FEDERICI
COMMANDINI
VRBINATISLIBER DE CENTRO
GRAVITATIS
SOLIDORVM.
CVM PRIVILEGIO IN ANNOS X.
BONONIAE,
Ex Officina Alexandri Benacii.
MDLXV.
ALEXANDRO FARNESIO
CARDINALI AMPLISSIMO.
ET OPTIMO.
Cvm multæ res in mathematicis
di&longs;ciplinis nequaquam &longs;atis ad
huc explicatæ &longs;int, tum perdif
ficilis, & perob&longs;cura quæ&longs;tio
e&longs;t de centro grauitatis corpo
rum &longs;olidorum; quæ, & ad co
gno&longs;cendum pulcherrima e&longs;t,
& ad multa, quæ à mathematicis proponuntur, præ
clare intelligenda maximum affert adiumentum. de
qua neminem ex mathematicis, neque no&longs;tra, neque
patrum no&longs;trorum memoria &longs;criptum reliqui&longs;&longs;e &longs;ci
mus. & quamuis in earum monumentis literarum
nulla reperiantur, ex quibus in hanc &longs;ententiam addu
ci po&longs;&longs;umus, vt exi&longs;timemus hanc rem ab
rime tractatam e&longs;&longs;e; tamen ne&longs;cio quo fato adhuc
in eiu&longs;modi librorum ignoratione ver&longs;amur. Archi
medes quidem
cuius in&longs;criptio e&longs;t,
norum copio&longs;i&longs;sime, atque acuti&longs;sime con&longs;crip&longs;it: &
in eo explicando
Sed de cognitione
non mul
tos abhinc annos MARCELLVS II. PONT. MAX.
manitas, libros eiu&longs;dem Archimedis de ijs, quæ ve
huntur in aqua, latine redditos dono dedit. hos cum
ego, ut aliorum &longs;tudia incitarem,
mentariis
tari non po&longs;&longs;e, quin Archimedes vel de hac materia
&longs;crip&longs;i&longs;&longs;et, vel aliorum mathematicorum &longs;cripta per
legi&longs;&longs;et. nam in iis tum alia nonnulla, tum maxime
illam propo&longs;itionem, ut euidentem, & aliàs proba
tam a&longs;&longs;umit,
dis rectanguli axem ita diuidere, vt pars, quæ ad verti
cem terminatur, alterius partis, quæ ad ba&longs;im dupla
&longs;it. Verum hæc ad cam partem mathematicarum
di&longs;ciplinarum præcipue refertur, in qua de centro
grauitatis corporum &longs;olidorum tractatur. non e&longs;t au
tem con&longs;entaneum Archimedem illum admirabilem
virum hanc propo&longs;itionem &longs;ibi argumentis con
firmandam exi&longs;timaturum non fui&longs;&longs;e, ni&longs;i eam vel
aliis in locis probaui&longs;&longs;et, vel ab aliis probatam e&longs;&longs;e
comperi&longs;&longs;et. quamobrem nequid in iis libris intel
ligendis de&longs;iderari po&longs;&longs;et, &longs;tatui hanc etiam partem
vel à veteribus prætermi&longs;&longs;am, vel tractatam quidem,
fed in tenebris iacentem, non intactam relinquere;
atque ex a&longs;sidua mathematicorum, præ&longs;ertim Archi
medis lectione, quæ mihi in mentem venerunt, ea in
medium afferre; ut centri grauitatis corporum &longs;oli
dorum, &longs;i non perfectam, at certe aliquam noti-Quem meum laborem
maticis&longs;olum, verum iis etiam, qui naturæ ob&longs;curi
tate delectantur,
enim
que
neque id vlli mirandum videri debet. vt enim in cor
poribus no&longs;tris omnia membra, ex quibus certa quæ
dam officia na&longs;cuntur, diuino quodam ordine inter
&longs;e implicata, & colligata &longs;unt: in
la con&longs;piratio, quam
ita tres illæ Philo&longs;ophiæ (ut Ari&longs;totelis verbo vtar)
quæ veritatem &longs;olam propo&longs;itam habent, licet qui
bu&longs;dam qua&longs;i finibus &longs;uis regantur: tamen
quæque per &longs;e ip&longs;am quodammodo imperfecta e&longs;t:
neque altera &longs;ine alterius auxilio plene comprehen
di pote&longs;t. complures præterca mathematicorum no
di ante hac explicatu difficillimi nullo negotio expe
diti e&longs;&longs;ent: atque (ut vno verbo complectar) ni&longs;i
mea valde amo, tractationem hanc meam &longs;tudio&longs;is
non mediocrem vtilitatem, & magnam volupta
tem allaturam e&longs;&longs;e mihi per&longs;ua&longs;i. cum autem ad hoc
&longs;cribendum aggre&longs;&longs;us e&longs;sem, allatus e&longs;t ad me liber
Franci&longs;ci Maurolici Me&longs;&longs;anen&longs;is, in quo vir ille do
cti&longs;simus, & in iis di&longs;ciplinis exercitati&longs;simus af
firmabat &longs;e de centro grauitatis corporum &longs;olido
rum con&longs;crip&longs;i&longs;&longs;e. cum hoc intellexi&longs;&longs;em, &longs;u&longs;tinui
me pauli&longs;per: tacitus'que expectaui, dum opus cla-
in lucem proferretur: mihi enim exploratis&longs;imum
erat: Franci&longs;cum Maurolicum multo doctius, &
exqui&longs;itius hoc di&longs;ciplinarum genus &longs;criptis &longs;uis tra
diturum. &longs;ed cum id tardius fieret, hoc e&longs;t, ut ego
interpretor, diligentius, mihi diutius hac &longs;criptione
non &longs;uper&longs;edendum e&longs;&longs;e duxi, præ&longs;ertim cum iam li
bri Archimedis de iis, quæ uehuntur in aqua, opera
mea illu&longs;trati typis nec me alia cau&longs;
&longs;a impuli&longs;&longs;et, ut de centro grauitatis corporum &longs;oli
dorum &longs;criberem, ni&longs;i ut hac etiam ratione lux eis
quàm maxime fieri po&longs;&longs;et afferretur.
