniam enim BC
AC in punctis DE bifariam &longs;ecantur; erit
CD ad DB, vt CE ad EA. linea igitur
DE ip&longs;i AB e&longs;t æquidi&longs;tans. quare
gulum ABC &longs;imile e&longs;t triangulo EDC.
ac propterea ita e&longs;t BC ad CD, vt AB
ad DE. e&longs;t autem. BC dupla ip&longs;ius CD
(&longs;iquidem punctum D bifariam diuidit
BC) erit igitur AB dupla ip&longs;ius DE. At
vero quoniam AB DE &longs;unt parallelæ, erit triangulum AFB
triangulo EFD &longs;imile. & vt AB ad ED, ita AF ad FD, e&longs;t
autem AB ip&longs;ius ED dupla, ergo AF ip&longs;ius FD dupla
exi&longs;tit. quod demon&longs;trare oportebat.
Exijs, quæ demon&longs;trata &longs;unt, o&longs;tendemus, quod paulò an
te propoiuimus, nempè cùm lineæ AD BE bifariam &longs;ecent
BC CA. Dico lineam CF productam bifariam quoque &longs;e
care ip&longs;am AB.
Producatur enim (ijsdem po&longs;itis) CFGH; quæ lineam
AB &longs;ecetin G. & à puncto B
ip&longs;i AD æquidi&longs;tans ducatur
BH. quæ ip&longs;i CG occuriat in
H. Quoniam igitur FD, e&longs;t i
p&longs;i BH &ecedil;quidi&longs;tans, erit CD
ad DB, vt CF ad FH. CD
rò e&longs;t æqualis BD; ergo CF ip&longs;i
FH æqualis exi&longs;tit. ac propterea
CH dupla e&longs;t ip&longs;ius (F. At ve
rò quoniam ob &longs;imilitudinem
HC ad CF, vt BH ad DF; erit & BH ip&longs;ius FD duplex.
exi&longs;tit. erunt igitur BH FA inter &longs;e &ecedil;quales. Quoniam autem
BH e&longs;t &ecedil;quidi&longs;tans ip&longs;i AF, æquiangula erunt triagula GBH
ip&longs;i AF æqualis; erit & BG ip&longs;i GA æqualis. ergo recta li
nea EFG bifariam diuidit AB. quod demon&longs;trare oporte
bat.
Reliquum e&longs;t, vt ob &longs;equentem dem on&longs;trationem alteram
propo&longs;itionem o&longs;tendamus.
PROPOSITIO.
Centrum grauitatis cuiu&longs;libet trianguli e&longs;t in recta linea
ba&longs;i ducta æquidi&longs;tante, quæ latus ita diuidat, vt pars ad an
gulum reliquæ ad ba&longs;im &longs;it dupla.
In trianagulo enim ABC ducta
&longs;it DE ba&longs;i BC æquidi&longs;tans, quæ
latus AB diuidatin D, ita vt DA
ip&longs;ius DB &longs;it duplex. Dico in linea
DE centrum e&longs;&longs;e grauitatis triangu
li ABC. Ducatur ab angulo A ad
dimidiam BC linea AF, quæ di
vt AG ad GF, ac propterea erit
AG ip&longs;ius GF dupla. punctum er
go G centrum e&longs;t grauitatis trian
guli ABC. Quare con&longs;tat
e&longs;&longs;e in linea DE. quod demon&longs;tra
re oportebat
COROLLARIVM.
Ex hoc elici pote&longs;t centrum grauita tis cuiu&longs;libet trianguli
e&longs;&longs;e in medio ductæ lineæ ba&longs;i æquidi&longs;tantis, qu&ecedil; latus diui
datita, vt portio ad verticem &longs;it reliqu&ecedil; ad ba&longs;im dupla.
E&longs;t enim DG ad GE, vt BF ad FC. &longs;unt verò BF FC
quales; ergo & DG GE inter &longs;e &longs;unt æquales. quare grauita
tis centrum G e&longs;t medium line&ecedil; DE.
2.
&longs;tratic
13.
PROPOSITIO. XV.
Omnis trapezij duo latera inuicem habentis æ
quidi&longs;tantia centrum grauitatis e&longs;t in recta linea,
quæ latera æquidi&longs;tantia bifariam &longs;ecta
ita diui&longs;a, vt ip&longs;ius portio terminum habens mino
rem parallelam bifariam diui&longs;am ad
tionem eandem habeat proportionem, quam ha
bet vtraque &longs;imul, quæ &longs;it æqualis duplæ maioris
parallelarum cum minore ad
maiore.
linea
verò EF bifariam diuidat AD BC. Quòd igitur in linea EF &longs;it cen
trum grauitatis trapezii, per&longs;picuum est. productis enim CDG FEG
BAG, liquet in idem punctum,
cùm &longs;it AD æquidi&longs;tans ip&longs;i BC, nece&longs;&longs;e e&longs;t proportionem
ip&longs;ius BA ad AG, ip&longs;iusquè FE ad EG, & CD ad DG, quæ
mirum
trianguli GBC centrum grauitatis in linea GF. &longs;imiliterque trianguli
centrum grauitatis erit in linea EF. iungatur itaque BD, quæ int
æquain punctisac per ea
BC æquidi&longs;tantes
verò EB &longs;ecet NT in O. Iungaturquè
cùm &longs;it HB tertia pars ip&longs;ius B D
HB dupla.
e&longs;t ab angulo D ad dimidiam BC ducta.
centrum grauitatis est punctum X. Eademquè ratione
tertia pars ip&longs;ius DB, ac proptcrea &longs;it BK ip&longs;ius KD dup
&longs;itquè KN æquidi&longs;tans ip&longs;i AD; erit centrum grauitatis tri
guli ABD in linea KN; idem verò centrum reperitur quo
in linea BE, cùm &longs;it ab angulo B ad dimidiam AD duc
ergo
guli ABD. magnitudinis igitur ex vtri&longs;que triangulis ABD BI
compo&longs;itæ, quæ e&longs;t trapezium
nea EF, quare trapezii ABCD centrum grauitatis est punctum
P. At verò triangulum BCD ad ABD proportionem habet eam, quam
OP ad P
tatis, ac punctum P vtrorumque commune centrum.
triangulum BDC adtriangulum ABD, ita e&longs;t
&longs;iquidem &longs;unt in ijsdem parallelis AD BC. quare vt BC ad
AD, ita OP ad PX.
ticem &longs;unt &ecedil;quales, & angulus PRO ip&longs;i PSX, veluti angulus
ROP angulo PXS e&longs;t &ecedil;qualis, erit triangulum OPR triangu
lo XPS &longs;imile; quare
BC ad AD, vt OP ad PX
& antecedentium dupla, duæ &longs;cilicet BC ad AD, vt duæ PR
ad PS. & componendo duæ BC cum AD ad AD; vt duæ
PR cum PS ad PS. & ad con&longs;equentium dupla, vt &longs;cilicet
duæ BC cum AD ad duas AD, ita duæ PR cum PS ad duas
PS. dictum e&longs;t autem BC ad AD ita e&longs;&longs;e, vt PR ad PS. quare
conuerrendo AD ad BC erit, vt PS ad PR. & antecedentium
dupla. hoc e&longs;t duæ AD ad BC, vt duæ PS ad PR. Itaque in
eadem &longs;unt proportione duç BC cum AD ad duas AD, vt
du&ecedil; PR
PS ad PR. antecedentes igitur ad &longs;uas &longs;imul con&longs;equentes in
eadem erunt proportione.
AD cum BC, ita duæ RP cum PS ad duas P S cum PR,
verùm duæ quidem RP cum PS e&longs;t vtraque &longs;imul SR RP.
enim a&longs;&longs;umitur PR, &longs;emel verò PS. Cum autem lineæ DH ES
à lineis diuidantur &ecedil;quidi&longs;tantibus ED OT HM, erit DK ad
KH, vt ER ad CS; kD verò e&longs;t æqualis KH, erit ER ip&longs;i
RS &ecedil;qualis. erit igitur ER cum RP,
&ecedil;qualis.
&longs;umitur PS, &longs;emel què PR. & quoniam FS e&longs;t &ecedil;qualis ip&longs;i SR.
quod quidem eodem modo o&longs;tendetur, cùm &longs;it FS ad SR, vt
BH ad Hk. erit FS cum SP,
Quare ita &longs;ehabet PE ad PF, vt duæ BC cum AD ad duas
AD cum BC. Centrum igitur grauitatis P trapezij ABCD
in linea e&longs;t EF, quæ
reliquampartem PF eam habet proportionem, quam du
ip&longs;ius BC, quæ e&longs;t maior æquidi&longs;tautium, vna cum min
AD, ad duplam minoris AD cum maiore BC,
ta &longs;unt, quæ propo&longs;ita fuerant.
me demon
&longs;tratis.
13.
quint
huius
in
SCHOLIVM.
habet
quidem verba illa
ponerem
Hæc &longs;unt, quæ de centro grauitatis figurarum rectiline
Archimedes &longs;cripta reliquit. Ex quibus maxima certè vtil
habetur; neque ampliùs de rectilineis figuris Archimedes p
tractare voluit. ex dictis enim alia omnia dependent. Nan
tra grauitatis rectilinearum figurarum, quæ æquales angu
lateraque æqualia habent, ex his in uenire poterimus. quæ
dem figur&ecedil; in circulo in&longs;cribi po&longs;&longs;unt. Quod &longs;anè Federi
Comandinus in eius libro de centro grauitatis &longs;olidorum
prioribus propo&longs;itionibus præ&longs;titit. aliaquè nonnulla, vt
tragrauitatis rectilinearum figurarum in ellip&longs;i, deindè ip
circuli, & ellip&longs;is centra grauitatis in uenit. omne&longs;què dem
&longs;trationes in ijs, quæ in hoc libro iam demon&longs;trata &longs;unt,
dauit. præterea ex his etiam idem Commandinus in com
tarijs libri Archimedis de quadratura paraboles, (quo ad p
xim) grauitatis centrum cuiu&longs;libet figur&ecedil; rectilineæ adin
nit. Quod quidem nos quoque, vt initio polliciti fuimus,
nullis mutatis idem o&longs;tendemus. hoc prius &longs;uppo&longs;ito.
Triangula in eadem ba&longs;i con&longs;tituta eam inter &longs;e propo
nem habent, quam eorum altitudines.
Hoc autem demon&longs;tratum e&longs;t ab excell enti&longs;simis viris,
ri&longs;què Euclidis interpretibus, Federico
&longs;tophoro Clauio; qui hanc propo&longs;itionem po&longs;t primam
ti libri Euclidis demon&longs;trarunt.
PROBLEMA.
Cuiu&longs;libet rectiline&ecedil; figur&ecedil; centrum grauitatis inuenire.
Triangulorum centrum grauitatis iam ab Archimede de
mon&longs;tratum e&longs;t.
Sit itaque primùm quadri
laterum ABCD, cuius opor
teat centrum grauitatis inue
nire. Ducatur AC, quæ qua
drilaterum in duo triangula
ABC ACD diuidet. à
què
lares ducantur BE DF. In
ueniantur deinde ex dictis
tra grauitatis triangulorum
ABC ACD. &longs;intquè puncta
GH. iungaturquè GH, quæ diuidatur in K, ita vt GK
ad KH &longs;it, vt DF ad BE. Dico punctum K centrum
e&longs;&longs;e grauitatis quadrilateri ABCD. Quoniam enim triangu
la ABC ACD in eadem &longs;unt ba&longs;i AC, erunt inter &longs;e&longs;e, vt al
titudines. quare triangulum ACD ita &longs;e habet ad
ABC, vt DF ad BE. hoc e&longs;t GK ad KH.
trum e&longs;t grauitatis magnitudinisex vtril què triangulis ABC
ACD compo&longs;itæ; hoc e&longs;t quadrilateri ABCD.
Sit autem pentagonum
ABCDE.
AD. inueniaturquè
li ABC centrum grauitatis
H. quadrilateri verò ACDE
ex proximè
trum
Iam vtique con&longs;tat (du
cta HK) centrum grauita
tis totius ABCDE in linea
Rurilus trianguli ADE centrum inueniatur F
quadrilateri verò ADCB punctum G. iungaturquè GF. e
eodem modo centrum grauitatis totius ABCDE in linea F
&longs;ed e&longs;t quoque in linea HK, ergo vbr&longs;e inuicem &longs;ecant, vt
L, centrum erit grauitatis pentagoni ABCDE.
In hexagonis &longs;imiliter.
vt ABCDEF iungantur
AC AE, deinceps inuenia
tur trianguli ABC
grauitatis G, pentagoni
verò ACDEF ex dictis cen
trum &longs;it H. ductaquè GH
centrum grauitatis totius
ABCDEF erit in linea GH
&longs;imiliter centrum grauita
tis trianguli AFE &longs;it K,
tagoni verò AEDCB &longs;it L, iunctaquè KL, erit centrum gr
uitatis totius hexagoni in linea KL. verùm e&longs;t etiam in lin
GH. ergo errt in M. in quo GH
Nequè aliter in heptago
no ABCDEFG, in quo du
cantur BG CE. trianguli
verò ABG centrum graui
tatis &longs;it H. hexagoni
GBCDEF, &longs;it K. deinde
trianguli CDE
uitatis &longs;it L, hexagoni ve
rò CEFGAB &longs;it M. iun
cti&longs;què HK ML, eadem ra
tione centrum grauitatis
Eodemquè pror&longs;us modo in octagono, & in alijs demc
figuris centrum graui ta tis inuenietur. quæ quidem facere
portebat.
Curautem hoc modo centra grauitatum in præfatis figu
ris po&longs;itione tantùm, & non determinatè ea in determinata,
linea, & in tali &longs;itu exi&longs;tere inuenerimus, vt in parallelogram
mis & in triangulis factum fuitab Archimede; explicabitur in
&longs;ecundo libro po&longs;t tertiam proportionem; vbi o&longs;tendemus,
in quibus figuris determinatè inueniri pote&longs;t centrum graui
tatis.
Antequam autem finem primolibro imponamus,
e&longs;t; vt ea quæ in præfatione &longs;uppo&longs;uimus, o&longs;tendamus. pri
mùm què quando &longs;ecundùm rectam lineam aliqua diuiditur
figura per centrum grauitatis, aliquando diuidi in partes &longs;em
per &ecedil;quales, & aliquando in partes inæquales.
PROPOSITIO.
Figura dari pote&longs;t, qu&ecedil; per centrum grauitatis recta li
nea diui&longs;a, &longs;emperin partes diuidatur æquales.
Sit
ABCD, cuius
uitatis E. Ducaturquè per
E
vel diameter e&longs;t, vel min^{9}.
&longs;i e&longs;t diameter, iam
&ecedil;qua e&longs;&longs;e diui&longs;um. Si verò non e&longs;t diameter,
AC BD, quæ per E tran&longs;ibunt. Quoniam igitur AF e&longs;t æqui
diftans ip&longs;i CG, eritangulus EAF ip&longs;i ECG, & EFA ip&longs;i EGC
æqualis, e&longs;t autem AEF ip&longs;i GEC ad verticem æqualis,
AE ip&longs;i EC æquale; erit triangulum AEF triangulo CEG &ecedil;qua
le. eodemquè modo o&longs;tendetur triangulum FEB triangulo
EGD. & triangulum AED ip&longs;i BEC æquale. Ex quibus patet.
figuram ex tribus triangulis compo&longs;itam, hoc e&longs;t figuram
FGDA ip&longs;i FGCB æqualem e&longs;&longs;e. diuiditurergo
mum
quales. quod demon&longs;trare oportebat.
Hoc idem multis alijs figuris accidet, vt pentagonis, he
gonisæquiangulis, & æquilateris, & alijs.
PROPOSITIO.
Figura dari pote&longs;t, quæ per centrum grauitatis recta li
diui&longs;a, non &longs;emper in partes diuidatur &ecedil;quales.
Habeat triangulum ABC
latera AB AC æqualia. trian
guliverò centrum grauitatis &longs;it
D. à quo ip&longs;i BC &ecedil;quidi&longs;tans
Ducatur FDG. Dico partem
AFG
ducatur ADE, quæ bifariam
& à puncto G
ip&longs;i AE &ecedil;quidi&longs;tans ducatur
HGK. compleanturque figur&ecedil;
EH KF. Quoniam enim FG
& e&longs;t BE ip&longs;i EC æqualis. eritigitur FD ip&longs;i DG &ecedil;qua
vt etiam paulò ante 15. huius o&longs;tendimus. quare FG ip
DG dupla. e&longs;t. ac propterea
e&longs;t parallelogrammi DK. quia verò AD ip&longs;ius DE du
exi&longs;tit, erit quoquè parallelogrammum DH ip&longs;ius DK
plum. Quare DH ip&longs;i FK e&longs;t æquale. At verò quoni
mo DH æquale. triangulum igitur AFG parallelog
FK e&longs;t æquale. Quare pars AFG parte BFGC minor
&longs;tit. quod demon&longs;trare oportebat.
te
tionem
ius.
mi.
Hinc per&longs;picuum e&longs;t, eandem figuram per centrum gra
tatis diui&longs;am, aliquando in partes in æquales, aliquando in
tes æquales diuidi po&longs;&longs;e. in partes in&ecedil;quales iam o&longs;ten&longs;um
hocaccidere
neam ADE, quæ triangulum ABC in duo &ecedil;qua diuidi
eadem altitudine, ba&longs;e&longs;què BE EC inter &longs;e &longs;int æquales.
Adhuc (veluti initio quoque diximus) &longs;i fuerit prisma, vt
AB, cuius altera ba&longs;is &longs;it AC. tale verò &longs;it prisma, vt pl mum
AC planis CH CK &c. &longs;it erectum. &longs;it autem ip&longs;ius ba&longs;is
AC centrum grauitatis E. Dico &longs;i prima &longs;u&longs;pendatur ex pu
cto E, ba&longs;im AC horizonti æquidi&longs;tantem permanere. vt co
gno&longs;camusea, quæ his libris pertractantur, ad praxim po&longs;&longs;e
reduci. & ne aliquid ab&longs;que demon&longs;tratione confirmatum re
linquamus. hoc quoque o&longs;tendemus. hoc pacto.
Primùm quidem exijs, quæ demon&longs;trata &longs;unt, rectilineæ
figuræ AC centrum granitatis inueniatur E. eodemquè mo
do figuræ BD centrum grauitatis &longs;it F. Iungaturquè EF,
quæ bifariam diuidatur in G. Iam patet punctum G cen
trum e&longs;&longs;e grauitatis pri&longs;matis AB, ex octaua propo&longs;itione Fe
derici
lario quintæ propo&longs;itionis eiu&longs;dem libri, lineam EF late
ribus AD CB &ecedil;quidi&longs;tantem e&longs;&longs;e. quoniam
CK ad rectos &longs;untangulos plano AC, erit CB eorum commu
nis&longs;ectio eidem plano AC perpendicularis. acpropterea EF
ip&longs;i CB æquidi&longs;tans plano AC perpendicularis exi&longs;tit.
ma propo&longs;itione de libra no&longs;trorum mechanicorum pon
AB ex E &longs;u&longs;pen&longs;um
horizonti perpendicularis. Quando autem EF erit horizc
ti perpendicularis, erit planum AC horizonti æquidi&longs;tan
exi&longs;tet. Inuento igitur centro grauitatis E ip&longs;ius ba&longs;is A
&longs;i AB &longs;u&longs;pendatur ex E, linea EGF in centrum mundi to
det; planumquè AC horizonti erit æquidi&longs;tans. quod de
&longs;trare oportebat.
mi.
mi.
PRIMI LIBRI FINIS.
GVIDIVBALDI
E MARCHIONIBVS
MONTIS.
In Secundum Archimedis æqueponderan
tium Librum.
PRÆFATIO.