faciendum exi&longs;timaui, quòd in &longs;pem ueniebam fore,
ut cum ego ex omnibus mathematicis primus, hanc
materiam explicandam &longs;u&longs;cepi&longs;&longs;em; &longs;i quid errati for
te à me commi&longs;&longs;um e&longs;&longs;et, boni uiri potius id meæ de
&longs;tudio&longs;is hominibus bene
arrogantiæ a&longs;criberent. re&longs;tabat ut con&longs;iderarem, cui
potis&longs;imum ex principibus uiris contemplationem
hanc, nunc primum memoriæ, ac literis proditam de
dicarem. harum mearum cogitationum &longs;umma fa
cta, exi&longs;timaui nemini conuenientius de centro graui
tatis corporum opus dicari oportere, quàm ALE
XANDRO FARNESIO grauis&longs;imo, ac prudentis&longs;i
mo Cardinali, quo in uiro &longs;umma fortuna &longs;emper
&longs;umma uirtute certauit. quid enim maxime in te ad
mirati debeant homines, ob&longs;curum e&longs;t; u&longs;um'ne re-
bui&longs;ti, &
norificenti&longs;simarum legationum; an excellentiam
in omni genere literarum, qui vix
homines iam confirmata ætate &longs;ummo &longs;tudio,
turnisque
comprehendi&longs;ti: an con&longs;ilium, & &longs;apientiam in re
gendis, &
&longs;ententiæ in &longs;ancti&longs;simo Reip. Chr&longs;tianæ con&longs;ilio di
ctæ, potius diuina oracula, quàm &longs;ententiæ habitæ
&longs;unt, & habentur. prætermitto liberalitatem, & mu
nificentiam tuam, quam in &longs;tudio&longs;i&longs;simo quoque ho
ne&longs;tando quotidie magis o&longs;tendis, ne videar auribus
tuis potius, quàm veritati &longs;eruire. quamuis à te in tot
præclaros viros tanta beneficia collata &longs;unt, &
runtur
nihil iucundius, quàm eximia tua liberalitate homi
nes ad amplexandam virtutem, licet currentes incita
re. nihil dico de ceteris virtutibus tuis, quæ tantæ
&longs;unt, quantæ ne cogitatione quidem comprehendi
po&longs;&longs;unt. Quamobrem hac præcipue de cau&longs;&longs;a te hu
ius meæ lucubrationis patronum e&longs;&longs;e volui, quam ea,
qua &longs;oles, humanitate accipies. te enim &longs;emper ob
diuinas virtutes tuas colui, & ob&longs;eruaui:
hi fuit optatius; quàm tibi per&longs;pectum e&longs;&longs;e meum
erga te animum; cœ
lum igitur digito attingam, &longs;i po&longs;t graui&longs;simas oc-
impertiri temporis non grauaberis:
tibi &longs;emper addicti erunt, numerare. Vale.
Federicus Commandinus.
FEDERICI COMMANDINI
VRBINATIS LIBER DE CENTRO
GRAVITATIS SOLIDORVM.
DIFFINITIONES.
Centrvm grauitatis, Pappus
Alexandrinus in octauo ma
thematicarum collectionum
libro ita diffiniuit.
matos e)=inai shme=ion ti kei/menon e)nto/s, a/fÕ
o(/u katÕe/poi/nian a\rtnqe/n to/ ba/ros n(mere=i
fero/menon, kai\ fula/ssei th/n e)c a)rxh=s qe/
sin, o)u mh\ peritrepo/menon e)n th= fora=
Dicimus autem centrum grauitatis uniu&longs;cu
iu&longs;que corporis punctum quoddam intra po&longs;i
tum, à quo &longs;i graue appen&longs;um mente concipia
tur, dum fertur quie&longs;cit; & &longs;eruat eam, quam in
principio habebat po&longs;itionem: neque in ip&longs;a la
tione circumuertitur.
Po&longs;&longs;umus etiam hoc modo diffinire.
Centrum grauitatis uniu&longs;cuiu&longs;que &longs;olidæ figu
ræ e&longs;t punctum illud intra po&longs;itum, circa quod
undique partes æqualium momentorum con&longs;i
&longs;tunt. &longs;i enim per tale centrum ducatur planum
figuram quomodocunque &longs;ecans &longs;emper in par
Pri&longs;matis, cylindri, & portionis cylindri axem
appello rectam lineam, quæ oppo&longs;itorum plano
rum centra grauitatis coniungit.
Pyramidis, coni, & portionis coni axem dico li
neam, quæ à uertice ad centrum grauitatis ba&longs;is
perducitur.
Si pyramis, conus, portio coni, uel conoidis &longs;e
cetur plano ba&longs;i æquidi&longs;tante, pars, quæ e&longs;t ad ba
&longs;im, fru&longs;tum pyramidis, coni, portionis coni, uel
conoidis dicetur; quorum plana æquidi&longs;tantia,
quæ opponuntur &longs;imilia &longs;unt, & inæqualia: axes
uero &longs;unt axium figurarum partes, quæ in ip&longs;is
comprehenduntur.
PETITIONES.
Solidarum figurarum fimilium centra grauita
tis &longs;imiliter &longs;unt po&longs;ita.
Solidis figuris &longs;imilibus, & æqualibus inter &longs;e
aptatis, centra quoque grauitatis ip&longs;arum inter &longs;e
aptata crunt.
THEOREMA I. PROPOSITIO I.
Omnis figuræ rectilineæ in circulo de&longs;criptæ,
quæ æqualibus lateribus, & angulis contine
trum.
Sit primo triangulum æquilaterum abc in circulo de
&longs;criptum: & diui&longs;a ac bifariam in d, ducatur bd. erit in li
nea bd centrum grauitatis
primi libri Archimedis de centro grauitatis planorum. Et
quoniam linea ab e&longs;t æqualis
lineæ bc; & ad ip&longs;i dc;
bd utrique communis: trian
gulum abd æquale erit trian
gulo cbd: & anguli angulis æ
quales, qui æqualibus lateri
bus &longs;ubtenduntur. ergo angu
li ad d quòd
cum linea bd &longs;ecet ae bifa
riam, & ad angulos rectos; in
ip&longs;abd e&longs;t centrum circuli.
quare in eadem bd linea erit
centrum grauitatis trianguli, & circuli centrum. Similiter
diui&longs;a ab bifariam in e, & ducta ce, o&longs;tendetur in ip&longs;a
que centrum contineri. ergo ea erunt in puncto, in quo li
neæ bd, ce conueniunt. trianguli igitur abc centrum gra
uitatis e&longs;t idem, quod circuli centrum.
mæ tertii
Sit quadratum abcd in cir
culo de&longs;criptum: & ducantur
a c, bd, quæ conueniant in e. er
go punctum e e&longs;t centrum gra
uitatis quadrati, ex decima eiu&longs;
dem libri Archimedis. Sed cum
omnes anguli ad abcd recti
&longs;int; erit abc femicirculus:
neæ ac, bd diametri circuli: idem igitur e&longs;t centrum
grauitatis quadrati, & circuli centrum.
Sit pentagonum æquilaterum, & æquiangulum in circu
lo de&longs;criptum abcd e. & iun
cta bd,
ducatur cf, & producatur ad
circuli circumferentiam in g;
quæ lineam ae in h &longs;ecet: de
inde iungantur ac, cc. Eodem
modo, quo &longs;upra demon&longs;tra
bimus angulum bcf æqualem
e&longs;&longs;e. angulo dcf; & angulos
ad f utro&longs;que rectos: & idcir
co lineam cfg per circuli cen
trum tran&longs;ire. Quoniam igi
tur latera cb, ba, & cd, de æqualia &longs;unt; & æquales anguli
cba, cde: erit ba&longs;is ca ba&longs;i: ce, & angulus bca angulo
dce æqualis. ergo & reliquus ach, reliquo ech. e&longs;t au
tem ch utrique triangulo ach, ech communis. quare
ba&longs;is ah æqualis e&longs;t ba&longs;i hc: & anguli, qui ad h recti:
recti, qui ad f. ergo lineæ ae, bd inter &longs;e &longs;e æquidi&longs;tant.
Itaque cum trapezij abde latera bd, ae æquidi&longs;tantia à li
nea fh bifariam diuidantur; centrum grauitatis ip&longs;ius erit
in linea fh, ex ultima eiu&longs;dem libri Archimedis. Sed trian
guli bcd centrum grauitatis e&longs;t in linea cf. ergo in eadem
linea ch e&longs;t centrum grauitatis trapezij abde, & trian
guli bcd: hoc e&longs;t pentagoni ip&longs;ius centrum: & centrum
circuli. Rur&longs;us &longs;i iuncta ad,
tur ekl: demon&longs;trabimus in ip&longs;a utrumque centrum in
e&longs;&longs;e. Sequitur ergo, ut punctum, in quo lineæ cg, el con
ueniunt, idem &longs;it centrum circuli, & centrum grauitatis
pentagoni.
medis.
Sit hexagonum abcdef æquilaterum, & æquiangulum
in circulo de&longs;ignatum:
u&longs;que circumferentiam; quæ &longs;ecet ae in h. Similiter conclu
demus cg per centrum circuli tran&longs;ire: & bifariam &longs;ecate
lineam ae;
Cum igitur cg per centrum circuli tran&longs;eat; & ad
f perueniat nect&longs;&longs;e e&longs;t: quòd cdef &longs;it dimidium circumfe
rentiæ circuli. Quare in eadem
diametro cf erunt centra gra
uitatis triangulorum bcd,
afe, & quadrilateri abde, ex
quibus con&longs;tat hexagonum ab
cdef. per&longs;picuum e&longs;t igitur in
ip&longs;a cf e&longs;&longs;e circuli centrum, &
centrum grauitatis hexagoni.
Rur&longs;us ducta altera diamctro
ad, ei&longs;dem rationibus o&longs;tende
mus in ip&longs;a utrumque
ine&longs;&longs;e. Centrum ergo grauita
tis hexagoni, & centrum circuli idem erit.
medis.
9.
m
Sit heptagonum abcdefg æquilaterum atque æquian
gulum in circulo de&longs;criptum:
& iungantur ce, bf, ag: di
ui&longs;a autem ce bifariam in
cto h: & iuncta dh produca
tur in k. non aliter demon
&longs;trabimus in linea dk e&longs;&longs;e cen
trum circuli, & centrum gra
uitatis trianguli cde, & tra
peziorum bcef, abfg, hoc
e&longs;t centrum totius heptago
ni: & rur&longs;us eadem centra in
alia diametro cl &longs;imiliter du
cta contineri. Quare & centrum grauitatis heptagoni, &
centrum circuli in idem punctum conueniunt. Eodem mo
culo de&longs;cribuntur, probabimus
& centrum circuli idem e&longs;&longs;e. quod quidem demon&longs;trare
oportebat.
Ex quibus apparet cuiuslibet figuræ rectilincæ
in circulo plane de&longs;criptæ centrum grauitatis
e&longs;&longs;e, quod & circuli centrum.
Figuram in circulo plane de&longs;criptam appella
mus, cuiu&longs;modi e&longs;t ea, quæ in duodecimo elemen
torum libro, propo&longs;itione &longs;ecunda de&longs;cribitur.
ex æqualibus enim lateribus, & angulis con&longs;tare
per&longs;picuum e&longs;t.
THEOREMA II, PROPOSITIO II.
Omnis figuræ rectilineæ in ellip&longs;i plane de&longs;cri
ptæ centrum grauitatis e&longs;t idem, quod ellip&longs;is
centrum.
Quo modo figura rectilinea in ellip&longs;i plane de&longs;cribatur,
docuimus in commentarijs in quintam propo&longs;itionem li
bri Archimedis de conoidibus, & &longs;phæroidibus.
Sit ellip&longs;is abcd, cuius maior axis ac, minor bd:
ganturque
ctis efgh. à centro autem, quod &longs;it k ductæ lineæ ke, kf,
kg, kh u&longs;que ad &longs;ectionem in puncta lmno protrahan
tur: & iungantur lm, mn, no, ol, ita ut ac &longs;ecet li
neas lo, mn, in z
erunt lk, kn linea una,
& lineæ ba, cd æquidi&longs;tabunt lineæ mo: & bc, ad ip&longs;i
ln. rur&longs;us lo, mn axi bd æquidi&longs;tabunt: & lm, Quoniam enim triangulorum abk, adk, latus
bk e&longs;t æquale lateri kd, & ak utrique commune;
ad k recti. ba&longs;is ab ba&longs;i ad; & reliqui anguli reliquis an
gulis æquales erunt. eadem quoqueratione o&longs;tendetur bc
æqualis cd; & ab ip&longs;i
bc. quare omnes ab,
bc, cd, da &longs;unt æqua
les. & quoniam anguli
ad a æquales &longs;unt angu
lis ad c; erunt anguli b
ac, acd coalterni inter
&longs;e æquales;
acb. ergo cd ip&longs;i ba;
& ad ip&longs;i bc æquidi
&longs;tat. Atuero cum lineæ
ab, cd inter &longs;e æquidi
&longs;tantes bifariam &longs;ecen
tur in punctis eg; erit li
nea lekgn diameter &longs;e
ctionis, & linea una, ex
demon&longs;tratis in uige&longs;i
maoctaua &longs;ecundi coni
corum. Et eadem ratione linea una mfkho. Sunt
bc inter &longs;e &longs;e æquales, & æquidi&longs;tantes. quare & earum di
midiæ ah, bf;
lineæ æquales, & æquidi&longs;tantes erunt.