Secundus Archimedisliber, vtinitio primi
libri præfati &longs;umus, &longs;ubtili&longs;&longs;ima theo
remata &longs;peculatur. Vultenim Archimedes
inue&longs;tigare centrum grauita tis plani coni
cæ&longs;ectionis, quæ parabole pa&longs;&longs;im vocatur.
quamuis Archimedes alio nomine, ac po
tiùs de&longs;criptione quadam
cuparit
ta. Refert enim Eutocius A&longs;calonita in principio &longs;ui
tarij
mini (cui Pappus etiam ex Ari&longs;t&ecedil;i &longs;ententia a&longs;&longs;entire videtur)
quòd qui ante Apoll onium fuerunt, perfectam, & ab&longs;olutam
conorum
non habuerunt; inter
quos re&longs;po&longs;uit Archime
de.
nientes, ip&longs;um per
guli
lutionem manente vno
eorum, quæ circa
derarunt
definitionibus Euclidis
vndecimi libri elem
torum
ex
ex triangulo EDC, & conus FBC ex rectangulo triangulo
p&longs;i DC æqualis, conus
ABC vocabit rectan
gulus. nam vtcumquè
ducto plano per axem,
ABC; erit angulus BAC
ad coniverticem rectus:
&longs;iquidem DAC recti di
midius exi&longs;tit, veluti
DAB. pari ratione &longs;i ED
fuerit ip&longs;a DC minor;
erit conus EBC obtu&longs;i
angulus:nam ducto per axem plano, habebit triangulum
EBC angulum ad verticem coni BEC obtu&longs;um; cùm &longs;it
nus FBC acutiangulus nuncupabitur; quoniam
per axem FBC angulum ad verticem coni F acutum po&longs;&longs;ide
bit; &longs;iquidem minor e&longs;t BFC, quam BAC. Refert deinde,
quòd vnumquemquè
horum conorum
dem
runt; vt &longs;it rectangu
lus conus ABC; trian
gulum verò per axem
&longs;it ABC. in latere au
tem AC quoduis &longs;u
matur punctum D;
ducaturquè DE ad
AC perpendicularis;
& per DE ducatur pla
num plano ABC ere
ctum, quod quidem conum &longs;ecet, &longs;ectio autem &longs;it FDG. qu&ecedil;
&longs;anè e&longs;t &longs;e ctio, quæ abip&longs;is vocatur rectanguli coni &longs;ectio,
quippè quæ &longs;i intelligatur terminata recta linea FG, nuncupa
tur portio recta linea, rectangulique coni &longs;ectione contenta.
mcorum A
pol.
Si verò conus
ABC fuerit obtu
&longs;iangulus, &longs;itquè
triangulum per
axem ABC,
dem
uis puncto D, du
cta DE ad re
ctos angulos ip&longs;i
AC, acper DE
ducto plano ad
planum ABC erecto, quod conum &longs;ecet, vt FDG; erit FDG
obtu&longs;ianguli coni &longs;ectio, quæ vnà cum recta FG vocatur por
tio recta linea, obtu&longs;ianguliquè coni &longs;ectione contenta.
Similiter
no acutiangulo ABC,
cuius triangulum per a
xem &longs;it ABC. & à
D ducta &longs;it DE perpen
dicularis ip&longs;i AC, du
ctoquè plano per DE ad
planum ABC erecto, e
rit DFEG acutianguli
coni &longs;ectio.
Apollonius au-
tem Perg&ecedil;us, qui ab
&longs;oluti&longs;&longs;ima commenta
ria de conicis &longs;crip&longs;it,
huiu&longs;modi conos omnesvocauit rectos; ad differentiam coni
&longs;caleni. coni enim rectiaxes habent ba&longs;ibus erectos. &longs;caleni ve
rò nequaquam. & in &longs;calenis latera triangulorum per axem
non &longs;unt &longs;emper æqualia. quod &longs;emper conis rectis contingit.
Preterea &longs;ectionem rectanguli coni parabolen nominauit;
obtu&longs;ianguli verò coni &longs;ectionem hyperbolen; &longs;ectionem au
tem acutianguli coni ellip&longs;im nuncupauit. & in vnoquoque
cono tàm recto, quàm &longs;caleno has tres ine&longs;&longs;e &longs;ectiones Ex quibus colligit Geminus (quem Eutocius, alijquè
complures &longs;ecuti &longs;unt) eos, qui ante Apollonium extitere,
conostantùm rectos cognoui&longs;&longs;e. & in vnoquoque cono
tantùm &longs;ectionem animaduerti&longs;&longs;e. quod quidem &longs;i de ijs, qui
ante Archimedem fuere intelligatur; ad mitti forta&longs;&longs;e poterit;
ac præ&longs;ertim de Euclide. vt patet ex definitione coni abeo
tradita. At verò de Archimede, qui po&longs;t Euclidem, ante verò
Apollonium fuit, non ita facilè concedendum videtur.
ijs, quæ &longs;cripta reliquit. eum non &longs;olùm notitiam ha-
bui&longs;&longs;e de conis rectis; verùm
p&longs;ius &longs;criptis conijci pote&longs;t. In primo enim librode &longs;phæ
ra, & cylindro multis in locis, vtin &longs;eptima, octaua, no
na, decimaquarta, decimaquinta propo&longs;itione; alijsquè in
locis conos nominat &ecedil;quicrures, quod quidem &longs;ecundum i
p&longs;um &longs;unt, qui in eius &longs;uperficie æquales habent rectas lineas
à vertice coni ad ba&longs;im ductas. item in epi&longs;tola quoque libri
de conoidibus & &longs;ph&ecedil;roidibus, quam Archimedes De&longs;itheo
&longs;cribit. cùm de obtu&longs;iangulo conoideverba facit, conum vo
catæquicrurem. Quòd &longs;i Archimedes hos conos vocauit æ
quicrures, cui dubium, ip&longs;um eosad differentiam eorum, qui
non &longs;unt æquicrures ita nuncupa&longs;&longs;e? qui verò non &longs;unt æ
quicrures ex ip&longs;omet Apollonio &longs;unt &longs;caleni; nam æquicrures
hoc modo coni axes habent ba&longs;ibus erectos. qui igitur non
erunt æquicrures, eorum axes &longs;uis ba&longs;ibus nunquàm erunt e
recti. Præterea idem quoque confirmari pote&longs;t ex demon
&longs;tratione vige&longs;imæquintæ propo&longs;itionis eiu
cùm nominet Archimehes conum rectum proculdubiò ad
differentiam eorum, qui non &longs;untrecti ita eum nuncupauit.
nam &longs;i Aichimedes (ex illorum &longs;ententia) conos tan ùm re
ctos cognoui&longs;&longs;et; quor&longs;um his in locis conum rectum, vel æ
quicrurem nomina&longs;&longs;et? &longs;at &longs;ibi fui&longs;&longs;et conum tan tum dixi&longs;&longs;e.
Neque verò dicendum e&longs;t Archimedem per cono recto intel
lexi&longs;&longs;e conum rectangulum eo modo, quem &longs;upra expo&longs;ui
mus. nam in ea propo&longs;itione, dum con&longs;tituit hunc conum,
non con&longs;urgit conus rectangulus, &longs;ed obtu&longs;iangulus quapro
pter conum rectum nominatad differentiam coni &longs;caleni. C&ecedil;
terùm ut manife&longs;tè o&longs;ten damus Archimedem conos cogno-
noidibus, & &longs;ph æroidibus, in qua proponit Archimedes co
num con&longs;tituere, & inuenire, in quo &longs;it&longs;ectio ellip&longs;is data, ver
tex autem coni in linea exi&longs;tat a centro ellip&longs;is ad
los ellip&longs;is plano erecta. Exqua con&longs;tructione planè apparet,
Archimedem (vt ex eius demon&longs;tratione con&longs;tat) hocin lo
co querere, & inuenire conum proculdubio &longs;calenum. vt
ex nona eiu&longs;dem libri propo&longs;itione per&longs;picuum e&longs;&longs;e pote&longs;t; in
qua vt plurimùm conus inuenitur &longs;calenus. Ex quibus mani
fe&longs;ti&longs;&longs;imè patet Archimedem non &longs;olùm de conis rectis,
etiam de conis &longs;calenis notitiam habui&longs;&longs;e. Porrò ea verba, qu&ecedil;
refert Eutocius ex &longs;ententia Heraclij, qui Archimedis vitam
literis mandauit; idip&longs;um &longs;atis manife&longs;tant. Heraclius enim
inquit Archimedem quidem
aggre&longs;&longs;um; Apollonium verò, cùm ea inueni&longs;&longs;etab Archime
de nondum edita; tanquam eius propria edidi&longs;&longs;e. quod qui
dem etiam exip&longs;iusmet Archimedis &longs;criptis
in libro namque de conoidibus, & &longs;phæroidibus ante
propo&longs;itionem vbi Archimedes theorema proponit alibi de
mon&longs;tratum, inquit,
principio etiam libri de quadratura paraboles, cùm nonnulla
propo&longs;ui&longs;&longs;et; po&longs;t tertiam propo&longs;itionem &longs;cilicet, inquit
mon&longs;trata autem &longs;unthæc in elementis conicis.
Archimedem Obijciet verò aliquis,
non propterea con&longs;tare, h&ecedil;c elementa eonica, quorum me
minit Archimedes, ip&longs;iusmet e&longs;&longs;e Archimedis; cùm non affir
met, hæcfui&longs;&longs;e ab ip&longs;o dem on &longs;trata. verùm illud in primis ma
nife&longs;tum e&longs;t, tempore Archimedis conica elementa extiti&longs;&longs;e.
vt nonnulli Euclidem quatuor conicorum libros edidi&longs;&longs;e
firmant
libro a&longs;&longs;erit. Sed ex modo loquendi Archimedis planè
hæc fui&longs;&longs;e ab ip&longs;o con&longs;cripta. Nam quando Archimedes ali
qua &longs;upponitab alijs demon&longs;trata,
ab alijs demon&longs;trata e&longs;&longs;e; vtin vndecima propo&longs;itionedeco
noidibus, & &longs;phæroidibus; cùm inquit.
portionem compo&longs;itam e&longs;&longs;e ex proportione ba&longs;ium, & proportione altitu
dinum,
in libro de &longs;ph&ecedil;ra, & cylindro an te propo&longs;itionem decimam
&longs;eptimam, cùm nonnulla &longs;uppo&longs;uerit ab alijs demon &longs;trata in
quit.
verò parte
mon&longs;tratum e&longs;t enim aliis in locis portiones &longs;e&longs;quitertias e&longs;&longs;e
quod quia ip&longs;emet a&longs;&longs;ecutus e&longs;t in libro de quadratura para
boles, idcircò non addit ab ip&longs;omethoc o&longs;ten&longs;um fui&longs;&longs;e. A
liaquè huiu&longs;modi loca breuita tis &longs;tudio omitto o&longs;tendentia
ea, quæ Archimedes &longs;upponit tanquam demon&longs;trata,
non additab alijs o&longs;ten&longs;a e&longs;&longs;e, à &longs;e ip&longs;o demon&longs;trata fui&longs;&longs;e, vt
in demon&longs;tratione decimæ quart&ecedil; propo&longs;itionis primi libri,
nec non ex octaua huius &longs;ecundi libri dem on&longs;tratione; alij&longs;
què locis per&longs;picuum e&longs;&longs;e pote&longs;t. Quare tùm ex præfntis Archi
medis locis, tùm Heraclij te&longs;tim onio manife&longs;tè elicipote&longs;t,
Archimedem elementa conica &longs;crip &longs;i&longs;&longs;e. Neque verò quicqua
nos turbare debet, quòd Apollo nius coni &longs;ectionibus nomina
impo&longs;uerit; &longs;i tamen ip&longs;e prim us fuit; cùm eas proprijs nomi
nibus, vt potè parabolen, hyperbolen, & ellip&longs;im nuncupet;
& in quolibet cono omnes agnouerit &longs;ectiones. Nam quam
uis v&longs;que ad Archimedis tempus hi termini nondum extite
rint; & in &longs;ingulis conis pri&longs;ci illi vnicam
&longs;ectionem; Archimedes tamen vlteriùs progre&longs;&longs;us e&longs;t. etenim
hæc quoque
runt: quandoquidem in demon&longs;tratione nonæ propo&longs;itio
nis de conoidibus, & &longs;ph&ecedil;roidibus ellip&longs;im nominat. Pr&ecedil;te
rea non &longs;olùm cognouit Archimedes conos &longs;ecari po&longs;&longs;e pla
nis lateribus coni erectis, verùm etiam alijs modis: quod qui
dem exemplo ellip&longs;is manife&longs;tari optimè pote&longs;t. Nam in o
ctaua propo&longs;itione eiu&longs;dem libri ellip&longs;es latus coni ad angu
los rectos minimè &longs;ecant. veluti quoque in nona propo&longs;itione
po&longs;itionem inquit Archimedes.
eius lateribus coeunti, &longs;ectio vel erit circulus, vel acutianguli coni &longs;e
ctio.
gulo, verùm in omnibus conis&longs;ectionem ellip&longs;is cognoui&longs;&longs;e.
Præterea ex hocloquendi modo liquet ip&longs;um &longs;ectionem quo
&longs;ita,
propter &longs;i in omnibus conis ellip&longs;is nouit &longs;ectionem; cur in i
p&longs;is, & parabolas, & hyperbolas minùs animaduertit? cùm
&longs;it manife&longs;tum ex dictis in cono obtu&longs;iangulo &
& ellip&longs;im; in rectangulo autem parabolem, ellip&longs;imquè co
gnoui&longs;&longs;e? hòc certè non e&longs;t a&longs;&longs;erendum. Ex hoc enim per&longs;pi
cuum e&longs;t Archimedem cognoui&longs;&longs;e conos &longs;ecari po&longs;&longs;e planis,
quæ non &longs;int &longs;emper ad coni latus erecta. dormita&longs;&longs;equè Eu
tocium Geminum, & alios &longs;ecus hac in parte de Archimede
&longs;entientes. Ampliùs
ri po&longs;&longs;e rectangulos conoides, itidemquè &
nis, quæ neque &longs;int per axem ducta, neque axi æquidi&longs;tantia;
neque &longs;uper axem erecta. vt in duodecima, decimatertia, &
decima quarta propo&longs;itione eiu&longs;dem libri patet. quomodo i
taque his quoque modis quemlibet conum &longs;ecari po&longs;&longs;e igno
rauit? Non e&longs;t igitur ambigendum Archimedem cognoui&longs;
&longs;e conos &longs;ecari po&longs;&longs;e planis ad latus coni differentem inclina
tionem habentibus. Ex quibus per&longs;picuum e&longs;t, ip&longs;um in om
nibus conis omnes ine&longs;&longs;e &longs;ectiones omnino animaduerti&longs;&longs;e.
At &longs;i concedamus etiam &longs;ua tempe&longs;tate nondum &longs;ectioni
bus ip&longs;is propria fui&longs;&longs;e impo&longs;ita nomina; tam eam parabo
lem, quæ erat rectanguli coni &longs;ectio; quàm quæ erat &longs;ectio
alterius coni, cùm &longs;it eadem &longs;ectio, eodem nomine nuncu
pabat; nempè rectanguli coni &longs;ectionem. Et hoc, quia
priùs hæc &longs;ectio cognita &longs;uit in cono rectangulo (vnde &longs;i
bi nomen vindicauit) quam in alio. quod idem dicen
dum e&longs;t de alijs &longs;ectionibus. Vt manife&longs;tum e&longs;&longs;e pote&longs;t
exemplo &longs;ectionis acutianguli coni. Archimedes enim eo
dem loco, anteprimam &longs;cilicet propo&longs;itionem de conoidi
bus, & &longs;ph&ecedil;roidibus inquit,
stantibus &longs;ecetur; quæ cum omnibus ip&longs;ius lateribus coeant, &longs;ectio
nes, uelerunt circuli; uel conorum acutiangulorum &longs;ectiones.
catigitur Archimedes acutianguli coni &longs;ectionem, tam coni veluti
& decimaquarta propo&longs;itione
ctio ab ip&longs;o ea dum modo non &longs;it
ad axem erecta. nullaquè alia de cau&longs;a hæ &longs;ectiones omnes i
dem acutianguli coni &longs;ectionis nomen obtiuerunt; ni&longs;i quia
priùs hæc &longs;ectio à cono acutiangulo nomen accepit, quan do
quidem in ip&longs;o forta&longs;se primùm cognita fuit, quaàm in alijs.
Ex dictis itaque manife&longs;tum e&longs;t, &longs;ententiam Heraclij veram
e&longs;&longs;e po&longs;&longs;e, & rationi valdè con&longs;entaneam; Archimedem &longs;cili
cet elementa conica &longs;crip&longs;i&longs;&longs;e; Apollonium què, cùm ea ab Ar
chimede nondum edita inueni&longs;&longs;et, &longs;icut propria &longs;ua edidi&longs;&longs;e.
Omitto interim multa ab Archimede in eius libris &longs;upponi,
quæ non ni&longs;i in conicis e&longs;&longs;e dcbebant, quæ quidem
&longs;olùm in conicis Apolloni. Negandum tamen non e&longs;t, vt
Eutocius quoque affirmat, ip&longs;um Apollonium multa auxi&longs;&longs;e,
multaquè ad conica &longs;pectantia adinueni&longs;&longs;e. vt ip&longs;emet Apol
lonius in epi&longs;tola ad Eudemum fatetur. cùm tamen non &longs;it
&longs;emperfacilè inuentis addere. Sed de his hactenus. &longs;at &longs;it au
tem noui&longs;&longs;e, Archimedem,
tionem recta linea, rectanguliquè coni &longs;ectione contentam,
eam &longs;ignificare fectionem, quæ parabole nuncupatur.
poll.
GVIDIVBALDI
E MARCHIONIBVS
MONTIS.
IN SECVNDVM ARCHIMEDIS
ÆQVEPONDERANTIVM
LIBRVM.
PARAPHRASIS
SCHOLIIS ILLVSTRATA.
PROPOSITIO. I.
Si duo &longs;pacia recta linea, & re
ctanguli coni &longs;ectione conten
ta, quæ ad datam rectam
applicare po&longs;&longs;umus, non ha
beantidem grauitatis
magnitudinis ex vtri&longs;que i
p&longs;orum compo&longs;itæ centrum
grauitatis erit in recta linea, quæ ip&longs;orum centra
grauitatis coniungit; ita diuidens dictam rectam li
neam, vt ip&longs;ius portiones permutatim eandem ad
inuicem proportionem habeant, vt &longs;pacia.
ip&longs;orum autem centra
grauitatis &longs;int puncta EF.
H;
ad HE. o&longs;tendendum e&longs;t magnitudmis ex utri&longs;què AB CD &longs;pa
ciis compo&longs;itæ centrum grauitaias e&longs;&longs;e punctum H. &longs;it quidemip&longs;i EH
utraque ip&longs;arum FG FK æqualis; ip&longs;i autem FH, hocest GE
(&longs;untenim EH GF æquales, à quibus dempta communi
GH remanent EG HF &ecedil;quales)
FH e&longs;t æqualis LE, & FK ip&longs;i EH,
æqualis.
verò FH vtraque &longs;it æqualis LE EG. ip&longs;i autem HE vtra
que æqualis GF FK,
cùm &longs;it LG ad GK, vt FH ad HE;
EG GK
AB,
ip&longs;i LG æquidistantes ducantur
LG di&longs;tent, ductis &longs;cilicet MQ ON æquidi&longs;tantibus, &longs;int
LM LQ GO GN inter &longs;e æquales;
&longs;pacio AB æquale
ctum
oppo&longs;ita MQ ON parallelogrammi MN.
que &longs;pacium NX. habebit quidem MN. ad NX proportionem,
AB ad CD proportionem ip&longs;ius LG ad G
CD ad NX.
ip&longs;i NX æquale. Centrum autem grauitatisip&longs;ius
F.
grammi NX oppo&longs;ita latera ON XP bifariam &longs;ecat.
quoniam æqualis e&longs;t LH ip&longs;i HK, totaquè LK appa&longs;ita latera
XP
Verùm ip&longs;um MP æquale est utri&longs;que MN NX,
&longs;int centra grauitatis EF, æquepondera bunt &longs;pacia MN
NX ex di&longs;tantijs FH HE. &longs;i igitur loco parallelo gram mo
rum MN NX ponatur AB in E, & CD in F, cùm &longs;it
AB ip&longs;i MN, & CD ip&longs;i NX æquale; &longs;pacia AB CD ex
di&longs;tantijs FH HE æqueponderabunt.
nis ex utri&longs;que AB CD
H.
SCHOLIVM.
Cùm &longs;it intentio Archimedis non nulla pertractare ad pa
rabolen &longs;pectantia; primùm iacit fundamentum, parabolas
nempe ita &longs;e habere, vt permutatim di&longs;tantiæ, ex quibus
&longs;untcollocatæ, &longs;e habent. &
nibus mutuam hanc conuenientiam ex dictis ex primo libro
depræhendere liceat, hoc tamen loco peculiariter voluitad
huberiorem do ctrinam id ip&longs;um in parabolis demon&longs;trare.