cd diametro mo: & pariter ad, bc ip&longs;i ln æquidi&longs;tare o
&longs;tendemus. Si igitur
portio ellip&longs;is ad portionem adc moueri, cum primum b
applicuerit ad d,
ba lineæ ad; & bc ip&longs;i cd congruet: punctum uero e ca
det in h; f in g: & linea ke in lineam kh: & kf in kg. qua
re & el in ho, et fm in gn. At ip&longs;a lz in zo; et m
cadet. congruet igitur triangulum lkz triangulo okz: et
m quòd cum etram recti
&longs;int, qui ad k; æquidi&longs;tabunt lineæ lo, mn axi bd. & ita
demon&longs;trabuntur lm, on ip&longs;i ac æquidi&longs;tare. Rurfus &longs;i
iungantur al, lb, bm, mc, cn, nd, do, oa: & bitariam di
uidantur: à centro autem k ad diui&longs;iones ductæ lineæ pro
trahantur u&longs;que ad &longs;ectionem in puncta pqrstuxy: & po
&longs;tremo py, qx, ru, st, qr, ps, yt, xu coniungantur. Simili
ter o&longs;tendemus lineas
py, qx, ru, st axi bd æ
quidi&longs;tantes e&longs;&longs;e: & qr,
ps, yt, xu æquidi&longs;tan
tes ip&longs;i ac. Itaque dico
harum figurarum in el
lip&longs;i de&longs;criptarum cen
trum grauitatis e&longs;&longs;e
ctum
lip&longs;is centrum. quadri
lateri enim abcd cen
trum e&longs;t k, ex decima e
iu&longs;dem libri Archime
dis, quippe
nes diametri
Sed in figura albmcn
do, quoniam trianguli
alb centrum grauitatis
e&longs;t in linea le:
zijomcd in kg: & trianguli cnd in ip&longs;a gn: erit magnitu
dinis ex his omnibus con&longs;tantis, uidelicet totius figuræ cen
trum grauitatis in linea ln: & ob eandem cau&longs;&longs;am in linea
om. e&longs;t enim trianguli aod centrum in linea oh: trapezij
alnd in hk: trapezij lbcn in kf: & trianguli bmc in fm.
cum ergo figuræ albmcndo centrum grauitatis &longs;it in li
nea ln, & in linea om; erit centrum ip&longs;ius punctum k, in Po&longs;tremo in figura
aplqbrmsctnudxoy centrum grauitatis trian
guli pay, & trapezii ploy e&longs;t in linea az: trapeziorum
uero lqxo, qbdx centrum e&longs;t in linea zk: &
brud, rmnu in k
li sct in quare magnitudinis ex his compo&longs;itæ
in linea ac con&longs;i&longs;tit. Rur&longs;us trianguli qbr, & trapezii ql
mr centrum e&longs;t in linea b
aytc, yont in linea
xdu centrum in totius ergo magnitudinis centrum
e&longs;t in linea bd. ex quo &longs;equitur, centrum grauitatis figuræ
aplqbrmsctnudxoy e&longs;&longs;e
bd commune, quæ omnia demon&longs;trare oportebat.
medis.
THEOREMA III. PROPOSITIO III.
Cuiuslibet portio
nis circuli, & ellip&longs;is,
quæ dimidia non &longs;it
maior, centrum graui
tatis in portionis dia
metro con&longs;i&longs;tit.
HOC eodem pror&longs;us
modo demon&longs;trabitur,
quo in libro de centro gra
uitatis planorum ab Ar
chimede
in portione
linea, & rectanguli coni &longs;e
ctione grauitatis
e&longs;&longs;e in diametro portio
nis. Etita demon&longs;trari po
ctione, &longs;eu hyperbola continetur.
THEOREMA IIII. PROPOSITIO IIII.
IN circulo & ellip&longs;i idem e&longs;t figuræ & graui
tatis centrum.
SIT circulus, uel ellip&longs;is, cuius centrum a. Dico a gra
uitatis quoque centrum e&longs;&longs;e. Si enim fieri pote&longs;t, &longs;it b cen
trum grauitatis: & iuncta ab extra figuram in c produca
tur: quam uero proportionem habet linea ca ad ab, ha
beat circulus a ad alium circulum, in quo d; uel ellip&longs;is ad
aliam ellip&longs;im: & in circulo, uel ellip&longs;i figura rectilinea pla
ne de&longs;cribatur adco, ut tandem relinquantur portiones
quædam minores circulo, uel ellip&longs;i d; quæ figura &longs;it abcefg
hklmn. Illud uero in circulo fieri po&longs;&longs;e ex duodecimo
elementorum libro, propo&longs;itione &longs;ecunda manife&longs;te con
&longs;tat; at in ellip&longs;i nos demon&longs;tra
uinms in commentariis in quin
tam propo&longs;itionem Archimedis
de conoidibus, & &longs;phæroidibus.
erit igitur a centrum grauitatis
ip&longs;ius figuræ, quod proxime
dimus. Itaque quoniam circulus
a ad circulum d, uel ellip&longs;is a ad
ellip&longs;im d eandem
habet, quam linea ca ad ab:
portiones uero &longs;unt minores cir
culo uel ellip&longs;i d: habebit circu
lus, uel ellip&longs;is ad portiones ma
iorem proportionem, quàm ca
ad ab: & diuidendo figura recti
linea abcefghklmn ad portiones
habebit maiorem
quam cb ad ba. fiat ob ad ba,
ut figura rectilinea ad portio
nes. cum igitur à circulo, uel el
lip&longs;i, cuius grauitatis centrum
e&longs;t b, auferatur figura rectilinea
efghklmn, cuius centrum a;
reliquæ magnitudinis ex portio
nibus compo&longs;itæ centrum graui
tatis erit in linea ab producta,
& in puncto o, extra figuram po
&longs;ito. quod quidem fieri nullo mo
do po&longs;&longs;e per&longs;picuum e&longs;t. &longs;equi
tur ergo, ut circuli & ellip&longs;is cen
trum grauitatis &longs;it punctum a,
idem quod figuræ centrum.
apud
panum.
medis.
ALITER.
Sit circulus, uel ellip&longs;is abcd,
cuius diameter db, & centrum e:
nea ac, &longs;ecans ip&longs;am db ad rectos angulos. erunt adc,
abc circuli, uel ellip&longs;is dimidiæ portiones. Itaque quo
niam por
uitatis e&longs;t
in diame
tro de: &
portionis
abc cen
trum e&longs;t
ip&longs;a eb: to
tius circu
li, uel ellip&longs;is grauitatis centrum erit in diametro db.
Sit autem portionis adc erit g por
tionis abc centrum. nam &longs;i hæ portiones, quæ æquales
& &longs;imiles &longs;unt, inter &longs;e &longs;e aptentur, ita ut be cadat in de,
& punctum b in d cadet, & g in f: figuris autem æquali
bus, & &longs;imilibus inter &longs;e aptatis, centra quoque grauitatis
ip&longs;arum inter &longs;e aptata erunt, ex quinta petitione Archi
medis in libro de centro grauitatis planorum. Quare cum
portionis adc centrum grauitatis &longs;it f: & portionis
abc centrum g: magnitudinis; quæ ex utri&longs;que efficitur:
hoc e&longs;t circuli uel ellip&longs;is grauitatis centrum in medio li
neæ fg, quod e&longs;t e, con&longs;i&longs;tet, ex quarta propo&longs;itione eiu&longs;
dem libri Archimedis. ergo circuli, uel ellip&longs;is centrum
grauitatis e&longs;t idem, quod figuræ centrum. atque illud e&longs;t,
quod demon&longs;trare oportebat.
Ex quibus &longs;equitur portionis circuli, uel ellip
&longs;is, quæ dimidia maior &longs;it, centrum grauitatis in
diametro quoque ip&longs;ius con&longs;i&longs;tere.
Sit enim maior portio abc, cu
pleatur circulus, uel ellip&longs;is, ut portio reliqua fit aec, diaQuoniam igitur circuli uel ellip&longs;is
aecb grauitatis centrum e&longs;t in diametro be, & portio
nis aec centrum in linea ed: reliquæ portionis, uidelicet
abc centrum grauitatis in ip&longs;a bd con&longs;i&longs;tat nece&longs;&longs;e e&longs;t, ex
octaua propo&longs;itione eiu&longs;dem.
THEOREMA V. PROPOSITIO V.
SI pri&longs;ma &longs;ecetur plano oppo&longs;itis planis æqui
di&longs;tante, &longs;ectio erit figura æqualis & &longs;imilis ei,
quæ e&longs;t oppo&longs;itorum planorum, centrum graui
tatis in axe habens.
Sit pri&longs;ma, in quo plana oppo&longs;ita &longs;int triangula abc,
def; axis gh: & &longs;ecetur plano iam dictis planis
te; quod faciat &longs;ectionem klm; & axi in
Dico klm triangulum æquale e&longs;&longs;e, & fimile triangulis abc
def; atque eius grauitatis centrum e&longs;&longs;e punctum n. Quo
niam enim plana abc
Klm æquidi&longs;tantia
tur a plano ae; rectæ li
neæ ab, Kl, quæ &longs;unt ip
&longs;orum
nes inter &longs;e &longs;e æquidi
&longs;tant. Sed æquidi&longs;tant
ad, be; cum ae &longs;it para
lelogrammum, ex pri&longs;
matis diffinitione. ergo
& al
erit; & propterea linea
kl, ip&longs;i ab æqualis. Si
militer demon&longs;trabitur
lm æquidi&longs;tans, & æqua
lis bc; & mk ip&longs;i ca.Itaque quoniam duæ lineæ Kl, lm &longs;e &longs;e tangentes, duabus
lineis &longs;e &longs;e tangentibus ab, bc æquidi&longs;tant; nec &longs;unt in e o
dem plano: angulus klm æqualis e&longs;t angulo abc: & ita an
gulus lmk, angulo bca, & mk lip&longs;i cab æqualis probabi
tur. triangulum ergo klm e&longs;t æquale, & &longs;imile triang ulo
abc. quare & triangulo def. Ducatur linea cgo, & per ip
&longs;am, & per cf ducatur planum &longs;ecans pri&longs;ma; cuius & paral
lelogrammi ae communis &longs;ectio &longs;it opq. tran&longs;ibit linea
fq per h, & mp per n. nam cum plana æquidi&longs;tantia &longs;ecen
tur à plano cq, communes eorum &longs;ectiones cgo, mp, fq
&longs;ibi ip&longs;is æquidi&longs;tabunt. Sed & æquidi&longs;tant ab, kl, de. an
guli ergo aoc, kpm, dqf inter &longs;e æquales &longs;unt: & &longs;unt
æquales qui ad puncta akd con&longs;tituuntur. quare & reliqui
reliquis æquales; & triangula aco, Kmp, dfq inter &longs;e &longs;imi
lia erunt. Vt igitur ca ad ao, ita fd ad dq: & permutando
ut ca ad fd, ita ao ad dq.e&longs;t autem ca æqualis fd. ergo &
ao ip&longs;i dq. eadem quoque ratione & ao ip&longs;i Kp æqualis
demon&longs;trabitur. Itaque &longs;i triangula, abc, def æqualia &
&longs;imilia inter &longs;e
cadet linea fq in lineam
cgo. Sed &
uitatis h in g
det.
fq per h: & planum per
co & cf
gh ducetur:
neam mp
&longs;ire nece&longs;&longs;e erit. Quo
niam ergo fh, cg æqua
les &longs;unt, &
neæ, quæ ip&longs;as
cmf, gnh, opq æqua
les æquidi&longs;tantes æquidi&longs;tant autem cgo, mnp. ergo
on, gm, & linea mn æqualis cg; & np ip&longs;i go. aptatis igi
tur klm, abc
in co, & punctum n in g cadet. Quòd
uitatis trianguli abc, & n trianguli klm grauitatis cen
trum erit id, quod demon&longs;trandum relinquebatur. Simili
ratione idem contingere demon&longs;trabimus in aliis pri&longs;ma
tibus, &longs;iue quadrilatera, &longs;iue plurilatera habeant plana,
quæ opponuntur.
cimi
cimi
cimi
titionem
Archime
dis.
COROLLARIVM.
Ex iam demon&longs;tratis per&longs;picue apparet, cuius
Iibet pri&longs;matis axem, parallelogrammorum lat eri
bus, quæ ab oppo&longs;itis planis
THE OREMA VI. PROPOSITIO VI.
Cuiuslibet pri&longs;matis centrum grauitatis e&longs;t in
plano, quod oppo&longs;itis planis æquidi&longs;tans, reli
quorum planorum latera bifariam diuidit.
Sit pri&longs;ma, in quo plana, quæ opponuntur &longs;int trian
gula ace, bdf: & parallelogrammorum latera ab, cd,
ef bifariam
tem planum ducatur; cuius &longs;ectio figura gh K. erit linea
gh æquidi&longs;tans lineis ac, bd & hk ip&longs;is ce, df. quare ex
decimaquinta undecimi elementorum, planum illud pla
nis ace, bdf æquidi&longs;tabit, & faciet &longs;ectionem figu
ram ip&longs;is æqualem, & &longs;imilem, ut proxime demon&longs;tra
uimus. Dico centrum grauitatis pri&longs;matis e&longs;&longs;e in plano
ghk. Si enim fieri pote&longs;t, &longs;it eius centrum l: & ducatur
lm u&longs;que ad planum ghk, quæ ip&longs;i ab æquidi&longs;tet.