& quamuis in primo libro dixerit Archimedes magnitudi
nes æqueponderare, quan do ita &longs;e habent inter &longs;e, ut di&longs;tan
tiæ permutatim &longs;e habent; hocautem loco quærit
uitatis magnitudinis ex parabolis compo&longs;itæ; non &longs;unt
propo&longs;itiones diuer&longs;æ. nam & in primo libro dum in demon
&longs;tratio ne quærit proportionem di&longs;tantiarum, o&longs;tendit, vbi
nam &longs;it centrum grauitatis magnitudinum. quare
po&longs;itiones videantur diuer&longs;æ, non &longs;unt tamen diuer&longs;æ, ete
nim vt po&longs;t tertiam primi libri propo&longs;itionem adnotauimus,
ctum H centrum e&longs;t grauitatis magnitudinis ex vtri&longs;que
AB CD compo&longs;itæ. ergo AB, & CD ex di&longs;tantijs HEHF
æqueponderant. & è contra. hoc e&longs;t AB CD æqueponde
rant ex di&longs;tantijs EH HF. ergo punctum H centrum e&longs;t
grauitatis magnitudinis ex vtri&longs;que AB CD compo&longs;rtæ;
&longs;it EHF recta linea. Solent autem mathematici aliquan do
eandem propo&longs;itionem pluribusmedijs demon&longs;trare; idcirco
con&longs;iderandum e&longs;t, Archimedem in hac propo&longs;itione alio v
ti medio ad o&longs;tendendum punctum H centrum e&longs;ie graui
tatis, quo u&longs;us e&longs;t in &longs;exta propo&longs;itione primi libri. cùm in pri
mo libro per diui&longs;ionem magnitudinum, diui&longs;io nem què di
&longs;tantiarum vniuer&longs;aliter domon&longs;tret centrum grauitatis ma
gnitudinum. hoc autem loco per parallelogramma MN
NX parabolis æqualia, & circa centra grauitatis EF con&longs;ti
tuta, in uenit centrum grauitatis magnitudinis ex vtri&longs;que pa
quod e&longs;t
ctum H. medium nempè totius parallelogrammi MP.
quod idem punctum H centrum e&longs;t grauitatis vtriu&longs;que pa
raboles AB CD in EF collocatæ.
huius.
Ex his ob&longs;eruandum occurrit, hanc e&longs;&longs;e peculiarem metho
dum, qua po&longs;&longs;umus quorumlibet planorum æquepondera
tionem o&longs;tendere; hoc e&longs;t plana ex di&longs;tantijs eandem permu
tatim proportionem habentibus, vt eadem met plana, æque
ponderare; dum modo ip&longs;is æqualia parallelogramma con&longs;ti
tuere po&longs;&longs;imus. ac propterea &longs;upponit Archimedes, nos po&longs;&longs;e
applicare ad rectam lineam &longs;pacium æquale &longs;pacio recta li
nea, rcctanguliquè coni &longs;ectione contento. quod
cium &longs;upponit parallelogram mum exi&longs;tere, cùm pun
ctum E centrum &longs;it grauitatis &longs;pacij MN, e&longs;t F
&longs;pacij NX. punctum verò H totius PM. quòd &longs;i MN
NX & MP non e&longs;&longs;ent parallelogramma, neque puncta EFH
eorum centra grauitatis exi&longs;terent. vt ex demon&longs;tranone pa
tet. &longs;uppo&longs;uit tamen Archimedes nos po&longs;&longs;e applicare ad re
ctam lineam parallelogrammum æquale &longs;pacio recta linea,
rectanguliquè coni&longs;ectione contento; quia duplici medio in
ptima, & vige&longs;imaquarta, docuit quamlibet portionem recta
linea, rectanguliquè coni &longs;ectione contentam &longs;e&longs;quitertiam
e&longs;&longs;e trianguli eandem ip&longs;i ba&longs;im habentis, &
lem. Ex qua propo&longs;itione facilè con&longs;tat nos parabol&ecedil;
ad rectam lineam applicare po&longs;&longs;e, vt propo&longs;itum fuit hoc
modo.
PROBLEMA.
Ad datam rectam lineam dat&ecedil; parabol&ecedil; &ecedil;quale parallelo
grammum applicare, ita vt data linea oppo&longs;ita
mi
Data &longs;it parabole
ABC, &longs;itquè data recta
linea GK. oportet ad
GK
applicare æquale por
tioni ABC, ita vt GK
bifariam diuidat oppo
&longs;ita parallelogram mi
latera. Con&longs;tituatur &longs;u
per AC
qd ba&longs;im habeat AC,
eandemque portionis
fiet,
AB BC. eritvtique parabole ABC trianguli ABC &longs;e&longs;quitertia.
Itaque diuidatur AC in tria &ecedil;qualia, quarum vna pars &longs;it CH.
producaturquè AC. fiatquè CL ip&longs;i CH &ecedil;qualis
ip&longs;ius AC &longs;e&longs;q uitertia. Et obid (iuncta BL) erit triangulum
ABL trianguli ABC &longs;e&longs;quitertium. &longs;unt quippè triangula ABL
ABC inter &longs;e, vt ba&longs;es AL AC. ac per con&longs;equens triangulum
ABL patabol&ecedil; ABC exi&longs;tit &ecedil;quale. Applicetur itaque ad linea
GK
læ ABC &ecedil;quale. deinceps ducatur NP ip&longs;i GK
&ecedil;quidi&longs;tans, qu&ecedil; bifariam diuidat oppo&longs;ita latera GR
KS. producanturquè RG SK. fiantquè GO KX &ecedil;
quales ip&longs;is GN KP. iungaturquè OX; erit nimi-
rum parallelogram mum OP ip&longs;i GS &ecedil;quale. qua
re parallelogram mum OP parabol&ecedil; ABC exi&longs;tit &ecedil;
quale. Applicatum e&longs;t igitur ad GK parallelogram
mum expo&longs;it&ecedil; parabol&ecedil; &ecedil;quale. lineaquè GK paralle
logrammi OP bifariam diuidit oppo&longs;ita latera ON
XP. quod fieri oportebat.
conicorum
Apoll.
ch. dquad.
patab.
mi.
Si in portione recta linea rectanguliquè coni
&longs;ectione contenta triangulum in&longs;cribatur,
ba&longs;im cum portione habens, & altitudinem æqua
lem: & rur&longs;us in reliquis portionibus triangula in
&longs;cribantur, quæ ea&longs;dem ba&longs;es cum portionibus
habeant, & altitudinem æqualem; &longs;emper què in
re&longs;iduis portionibus triangula eodem modo
in&longs;cribantur: figura, quæ in portione oritur,
planè in&longs;cribi dicatur. Patet quidem lineas
portionis proximi, eo&longs;què deinceps coniungen
tes, ba&longs;i portionis æquidi&longs;tantes e&longs;&longs;e; bifariamquè
à diametro portionis diuidi; diametrum verò in
proportione diuidere numeris deinceps impari
bus. vno deno minato ad verticem portionis. Hoc
autem ordinate o&longs;ten&longs;um e&longs;t.
SCHOLIVM.
Scopus Archimedis in hoc &longs;ecundo libio, vt initio primi
diximus, e&longs;t inuenire centrum grauitatis paraboles. & vt de
ducatnosin hanc cognitionem, quadam vtitur figura rectili
nea in parabole in&longs;cripta, qu&ecedil; plurimùm conducit, & e&longs;t
quam medium ad inueniendum hoc grauitatis centrum. his
igitur verbis docet, quo modo in parabole in &longs;criben da &longs;it h&ecedil;c
figura; in quibus multa quo que proponit tanquam &longs;it pro
po&longs;itio quædam; in qua multa &longs;int o&longs;ten denda. quorum ta
m&ecedil;n demon&longs;trationem omi&longs;it, ac tanquam ab eo alibi de
mon&longs;tratam. Horum autem ex Apollonij Perg&ecedil;i conicis
demon&longs;trationem elicere quidem potui&longs;&longs;emus. at quoniam
Archimedes ip&longs;e non nulla ad hæ c&longs;pectantia alijs in locis de
mon&longs;trauit ideo Archimedem per Archimedem declarare o
portunum magis nobis vi&longs;um e&longs;t.
Sit portio contenta recta linea, rectanguliquè coni &longs;ectio
ne ABC, cuius diameter BD. Iunganturquè AB BC, diuida
tur deinde AB bifariam in E, a quo ip&longs;i BD æquidi&longs;tans
vt Archimedes demon&longs;trauit in libro de quadratura parabo
les propo&longs;itione decimaoctaua. iunganturque AF FB. rur
fus bifariam diuidantur AF FB in punctis GH, à quibus
ip&longs;i BD ducantur æquidi&longs;tantes GI HK
&longs;am erit punctum I vertex portionis AIF. K verò portio
nis FKB. connectanturquè AI IF FK KB. eademquè pror
fus ratione ad alteram partem in&longs;cribantur triangula CLB
CML, & LNB. Primùm
planè in&longs;criptum, vt Archimedes ip&longs;e infra in demon&longs;tratio
nibus quintæ, &longs;extæ, & octauæ propo&longs;itionis nominat. Dein
de figura AFBLC, figuraquè AIFKBNLMC dicuntur in
portione planè in&longs;criptæ. figuraquè AFBLC vna cum AC
des in &longs;ecunda parte demon&longs;trationis quintæ propo&longs;itionis
huius libri nuncupat. ideòquè erit AIFKBNLMC nonago
num in portione planè in&longs;criptum. & ita in alijs.
STV. o&longs;tendendum e&longs;t, lineas KN FL IM ba&longs;i AC &ecedil;qui
di&longs;tantes e&longs;&longs;e. deinde diametrum BD lineas KN FL IM
bifariam in punctis STV diuidere po&longs;tremo lineas KN F
IM ita diametrum BD di&longs;pe&longs;cere, vt po&longs;ito vno BS, linea ST
&longs;it tria, TV quinque; & VD &longs;eptem. Producantur FE KH
ad RX. quoniam enim FR e&longs;t æquid
EB, vt AR ad RD; e&longs;tque AE ip&longs;i EB æqualis ergo AR i
p&longs;i RD æqualis exi&longs;tit. eodem què modo o&longs;tendetur FX æ
qualem e&longs;&longs;e XT. quandoquidem e&longs;t FX ad XT, vt FH ad
HB. &longs;imiliterquè ad alteram partem, exi&longs;tentibus LO NP i
p&longs;i BD æquidi&longs;tantibus, erit DO ip&longs;i OC æqualis, & TP
ip&longs;i PL. quod quidem eodem pror&longs;us modo demon&longs;trabi
tur. Quoniam autem AC bifariam à diametro diuiditur in
puncto D, erit DR ip&longs;i DO æqualis, cùm vnaquæque &longs;it
dimidia ip&longs;arum AD DC æqualium. e&longs;t igitur RD dimidia
ip&longs;ius AD, quæ dimidia e&longs;t ba&longs;is AC. quod idem euenit ip&longs;i
DO. quare BD &longs;e&longs;quitertia e&longs;t ip&longs;ius FR, & ip&longs;ius LO, ex de
cimanona Archimedis de quadratura paraboles. ac propterea
eandem habet proportionem BD ad FR, quam ad LO. vnde
&longs;equitur FR æqualem e&longs;&longs;e ip&longs;i LO. & obid FL ip&longs;i AC
quidi&longs;tantem& FT ip&longs;i RD, & TL ip&longs;i DO &ecedil;qualem.
vnde FT ip&longs;i TL &ecedil;qualis exi&longs;tit. eadem quèratione pror&longs;us in
portione FBL o&longs;tendetur KN ip&longs;i FL, ac per con&longs;equens i
p&longs;i AC &ecedil;quidi&longs;tantem e&longs;&longs;e. & KS ip&longs;i SN æqualem exi&longs;te
re. Producatur IG ad Z, quæ ip&longs;am AB &longs;ecet in 9. linea ve
rò LO &longs;ecet BC in
ip&longs;am &longs;ecet BC in
rit AG ad GF, ut A9 ad 9E, & AZ ad ZR. & e&longs;t AG ip&longs;i
GF æqualis, erit igitur A9 ip&longs;i 9E, & AZ ip&longs;i ZR æquaiis.
Eodemquè modo o&longs;tendetur C
qualem e&longs;&longs;e. quo niam autem in portione AFB a dimidia ba&longs;i
ducta e&longs;t LF, à pun cto autem 9, hoc e&longs;t à dimidia dimidi&ecedil; ba
&longs;is AB (e&longs;t enim E9 dimidia ip&longs;ius AE, quæ dimidia e&longs;t ba&longs;is
AB) ducta e&longs;t 9I diametro æquidi&longs;tans, erit EF &longs;e&longs;quitertiai
p&longs;ius I9 parique ratione o&longs;tendetur QL &longs;e&longs;quitereiam e&longs;&longs;e i
p&longs;ius M
ad AR. eademqueiatione ita &longs;ehabet BD ad QO, vt DC
ad CO. Sed vt DA ad AR, ita e&longs;t DC ad CO, e&longs;t quip
pe DA ip&longs;ius AR dupla, veluti DC ip&longs;ius CO. quare i
o&longs;ten&longs;a verò e&longs;t RF &ecedil;qualis OL, reli
quaigitur EF reliquæ QL e&longs;t æqualis, quia verò ita e&longs;t FE
ad M
lis exi&longs;tit. quoniam autem ob triangu&longs;oium &longs;imilitudinem
AER A9Z, ita e&longs;t AR ad AZ, vt ER ad 9Z. ob &longs;imili
tudinem vero triangulorum QOC
vt QO ad
QO ad
vero ER ip&longs;i QO, æqualis, ergo 9Z ip&longs;i at
vero o&longs;ten&longs;a e&longs;t I9 &ecedil;qualis M
&longs;e parallelæ. quare IM ip&longs;i AC e&longs;t æquidi&longs;tans. Quoniam
que AR e&longs;t æqualis CO, & horum dimidia, hoc e&longs;t RZ ip&longs;i
OY æqualis erit. atqui DR e&longs;t ip&longs;i DO æqualis; ergo DZ ip&longs;i
DY exi&longs;tit æqualis. ip&longs;i verò DZ e&longs;t æqualis IV, & ip&longs;i DY æ
qualis VM. eruntigitur IV VM inter &longs;e equales. Iam itaque
o&longs;ten&longs;um e&longs;t, lineas KN FL IM, qu&ecedil; coniunguntangulos fi
guræ in parabole planè in&longs;criptæ, ip&longs;i AC æquidi&longs;tantes e&longs;&longs;e.
Diametrum què BD ip&longs;as in punctis STV bifariam di&longs;pe&longs;cere.
ex
ti
Quoniam itaque in portione FBL à dimidia ba&longs;i ducta e&longs;t
TB, a dimidia verò dimidiæ ba&longs;is ducta e&longs;t XK, erit BT
quitertia ip&longs;ius KX, hoc e&longs;t ip&longs;ius ST. e&longs;t enim KT parallelo
grammum, & ST ip&longs;i KX æqualis. Si igitur ponatur BT
quattuor, erit ST tria, & BS vnum. &longs;imiliter quoniam BD
&longs;e&longs;quitertia e&longs;t ip&longs;ius FR, hoc e&longs;t ip&longs;ius TD, cùm &longs;it TD ip&longs;i
FR &ecedil;qualis. &longs;i ita que ponatur BD &longs;exdecim, erit vnaquæque
FR TD duodecim. & TB quattuor, vt po&longs;itum fuit.
autem (vt diximus) e&longs;t BD ad ER, vt DA ad AR, erit BD du
pla ip&longs;ius RE. quare &longs;i BD e&longs;t &longs;exdecim, erit RE octo. & quo
niam e&longs;t FR duodecim, erit EF quatuor. e&longs;t autem FE ip&longs;ius
I9 &longs;e&longs;quitertia, erit igitur I9 tria. & quoniam e&longs;t ER ad 9Z, vt
RA ad AZ, erit ER dupla ip&longs;ius 9Z. ac propterea erit 9Z quat
tuor, cum &longs;it ER octo, & e&longs;t 9I tria, tota ergo IZ, hoc e&longs;t DV,
&longs;eptem exi&longs;tet. &longs;ed quoniam e&longs;t DT duodecim, cuius pars
DV e&longs;t &longs;eptem, eritreliqua VT quinque. Po&longs;ito igitur BS v
no, erit ST tria, TV quinque, & VD &longs;eptem. quod erat quo
que demon&longs;trandum. Et hæc &longs;unt qu&ecedil; ab Archimede pro
po&longs;ita fucrant.
medis de
quad. pa
rab.
Ex his tamen nonnulla quoque colligemus ad ea, quæ &longs;e
quuntur nece&longs;&longs;aria. ac primùm quidem con&longs;tat BD quadru
plam e&longs;&longs;e ip&longs;ius BT, & ip&longs;ius FE.
O&longs;ten&longs;um e&longs;t enim BD &longs;exdecim e&longs;&longs;e, & BT quatuor, & FE
itidem quatuor exi&longs;tere. Ex demon&longs;tratione autem Archime
dis decimæ nonæ ptopo&longs;itionis de quadratura paraboles cla
rè elicitur BD quadruplam e&longs;&longs;e ip&longs;ius BT.
Ex quibus etiam &longs;equitur FE QL inter &longs;e æquales e&longs;&longs;e. am
bo enim &longs;unt, vt quatuor.
Præterea o&longs;tendendum e&longs;t triangulum AFB
&ecedil;quale e&longs;&longs;e, portionem què paraboles AFB portiom BLC &ecedil;qua
lem. Ampliùs triangulum AIF triangulo CML, & portio
nem AIF portioni CML æqualem e&longs;&longs;e, & reliqua triangula
reliquis triangulis, acportiones portionibus &ecedil;quales e&longs;&longs;e.
Ex vige&longs;ima prima propo&longs;itione Archimedis de quadratu
ra paraboles triangulum ABC vniu&longs;cuiu&longs;que trianguli AFB
re triangula AFB BLC inter &longs;e &longs;unt &ecedil;qualia. At vero
portio BLC trianguli BLC, eritportio AFB ad triangulum
AFB, vt portio CLB ad triangulum CLB, & permutando
portio AFB ad portionem CLB, vt triangulum AFB ad
ip&longs;um CLB
CLB inter &longs;e &longs;unt æquales. Eademquè ratione
octuplum e&longs;t trianguli AIF, & triangulum CLB octuplum
ip&longs;ius CML. vnde triangula AIF CML &longs;unt æqualia. et ea
rum quoque portiones AIF CML &longs;unt æquales, &longs;iquidem
&longs;unt triangulorum &longs;e&longs;quitertiæ. Et hoc modo reliqua trian
gula FKB LNB, & portiones FKB LNB
les. cùm &longs;it triangulum FBL dictorum triangulorum octu
plum. quod oportebat quoque demon&longs;trate.
chimedis
de quad.
parab.
21.
medis de
quad. pa
rab.
His demon&longs;tratis &longs;equitur Archimedes qua&longs;i connectens &longs;e
quentem propo&longs;itionem cumijs, quæ &longs;uppo&longs;ita &longs;unt, inqui
ens,
PROPOSITIO. II.
Si autem & in portione rectalinea, rectangu
lique coni &longs;ectione contenta, figura rectilinea pla
ne in&longs;cribatur, in&longs;criptæ figuræ centrum grauita
tis erit in diametro portionis.
linea figura AEFGBHIKC. portionis verò diameter &longs;it BD.
gantur GH FI EK. qu&ecedil; ip&longs;i AC, & inter &longs;e &ecedil;quidi&longs;tantes
erunt. h&ecedil; verò lineæ diametrum BD &longs;ecentin punctis NML
tro BD diui&longs;æ in punctis NML, trapezium AEKC duas
DL, quare
eandem cau&longs;am
FGHI centrum est in MN.
GBH centrum grauitatis e&longs;t in BN.
GH bifariam diuidat.
AEFGBHIKC
mon&longs;trare oportebat.
stratis.
huius.
huius.
SCHOLIVM.
Ecce qúo Archimedes incipit inue&longs;tigare centrum graui
tatis paraboles. nam ex hoc, quod o&longs;tendit centrum grauita
tis figuræ in portione planè in&longs;criptæ e&longs;&longs;e in diametro por
tionis, &longs;tatim colliget in quarta propo&longs;itione centrum graui
tatis paraboles in diametro quoque ip&longs;ius portionis exi&longs;tere.
interponitautem Archimedes &longs;equentem propo&longs;itionem.
antequam inueniat centrum grauitatis paraboles, opus habet
prius o&longs;tendere centra grauitatis duarum, & vt ita dicam om
nium parabol
ad quod demon&longs;trandum, hanc
ptis priùs accidere
tam propo&longs;itionem o&longs;tendere, quam tertiam; &longs;equentem ve
rò propo&longs;itionem immediatè po&longs;uit po&longs;t &longs;ecundam, ordo e
nim &longs;ic po&longs;tulat. etenim ambæ deijs pertractant, quæ rectili
neis figuris plane in&longs;criptis accidunt. Pr&ecedil;terea earum demon
&longs;trationes ferè circa eadem ver&longs;antur, cùm ijsdem rectis lineis
in portionibus eodem modo ductis vtantur; ob &longs;equentis ve
rò propo&longs;itionis in telligentiam h&ecedil;c priùs o&longs;tendemus.
LEMMA I.
Eandem habeat proportionem AB ad CD, quam habet
GH ad KL. CD verò ad EF &longs;imiliter GH KL MN
æquidi&longs;tantes, &longs;intautem ductæ BDF HLN rectæ lineæ; &longs;it
què BD ad DF, vt HL ad LN. &longs;itquè maior AB quàm
CD, & CD, quàm EF. vnde erit quoquè GH maior KL,
& KL, quam MN. iuncti&longs;què AC CE, & GK KM.
Dico &longs;pacium ACDB ad &longs;pacium CEFD eandem habere
proportionem, quam &longs;pacium GKLH ad &longs;pacium KMNL.