&longs;a, relinquetur
Vtraque uero linearum ag, gb diuidatur in partes æqua
les ip&longs;i ng: & per puncta diui&longs;ionum plana oppo&longs;itis pla
nis æquidi&longs;tantia ducantur. erunt &longs;ectiones figuræ æqua
les, ac &longs;imiles ip&longs;is ace, bdf: & totum pri&longs;ma diui&longs;um erit
in pri&longs;mata æqualia, & &longs;imilia: quæ cum inter &longs;e
& grauitatis centra &longs;ibi ip&longs;is congruentia,
habebunt. Itaq:
&longs;unt magnitudi
nes
les ip&longs;i nh, & nu
mero pares, qua
rum centra gra
uitatis in
cta linea con&longs;ti
tuuntur: duæ ue
ro mediæ æqua
les &longs;unt: & quæ ex
utraque parte i
p&longs;arum &longs;imili-
ter æquales: & æ
quales rectæ li
neæ, quæ inter
grauitatis centra
interiiciuntur.
quare ex corolla
rio quintæ pro
po&longs;itionis primi
libri Archimedis
de centro graui
tatis planorum; magnitudinis ex his omnibus compo&longs;itæ
centrum grauitatis e&longs;t in medio lineæ, quæ magnitudi
num mediarum centra coniungit. at qui non ita res ha
Con&longs;tat igitur centrum grauitatis pri&longs;matis e&longs;&longs;e in plano
ghk, quod nos demon&longs;trandum propo&longs;uimus. At&longs;i op
po&longs;ita plana in pri&longs;mate &longs;int quadrilatera, uel plurilatera,
eadem erit in omnibus demon&longs;tratio.
THEOREMA VII. PROPOSITIO VII.
Cuiuslibet cylindri, & cuiuslibet cylindri por
tionis centrum grauitatis e&longs;t in plano, quod ba&longs;i
bus æquidi&longs;tans, parallelogrammi per axem late
ra bifariam &longs;ecat.
SIT cylindrus, uel cylindri portio ac: & plano per a
xem ducto &longs;ecetur; cuius fectio &longs;it parallelogrammum ab
c d: & bifariam diui&longs;is ad, bc parallelogrammi lateribus,
per diui&longs;ionum puncta ef planum ba&longs;i æquidi&longs;tans duca
tur; quod faciet &longs;ectionem, in cylindro quidem circulum
æqualem iis, qui &longs;unt in ba&longs;ibus, ut demon&longs;trauit Serenus
in libro cylindricorum, propo&longs;itione quinta: in cylindr
uero portione ellip&longs;im æqualem, & &longs;imilem eis, quæ &longs;unt
in oppo&longs;itis planis, quod nos
demon&longs;trauimus in commen
tariis in librum Archimedis
de conoidibus, & &longs;phæroidi
bus. Dico centrum grauita
tis cylindri, uel cylindri por
tionis e&longs;&longs;e in plano ef. Si
fieri pote&longs;t, fit centrum g: &
ducatur gh ip&longs;i ad æquidi
&longs;tans, u&longs;que ad ef planum.
Itaque linea ae continenter
diui&longs;a bifariam, erit tandem
pars aliqua ip&longs;ius ke, minor
gh. Diuidantur ergo lineæ
ae, ed in partes æquales ip&longs;i
ke: & per diui&longs;iones plana ba
&longs;ibus æquidi&longs;tantia
erunt iam &longs;ectiones, figuræ æ
quales, & &longs;imiles eis, quæ &longs;unt
in ba&longs;ibus: atque erit cylindrus in cylindros diui&longs;us: & cy
lindri portio in portiones æquales, & &longs;imiles ip&longs;i kf. reli
qua &longs;imiliter, ut &longs;uperius in pri&longs;mate concludentur.
THEOREMA VIII. PROPOSITIO VIII.
Cuiuslibet pri&longs;matis, & cuiuslibet cylindri, uel
cylindri portionis grauitatis centrum in medio
ip&longs;ius axis con&longs;i&longs;tit.
Sit primum af pri&longs;ma æquidi&longs;tantibus planis
quod &longs;olidum parallelepipedum appellatur: & oppo&longs;ito
rum planorum cf, ah, da, fg latera bifariam diuidantur in
punctis klmnopqrstux: & per diui&longs;iones ducantur
plana kn, or, sx. communes autem eorum planorum &longs;e
ctiones &longs;int lineæ yz,
erit ex decima eiu&longs;dem libri Archimedis parallelogrammi
cf centrum grauitatis punctum y; parallelogrammi ah
parallelogrammi dh,
parallelogrammi cg
dium uniu&longs;cuiu&longs;que axis, ui
delicet eius lineæ quæ oppo
&longs;itorum
iungit. Dico
grauitatis ip&longs;ius &longs;olidi. e&longs;t
enim, ut demon&longs;trauimus,
&longs;olidi af centrum grauitatis
in plano Kn; quod oppo&longs;i
tis planis ad, gf æquidi&longs;tans
reliquorum planorum late
ra bifariam diuidit: & &longs;imili
ratione idem centrum e&longs;t in plano or, æquidi&longs;tante planis
ae, bf oppo&longs;itis. ergo in communi ip&longs;orum &longs;ectione: ui
delicet in linea yz. Sed e&longs;t etiam in plano tu, quod
yz &longs;ecatin
punctum
planorum oppo&longs;itorum centra coniungunt.
Sit aliud prima af; & in eo plana, quæ opponuntur, tri
angula abc, def:
lateribus ad, be, cf in punctis ghk, per diui&longs;iones
ducatur, quod oppo&longs;itis planis æquidi&longs;tans faciet
triangulum ghx æquale, & &longs;imile ip&longs;is abc, def. Rur&longs;us
diuidatur ab bifariam in l: & iuncta cl per ip&longs;am, & per
cKf planum ducatur pri&longs;ma &longs;ecans, cuius, &
mi ae communis &longs;ectio &longs;it lmn. diuidet punctum m li
neam gh bifariam; & ita n diuidet lineam de: quoniam
triangula acl, gkm, dfn æqualia &longs;unt, & &longs;imilia, ut &longs;upra
demon&longs;trauimus. Iam ex iis, quæ tradita &longs;unt, con&longs;tat cen
trum grauitatis pri&longs;matis in plano ghk contineri. Dico
ip&longs;um e&longs;&longs;e in linea km. Si enim fieri potc&longs;t, &longs;it o centrum; Itaquce li
nea hm
quædam qm, minor op. deinde hm, mg diuidantur in
partes æquales ip&longs;i mq: & per diui&longs;iones lineæ ip&longs;imK
æquidi&longs;tantes ducantur. puncta uero, in quibus hæ trian
gulorum latera &longs;ecant, coniungantur ductis lineis rs, tu,
xy; quæ ba&longs;i gh æquidi&longs;tabunt. Quoniam enim lineæ gz,
h
ita erit mh, ad h
Sed ut mz ad zg, ita kr ad rg: & ut m
ad sh. quare ut kr ad rg, ita ks ad sh. æquidi&longs;tant igitur
inter &longs;e &longs;e rs, gh. eadem quoque ratione demon&longs;trabimus Et quoniam triangula, quæ
fiunt à lineis Ky, yu, us, sh æqualiz &longs;unt inter &longs;e, & &longs;imilia
triangulo Kmh: habebit triangulum Kmh ad
K
&longs;ed Kh po&longs;ita e&longs;t quadrupla ip&longs;ius ky. ergo triangulum
kmh ad triangulum K
quam &longs;exdecim ad
us, s
e&longs;t quam hK ad ky: & &longs;imiliter eandem habere demon&longs;tra
bitur trian
gulum kmg
ad quatuor
x, x
rzg. quare
totum trian
gulum Kgh
ad omnia tri
angula gzr,
r
K, K
us, s
erit, ut hk ad
ky, hoc e&longs;t
ut hm ad m
q. Si igitur in
triangulis abc, def de&longs;cribantur figuræ &longs;imiles ei, quæ de
&longs;cripta e&longs;t in ghK triangulo: & per lineas &longs;ibi re&longs;ponden
tes plana ducantur: totum pri&longs;ma af diui&longs;um erit in tria
&longs;olida parallelepipeda y
les & &longs;imiles ip&longs;is parallelogrammis y
pri&longs;mata gzr, r
K, k
item ba&longs;es æquales, & &longs;imiles &longs;unt dictis triangulis; altitu
do autem in omnibus, totius pri&longs;matis altitudini æqualis.
linea
nea
&longs;itorum centra coniungant. ergo magnitudinis ex his &longs;oli
dis compo&longs;itæ centrum grauitatis e&longs;t in linea
ip&longs;i mk æquidi&longs;tans, quæ cum triangu
lum igitur ghk ad omnia triangula gzr,
k
ad mq; hoc e&longs;t, quam
telligantur, quou&longs;que coeant; erit ob linearum qy, mk æ
quidi&longs;tantiam, ut hq ad qm, ita
do, ut hm ad mq, ita linea uero
quàm
nem, quàm ad quare triangulum etiam ghk ad omnia
iam dicta triangula maiorem
&longs;ed ut
tum
yu, us, s
ta, eandem inter &longs;e proportionem habent, quam ba&longs;es; ut
ex trige&longs;ima&longs;ecunda undecimi elementorum con&longs;tat. &longs;unt
autem &longs;olida parallelepipeda pri&longs;matum triangulares ba
&longs;es habentium dupla: &longs;equitur, ut etiam huiu&longs;modi pri&longs;
mata inter &longs;e &longs;int, &longs;icut eorum ba&longs;es. ergo totum pri&longs;ma ad
omnia pri&longs;mata maiorem proportionem habet, quam
ad
mnia pri&longs;mata proportionem habent maiorem, quàm
ad ofiat
omnia pri&longs;mata. Itaque cum à pri&longs;mate af, cuius
grauitatis e&longs;t o, auferatur magnitudo ex &longs;olidis parallelepi
pedis y
&longs;it
con&longs;tat, grauitatis centrum erit in linea
in puncto ergo punctum
erit magnitudinis
r
te&longs;t. e&longs;t enim ex diffinitione centrum grauitatis &longs;olidæ figu
ræ intra ip&longs;am po&longs;itum, non extra. quare relinquitur, ut
trum grauitatis pri&longs;matis &longs;it in linea Km. Rur&longs;us bc bifa
riam in diuidatur: & ducta a
agd plan um ducatur; quod pri&longs;ma &longs;ecet:
lelogrammo bf &longs;ectionem
quoque cf bifariam: & erit plani eius, & trianguli ghK
communis &longs;ectio gu; quòd
po&longs;itum &longs;it. Similiter demon&longs;trabimus centrum grauita
tis pri&longs;matis in ip&longs;a gu ine&longs;&longs;e. &longs;it autem planorum cfnl,
ad
axis erit, cum tran&longs;eat per centra grauitatis triangulorum
abc, ghk def, ex quartadecima eiu&longs;dem. ergo centrum
grauitatis pri&longs;matis af e&longs;t punctum
12 quinti.
quinti.
cimi
apud
panum
Sit pri&longs;ma ag, cuius oppo&longs;ita plana &longs;int quadrilatera
abcd, efgh:
ui&longs;iones planum ducatur; quod &longs;ectionem faciat quadrila
terum Klmn. Deinde iuncta ac per lineas ac, ae ducatur
planum
triangulares ba&longs;es habentia abcefg, adcehg. Sint
triangulorum abc, efg gra
uitatis centra op: & triangu
lorum adc, ehg centra qr:
no klmn occurrant in pun
ctis st. erit ex iis, quæ demon
&longs;trauimus, punctum s grauita
tis centrum trianguli klm; &
ip&longs;ius pri&longs;matis abcefg: pun
ctum uero t centrum grauita
tis trianguli Knm, & pri&longs;ma
tis adc, ehg. iunctis igitur
oq, pr, st, erit in linea oq
trum grauitatis quadrilateri
abcd, quod &longs;it u: & in linea
pr
&longs;it autem x. denique iungatur
u x, quæ &longs;ecet lineam &longs; t in y. &longs;e
cabit enim cum &longs;int in eodem
plano:
Dico idem punctum y centrum quoque gra uitatis e&longs;&longs;e to
tius pri&longs;matis. Quoniam enim quadrilateri klmn graui
tatis centrum e&longs;t y: linea sy ad yt ean dem proportionem
habebit, quam triangulum knm ad triangulum klm, ex 8
Archimedis de centro grauitatis planorum. Vt autem
gulum knm ad ip&longs;um klm, hoc e&longs;t ut triangulum adc ad
triangulum abc, æqualia enim &longs;unt, ita pri&longs;ma adcehg quare linea sy ad yt eandem propor
tionem habet, quam pri&longs;ma adcehg ad pri&longs;ma abcefg.
Sed pri&longs;matis abcefg centrum grauitatis e&longs;t s: & pri&longs;ma
tis adcehg centrum t. magnitudinis igitur ex his compo
&longs;itæ hoc e&longs;t totius pri&longs;matis ag centrum grauitatis e&longs;t pun
ctum y; medium &longs;cilicet axis ux, qui oppo&longs;itorum plano
rum centra coniungit.