Producantur AC CE, quæ cum BF conueniantin OP.
productæquè GK KM cum HN conueniant in QR.
concurrentenim, quoniam CD KL &longs;unt minores ip&longs;is AB
ad CD, ita CD ad V. & vt GH ad kL, ita KL ad X.
deinceps CD ad EF, ita EF ad Y. & vt KL ad MN,
ita MN ad Z. Quoniam igitur triangulum ABO &longs;imile
e&longs;t triangulo CDO, cùm &longs;it CD æquidi&longs;tansip&longs;i AB. ha
bet AB ad CD duplicatam. hoc e&longs;t quam hab et AB ad
V. Eodemquè modo o&longs;ten detur
ita e&longs;&longs;e, vt GH ad X
triangulum igitur ABO eandem habet proportionem ad
do &longs;pacium ACDB ad triangulum CDO e&longs;t, vt &longs;pacium
GKLH ad triangulum
lorum &longs;imilitudinem ABO CDO, ita e&longs;t AB ad CD, vt
BO ad OD. &longs;imiliter ob &longs;imilitudinem
KLQ ita e&longs;t GH ad kL, vt HQ ad QL. & e&longs;t AB ad CD,
vt GH ad KL, erit BO ad OD, vt HQ ad QL. &
dendo BD ad DO, vt HL ad
ad DB, vt LQ ad LH. & e&longs;t BD ad DF, vt HL ad LN, erit
ex &ecedil;quali DO ad DF, vt LQ ad LN. Quoniam autem &longs;imi
lium triangulorum CDP EFP latus CD ad latus EF ita &longs;e
habet, vt DP ad PF. &longs;imiliter exi&longs;tentibus &longs;imilibus triangu
lis KLR MNR ita e&longs;t KL ad MN, vt LR ad RN, & vt CD
ad EF, ita e&longs;t KL ad MN, erit DP ad PF, vt LR ad RN.
& per conuer&longs;ionem rationis PD ad DF, vt RL ad LN. &
conuertendo DF ad DP, vt LN ad LR. diximus
ad DF ita e&longs;&longs;e, vt QL ad LN, & e&longs;t DF ad DP, vt LN ad
LR. ergo ex &ecedil;quali erit OD ad DP, vt QL ad LR. At verò
quoniam ita e&longs;t OD ad DP, vt triangulum OCD ad PCD,
& vt QL ad LR, ita e&longs;t triangulum QKL ad
erit OCD ad PCD, vt QKL ad RKL. Quoniam
gula CDP EFP &longs;unt &longs;imilia, triangulum CDP ad triangulum
EFP proportionem habebit, quam CD ad EF duplicatam,
hoc e&longs;t quam habet CD ad Y, cùm &longs;int CD EF Y propor
tionales. &longs;imiliter ob triangulorum KLR MNR &longs;imilitudi
nem triangulum KLR ad MNR, ita erit vt KL ad Z, e&longs;t au
tem CD ad Y, vt KL ad Z, erit igitur
EFP, vt KLR ad MNR, & diuiden do
gulum EFP, vt &longs;pacium KMNL ad triangulum MNR. &
uertendo triangulum EFP ad &longs;pacium CEFD, vt
MNR ad &longs;pacium KMNL. Itaque quoniam o&longs;ten&longs;um e&longs;t i
ta e&longs;&longs;e &longs;pacium ACDB ad triangulum CDO, vt &longs;pacium
GKLH ad triangulum
gulum CDP, ita triangulum KLQ ad
de, vt triangulum CDP ad triangulum EFP, ita
KLR ad triangulum MNR; deniquè vt triangulum EFP ad
&longs;pacium CEFD, ita triangulum MNR ad &longs;pacium kMNL,
CEFD, vt &longs;pacium GKLH ad &longs;pacium KMNL. quod
&longs;trare oportebat.
cor.
ti.
ti.
cor.
ex
ti.
LEMMA II.
&longs;tantes EF GH, &longs;itquè maior AB, quàm CD, & EF, quam
GH. & &longs;uper CD GH &longs;int triangula CDP GHR,
FHR rectæ lineæ, & vt BD ad DP, ita &longs;it FH ad HR.
AC EG. Dico &longs;pacium ACDB ad
&longs;pacium EG HF ad triangulum GHR.
Eadem enim pror&longs;us ratione productis AC EG, quæ cum
BP FR conueniantin OQ, o&longs;tendetur &longs;pacium AD ad trian
gulum CDO ita e&longs;&longs;e, vt &longs;pacium EH ad triangulum
e&longs;&longs;e OD ad DB, ut QH ad HF. & quoniam e&longs;t BD ad DP, vt
ad DP, ita e&longs;t triangulum CDO ad triangulum CDP, & vt
QH ad HR, ita triangulum GHQ ad GHR. cùm itaque &longs;it
AD ad CDO, vt EH ad GHQ, & vt CDO ad CDP, ita
CDP, vt &longs;pacium EH ad triangulum GHR. quod demon&longs;tra
re oportebat.
LEMMA. III.
Sit A ad CD, vt E ad FG, diuidan
ita vt &longs;it CH ad HD, vt FK ad KG.
Dico A ad DH ita e&longs;&longs;e, vt E ad KG.
A verò ad CH, vt E ad Fk.
Quoniam enim ita e&longs;t CH ad HD, vt FK ad kG; e
rit componendo CD ad DH, vt FG ad GK. e&longs;t autem A
ad CD, vt E ad FG; CD verò e&longs;t ad DH, vt FG ad G
go ex æquali A erit ad DH, vt E ad GK. Deinde
niam e&longs;t GH ad HD, vt FK ad kG; erit conuertendo
DH ad HC, vt GK ad KF. rur&longs;us igitur ex æquali A e
rit ad CH, vt E ad FK. quod o&longs;tendere oportebat.
ti
PROPOSITIO. III.
Siin
ctanguliquè coni &longs;ectione contentarum rectili
neæ figuræ planè in&longs;cribantur; figuræ verò in&longs;cri
ptæ latera inter &longs;e multitudine æqualia habeant;
rectilinearum centra grauitatum portionum dia
metros &longs;imiliter &longs;ecabunt.
lineæ figuræ
inter &longs;e numero æqualia habeanta, Diametri verò portionum &longs;int BD
erunt; bifariam què à diametro BD in punctis LMN diui&longs;æ e
runt. Iungantur &longs;imiliter
meter OR in punctis 9eruntquè ductæ lineæ ip&longs;i
XP, & inter &longs;e æquidi&longs;tantes.
æquidi&longs;tantibus
ribus;
& LD &longs;eptem. &longs;ed
portionibus diuiditur numeris deinceps imparibus,
ratione &longs;i ponatur O
&longs;eptem.
quales.
tet diametrorum portiones in eadem e&longs;&longs;e proportione
e&longs;t BN ad NM, & NM ad ML, & ML ad LD, ita e&longs;&longs;e O
vt RO ad O9; (&longs;unt.n.ut &longs;exdecim ad nouem) & ut DB ad BL,
ita e&longs;t quadratum ex AD ad
ita e&longs;t
AD ad
ergo ut AD ad EL, ita XR ad S9. & horum dupla
EK, vt XP ad ST:
ad BM, vt 9O ad O
EL ad FM ita e&longs;&longs;eut S9 ad Y
ita e&longs;&longs;e, ut ST ad YV.
licet quatuor ad vnum; &longs;imiliter o&longs;tendetur FM ad GN ita e&longs;&longs;e
vt Y
&longs;olùm portiones diametrorum (ut dixim us) in eadem e&longs;&longs;e pro
portione, &longs;ed & T rapeziorum ip&longs;ius quidem AE
eandem habeant proportionem AC EK, quam XP ST.
LD 9R bifariam diuidant &longs;uas æquidi&longs;tantes AC EK.
& XP ST. etenim &longs;i ponatur trapezij AK centrum graui
tatis
vt dupla ip&longs;ius AC cum EK ad duplam ip&longs;ius EK
cum AC. & 9
ST ad duplam ST cum XP. quoniam autem ita e&longs;t AC ad EK,
p&longs;ius AC ad EK erit, vt dupla ip&longs;ius XP ad ST.
& componendo dupla ip&longs;ius AC cum EK, vt dupla
p&longs;ius XP cum ST ad ST. At verò EK ad duplam
ip&longs;ius EK, ita e&longs;t, vt ST ad duplam ip&longs;ius ST, &longs;ed EK
ad AC e&longs;t, vt ST ad XP, erit EK ad vtra&longs;que con&longs;e
quentes &longs;im ul &longs;umptas, hoc e&longs;t ad duplam ip&longs;ius EK cum
AC, vt ST ad &longs;uas con&longs;equentes, nempe ad duplam ip&longs;ius
ST cum XP. Itaque quoniam ita e&longs;t dupla ip&longs;ius AC
ad duplam ip&longs;ius EK cum AC, vt ST ad duplam ip&longs;ius
ST cum XP. erit ex &ecedil;quali dupla ip&longs;ius AC cum EK ad du
plam ip&longs;ius EK cum AC, vt dupla ip&longs;ius XP cum ST ad
duplam ip&longs;ius ST cum XP. ac propterea ita e&longs;t L
vt 9
militer po&longs;ita.
tur)
militer
ta vt &longs;it L
grauitatum
O
tem habent proportionem Trapezia, & triangula:
&longs;it AD ad EL, vt XR ad S9, & ut EL ad FM, ita S9 ad Y;
e&longs;tquè DL ad LM, ut R9 ad 9
que; erit &longs;pacium AL ad &longs;pacium EM, vt &longs;pacium X9 ad
cium S. &longs;imiliterquè o&longs;tendetur DK ad LI ita e&longs;&longs;e, vt RT
ad 9V. quare totum trapezium AK ad EI e&longs;t, vt XT ad SV.
pariquè ratione o&longs;tendeturita e&longs;&longs;e trapezium EI ad FH, vt
SV ad YZ. quia verò ita e&longs;t FM ad GN, vt Y
e&longs;t autem MN ad NB, vt
vnum, erit &longs;pacium FN ad triangulum GBN, vt &longs;pacium
Y
e&longs;&longs;e &longs;pacium IN ad triangulum BNH, vt &longs;pacium V
triangulum O
ad triangulum BGH, vt trapezium YZ ad
ad EI. erit punctum
&longs;imiliquè modo diuidatur
zium XT ad SV; erit punctum
XSYVTP. quia verò ita e&longs;t AK ad EI, vt XT ad SV, erit
ad
FH ad triangulum BGH, erit punctum
figuræ FGBHI. eademquè ratione diuidatur
trum grauitatis figuræ YQOZV. &longs;ed e&longs;t FH ad BGD, vt YZ
ad OQZ, eritigitur
ita e&longs;t Ak ad EI, vt XT ad SV, erit componendo AEFIKC
ad EI, vt figura XSYVTP ad SV; & e&longs;t EI ad FH, vt SV ad
YZ. ergo ex æquali figura AEFIKC erit ad FH, vt figura
XSYVTP ad YZ. e&longs;t autem FH ad BGH, vt YZ ad OQZ. e
ritigitur figura AEFIKC ad &longs;uas con&longs;equentes, ad figuram
&longs;cilicet FGBHI, vt figura XSYVTP ad &longs;uas con&longs;equentes, hoc
e&longs;t ad figuram YQOZV. Diuidatur itaque
ad
uidatur
ram YQOZV, erit punctum
guræ XSYQOZVTP. quia verò ita e&longs;t figura AEFIKC ad fi
guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e
rit
ad R9, cùm &longs;in^{4} ut&longs;exdecim ad &longs;eptem. & e&longs;t L
ad
decim ad quinque; & e&longs;t L
vt OR ad 9
vtram que &longs;imul
e&longs;t
ad D
ad R
Cùm itaque &longs;it BD ad DR, & ad B
rit BD ad DR B
do tota BD ad totam OR, vt ablata D
rur&longs;u&longs;què permutando
BD ad
eritigitur BD ad B
ita &longs;e habet ad
lineæ figuræ in portione ABC in&longs;criptæ centrum grauitatis
proportione diuidere BD, veluti centrum grauitatis
in portione XOP
re oportebat.
po&longs;t
mi huius
demon&longs;tra
ta &longs;unt.
de quad.
parab. &
20,
conicorum
Apoll.
huius.
buius.
te
mi huius.
mi huius.
mi huius.
ma m
mi huius.
te
mi huius.
3.
2.
te
mi huius
16.
co.
3.
ante
18.
SCHOLIVM.
Hinc colligere licet parabolas omnes inter &longs;e &longs;imiles e&longs;&longs;e. Re
fert enim Eutocius hoc in loco, Apollonium perg&ecedil;um in &longs;ex
to Conicorum libro. (qui nondum in lucem prodijt) &longs;imiles
coni &longs;ectiones dixi&longs;&longs;e eas e&longs;&longs;e, quando in vnaquaque &longs;ectione
line&ecedil;
na, quotin alia; vt in &longs;uperioribus figuris ductæ fuerunt, in v
na quidem EK FI GH ip&longs;i AC æquidi&longs;tan tes; & in altera ST
YV QZ ip&longs;i PX æquidi&longs;tantes; qu&ecedil; quidem efficiant, vt dia
metri in eadem proportione diui&longs;æ proueniant; vt&longs;unt BN
NM ML LD; & O
FI GH in eadem &longs;int proportione ip&longs;arum XP ST YV QZ.
& quoniam hæ conditiones in omnibus po&longs;&longs;unt accidere pa
rabolis; vt ex ijs, quæ demon&longs;trata &longs;unt, manife&longs;tum e&longs;t; id
circo parabolæ omnes &longs;unt &longs;imiles. Neque verò
e&longs;t, quoniam parabolæ &longs;unt &longs;imiles, figur as quoque planè
in&longs;criptas, vt AEFGBHIKC & XSYQOZVTP &longs;imiles e&longs;&longs;e in
ter &longs;e, ea præ&longs;ertim &longs;imilitudine, qua &longs;unt figuræ rectilineæ;
vt&longs;cilicet anguli &longs;int æquales, & circum &ecedil;quales angulos late
ra proportionalia. in parabolis
&longs;atenim e&longs;t, vt præfatæ ad&longs;int conditiones; ex quibus &longs;equi
tur (vt o&longs;tendimus) trapezia AK EI FH, triangulum què
BGH in eadem e&longs;&longs;e proportione trapeziorum XT SV YZ, ac
&longs;itione inquit
liquè coni &longs;ectione contentarum,
reperiri po&longs;&longs;e aliquas parabolas recta linea terminatas no e&longs;&longs;e
&longs;imiles inter &longs;e; ea nimirumiam explicata &longs;imilitu dine. &longs;unte
nim Archimedis verba hoc modo intelligenda, nempè, &longs;i in
vtraque portionum recta linea rectanguliquè coni &longs;ectione
contentarum, quæ omnes &longs;unt &longs;imiles, & c. veluti &longs;i dicere
mus. In &longs;imilibus &longs;emicirculis anguli omnes &longs;untrecti. non
e&longs;t in telligendum nonnullos &longs;emicirculos inter &longs;e di&longs;&longs;imiles
exi&longs;tere po&longs;&longs;e. &longs;ed hoc modo; in &longs;emicirculis, qui omnes &longs;unt
&longs;imiles, anguli&longs;unt recti. Et hoc modo &longs;emperintelligere o
portet, quando in &longs;equentibus Archimedes parabolas &longs;imiles
nominat. Nam & Archimedes cognouit omnes parabolas
inter &longs;e &longs;imiles e&longs;&longs;e; vt ip&longs;e in demon&longs;tratione octauæ propo&longs;i
tionis huius &longs;upponere videtur. Oportebatenim aliquam in
parabolis demon&longs;trare &longs;imilitudinem, vt dem on&longs;trari po&longs;&longs;et
centrum grauitatis in omnibus parabolis e&longs;&longs;e in certo, ac de
terminato &longs;itu ip&longs;ius figuræ. in figuris enim, quæ aliquam in
ter&longs;e non habent &longs;imilitudinem, in ip&longs;is centrum grauitatis
determinari minimè po&longs;&longs;e videtur. Dicet autem forta&longs;&longs;e ali
quis, determinatur tamen centrum grauitatis in omnibus
gulis, quæ quidem in ter&longs;e non &longs;unt&longs;imilia. Cui re&longs;ponden
dum; triangula omnia inter &longs;e &longs;imilia non e&longs;&longs;e &longs;imilitudine
rectilinearum figurarum, nempè vt anguli &longs;intæquales, & cir
cum æqualesangulos latera proportionalia. quòd tamen nul
lam inter &longs;e&longs;e habeant conuenientiam, omnino negatur.
triangula omnia &longs;imul quodam modo illam habent conue
nientiam, & &longs;imilitudinem; quæ parabolis accidit.
In triangulis enim ABC DEF duct&ecedil; &longs;int AG DH ab angu
lis ad dimidias ba&longs;es. &longs;intquè diui&longs;a triangulorum latera in ea
dem proportione, in punctis kL, OP. & vt AK KL LB, ita &longs;it
AM MN NC, & DQ QR RF. ducti&longs;què KM LN OQ PR,
quæ lineas AG DH &longs;ecentin punctis ST VX; primùm
erunt KM LN OQ PR ba&longs;ibus BC EF æquidi&longs;tantes; quas
lineæ AG DH in punctis ST VX bifariam diuident, cùm &longs;it
PX ad XR, & OV ad
LN OQ PR in eadem proportione diui&longs;æ; &longs;iquidem ita e&longs;t
AS ST TG, ut DV VX XH. cùm &longs;int, ut expo&longs;itæ propor
tiones AK KL LB, & DO OP PE. Præterea erit &longs;pacium,
BN ad LM, vt ER ad PQ, & LM ad triangulum AK M,
vt PQ ad triangulum
&longs;imile e&longs;t triangulo ALN, oblatus LN ip&longs;i BC æquidi&longs;tans;
erit ABC ad ALN, ut AB ad AL duplicata. eodemquè modo
erit DEF ad DPR, vt DE ad DP duplicata; eandem aut
habet proportionem AB ad AL, quam DE ad DP: quadoqui
dem latera AB DE in eadem &longs;unt proportione diui&longs;a; eritigi
tur triangulum ABC ad ALN, vt triangulum DEF ad DPR.
&longs;imiliterquè o&longs;tendetur ALN ad AkM ita e&longs;&longs;e, ut DPR ad
ALN ad AKM e&longs;t, vt DPR ad
nem rationis ALN ad LM, vt DPR ad
ALN ad AKM, ut DPR ad
AKM, vt PQ ad
LM, vt ER ad PQ, & LM ad triangulum AKM,
vt PQ ad triangulum
e&longs;t omnia triangula aliquam inter &longs;e habere &longs;imilitudinem,
ex qua po&longs;&longs;ibile fuit determinare in omnibus &longs;itum, vbQuòd &longs;i figur&ecedil; nullam conuenien
tiam, nullamquè &longs;imilitudinem inter &longs;e habuerint; ut in qua
drilateris, pentagonis, & reliquis figuris, quæ inter &longs;e neque
latera neque angulos &ecedil;quales
ter&longs;e conuenientiam, & &longs;imilitudinem habere po&longs;&longs;unt; im
po&longs;&longs;ibile quidem e&longs;&longs;et in ip&longs;is determinare &longs;itum
tis; ita vt omnibus quadrilateris, ac omnibus pentagonis quo
modo cunque factis, & ita c&ecedil;teris figuris de&longs;eruire po&longs;&longs;it. Cum
exempli gratia in pentagonis modò in vno, modò in alio &longs;i
tu centrum reperiatur; prout &longs;unt diuer&longs;&ecedil; figuræ. Po&longs;&longs;umus
quidem in vnaquaque figura reperire punctum po&longs;itione,
quod &longs;it quidem centrum grauitatis illius determinatæ figu
r&ecedil;t. vtin fine primilibri o&longs;ten dimus. e&longs;&longs;et tamen impo&longs;&longs;ibile
in omnibus proprium certum, ac determinatum &longs;itum repe
rire; vt &longs;cilicet &longs;it in tali linea, taliquè modo diui&longs;a, vtomnib^{9}
pentagonis, & hexagonis, cæteri&longs;què huiu&longs;modi de&longs;eruire
po&longs;&longs;it. vt determinatur in triangulis, & vt determinari pote&longs;t
in quadrilateris; quæ vel &longs;int parallelogramma, vel duo
latera &longs;int æquidi&longs;tantia. cùm in his conuenientia, quàm
triangulis accidere o&longs;tendimus, reperiatur; quandoquidem
&longs;unt &longs;imiliter in parallelogrammis fa
cilè erit o&longs;tendere aliquam inter &longs;e &longs;imilitudinem exi&longs;tere.
tagona
les, & æqualia latera habent; iam con&longs;tat &longs;imilia e&longs;&longs;e inter &longs;e.
præterea circuliomnes &longs;unt &longs;imiles. Ellip&longs;es quoque inter &longs;e
aliquam habent &longs;imilitudinem, in quibus de&longs;cribitur figura,
planè in&longs;cripta. vt per&longs;picuum e&longs;t in libro Federici Comman
dini de centro grauitatis &longs;olidorum. ac propterea in his, & in
alijs, quibus inter &longs;e aliqua &longs;imililudo reperiri pote&longs;t, centrum
quoque grauitatis determinari poterit.
ex lèmate
ne
mi huius.
coro.
LEMMA.
Sint quatuor magnitudines ABCD. &longs;itquè A maior B;
&C maior D. Dico A ad D maiorem habere proportio
nem, quàm habet B ad C.
Hoc à nobis o&longs;ten&longs;um fuitinitio tractatus devecte in no
&longs;tris mechanicishoc pacto.
portionem, quam B ad C; & A ad D maiorem
quoque habet proportionem, quàm habetad C;
A igitur ad D maiorem habebit, quàm B ad C.
quod demon&longs;trare oportebat.
PROPOSITIO. IIII.
Omnis portionis recta linea, rectanguliquè co
ni &longs;ectione contentæ, centrum grauitatis e&longs;tin dia
metro portionis.
&longs;trandum est dictæ portionis centrum grauitatis e&longs;&longs;e in linea BD. &longs;i.n.
non, &longs;it punctum E. & ab ip&longs;o ducatur ip&longs;i BD aquidistans EF; at
que in portione in&longs;cribatur triangulum ABC eandem ba&longs;im& quam proportionem
habet CF ad FD, eandem habeat triangulum ABC ad &longs;pacium
vt relictæ portiones
B D. &longs;it punctum H. connectaturquè HE, & producatur; &
cto C
tiones AOG GPB BQN NRC &longs;imul &longs;unt ip&longs;o K mino
res; maiorem habebit proportionem triangulum ABC ad
ctas portiones, quàm ad K; in&longs;cripta verò figura AGBNC ma
ior e&longs;t triangulo ABC, K verò maius e&longs;t reliquis portionibus.
BQN, NRC,
ABC ad K, ita est CF ad FD; figura igitur in&longs;cripta ad reliquas por
tiones maiorem habebit proportionem, quam CF ad FD; hoc e&longs;t LE ad
EH.
HD drui&longs;æ. quare cùm figura in&longs;cripta ad reliquas portio
nes maiotem habeat proportionem, quàm LE ad EH; linea,
quæ ad EH eandem habeat
pta ad reliquas portiones, maior erit, Quoniam igi
tur punctum E centrum e&longs;t grauitatis totius portionis, figuræ
in&longs;criptæ
tudinis ex circumrelictis portionibus compo&longs;itæ centrum grauitatis e&longs;&longs;e in
linea HE producta; ita vt a&longs;&longs;umpta aliqua recta linea
nem habeat ad EH, quam figura in&longs;cripta ad circumrelictas portiones.
Quare magnitudinis ex circumrelictis portionibus compo&longs;itæ centrum gra
uitatis e&longs;t punctum M. quod est ab&longs;urdum. Ducta enim linea
punctum M ip&longs;i BD æquidi&longs;tante, in ea omnes circumrelictæ portiones
centra grauitatis habebunt.
bus BPG-BQN compo&longs;itæ centrum grauitatis e&longs;&longs;et in parte
MS. centrum verò grauitatis portionum AOG CRN e&longs;&longs;etin
parte MX; ita ut M omnium dictarum portionum e&longs;&longs;et gra
uitatis centrum. quæ &longs;untquidem inconuenientia. quippè
quæ etiam eodem modo &longs;equentur, &longs;i ST ip&longs;i BD
non e&longs;&longs;et.
linea BD.
huius.
SCHOLIVM.
In hac demon&longs;tratione ob&longs;eruandum e&longs;t; quòd
chimedes inquit, in
telligendum e&longs;t, in&longs;cribatur primò pentagonum AGBNC
in portione planè in&longs;criptum; quod quidem relinquet por
tiones AOG GPB BQN NRC, quæ &longs;imul uel erunt minores
&longs;pacio K, vel minùs. &longs;i non, rur&longs;us planè adhuc in&longs;cribatur
in portione ABC nonagonum; deinde alia figura; idquè &longs;em
per fiat, donec circumrelictæ portiones &longs;imul &longs;int &longs;pacio K
minores. quod quidem fieri po&longs;&longs;e ex prima decimi Euclidis
Aufertur enim &longs;emper maius, Cùm quæ
libet portio paraboles trianguli plane in ip&longs;a in&longs;eripti &longs;it &longs;e&longs;
quitertia. Vnde triangulum ABC maius e&longs;t, quàm
portionis ABC. triangulum què AGB maius, quàm
portionis AGB. &longs;imiliter triangulum BNC portionis BNC &
ita in alijs. Quæ quidem omnia &longs;untquoque manife&longs;ta ex vi
ge&longs;ima propo&longs;itione, eiu&longs;què demon&longs;tratione de quadratura
paraboles Archimedis.
de quad.
parab.
Demon&longs;trato centro grauitatis cuiu&longs;libet paraboles in eius
diametro exi&longs;tere; o&longs;tendet Archimedes, (vt diximus) in pa
rabolis grauitatum centra in eadem proportione diametros
di&longs;pe&longs;cere. antequam autem hoc demon&longs;tret, duas pr&ecedil;mittit
&longs;equentes propo&longs;itiones ad demon&longs;trationem nece&longs;&longs;arias.
PROPOSITIO. V.
Si in portione recta linea, rectanguliquè coni
&longs;ectione contenta rectilinea figura planè in&longs;criba
tur, totius portionis
e&longs;t vertici portionis,
primùmquè in ip&longs;a planè in&longs;eribatur triangulum ABC. & diuidatur
centrum grauitatis punctum E. Diuidatur itàque bi&longs;ariam vtraque
AB BC in punctis FG. &
&longs;tantes FK GL. erit &longs;anè portionis A
que puncta HI. connectanturquè HI FG.
erit vtique punctum Q vertici B propinquius, quàm N. quia
verò e&longs;t BF ad FA, vt BG ad GC, erit FG
eritquè FN ad NG, vt AD ad DC. e&longs;t verò AD ip&longs;i DC æqua
lis, ergo FN NG inter &longs;e &longs;unt æquales. quoniam autem FN
e&longs;t ip&longs;i AD æquidi&longs;tans, erit AF ad FB, vt DN ad NB. e&longs;t au
tem AF dimidia ip&longs;ius AB; cùm &longs;int AF FB &ecedil;quales ergo &
DN dimidia e&longs;t ip&longs;ius DB. at verò quoniam DE terria e&longs;t
pars ip&longs;ius DB, &longs;iquidem e&longs;t BE ip&longs;ius ED dupla, erit pun
ctum N propinquius vertici B portionis, quàm pun
ctum E.
FN ip&longs;i NG, erit QH ip&longs;i QI æqualis. ac propterea magnitudinis ex
vtri&longs;que Aerit &longs;cilicet
punctum
punctum E, magnitudinis verò ex vtri&longs;què A
grauitatis propinquius e&longs;t vertici portionis, quam
mi huius.
lemma ta
aliter
buius.
ex its quæ
ante
ius demon
&longs;trata &longs;unt.
ex
mi huius.
batur. &longs;itquè totius portionis diameter BD, vtriu&longs;que autem portionis
AKB. BLC
AKB planè in&longs;cripta est figura rectilinea
tionis
centrum rectilineæ figuræ
grauitatis punctum H; trianguli verò punctum 1. Rur&longs;us autem &longs;it por
tionis BLC centrum grauitatis punctum M. trianguli verò
ctum N. iunganturquè HM JN
QT. erit vtique punctum Q vertici B propinquius,
T. & quoniam (&longs;i ducta e&longs;&longs;et FG) lineæ HM IN FG ab æ
portione. FG verò, vt o&longs;ten&longs;um e&longs;t, bifariam à linea BD di
uideretur; ergo & lineæ HM IN bifariam diui&longs;&ecedil;
AKB æquale est triangulum BLC; portio vero A
BLC e&longs;t æqualis. Demonstratum e&longs;t enim alis in loçis portiones
tudinis verò ex vtri&longs;que triangulis AKB BLC compo&longs;itæ punctum
T. Rur&longs;us itaque quoniam trianguli ABC centrum grauitatis e&longs;t
E, magnitudinis verò ex vtri&longs;que A
QE ita diui&longs;a
gulum ABC ad vtra&longs;que portiones A
minorem
ex triangulo ABC, trianguli&longs;què AKB BLC compo&longs;itæ
proportionem triangulum ABC ad triangula AKB BLC, eande ha
beat portio ip&longs;ius ad T terminata,
&longs;unt triangula portionibus. habebit TS ad SE
portio nem, quam QO ad OE ac propterea erit
propinquiusip&longs;i E, quàm O. Nam &longs;i punctum S primùm
e&longs;&longs;et in eodem puncto O, tunc TO ad OE, non quidem
maiorem, &longs;ed minorem haberet proportionem, quàm QO
ad OE, cùm &longs;it TO minor QO. &longs;imiliter ob eadem cau
&longs;am &longs;i punctum S e&longs;&longs;et inter OT, minorem haberet
portionem TS ad SE, quàm QS ad SE, quare & ad huc
maiorem haberet proportionem QO ad OE, quàm TS
ad SE. nece&longs;&longs;e e&longs;t igitur punctum S e&longs;&longs;e inter puncta OE.
Itaquè cùm punctum O &longs;it
punctum verò S centrum &longs;it grauitatis rectilineæ figuræ
AK BLC;
e&longs;&longs;e vertici B, quàm centrum rectilineæ figuræ in&longs;criptæ. Et in om
nibus rectilineis figuris in portionibus planè in&longs;criptis eadem e&longs;t ratio.
quod demon&longs;trare oportebat.
ma in
huius.
mi huius.
SCHOLIVM.
do
e&longs;t vertici portionis,
habet
tma/matos, h)/ tou= e)gg<10>afome/nou t<10>igw/nou gnw<10>i/mws
terpo&longs;ita &longs;unt, nullumquè cum alijs rectum &longs;en&longs;um habent,
quare horum loco ponerem
tou= tma/matos
Ob&longs;eruandum autem occurrit in demon&longs;trationibus, ab
Archimede allatis; quòd in prima demon&longs;tratione &longs;upponit
Archimedes, HFGI e&longs;&longs;e parallelogrammum. quòd vt &longs;it pa
rallelogrammum, nece&longs;&longs;e e&longs;t &longs;upponere centra grauitatis HI
&longs;ecare lineas KF LG in partes inuicem proportionales. quod
tamen &longs;upponi po&longs;&longs;e minimè videtur. Et &longs;i quis ex quinto
po&longs;tulato obijceret, centragrauitatis in æqualibus, &longs;imilibu&longs;
què figuris e&longs;&longs;e æqualiter po&longs;ita; admitti quidem pote&longs;t; quo-
vt
omnibus figuris rectilineis &ecedil;qualibus, & &longs;imilib^{9} accidere po
te&longs;t. Hoc tamé contingere po&longs;&longs;e in parabolis, vt AKB BLC, vi
detur in
ctilineæ figuræ; vtantea diximus. Quod etiam
hoc, quia non &longs;emper coaptari porei&longs;t portio AKB
ne BLC.
&longs;ectionis linea BLC &longs;ectionis line&ecedil; BKA &ecedil;qualis exi&longs;tet.
&longs;emper AC, & quæ &longs;untip&longs;i AC æquidi&longs;tates ad rectos &longs;int an
gulos diametro BD. &longs;i.n. &ecedil;quidi&longs;tantes line&ecedil; diametro fuerint
perpendiculares, tunc AB BC inter &longs;e &ecedil;quales e&longs;&longs;ent;
AKB
Quare centra grauiratis HI lineas KFLG in eadem proportio
ne &longs;ecare minimè&longs;upponi po&longs;&longs;e videtur; tùm exijs, quæ dicta
&longs;unt; tú quia hoc o&longs;tendet Archimedes in &longs;eptima propo&longs;itio
ne. quòd &longs;i adhuc non e&longs;t
ni; præ&longs;ertim cùm &longs;it demon&longs;trabile. ac propterea
tio
tú fuit. Huic
tione huiusloci dicendo, hoc &longs;upponere Archimedé, quia por
tiones AKBBLC &longs;unt&ecedil;quales, quarú diametri KFLG &longs;unt &ecedil;
quales, &
vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ip&longs;i IG Quæ
re&longs;pon&longs;io cùm ex dictis
demon&longs;tratiua, vtres mathematic&ecedil;
tenda e&longs;t.hac.n.ratione&longs;upponitur centra HI lineas KFLG in
eadem proportione &longs;ecare.quod nullo modo &longs;upponi pote&longs;t.
Quare dici poterit, & forta&longs;le rectiùs, quòd vis demon&longs;tratio
nis videtur in hoc e&longs;&longs;e con&longs;tituta, vt &longs;upponatur puncta HI
bicunque
&longs;iue etiam non fuerit ip&longs;i FG æquidi&longs;tans, demon&longs;tratio
&longs;uam &longs;emper habebit vim, Nam ex
ti patet centra grauitatis portionum AKB BLC e&longs;&longs;e in lineis
KF LG; hoce&longs;t inter puncta KF, & LG.
tra
d mon&longs;tratione &longs;upponit.
quidi&longs;tans erit, vel minùs: &longs;i e&longs;t æquidi&longs;tans,
e&longs;t HFGI, & vera e&longs;t demon&longs;tratio Archimedis. &longs;i verò
&longs;i HI ip&longs;i FG
quius e&longs;&longs;e vertici B portionis ABC,
&longs;equens,
Etquoniam lineæ HI FG à lineis diuiduntur KF BN LG &ecedil;
pGNG &ecedil;qualis, ergo HQ ip&longs;i QI &ecedil;qualis quoque erit. itaque
quoniam portiones AKBBLC &longs;unt æquales, erit magnitudi
nis ex vtri&longs;que AKB BLC portionibus compo&longs;it&ecedil;
uitatis in medio line&ecedil; HI. ergo eritpunctum
eadem demon&longs;tratio Archimedis o&longs;tendet centrum grauita
tis portionis ABC e&longs;&longs;e inter puncta
cùm ait,
punctum E magnitudinis verò ex vtri&longs;que AkB BLC compo&longs;icæ
est punctum
e&longs;&longs;e in in linea QE. hoc est inter puncta QE. Quare totius portionis
centrum grauitatis propinquius e&longs;t vertici portionis, quàm trian
guli planè in&longs;cripti.
tionis ABC, &longs;iuè &longs;it HI ip&longs;i FG æquidi&longs;tans, &longs;iue non æ.
quidi&longs;tans, propinquius e&longs;&longs;e vertici B portionis, quàm
nis, cùm inquit Archimedes,
HFGJ, & æqualisest FN ip&longs;i NG.immitando &longs;ecun
dam Archimedis demon&longs;trationem huius propo&longs;itionis, vel
delenda &longs;untverba,
ab aliquo ad dita; ita vt verba &longs;int hoc modo vniuer&longs;alia,
quoniam æqualis e&longs;t FN ip&longs;i NG,vel &longs;at for
ta&longs;&longs;e Archimedi vi&longs;um e&longs;t. &longs;e o&longs;tendi&longs;&longs;e hoc contingere exi
&longs;tente HI ip&longs;i FG æquidi&longs;tante. quòd &longs;i etiam non fuerit HI
æquidi&longs;tans FG, idem &longs;equi tanquam notum omi&longs;it. cùm per
facilis &longs;it demon&longs;tratio, vt dictum e&longs;t. Archimede&longs;què res val
dè notas &longs;&ecedil;pè prætermittere&longs;olet.
ius.
Hocidem etiam con&longs;iderari pote&longs;t in &longs;ecunda demon&longs;tra
tione quamuis verba hanc difficultatem non habeant.
dem &longs;equltur demon&longs;tratio, &longs;iuè&longs;it HM lineæ IN &ecedil;quidi&longs;tás,
vel non æquidi&longs;tans, vt ex verbis Archimedis per&longs;picuum e&longs;t.
etenim manife&longs;tum e&longs;t centra grauitatis portionum AKB
BLC e&longs;&longs;einlineis KF LG. &longs;imiliter centra grauitatis
gulorum AKB BLC in ijsdem e&longs;&longs;e lineis KF LG. vtin
ctis
tionales, vnde FI GN euadunt æquales. & quoniam por
tionum centra HM &longs;unt propinquiora verticibus KL, quam
triangulorum centra IN; ideo nece&longs;&longs;e e&longs;t
KI LN exi&longs;tere. quare &longs;int puncta HM vbicúque in lineis KI
LN con&longs;tituta;
&longs;iuenon æquidi&longs;tans, &longs;em per erit
tici B, quam T. eodem què modo erit punctum Q
neæ HM
BLC compo&longs;itæ. &longs;iquidem portiones &longs;unt &ecedil;quales. qu&ecedil;
omnia ex ip&longs;amet demon&longs;tratione &longs;unt manife&longs;ta. &longs;untquè
hæc
ius.
PROPOSITIO. VI.
Data portione rectalinea, rectanguliquè coni
&longs;ectione
ne in&longs;cribi pote&longs;t; ita vt linea inter centrum graui
nor &longs;it propo&longs;ita recta linea.
cuius centrum grauitatis &longs;it
punctum H. & in ip&longs;a planè in&longs;cribatur triangulum ABC. &longs;itquè pro
po&longs;ita recta linea F. & quam proportionem habet BH ad F, eandem
habeat triangulum ABC ad &longs;pacium
nes
ip&longs;iu&longs;què figuræ in&longs;criptæ centrum grauitatis &longs;it punctum E. Dico li
neam HE minorem e&longs;&longs;e ip&longs;a F. N am&longs;i non, vel æqualis est, vel
maior. Quoniam autem
quàm triangulum ABC, maius verò e&longs;t &longs;pacium K portio
nibus ANG GOB BPL LQC &longs;imul &longs;umptis, ideo
portionem, quàm triangulum ABC ad K. hoc est HB ad F. at ue
rò BH nonhabet minorem proportionem ad F, quàm habet ad HE.
cùmnon &longs;it minor HE ip&longs;a F.
ad F. quæ e&longs;t proportio trianguli ABC ad. K. vnde figu
ra rectilinea AGBLC ad circumrelictas portiones maiorem,
habebit proportionem, quàm BH ad HE. &longs;i verò ponatur
HE maior, quàm F, habebit BH ad F, hoc e&longs;t
ABC ad K maiorem proportionem, quàm BH ad HE.
circumrelictas portiones, quàm BH ad HE. Quare &longs;i fiat ut rectili
linea figura AGBLC ad circumrelictas portiones, &longs;ic alia quædam li
nea ad HE. erit maior, quàm BH. &longs;itquè HM. Cùm enim portio
nis ABC centrum grauitatis &longs;it H. figuræ verò rectilineæ AGBLC
punctum E. producta EH, a&longs;&longs;umptaquè aliqua recta linea proportione
babente ad EH, quam rectilineum AGBLC ad circumtelictas por
tiones; maior erit quàm HB. habeat igitur
HE
quas portiones,
nis ex circumrelictis portionibus compo&longs;itæ. quod e&longs;&longs;e non pote&longs;t. Ducta
enimrecta linea
grauitatis vnicuiquè portioni re&longs;pondentia
magnitudinis ex portionibus ANG GOB compo&longs;itæ &longs;it in
linea RS. &longs;ed in parte MR. in parteverò MS &longs;it grauitatis
centrum magnitudinis ex reliquis portionibus BPL LQC
compo&longs;itæ; ita vt punctum M magnitudinis ex omnibus
portionibus compo&longs;itæ centrum grauitatisexi&longs;tat. quæ
e&longs;&longs;e non po&longs;&longs;unt. quod idem accideret, &longs;i etiam RS ip&longs;i AC
æquidi&longs;tans non e&longs;&longs;et.
cùm neque maior, neque &ecedil;qualis e&longs;&longs;e po&longs;&longs;it.
mon&longs;trare oportebat.
ius.
SCHOLIVM.
In hac quoque demon&longs;tratione ob&longs;eruandum e&longs;t, quod
po&longs;t quartam huius adnotauimus; nimirum &longs;i pentagonum
AGBLC in portione planèin&longs;criptum relinqueret portiones
ANG GOB BPL LQC, quæ &longs;imul maiores, vel etiam æ-
portione ABC nonagonum, deinde altera figura, idquè &longs;em
per fiat, donec circumrelict&ecedil; portiones &longs;imul &longs;int &longs;pacio K
minores. quod quidem fieri po&longs;&longs;e ibidem o&longs;tendimus:
PROPOSITIO. VII.
Duabus portionibus &longs;imilibus recta linea, re
ctanguliquè coni &longs;ectione contentis, centra gra
uitatum diametros in eadem proportione di&longs;pe
&longs;cunt.
tri BD FH. &longs;itquè portionis ABC centrum grauitatis punctum K.
ip&longs;ius verò EFG punctum L. Demonstrandum est, puncta
eadem proportione diametros diuidere,
MH. & in portione EFG rectilineum planè in&longs;cribatur, ita vt linea
inter centrum
tatis punctum X.
F, quàm punctum X. & quoniam LX minor e&longs;t, quàm
LM, erit quoque punctum X vertici F propinquius, quàm
M.
ræin portione EFG in&longs;criptæ. hoc est &longs;imiliter planè,
figur&ecedil; latera multitudine &ecedil;qualia habeant)
tatis
nèin&longs;cript&ecedil; habentlatera multitudine æqualia, ip&longs;arum cen
tra grauitatis diametros BD FH in eadem proportione di&longs;pe
&longs;cent. quare erit BN ad ND, vt FX ad XH. po&longs;itum
fuitita e&longs;&longs;e FM ad MH, vt BK ad KD. &longs;i itaque
X propinquius e&longs;t ip&longs;i F, quàm M; erit & punctum N i
p&longs;i B propinquius, quàm K. e&longs;tverò punctum K
grauitatis portionis ABC, punctum verò N centrum figuræ
in&longs;cripte; ergo centrum grauitatis figur&ecedil; in&longs;criptæ
erit vertici portionis,
potest. Manife&longs;tum est igitur eandem habere proportionem BK ad KD.
quam FL ad LH.
SCHOLIVM.
Pr&ecedil;&longs;ens demon&longs;tratio ea tantùm ratione e&longs;&longs;icax e&longs;&longs;e vide
tur, quatenus &longs;upponitur punctum L vertici F propinqui^{9}
e&longs;&longs;e, quàm M. ex hoc enim &longs;equitur punctum X e&longs;&longs;e ip&longs;i F
propinquius, quàm M. vnde euenitab&longs;urdum, nempè,
ctum N e&longs;&longs;evertici B propinquius, quàm K. Quòd &longs;i &longs;up
po&longs;itum fuerit Bk ad KD ita e&longs;&longs;e, vt FP ad PH; fuerit
autem P inter LF; tunc centrum grauitatis figur&ecedil; in EFG
figur&ecedil; in ABC &longs;imiliter planè in&longs;cript&ecedil; inter KD eueniret;
e&longs;&longs;etquè centrum grauitatis portionis ABC vertici B propin
quius, quam centrum figuræ planè in&longs;criptæ. ideoquè
accideret ab&longs;urdum. Quare &longs;i &longs;uppo&longs;itum fuerit FP ad PH
e&longs;&longs;e, vt BK ad KD, tunc (vt eadem demon&longs;tratio rei propo
&longs;itæ in&longs;eruire po&longs;&longs;et) diuidenda e&longs;&longs;et diameter BD in
ta vt BQ ad QD &longs;it, vt FL ad LH. & quoniam maio
dem maior e&longs;t FL, quàm FP, & PH maior, quàm LH. Vtverò
FL ad LH, ita e&longs;t BQ ad QD; & vt FP ad PH. ita BK ad KD;
maiorem quoque habebit proportionem BQ ad QD, quàm
Quare maior e&longs;t DK, quàm
punctum K propinquius erit vertici B, quàm
planè in&longs;cribenda e&longs;&longs;et figura in portione ABC, ita vt linea
inter centrum figuræ in&longs;criptæ, & centrum portionis minor
e&longs;&longs;et, quàm
facta &longs;unt in EFG, fiant in ABC; & quæ in ABC,
o&longs;tendeturquè centrum figur&ecedil; in&longs;cript&ecedil; in portione EFG pro
pinquius e&longs;&longs;e vertici F, quàm centrum grauitatis ip&longs;ius portio
nis EFG. quod quidem fieri non pote&longs;t. Ex quibus perlpi
cuum fit demon&longs;trationem e&longs;&longs;e vniuer&longs;alem. & hanc
&longs;trationis partem Archimedem omi&longs;i&longs;&longs;e, vt notam. Etvt an
tea admonuimus, quòd centra grauitatis diametros in eadem
proportione diuidunt, omnibus parabolis competere intelli
gendum e&longs;t. &longs;iquidem omnes &longs;unt&longs;imiles. quo demon&longs;trato,
in &longs;equenti, quo in loco, & in qua diametri parte reperitur
trum grauitatis paraboles demon&longs;trat, quòd vt res per&longs;picua
reddatur; hæc priùs demon&longs;trabimus.
addi.
LEMMA. I.
Si magnitudo magnitudinis fuerit quadrupla, minorverò
magnitudo alterius magnitudinis &longs;it tripla, maior magnitu
do vtrarum què &longs;imul magnitudinum tripla erit.
Quadrupla &longs;it magnitudo A magnitudinis BC.
&longs;it verò BC alterius magnitudinis CD tripla. Di
co magnitudinem A vtrarumquè &longs;imul BC CD,
hoc e&longs;t BD triplam e&longs;se. Quoniam enim BC tri
pla e&longs;t ip&longs;ius CD, erit componendo BC cum CD,
hoc e&longs;t BD ip&longs;ius CD quadrupla. &longs;ed magnitudo
quoque A quadrupla e&longs;t ip&longs;ius BC, eandem igitur
habetproportionem A ad BC, vt BD ad CD. &
permutando A ad BD, vt BC ad CD. & e&longs;t
dem BC tripla ip&longs;ius CD, ergo A ip&longs;ius BD tri
pla exi&longs;tit. quod demon&longs;trare oportebat.
LEMMA. II.
Si magnitudo magnitudinis fuerit &longs;e&longs;quitertia, erit magni
tudo minor ip&longs;ius exce&longs;&longs;us tripla.
Sit magnitudo AB magnitudinis C &longs;e&longs;quiter
tia; exce&longs;&longs;us verò, quo AB &longs;uperat C, &longs;it BD. Dico quod qui
dem ex &longs;e patet. Nam quoniam BD e&longs;t exce&longs;
&longs;us, quo AB &longs;uperat C. magnitudo autem AB i
p&longs;am C &longs;uperat tertia ip&longs;ius C parte, cum &longs;it AB
ip&longs;ius C &longs;e&longs;quitertia. erit igitur BD tertia pars i
&longs;ius C. quare magnitudo C ip&longs;ius BD tripla
exi&longs;tit. quod o&longs;tendere oportebat.
LEMMA III.
Sit magnitudo AB ip&longs;ius BC &longs;extupla. &longs;it verò AD tripla
ip&longs;ius AC. Dico BD ip&longs;ius BA &longs;e&longs;quialteram e&longs;se.
AD. ac propterea ip&longs;am AD metictur. rur&longs;us quoniam AB,
hoc e&longs;t AC vnà cum CB &longs;extupla e&longs;t ip&longs;ius BC, erit
AC ip&longs;ius CB quintupla. vndè CB ip&longs;am AC, ac propterea
ip&longs;am AB metietur. Vtautem AC ad AD, ita fiat
CB ad aliam
G pars tertia, cùm &longs;it AC ip&longs;ius AD pars quoque
tertia. Itaque quoniam CB ad G e&longs;t, vt AC ad AD,
ip&longs;am CA metitur, eiu&longs;què e&longs;t pars quinta; ergo
Gip&longs;am quoque AD metietur, eritquè ip&longs;ius pars
quinta. Quoniam autem BC ip&longs;am BA metitur,
eademquè BC ip&longs;am quoque G metitur, erit BC
ip&longs;arum AB G communis men&longs;ura. quia verò AB
&longs;extupla e&longs;t ip&longs;ius CB, G verò e&longs;t eiu&longs;dem CB tri
pla, erit AB ad G, ut &longs;extupla ad triplam. hoc e&longs;t
&longs;e habebunt in dupla proportione. quapropter
AB dupla e&longs;t ip&longs;ius G; ac per con&longs;equens Gip&longs;am
AB metitur. Quoniam igitur G totam AD metitur, &
ablatam AB quoque metitur; metietur G reliquam BD. G
igitur ip&longs;arum AB BD communis exi&longs;tit men&longs;ura. &
AB dupla e&longs;t ip&longs;ius G, tota verò AD eiu&longs;dem G quintupla
exi&longs;tit, erit reliqua BD tripla ip&longs;ius G. Ex quibus&longs;equitur
DB ad BA ita &longs;e habere, vt tripla ad duplam. Quare DB
ip&longs;ius BA &longs;e&longs;quialtera exi&longs;tit. quod o&longs;tendere oportebat.
PROPOSITIO. VIII.
Omnis portionis recta linea, rectanguliquè co
ni &longs;ectione contentæ centrum grauitatis diame
trum portionis ita diuidit, vt pars ip&longs;ius ad verti
cem portionis reliquæ ad ba&longs;im &longs;it &longs;e&longs;quialtera.
ip&longs;ius verò diameter &longs;it BD. cen
trum autem grauitatis &longs;it punctum H. o&longs;tendendum e&longs;t BH ip&longs;ius HD
&longs;e&longs;quialteram e&longs;&longs;e. Planè in&longs;cribatur in portione ABC triangulum ABC.
cuius centrum grauitatis &longs;it punctum E. bi&longs;ariamquè diuidatur vtra
què AB BC in punctis FG. & ip&longs;i BD æquidi&longs;tantes ducantur F
que portionis A
ctum N. connectanturque FG MN
cent in punctis OQS. Quoniam igitur puncta MN in
proportione diuidunt KF LG, erit KM ad MF, vt LN ad
NG. & componendo KF ad FM, vt LG ad GN. &
mutando KF ad LG, vt FM ad GN. &longs;untquè KF LG
æquales; erit FM ip&longs;i GN &ecedil;qualis; & reliqua Mk reliquæ
LN æqualis. & quoniam FM GN, & Mk NL &longs;unt
di&longs;tantes, erunt FG MN KL inter &longs;e &ecedil;quales, &
tes
qualis. quia verò KF BD LG &longs;unt æquidi&longs;tantes, erit MQ ad
QN, vt FO ad OG. Cùm autem &longs;it BF ad FA, vt BG ad GC,
& vt AD ad DC, ita FO ad
OG. &longs;untautem AD DC æquales, ergo FO OG, ac per con
&longs;equens MQ QN inter &longs;e &longs;unt æquales. itaque quoniam por
nibus
dem proportione diametros &longs;ecare nece&longs;&longs;e e&longs;t)
p&longs;ius SQ, quadrupla existit.
po&longs;ita e&longs;t ex BS QH, & SQ, quadrupla ip&longs;ius
SQ ab ip&longs;is BS QH SQ,
con&longs;tans
tota HQ cum SB ad totam QS e&longs;t, vt ablata BS ad ab
Et quoniam quadrupla est BD ip&longs;ius BS. hoc enim demon&longs;tratum
ip&longs;a verò BS ip&longs;ius SX e&longs;t triplaVerùm ED ip&longs;ius
DB parstertia existit. Cùm centrum grauitatis trianguli ABC &longs;it
p
At verò quoniam totius lineæ BD (quæ compo&longs;ita e&longs;t ex DE
EX XB) tertia pars e&longs;t ip&longs;a DE. & tertia quoque ip&longs;a BX;
tionis centrum grauitatis est punctum H; magnitudinis verò ex v
tr
ctum
ad circumrelictas portionesCùm totaportio
tertia &longs;it trianguli ABC
rat triangulum ABC, &longs;int portiones AKB BLC &longs;imul &longs;um
ptæ.
tripla ip&longs;ius QX.&
niam HQ e&longs;t tripla ip&longs;ius QX, erit HQ cum QX, hoc
e&longs;t tota BX quadrupla ip&longs;ius QX, hoc e&longs;t ip&longs;ius HE. &longs;i
militer quoniam XH quadrupla e&longs;t ip&longs;ius HE;
gitur e&longs;t
DE ip&longs;ius EH. inuicem enim &longs;unt æquales
&longs;um e&longs;t. Cùm itaque &longs;it DE ip&longs;ius EH quintupla; erit DE
cum EH &longs;extupla ip&longs;ius EH.
ip&longs;ius HE. & e&longs;t BD ip&longs;ius DE tripla; &longs;equialtera igitur e&longs;t BH
trum BD, vtpars BH ad HD &longs;e&longs;quialtera exi&longs;tit. quod de
mon&longs;trare oportebat.
huius.
in
mi huius
ter
mi huius
huius
ius.
mi huius.
ius.
huius.
huius.
SCHOLIVM.
Ea verba in demon&longs;tratione po&longs;ita nempè
da, quòd &longs;cilicet Archimedes alicubi, & in fine, &longs;iue huius, &longs;i
ue alicuius alterius demon&longs;trationis, demon&longs;trauerit lineain
erat, ibi quoque pro &longs;igno po&longs;ita fuerit littera H. quod qui
dem o&longs;ten&longs;um e&longs;t à nobis paulò ante &longs;ecundam huius propo&longs;i
tionem; vbi etiam appo&longs;uim us pro &longs;igno hanc literam H.
e&longs;&longs;e. &longs;upponit autem hoc tanquam demon&longs;tratum po&longs;t pri
mam
tuor, vt eodem in loco o&longs;ten&longs;um fuità nobis. Vel ad ea re
&longs;pexit Archimedes, quæ ab ip&longs;o in decimanona propo&longs;itione
de quadratura paraboles demon&longs;tra ta fuerunt. vbi circa
demon&longs;trationis o&longs;tendit BD quadruplam e&longs;&longs;e ip&longs;ius BS.
Inuento itaque centro grauitatis paraboles, vult Archime
des in ue&longs;tigare centrum grauita tis fru&longs;ti à parabole ab&longs;ci&longs;&longs;i.
quemadmodum in primo libro po&longs;t inuentionem centri gra
uitatis trianguli, adin uenit etiam centrum grauitatis trapezij.
quod e&longs;t tan quam fru&longs;tum à triangulo ab&longs;ci&longs;sum. quare duo
adhuc theoremata proponit, in quorum po&longs;tremo, vbi &longs;it
trum grauitatis fru&longs;ti demon&longs;trat. in &longs;equenri verò quædam
demon&longs;trat nece&longs;&longs;aria, vt huiu&longs;modi centrum determinare
po&longs;&longs;it. Quoniam autem &longs;equens theorema arduum, difficile
què &longs;e&longs;e offert; non nulla priùs quibu&longs;dam lemmatibus o&longs;ten
demus, ne &longs;i in demon&longs;tratione ea in&longs;ererentur, longa nimis
euaderet, ac tædio&longs;a demon&longs;tratio. quæ quidem &longs;umma in di
get attentione. quamquàm in hoc theoremate explicando ad
vitandam ob&longs;curitatem copio&longs;um &longs;ermonem adhibendum
curauimus; ne breuitati &longs;tudentes ob&longs;curiores e&longs;&longs;emus.
LEMMA. I.
Si qua tuor magnitudines in continua fuerint proportione,
& earum exce&longs;&longs;us in eadem erunt proportione
Sint quatuor magnitudines AF BH CL D in continua
proportione; vt &longs;cilicet &longs;it AF ad BH, vt BH ad CL; & CL
ad D. exce&longs;&longs;us verò, quo AF &longs;uperat BH, &longs;it EF. & exce&longs;&longs;us, quo
BH &longs;uperat CL, &longs;it GH. exce&longs;&longs;us denique, quo CL &longs;uperat
D, &longs;it KL. eruntutique AE BH inter &longs;e &ecedil;quales, itidemquè
BG CL æquales. Dico EF GH KL in eadem e&longs;&longs;e proportio
ne, vt &longs;unt magnitudines AF BH CL, & vt BH CL D. Quo
niam enim tota AF ad totam BH e&longs;t, vt BH ad CL; hoc e&longs;t
vt ablata EA ad ablatam GB. erit reliqua EF ad reliquam GH;
vt AF ad BH. Pariquè ratione o&longs;tendetur GH ad kL ita e&longs;
&longs;e, vt BH ad CL. ergo exce&longs;&longs;us EF GH KL in eadem &longs;unt
proportione, vt magnitudines AF BH CL. quæ cùm &longs;int, vt
magnitudines BH CL D, &longs;iquidem omnes in continua &longs;unt
proportione; exce&longs;&longs;us igitur EF GH KL in eadem quoque
&longs;unt proportione, vt magnitudines BH CL D. quæ quidem
demon&longs;trare oportebat.
LEMMA. II.
Si tres fuerint magnitudines, & aliæ ip&longs;is numero æquales,
& in eadem proportione, in primis magnitudinibus prima;
& &longs;ecunda ad tertiam erunt, vt in &longs;ecundis magnitudinibus
prima & &longs;ecunda ad tertiam.
Sint tres magnitudines ABC, & aliæ tres DEF in
portione. Dico AB &longs;imul ad C ita e&longs;&longs;e, vt DE &longs;imul ad F.
AB &longs;imul ad C e&longs;t, vt DE &longs;imul ad F. quod demon&longs;trare opor
tebat.
LEMMA. III.
Si fuerit AB ad AC, vt DE ad DF. Dico exce&longs;&longs;um BC ad
Quoniam enim e&longs;t AB ad AC, vt DE ad DF, erit con-
uertendo CA ad AB, vt FD ad DE. & per conuer
&longs;ionem rationis AC ad CB, vt DF ad FE. & rur&longs;us
conuertendo CB ad CA, vt FE ad FD. quod
&longs;trare
ALITER.
Quoniam enim AB e&longs;t ad AC, vt DE ad DF, erit conuer
tendo AC ad AB, vt DF ad DE. diuidendoquè CB ad BA, vt
FE ad ED. e&longs;t autem AB ad AC, vt DE ad DF, erit igitur
ex æquali BC ad CA, vt EF ad FD. quod demon&longs;trare opor
tebat.
LEMMA IIII.
Si fuerint quotcunque magnitudines ABC, & nli&ecedil; ip&longs;is nu
mero æquales DEF, & in Dico vtramque
&longs;imul AD ad vtramque &longs;imul BE, & vtramque &longs;imul BE ad v
tramque &longs;imul CF eandem habere proportionem, quam ha
bet A ad B, & B ad C.
BE &longs;imul, vt A ad B. &longs;imiliter quoniam B ad C e&longs;t, vt E ad
F, erit BE &longs;imul ad CF &longs;imul, vt B ad C. in eadem igitur &longs;unt
proportione AD &longs;imul, & BE &longs;imul, & CF &longs;imul, vt ABC.
quod demon&longs;trare oportebat.
LEMMA. V.
Si magnitudo magnitudinis fuerit &longs;e&longs;quialtera ad tres quin
tas eiu&longs;dem erit duplex &longs;e&longs;quialtera.
Sit AB ip&longs;ius CD &longs;e&longs;quialtera. &longs;it uerò CE tres quintæ
ip&longs;ius CD. Dico AB ad CE ita e&longs;&longs;e, vt quinque ad duo. Fiat EF
&ecedil;qualis EC, erit CF &longs;ex quin tæ ip&longs;ius CD. & quoniam AB i
p&longs;ius CD e&longs;t &longs;e&longs;quialtera, &longs;uperabit AB ip&longs;am CD dimidia
ip&longs;ius CD. erit igitur AB &longs;eptem quintæ cum dimidia i
p&longs;ius CD. quare CF minor e&longs;t AB. fiat igitur AG æqua
lis CF. erit vtique AG &longs;ex quin t&ecedil; ip&longs;ius CD. & ob id GB
ip&longs;ius CD quinta e&longs;t pars cum dimidia. & quoniam CE e&longs;t
eiu&longs;dem CD tres quintæ, erit BG dimidia ip&longs;ius CE. qua
re GB ip&longs;am CE bis metietur. Et quoniam EF e&longs;t æqua
lis ip&longs;i EC, ip&longs;a BG bis quoque metietur ip&longs;am EF. quare
at verò GB &longs;ei
p&longs;am &longs;emel metitur ip&longs;a igitur GB totam AB quinquies metie
tur. Ex quibus liquet GB ip&longs;arum ABCE communem e&longs;&longs;e
men&longs;uram. Et e&longs;t quidem AB quintupla ip&longs;ius BG; ip&longs;a verò
CE eiu&longs;dem BG dupla. erit AB ad CE, vt quintupla ad
hoc e&longs;t duplex &longs;e&longs;quialtera. quod demon&longs;trare oportebat.
PROPOSITIO. VIIII.
Si quatuor lineæ in continua fuerint proportio
ne, & quam proportionem habet minima ad exce&longs;
&longs;um, quo maxima minimam &longs;uperat; eandem ha
beat quædam a&longs;&longs;umpta linea ad tres quintas exce&longs;
&longs;us, quo maxima proportionalium tertiam exce
dit: quam verò proportionem habet linea æ qualis
duplæ maximæ proportionalium, & quadruplæ &longs;e
cundæ, & &longs;extuplæ tertiæ, & triplæ quartæ ad
æqualem quintuplæ maximæ, & decuplæ &longs;ecundæ,
& decuplæ tertiæ, & quintuplæ quartæ, ean-
dem habeat quædam a&longs;&longs;umpta linea ad ex ce&longs;&longs;um,
quo maxima proportionalium tertiam &longs;uperat;
vtræque &longs;imul a&longs;&longs;umptæ lineæ erunt duæ quin
tæ maximæ.
ad BC &longs;it, vt BC ad BD. & vt BC ad BD, ita &longs;it BD ad BE.
quam proportionem habet BE ad E A, eandem habeat FG adtres quin
tas ip&longs;ius AD. quam autem proportionem habet linea æqualis duplæ i
p&longs;ius AB, & quidruplæ ip&longs;ius BC, & &longs;extuplæ ip&longs;i^{9} BD, & triplæ ip&longs;i^{9}
BE, ad
ip&longs;i^{9} B D, & quintuplæ ip&longs;ius BE, eandem habeat GH ad AD. O&longs;teden
dum est FH duasquintas e&longs;&longs;e ip&longs;ius AB. Quoniam enim proportiona
les &longs;unt AB BC BD BE, &
&
in continua &longs;unt proportione, & earum exce&longs;&longs;us AC CD DE
in eadem erunt proportione. quia verò tres &longs;unt magnitudi
nes proportionales AB BC BD; & ali&ecedil; ip&longs;is numero çquales, &
dinibus prima, & &longs;ecunda ad tertiam, vt in &longs;ecundis magni
tudinibus prima, & &longs;ecunda ad tertiam; hoc e&longs;t
AB BC ad BD eandem habebit proportionem, quam
eodem modo proportionales BC BD BE; crit vtraque &longs;imul
& omnes adomnes,
ad BD, vt horum dupla; erit vtraque &longs;imul AB BC ad BD, vt
dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD. e&longs;t
vtraque &longs;imul AB BC ad BD, vt AD ad DE. eritigitur AD ad
DE, vt dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD.
quia veròita etiam e&longs;t AD ad DE, vtvtraque &longs;imul CB BD ad
BE; erit dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD, vt
vtraque &longs;imul CB BD ad BE. & vtraque an tecedentia ad
que con&longs;equentia in eadem erunt proportione: eruntquè in
antecedenti du&ecedil; AB, tres BC, & &longs;ola BD. in con&longs;equenti verò
erunt duæ BD cum &longs;ola BE. erit igitur dupla ip&longs;ius AB, & tri
pla ip&longs;ius CB cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE,
vt vtraque &longs;imul CB BD ad BE. vtraque verò &longs;imul CB BD
ad BE e&longs;t, vt AD ad DE.
tem linea compo&longs;ita ex dupla ip&longs;ius AB, & quadrupla ip&longs;ius
CB, & quadrupla ip&longs;ius BD, & dupla ip&longs;ius BE, maior e&longs;t ea,
quæ compo&longs;ita e&longs;t ex dupla ip&longs;ius AB, & tripla ip&longs;ius CB, &
&longs;ola BD; maiorem habebit proportionem compo&longs;ita ex
pla ip&longs;ius AB, & quadrupla ip&longs;ius CB, & quadrupla ip&longs;ius BD,
& dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum
&longs;ola BE, quam compo&longs;ita ex dupla ip&longs;ius AB, & tripla ip&longs;ius
CB cum &longs;ola BD ad eandem compo&longs;itam ex dupla ip&longs;ius BD
cum &longs;ola EB. compo&longs;ita verò ex dupla ip&longs;ius AB, & tripla
ip&longs;ius BC cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE ita
o&longs;ten&longs;a e&longs;t &longs;e habere AD ad DE. compo&longs;ita igitur ex dupla i
p&longs;ius AB, & quadrupla ip&longs;ius BC, & quadrupla ip&longs;ius BD, &
dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum &longs;ola
BE maiorem habebit proportionem, quam AD ad DE.
itaque proportionem habet linea æqualis duplæ ip&longs;ius AB, & quadruplæ
ip&longs;ius BC, & quadruplæ ip&longs;ius BD, & duplæ ip&longs;ius BE ad
duplæ ip&longs;ius DB, & ad EB, eandem habebit AD adminorem ip&longs;a DE.
&longs;ita ex dupla ip&longs;ius AB, & quadrupla ip&longs;ius BC, & quadrupla
ip&longs;ius BD, & dupla ip&longs;ius BE, hoc e&longs;t
enim a&longs;&longs;umitur AB, & bis BE, quater verò BC, & quater BD)
rendo, ut OD ad DA, ita compo&longs;ita ex dupla ip&longs;ius BD
la BE ad
bunt proportionem.
po&longs;ita
mul AB BE, & quadrupla vtriu&longs;que &longs;imul BC BD ad compo
&longs;itam ex dupla vtriu&longs;que &longs;imul AB BE, & quadrupla
&longs;imul BC BD. In hocautem an tecedente bis&longs;umitur AB, qua
ter BC, &longs;exies verò BD, & ter BE.
demproportionem, quam linea æqualis duplæip&longs;ius AB, et quadruplæi
p&longs;ius CB, et &longs;extuplæ ip&longs;ius BD, ettriplæ ip&longs;ius BE ad lineam compo&longs;i
tam ex dupla vtriu&longs;que &longs;imul AB EB, et quadrupla vtriu&longs;que &longs;imul
CB BD. babet autem
proportionem, quam linea æ qualis duplæ ip&longs;ius AB, & qua
druplæ ip&longs;ius BC, & &longs;extuplæ ip&longs;ius BD, & triplæ ip&longs;ius BE
ad lineam æ qualem quintuplæ ip&longs;ius AB, & decuplæ ip&longs;ius
CB, & decuplæ ip&longs;ius BD, & quintuplæ ip&longs;ius BE, hoc e&longs;t ad
CB BD. In
BE, decies CB, & decies BD. & conuettendo habebit
p&longs;ius AB, & quadrupla ip&longs;ius CB, & &longs;extuplaip&longs;ius BD, & triplai
p&longs;ius EB. Di&longs;similiter autem quàm in proportionibus ordinatis, hocest
in perturbata proportione
&longs;e habet an tecedens OA ad con&longs;equens AD, vt in &longs;ecundis ma
gnitudinibus antecedens compo&longs;ita nempè ex dupla ip&longs;ius
AB, & quadrupla ip&longs;ius BC, & &longs;extupla ip&longs;ius BD, & tripla
ip&longs;ius BE, ad con&longs;equens lineam &longs;cilic et compo&longs;itam ex du
pla vtriu&longs;que &longs;imul AB BE, & quadrupla vtriu&longs;que &longs;imul CB
BD: ut autem in primis magnitudinibus con&longs;equens AD ad
aliud quippiam GH, ita in &longs;ecundis magnitudinibus aliud
quippiam, nempèlinea compo&longs;ita ex quintupla vtriu&longs;que &longs;i
mul AB BE cum decupla vtriu&longs;que &longs;imul CB BD ad antece
dens, hoc e&longs;t ad compo&longs;itam ex dupla ip&longs;ius AB, & quadru
pla ip&longs;ius CB, & &longs;extupla ip&longs;ius BD, & tripla ip&longs;ius BE. quare
CB BD. At verò
eiu&longs;dem AB e&longs;t, vt quinque ad duo; &longs;imiliter quintupla ip&longs;i^{9}
BE ad duplam eiu&longs;dem BE e&longs;t, vt quinque ad duo, erit quin
tupla vtriu&longs;que &longs;imul AB BE ad duplam vtriu&longs;que &longs;imul AB
BE, vt quinque ad duo. pariquè ratione decupla vtriu&longs;que &longs;i
mul CB BD ad quadruplam vtriu&longs;que &longs;imul CB BD e&longs;t, vt
decem ad quatuor, hoc e&longs;t vt quinque ad duo. &
ad con&longs;equentia in eadem erunt proportione, hoce&longs;t
ta ex quintupla vtriu&longs;que &longs;imul AB BE cum decupla vtriu&longs;que &longs;imul
CB BD ad compo&longs;itam ex dupla vtriu&longs;que &longs;imul AB BE, & quadru
pla vtriu&longs;que &longs;imul CB BD proportionem habet, quam quinque ad duo
Quare OA ad GH proportionem habet, quam quinque ad duo. Rur&longs;us
factum fuit AD ad DO, vt compo&longs;ita ex dupla vtriu&longs;que &longs;i
mul AB BE cum quadrupla vtriu&longs;que &longs;imul CB BD ad
BE vnà cum dupla ip&longs;ius BD. conuertendo etiam
eandem habet proportionem, quam
tecedens
qualem lineæ compo&longs;itæ ex dupla vtriu&longs;que &longs;imul AB BE cum quadru
pla vtriu&longs;que &longs;imul CB BD; est autem
in primis magnitudinibus con&longs;equens
&longs;cilicet
ad
ip&longs;ius BC &
& quadrupla vtriu&longs;que &longs;imul CB BD.
DO exce&longs;&longs;u OE; linea verò
AB BE, & quadrupla vtriu&longs;que &longs;imul CB BD lineam excedit
compo&longs;itam ex dupla ip&longs;ius AB cum tripla ip&longs;ius BC, ac &longs;ola
BD, exce&longs;&longs;u lineæ, quæ &longs;it æqualis &longs;oli CB cum tripla ip&longs;ius
ip&longs;ius BD, & dupla ip&longs;ius EB ad duplam vtriu&longs;que &longs;imul AB BE,
& quadruplam vtriu&longs;que &longs;imul CB BD. est autem
tione, tertia in ordine BD ad quartam BE, vt prima AB ad
&longs;ecundam BC, quare diuidendo vt DE ad EB, ita AC ad
CB. Rur&longs;us quoniam in lineis proportionalibus ob eandem
cau&longs;am CB ad BD ita e&longs;t, vt DB ad BE; erit diuidendo, vt
CD ad DB, ita DE ad EB. ego
nemtripla ip&longs;ius CD, adtriplam ip&longs;ius DB
lam DB.
vt DE ad EB. e&longs;t verò CD ad DB, vt DE ad
EB, & AC ad CB; erit igitur AC ad CB, vt tripla ip&longs;ius
CD ad triplam ip&longs;ius DB; & vt dupla ip&longs;ius DE ad
duplam ip&longs;ius EB.
tria &longs;imul con&longs;equentia, hoc e&longs;t,
tripla ip&longs;ius CD, & dupla ip&longs;ius DE ad compo&longs;itam ex CB,
& tripla ip&longs;ius DB, & dupla ip&longs;ius EB
ad CB, hoc e&longs;t, DE ad EB.
quàm in proportionibus ordinatis, hoc est in perturbata proportione,
quoniam e&longs;t in primis magnitudinibus antecedens OE ad
con&longs;equens ED, ita in &longs;ecundis magnitudinibus an
compo&longs;ita &longs;cilicet ex CB, cum tripla ip&longs;ius BD, & dupla ip
&longs;ius EB, ad con&longs;equens nem pè compo&longs;itam ex dupla vtriu&longs;
que &longs;imul AB BE, cum quadrupla vtriu&longs;que &longs;imul CB BD:
in primis verò magnitudinibus con&longs;equens DE ad aliud quip
piam EB e&longs;t, vt in &longs;ecundis magnitudinibus aliud quippia,
hoc e&longs;t compo&longs;ita ex AC cum tripla ip&longs;ius CD, & dupla ip
&longs;ius DE ad an tecedens, lineam &longs;cilicet compo&longs;itam ex CB cum
tripla ip&longs;ius BD, & dupla ip&longs;ius EB.
pla ip&longs;ius CD, & dupla ip&longs;ius DE ad duplam vtriu&longs;
que &longs;imul AB BE cum qnadrupla vtriu&longs;que &longs;imul CB
BD.
cum tripla ip&longs;ius CD, & dupla ip&longs;ius DE, & dupla
vtriu&longs;que &longs;imul AB BE, & quadrupla vtriu&longs;que &longs;i
mul CB BD, ad duplam vtriu&longs;que &longs;imul AB BE
cum quadrupla vtriu&longs;que &longs;rmul CB BD. In hoc autem
bis BE, quater CB, & quater BD. Duæ verò AB vnà
cum &longs;ola AC, & &longs;ola. CB, ex quatuor vicibus, quibus ip
&longs;a CB &longs;umitur, &longs;unt æquales tribus AB. tres autem CB,
quæ relictæ &longs;unt, vnà cum tribus CD, & tribus BD
ex quatuor vicibus, quibus ip&longs;a BD &longs;umitur, &longs;unt æ
quales &longs;ex CB. &longs;ola verò BD, quæ relicta fuit, vnà
cum duabus DE, & duabus BE, e&longs;t æqualis tribus
BD. linea nimirum AC cum tripla ip&longs;ius CD, &
dupla ip&longs;ius DE, & dupla vtriu&longs;que &longs;imul AB BE,
& quadrupla vtriu&longs;que &longs;imul CB BD, æqualis erit tri
plæ ip&longs;ius AB, cum &longs;extupla ip&longs;ius CB, & tripla ip
&longs;ius BD.
nem, quam linea æqualis triplæ ip&longs;ius AB cum &longs;extupla ip
&longs;ius CB & tripla ip&longs;ius BD ad duplam vtriu&longs;que &longs;imul
AB BE cum quadrupla vtriu&longs;que &longs;imul CB BD. &
quoniam
e&longs;&longs;e proportione, vt &longs;unt quatuor lineæ continuè pro
portionales AB BC BD BE; erunt tres AC CD
DE, & tres AB BC BD, & tres BC BD BE
conuertendo igitur in eadem quo
que erunt proportione. quare tres
tres BE BD BC, & tres BD BC BA
vtraque &longs;imul BE BD advtramque &longs;imul BD BC, &
vtraque &longs;imul BD BC ad vtramque &longs;imul BC BA
ita &longs;e habebunt, vt BE BD BC. hæ verò
BC &longs;unt, vt ED DC CA. ergo
vnaqueque ip&longs;arum EB BD, DB BC, CB BA
hoc e&longs;t
ad &longs;uas con&longs;equentes, nempè
vt vtraque &longs;imul EB BD cum vtraque &longs;imul AB BC,
& vtraque &longs;imul CB BD
&longs;umitur EB, & &longs;emel AB, bis BD, & bis BC. in con&longs;equentive
rò &longs;umitur
ip&longs;arum EA AD e&longs;t eadem,
pla vtriu&longs;que &longs;imul DB BC ad vtramque &longs;imul BD BA cum dupla
ip&longs;ius BC. Quare & dupla ad duplam eandem habebit
est, vt EA ad AD, ita dupla vtriu&longs;que &longs;imul EB BA cum quadru
pla vtriu&longs;que &longs;imul CB BD ad duplam vtriu&longs;que &longs;imul AB BD cum
compo&longs;ita ex dupla vtriu&longs;que &longs;imul AB BE, & qua-
po&longs;itæ ex dupla vtriu&longs;que &longs;imul AB BD, & quadruplaip&longs;ius CB. Ve
rùm
tres quintas ip&longs;ius AD, erit conuertendo EA ad EB, vt
tres quintæ ip&longs;ius AD ad FG; permutandoquè
tres quintasip&longs;ius AD, &longs;ic e&longs;t EB ad FG, vtigitur EB ad FG,
&longs;ic dupla vtriu&longs;que &longs;imul AB BE cum quadrupla vtriu&longs;que
mul AB BD cum quadrupla ip&longs;ius CB. osten &longs;um e&longs;t aut
ita e&longs;&longs;e, vt
p&longs;ius BD ad duplam vtriu&longs;que &longs;imul AB BE cum quadrupla
vtriu&longs;que &longs;imul CB BD. Atin hoc antecedente ter a&longs;&longs;umpta
e&longs;t AB, terquè BD, & &longs;exies CB. erititaque in primis magni
tudinibus antecedens OB ad con&longs;equens EB, vtin &longs;ecundis
magnitudinibus an recedens
BD cum &longs;extupla ip&longs;ius CB ad
triu&longs;que &longs;imul AB BE, & quadruplam vtriu&longs;que &longs;imul CB BD.
in primis verò magnitudinibus e&longs;t con&longs;equens EB ad aliud
quippiam FG, ut in &longs;ecundis magnitudinibus con&longs;equens,
hoc e&longs;t dupla vtriu&longs;que &longs;imul AB BE cum quadrupla vtriu&longs;
que &longs;imul DB BC ad aliud quippiam, nempè ad tres quintas
lineæ
tæ ex dupla utri^{9}
pla ip&longs;ius AB ad &longs;imiliter
tripla ip&longs;ius BD ad duplam eiu&longs;dem BD e&longs;t, vt tria ad duo.
CB ita &longs;e habet, vt &longs;ex ad quatuor, hoce&longs;t tria ad duo, & om
nesad omnes, hoc e&longs;t
et &longs;extupla ip&longs;ius CB ad compo&longs;itam ex dupla vtriu&longs;que &longs;imul AB BD,
& quadrupla ip&longs;ius CB proportionem habet, quam tria ad duo.
pli gratia quindecim ad decem,
pla vtriu&longs;que &longs;imul AB BD, & &longs;extupla ip&longs;ius CB
tas eiu&longs;dem
drupla ip&longs;ius, CB, quæ po&longs;ita e&longs;t decem,
quinque ad duo.
ip&longs;ius decem &longs;unt &longs;ex. at verò proportio, quam habet linea
po&longs;ita ex tripla vtriu&longs;que &longs;imul AB BD, & &longs;extupla ip&longs;ius CB
ad tres quintas lineæ compo&longs;it&ecedil; ex dupla vtriu&longs;que &longs;imul AB
BD cum quadrupla ip&longs;ius CB, e&longs;t æqualis ei, quam habet OB
ad FG. ergo erit OB ad FG, vtquinque ad duo.
autem e&longs;t, & AO ad GH proportionem habere, quam quinque ad duo;
totaigitur BA ad totam FH proportionem habet, quam quinque ad duo.
demon&longs;trare.
ius.
buius.
ius.
huius.
ius.
ti.
ius.
in
mi huius.
16,
11.
huius.
SCHOLIVM.
Græcus codex po&longs;t ea verba,
non habet,
nece&longs;&longs;aria videntur. ideo po&longs;t gr&ecedil;ca verba,
ou)/tws a)/te ag w_<10>o\s, gb
Vbiautem &longs;untverba,
cus codex tantùm habet,
In quibus de&longs;ideratur illa particula,
hoc modo,
Præterea cùm inquit,
a
Similiter quando in quit
keim
deratur,
sugkeime\nan e)/ k te tas b sunamfote/<10>ou ta=s ab bd, kai\ d ta)/s *gb
Po&longs;tremum theorema, & &longs;i non habeat
veluti pr&ecedil;cedens, non e&longs;t tamen &longs;ine aliqua ob&longs;curitate, ob cu
ius intelligentiam hanc priùs propo &longs;itionem o&longs;tendemus.
PROPOSITIO.
Si duæ fuerint rectæ line&ecedil; in para bolc ad diametrum ordi
natim applicatæ, erit maior parabole ad
dimidia line&ecedil; maioris ad cubum ex dimidia minoris.
In parabole ABC, cuius diameter BF, duæ &longs;int rectæ lineæ
ad diametrum applicatæ AC DE. Dico parabolen ABC ad
parabolen DBE eandem habere proportionem, quam cub^{9}
ex AF ad cubum ex DG. lungantur AB BC DB BE; &longs;ecet-
&longs;e&longs;quitertia e&longs;t trianguli ABC, itidemquè parabole DBE
trianguli DBE &longs;e&longs;quitertia exi&longs;tit, erit parabole ABC ad trian
gulum ABC, vt parabole DBE ad triangulum DBE. &
mutando parabole ABC ad parabolen DBE, vt triangulum
ABC ad triangulum DBE. Quoniam autem AC ordina
tim e&longs;t applicata, vnde AF ip&longs;i FC e&longs;t æqualis, ac per con&longs;e
quens AF e&longs;t ip&longs;ius AC dimidia. erit triangulum ABF dimi
dium trianguli ABC. cùm vtraquè &longs;ub eadem &longs;int altitudine.
eademquè ratione triangulum DBG trianguli DBE dimi
dium exi&longs;tit. quare vt triangulum ABF ad triangulum
DBG, ita e&longs;t triangulum ABC ad DBE triangulum, ac pro
pterea triangulum ABF ad DBG triangulum e&longs;t, vt parabo
le ABC ad parabolen DBE. Cùm autem &longs;it HG æquidi&longs;tans
ip&longs;i AF, &longs;iquidem &longs;unt ordinatim applicatæ, ob
&longs;imilitudinem ABF HBG, ita erit FB ad BG, vt AF ad HG
vtautem FB ad BG, ita quadratum ex AF ad quadratum ex
DG, eritigitur quadratum ex AF ad quadratum ex DG, vt AF
ad HG. quare line&ecedil; AF DG HG &longs;unt proportionales. Pro
ducatur FB, ducaturquè à puncto D ip&longs;i AB æquidi&longs;tans
DK, erit vtiquè triangulorum ABF DKG anguli ABF
DHG æquales, & angulus AFB angulo DGK e&longs;t æqualis, erit
igitur, & reliquus BAF reliquo KDG æqualis, ac propterea
triangulum ABF e&longs;t triangulo DKG &longs;imile. quare triangu
lum ABF ad triangulum DKG eam habet proportionem,
quàm AF ad DG duplicatam, hoc e&longs;t quàm AF ad HG, qu&ecedil;
e&longs;t ea, quàm habet FB ad BG. atqui triangulum ABF ad
DKG eam quoque habet proportionem, quam FB ad GK
duplicatam. tres igitur line&ecedil; FB GK GB &longs;unt proportiona
les. ex quibus &longs;equiturita e&longs;&longs;e FB ad GK, vt AF ad DG; &
GK ad GB, vt DG ad GH. &longs;ed quoniam triangulum GDK
ad GDB (cùm &longs;int &longs;ub eadem altitudine) ita e&longs;t, vt KG ad
BG, &longs;i igitur fiat HG ad L, vt KG ad BG, erit triangulum
GDK ad triangulum GDB, vt HG ad L. Cùm autem &longs;it
gulum ABF ad DKG, vt AF ad HG, e&longs;tquè
ad DBG, vt HG ad L, erit ex &ecedil;quali triangulum ABF ad
triangulum DBG, vt AF ad L. ac propterea parabole ABC
AF ad lineam L. Quoniam autem ita e&longs;t KG ad GB, vt
HG ad L, & vt eadem KG ad GB, ita e&longs;t DG ad GH. vt
verò DG ad GH, ita e&longs;t AF ad DG; crunt quatuor lineæ AF
DG HG L in continua proportione. & quoniam cubi in tri
pla &longs;unt proportione laterum, erit cubus ex AF ad cubum ex
DG, vt AF ad L. cubus ergo ex AF ad cubum ex DG eam
habet proportionem, quam parabole ABC ad parabolen
DBE. quod demon&longs;trare oportebat.
ch.de qua.
par.
&longs;extt.
conicorum
Apoll. &
ex
de quad.
parab.
ex cor.
11.
Oportet autem banc quoquè
tam, nem pè quòd &longs;olida parallelepipeda in eadem ba&longs;i con&longs;ti
tuta eam inter &longs;e proportionem habent, quam ip&longs;arum alti
tudines.
Hoc quidem à Federico Commandino in eius libro decen
tro grauitatis &longs;olidorum propo&longs;itione decimanona demon
&longs;tratum fuit.
PROPOSITIO. X.
Omnis fru&longs;ti à rectanguli coni portione ab&longs;ci&longs;&longs;i
centrum grauitatis e&longs;t in recta linea, quæ fru&longs;ti dia
meter exi&longs;tit, ita po&longs;itum, vt diui&longs;a linea in quin
que partes æquales, &longs;it in quinta parte media; ita
vt ip&longs;ius portio propinquior minoriba&longs;i fru&longs;ti ad
reliquam portionem eandem habeat proportio
nem, quam habet &longs;olidum ba&longs;im habens quadra
tumex dimidia maioris ba&longs;is fru&longs;ti, altitudinem au
tem lineam æqualem vtrique &longs;imul duplæ mino
ris ba&longs;is, & maiori ad &longs;olidum ba&longs;im habens qua
dratum ex dimidia minoris ba&longs;is fru&longs;ti,
autem lineam æqualem vtrique duplæ maioris, &
minori.
æquidi&longs;tantes.
gaturquè fru&longs;tum ADEC à portione ABC ab&longs;ci&longs;&longs;um. om
nes vtique lineæ ip&longs;is AC DE æquidi&longs;tantes in fru&longs;to AD
EC ductæ, erunt à linea GF bifartam diui&longs;æ, ex quo
tet quidem & ip&longs;ius ADEC diametrum e&longs;&longs;e GF, lineasquè AC
DE lineæ portionem in B contingenti æquidistantes e&longs;&longs;e. Recta
dia &longs;it HK. atque
IK eandem habeat proportionem, quam habet &longs;olidum ba&longs;im habens
quadratum ex AF, altitudinem verò lineam æqualem vtri&longs;que
&longs;imul duplæ ip&longs;ius DG, & ip&longs;i AF, ad &longs;olidum, quod
ba&longs;im habeat quadratum ex DG, altitudinem autem lineam æqua-
dum est frusti ADEC centrum grauitatis e&longs;&longs;e punctum 1.
&longs;umaturquè ip&longs;arum MN NO media proportionalis NX.
quarta verò proportionalis TN.
NO NT in continua erunt proportione.
ad TN, ita
cunque perueniat alterum punctum
FG, &longs;iue inter GB cadat. & quoniam in portione rectanguli coni
ABC
cipalis est diameter portionis, vel ducta diametro æquidistans.
lineæ verò AF DG ad ip&longs;am ordinatim &longs;unt ap
plicatæ, cùm &longs;int æquidistantes contingenti portionem
vt autem MN ad NO longitudine, itaest MN ad Nx potentia.
quandoquidem treslineæ MN NX NO &longs;unt proportio
nales. quare, & longitudine in eadem &longs;unt proportione
cet AF ad DG, ita MN ad NX.
nem DBE.
portio ABC ad portionem DBE.
ad culum ex Nx, ita MN ad NT.
MN NX NO NT in continua proportione. ac propterea
eritportio ABC ad portionem DBE, vt MN ad NT.
vt FH ad IR, e&longs;t verò FH ip&longs;ius FG tresquintæ, erit fru
&longs;tum ADEC ad portionem DBE, vt FH ad IR
tres quintæ ip&longs;ius GF ad IR. & quoniam &longs;olidum ba&longs;im habens qua
dratum ex AF, altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius
DG, & ip&longs;a AF, ad cubum ex AF proportionem habet,
lidi altitudo ad altitudinem cubi, &longs;iquidem &longs;unt in eadem ba
&longs;i. tàm emm &longs;olidum, quàm cubus ba&longs;im habet quadratum
ex AF. idcirco &longs;olidum ba&longs;im habens quadratum ex AF,
altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius DG, &
ip&longs;a AF ad cubum ex AF eam proportio nem habebit,
&longs;olidi altitudo,
tudinem cubi, hoc e&longs;t
ita e&longs;&longs;e AF ad DG, vt MN ad NX, eritconuertendo DG
ad AF, vt NX ad MN, & antecedentium dupla, hoc e&longs;t du
pla ip&longs;ius DG ad AF, vt dupla ip&longs;ius NX ad MN. & com
ponendo dupla ip&longs;ius DG cum AF ad AF, vt dupla ip&longs;ius
NX cum MN ad MN.
quadratum ex AF, altitudinem verò lineam compo&longs;itam ex
dupla ip&longs;ius DG cum AF ad cubum ex AF, ita
cum linea NM ad NM. est autem
ex DG, vt cubus ex MN ad cubum ex NX, vt o&longs;ten&longs;um e&longs;t,
erit
&longs;icut autem cubus ex DG ad &longs;olidum ba&longs;im habens quadratum ex DG,
altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius AF, cum linea
DG,
dem ba&longs;i, quadrato nempè ex DG. erit igitur vt cubus ex
DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudi
nem verò lineam compo&longs;itam ex dupla ip&longs;ius AF cum linea
DG,
lineam &longs;cilicet
DG.
MN ad NX, vt verò MN ad NX, ita NO
ad NT. cùm &longs;int MN NX NO NT in continua proportio
NT, & componendo, dupla ip&longs;ius AF cum DG ad
DG, vt dupla ip&longs;ius NO cum NT ad NT. & conuer
tendo DG ad duplam ip&longs;ius AF cum DG, vt NT ad
plam ip&longs;ius NO cum NT.
DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitu
dinem verò compo&longs;itam ex dupla ip&longs;ius AF cum DG, ita
e&longs;t
que ex ijs, quæ dicta &longs;unt, ita &longs;e habet &longs;olidum ba&longs;im ha
bens quadratum ex AF, altitudinem verò lineam com
po&longs;itam ex dupla ip&longs;ius DG, & linea AF ad cubum
ex AF, vt dupla ip&longs;ius NX cum NM ad MN,
cubus verò ex AF ad cubum ex DG e&longs;t, vt MN ad
NT; ita deinde &longs;e habetcubus ex DG ad &longs;olidum ba
&longs;im habens quadratum ex DG, altitudinem verò lineam
compo&longs;itam ex dupla ip&longs;ius AF, & ip&longs;a DG, vt
NT ad compo&longs;itam ex dupla ip&longs;ius NO, & ip&longs;a NT.
ex AF, altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius
DG, & linea AF, & cubus ex AF, & cubus ex
DG, & &longs;olidum ba&longs;im habens quadratum ex DG, altitu
dinem verò lineam compo&longs;itam: ex dupla ip&longs;ius AF, & ip&longs;a
DG, quatuor magnitudinibus proportionales, duabus &longs;imul &longs;umptis
tineæ compo&longs;itæ ex dupla ip&longs;ius NX
ri magnitudini MN; aliiquè deinceps NT, ac tandem lineæ
compo&longs;itæ ex duplaip&longs;ius NO, & ip&longs;a NT. ex æquali igitur
erit, vt &longs;olidum ba&longs;im habens quadratum ex AF, altitudinem
&longs;olidum ba&longs;im habens quadratum ex DG, altitudinem verò lt
neam compo&longs;itam ex dupla ip&longs;ius AF, & ip&longs;a DG, ita
compo&longs;ita ex dupla ip&longs;ius NX, & ip&longs;a MN ad compo&longs;itam
ex dupla ip&longs;ius NO, & ip&longs;a NT &longs;ed vt præfatum &longs;oii
dum
lineam compo&longs;itam ex dupla ip&longs;ius DG, & ip&longs;a AF
dictum &longs;olidum
nem verò compo&longs;itam ex dupla ip&longs;ius AF & ip&longs;a DG,
dupla ip&longs;ius NX cum MN, & dupla ip&longs;ius NO cum NT ad
compo&longs;itam ex dupla ip&longs;ius NO cum NT, quia verò in hoc
antecedenti &longs;emel &longs;umitur MN, & &longs;emel NT, bis verò NX,
& bis NO, erit HK ad KI, vt vtraque &longs;imul MN NT, & du
pla vtriu&longs;que &longs;imul NX NO ad duplam ip&longs;ius NO, & ip&longs;am
NT.
HK e&longs;t FG, quintupla verò alterius antecedentis MN NT,
& duplæ vtriu&longs;que &longs;imul NX NO e&longs;t quintupla vtriu&longs;que &longs;i
mul MN NT, & decupla vtriu&longs;que &longs;imul NX NO. decu
pla enim e&longs;t quintupla duplæ.
vtriu&longs;que &longs;imul MN NT, & decupla vtriu&longs;que &longs;imul NX NO ad du
plam ip&longs;ius ON, & ip&longs;am NT. & vt FG ad FK, quæe&longs;t duæ quin
tæip&longs;ius
vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN NT,
&longs;equens &longs;itduæ quintæ ip&longs;ius antecedentis. etenim dupla v
triu&longs;que &longs;imul MN NT quintuplæ earumdem &longs;imul MN
NT duæ quintæ exi&longs;tit. & quadrupla vtriu&longs;que &longs;imul NX
NO e&longs;t duæ quintæ decuplæ earumdem NX NO. quadru
pla enim decuplæ e&longs;t duæ quintæ. Quoniam itaque ita e&longs;t FG
ad FK, vt quintupla vtriu&longs;que &longs;imul MN NT, & decupla
vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN
NT, & quadruplam vtriu&longs;que &longs;imul NX NO, & vt FG ad
KI, ita quintupla vtriu&longs;que &longs;imul MN NT, & decupla vtriu&longs;
que &longs;imul NX NO ad duplam ip&longs;ius ON, & ip&longs;am NT:
erit FG ad &longs;uas con&longs;equentes &longs;imul &longs;umptas FK KI, hoc
e&longs;t FI, vt quintupla vtriu&longs;que &longs;imul MN NT, & decupla
vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN
NT, & quadruplam vtriu&longs;que &longs;imul NX NO, & duplam
ip&longs;ius ON, & ip&longs;am NT. &longs;ed in hoc con&longs;equenti bis &longs;umi
tur MN, quater NX, &longs;exies NO, & ter NT.
FG æd FI, ita quintupla vtriu&longs;que &longs;imul MN NT, & decupla v
triu&longs;que &longs;imul NX NO ad compo&longs;itam ex dupla ip&longs;ius MN, & qua
drupla ip&longs;ius NX, & &longs;extupla ip&longs;ius NO, & tripla ip&longs;ius NT.
conuertendo FI ad FG, vt compo&longs;ita ex dupla ip&longs;ius MN,
& quadrupla ip&longs;ius NX, & &longs;extupla ip&longs;rus NO, & tripla ip
&longs;iús NT ad quintuplam vtriu&longs;que &longs;imul MN NT, & decu
plam vtriu&longs;que &longs;imul NX NO.
neæ MN NX NO NT &longs;unt continuè proportionales.
fuit MN æqualis ip&longs;i FB, & NO ip&longs;i GB; crit reliqua OM
ip&longs;i FG æqualis. & vt TM ad TN ita factum fuit FH,
hoc e&longs;t tres quintæ ip&longs;ius FG, tres &longs;cilicet quintæ ip&longs;ius MO
ad IR. quare & conuertendo
pta linea NI ad tres quintas ip&longs;ius FG, hoc e&longs;t ip&longs;ius MO. vt autem
compo&longs;ita ex dupla ip&longs;ius NM, & quadrupla ip&longs;ius NX, & &longs;extupla ip
&longs;ius NO & tripla ip&longs;ius NT ad lineam compo&longs;itam ex quintupla vtrius
que &longs;imul MN NT, & decupla vtriu&longs;que &longs;imul XN NO, &longs;ic altera quæ
dam a&longs;&longs;umpta linea IF ad FG, hoc est ad MO, erit ex &longs;uperioribus RF
erit tres quintæ ip&longs;ius FB. & obid BR ad. RF e&longs;t, vt tria ad
duo.
EC centrum grauitatis erit in linea QR
ADEC
cùm &longs;it tota FB ad totam BR, vt ablata BG ad ablatam
quam QR, vt FB ad BR. itaque
linea e&longs;i BR; ip&longs;ius verò GB tres quintæ linea est
igitur GF est tres quintæ QR. quoniamigitur est, vt fru&longs;tum AD
EC adportionem DBE, ita MT ad NT,
MN ad NT, &longs;ic
GF; quæ est QR ad RI. erit igitur vt fru&longs;tum ADEC adportionem
DBE, ita QR ad RI. & est quidem totius portionis
Q: manife&longs;tum est igitur fru&longs;ti ADEC centrum grauitatis e&longs;&longs;e
ctum
dratum ex AF, altitudinem autem duplam ip&longs;ius DG cum
AF ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem
verò duplam ip&longs;ius AF
quad. pa
rab. &
corum A
poll.
quad. pa
rab. &
corum A
poil.
mi.
ti.
in
mi huius.
ti.
denti.
ius.
ius.
SCHOLIVM.
In hoc Theoremate primùm ob&longs;eruanda occurrunt verba
propo&longs;itionis, quibus Archimedes pr&ecedil;cipit pottionem HK
in I ita diui&longs;am e&longs;&longs;e oportere, vt HI ad IK eam habeat pro
portionem, quam habet &longs;olidum ba&longs;im habens quadratum
ex dimidia maioris ba&longs;is fru&longs;ti, altitudinem autem lineam æ
qualem vtrique &longs;imul duplæ minoris ba&longs;is, & maiori ad &longs;oli
dum ba&longs;im habens quadratum ex dimidia minoris ba&longs;is fru
&longs;ti, altitudinem autem lineam æqualem vtri&longs;que, duplæ &longs;cili
cet ba&longs;is maioris, & minori. hoc e&longs;t &longs;it HI ad IK, vt &longs;olidum
ba&longs;im habens quadratum ex AF, altitudinem verò lineam æ
qualem duplæ ip&longs;ius DE cum AC ad &longs;olidum ba&longs;im habens
quadratum ex DG, altitudinem verò lineam æqualem
&longs;imul duplæ ip&longs;ius AC, & ip&longs;i DE. In con&longs;tructione autem
hunc propo&longs;itionis locum explicans, & in pergre&longs;&longs;u totius
mon&longs;trationis
re, quam habet &longs;olidum ba&longs;im habens quadratum ex AF, alti
tudinem verò lineam æqualem
& ip&longs;i AF ad &longs;olidum ba&longs;im habens quadratum ex DG, al
titudinem verò lineam æqualem vtrique &longs;imul duplæ ip&longs;ius
AF, & DG. Quoniam autem &longs;olida parallelepipeda (vt præ
fata &longs;olida &longs;unt) in eadem ba&longs;i exi&longs;tentia ita &longs;e habentin ter&longs;e,
vt corum altitudine; &longs;olidum, quod ba&longs;im habet quadratum
ex AF, altitudinem autem duplam ip&longs;ius DE cum AC, du
plum erit &longs;olidi ba&longs;im habentis quadratum ex AF, altitudi
nem verò duplam ip&longs;ius DG cum AF. Nam hæc &longs;olida can
dem habent ba&longs;im, quadratum nempè ex AF; ip&longs;orumquè
alterum habet altitudinem duplam. quia cùm &longs;it DE dupla
ip&longs;ius DG, erit dupla ip&longs;ius DE dupla ip&longs;ius duplæ DG;
in dupla &longs;unt proportione. hoc e&longs;t altitudo, linea &longs;cilicet du
pla ip&longs;ius DE cum AC altitudinis nempè lineæ duplæ ip&longs;ius
DG cum AF dupla exi&longs;tit. Quare &longs;olidum ba&longs;im habens qua
dratum ex AF, altitudinem verò duplam ip&longs;ius DE cum AC
duplum e&longs;t &longs;olidi, quod ba&longs;im habeatidem quadratum ex AF,
altitudinem verò duplam ip&longs;ius DG cum AF. cademquè ratio
neo&longs;tendetur
dinem verò duplam ip&longs;ius AC cum DE duplum e&longs;&longs;e &longs;olidi ba
&longs;im habentis quadratum ex eadem DG, altitudinem autem du
plam ip&longs;ius AF cum DG. &longs;olidum igitur ba&longs;im habens qua
dratum ex AF, altitudinem autem duplam ip&longs;ius DE cum AC
ad &longs;olidum quadtatum habens ba&longs;im ex AF, altitudinent verò
duplam ip&longs;ius DG cum AF eam habet proportionem, quam
habet &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem
verò duplam ip&longs;ius AC cum AE ad &longs;olidum ba&longs;im
dratum ex DG, altitudinem verò duplam ip&longs;ius AF cum DG.
ex AF, altitudinem verò duplam ip&longs;ius DE cum AC ad &longs;ecun
dum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem
autem duplam ip&longs;ius AC cum DE eandem habet proportio
nem, quam habet tertium &longs;olidum ba&longs;im habens quadratum
ex AF, altitudinem autem duplam ip&longs;ius DG cum AF ad quar
tum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem ve
rò duplam ip&longs;ius AF cum DG. Quapropter Archimedes loco
primi, & &longs;ecundi &longs;olidi in propo&longs;itione propo&longs;iti rectè potuit
in demon&longs;tratione accipere tertium, & quartum &longs;olidum. co
dem enim modo, & in eadem proportione linea HK in pun
cto I diui&longs;a prouenit: quod quidem punctum fru&longs;ti ACED
centrum grauitatis exi&longs;tit.
Secundi libri Finis.
Erratorum quorundam re&longs;titutio.
Pagina 8, ver&longs;u 18, Archimedes. <33> 10, 7, &longs;ione. <33> 18, 20, conducenti. <33> 21, 14, per
di&longs;cere ip&longs;um. <33> 39, 25, hoc e&longs;t AB. <33> 43, 19, lineam. <33> 47, 20, cúm inquit, <33> 63,
20, GD DK in. <33> 65, 21, DC. Ibidem, 27, ex DC. <33> 67, 29, in maiori. <33> 69, in
po&longs;til: ex proxima propo&longs;itione. <33> 70, 5, vt NL <33> 73, 1, de his, vel. <33> 84, 8, AEEB
CF FD. <33> 90, 17, totus. <33> 98, 1, quam VH. Ibidem, 7, aufertur. <33> 11
&longs;uit. <33> 124, 19,
Ibide, 25, &longs;ta S 9, ad Y
OR ad. Ibidem, 31, vt OR ad
ita. Ibidem, 35, &longs;it BD ad D
13, ræ, vt. <33> 157, in po&longs;till ante 15, primi Ibidem, 17, maiorem. <33> 161, 24, erit KH.
<33> 167, 34, efficax. <33> 170, 1, ip&longs;ius AC erit. <33> 181, 36, ex dupla ip&longs;ius AB, <33> 191,
21, erunt. Ibidem, 22, DKG æquales.
REGISTRVM.
<12> ABCDEFGHIKLMNOPQRSTVXYZ,
AA BB.
Omnes duerniones, præter, BB, ternionem.
PISAVRI.
Apud Hieronymum Concordiam,
M. D. LXXXVII.