Rur&longs;us &longs;it pri&longs;ma ba&longs;im habens pentagonum abcde:
& quod ei opponitur &longs;it fghKl: &longs;ec
dk, el bifariam: & per diui&longs;iones ducto plano, &longs;ectio &longs;it
planum ducatur,
ma ak in duo pri&longs;mata; in pri&longs;
ma &longs;cilicet al, cuius plana op
po&longs;ita &longs;int triangula ab abcefg l:
& in prima bk cuius plana op
po&longs;ita &longs;int quadrilatera bcde
ghkl. Sint autem triangulo
rum abe, fgl centra grauita
tis puncta r &longs;: & bcde, ghkl
quadrilaterorum centra tu:
tes plano mnopq in punctis
xy. & itidem
xy. erit in linca rt
uitatis pentagoni abcde;
quod &longs;it z: & in linea &longs;u cen
trum pentagoni fghkl :&longs;it au
tem
cto plano in
punctum x e&longs;t centrum graui
tatis trianguli mnq, ac pri&longs;
matis al: & y grauitatis centrum quadrilateri nopq, ac
pri&longs;matis bk. quare y centrum erit pentagoni mnopq. &
&longs;e centrum. Simili ratione & in aliis pri&longs;matibus illud
idem facile demon&longs;trabitur. Quo autem pacto in omni
figura rectilinea centrum grauitatis inueniatur, docuimus
in commentariis in &longs;extam propo&longs;itionem Archimedis de
quadratura parabolæ.
Sit cylindrus, uel cylindri portio ce cuius axis ab: &longs;ece
turq, plano per axem ducto; quod &longs;ectionem faciat paral
lelogrammum cdef: & diui&longs;is cf, de bifariam in punctis
gh, per ea ducatur planum ba&longs;i æquidi&longs;tans. erit &longs;ectio gh
circulus, uel ellip&longs;is, centrum habens in axe; quod &longs;it K at
que erunt ex iis, quæ demon&longs;trauimus, centra grauitatis
planorum oppo&longs;itorum puncta ab: & plani gh ip&longs;um k in
quo quidem plano e&longs;t centrum grauitatis cylindri, uel cy
lindri portionis. Dico punctum K cylindri quoque, uel cy
lindri portionis grauitatis centrum e&longs;&longs;e. Si enim fieri po
te&longs;t, &longs;it l centrum:
ducatur. quam ucro proportionem habet linea mK ad kl
& in cit culo, uel ellip&longs;i plane de&longs;cribatur rectilinea figura,
ita ut
&longs;it opgqrsht:
nis cd, fe, per lineas &longs;ibi ip&longs;is re&longs;pondentes plana
Itaque cylindrus, uel cylindri portio diuiditur in pri&longs;ma,
cuius quidem ba&longs;is e&longs;t figura rectilinea iam dicta, centrum
que grauitatis punctum K: & in multa &longs;olida, qaæ pro ba&longs;i
bus habent relictas portiones, quas nos &longs;olidas portiones
appellabimus. cum igitur portiones &longs;int minores &longs;pacio
u, circulus, uel ellip&longs;is gh ad portiones maiorem propor
tionem habebit, quàm linea mk ad Kl. fiat nk ad Kl, ut
circulus uel ellip&longs;is gh ad ip&longs;as portiones. Sed ut circulus
uel ellip&longs;is gh ad figuram rectilineam in ip&longs;a de&longs;cri
ptam, ita e&longs;t cylindrus uel cylindri portio ce ad pri&longs;ma,
quod rectilineam figuram pro ba&longs;i habet, & altitudinem
æqualem; id, quod infra demon&longs;trabitur. crgo per conuer
&longs;ionem rationis, ut circulus, uel ellip&longs;is gh ad portioncs re
lictas, ita cylindrus, uel cylindri portio ce ad &longs;olidas por
tiones, quate cylindrus uel cylindri portio ad &longs;olidas por
tiones eandem proportionem habet, quam linea nk ad k
& diuidendo pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura ad &longs;o
lidas portiones eandem proportionem habet, quam nl ad
lk & quoniam a cylindro uel cylindri portione, cuius gra
uitatis centrum e&longs;t l, aufertur pri&longs;ma ba&longs;im habens rectili
neam
dinis ex &longs;olidis portionibus
in linea kl protracta, & in puncto n; quod e&longs;t relin
quitur ergo, ut
tionis &longs;it quæ omnia
At uero cylindrum, uel cylindri
ad pri&longs;ma, cuius ba&longs;is e&longs;t rectilinea figura in &longs;pa
cio gh de&longs;cripta, & altitudo æqualis; eandem ha
figuram, hoc modo demon&longs;trabimus.
Intelligatur circulus, uel ellip&longs;is x æqualis figuræ rectili
neæ in gh &longs;pacio de&longs;criptæ. & ab x con&longs;tituatur conus, uel
coni portio,
lindri portio ce. Sit deinde rectilinea figura, in qua y eade,
quæ in &longs;pacio gh de&longs;cripta e&longs;t: & ab hac pyramis æqucalta
con&longs;tituatur. Dico
qualemni&longs;i enim &longs;it æqualis, uel maior, uel minor crit.
Sit primum maior, et exuperet &longs;olido z. Itaque in circu
lo, uel ellip&longs;i x de&longs;cribatur figura rectilinea; & in ea pyra
mis candem, quam conus, uel coni portio altitudinem ha
bens, ita ut portiones relictæ minores &longs;int &longs;olido a, quem
admodum docetur in duodecimo libro elementorum pro
po&longs;itione undecima. erit pyramis x adhuc pyramide y ma
ior. & quoniam piramides æque altæ inter &longs;e &longs;unt, &longs;icuti ba
&longs;es; pyramis x ad piramidem y eandem proportionem ha
bet, quàm figura rectilinea x ad figuram y. Sed figura recti
linea x cum &longs;it minor circulo, uel cllip&longs;i, e&longs;t etiam minor fi
gura rectilinea y. ergo pyramis x pyramide y minor erit.
Sed & maior; quod ficri At &longs;i conus, uel coni por
tio x ponatur minor pyramide y: &longs;it alter conus æque al
tus, uel altera coni portio X ip&longs;i pyramidi y æqualis. crit
eius ba&longs;is circulus, uel ellip&longs;is maior circulo, uel ellip&longs;i x,
quorum exce&longs;&longs;us &longs;it &longs;pacium
p&longs;i X figura rectilinea de&longs;cribatur, ita ut portiones relictæ
&longs;int
culo, uel ellip&longs;i x, hoc e&longs;t figura rectilinea y. & pyramis in
ca con&longs;tituta minor cono, uel coni portione X, hoc e&longs;t mi
nor pyramide y. e&longs;t ergo ut X figura rectilinea ad figuram
rectilineam y, ita pyramis X ad pyramidem y. quare cum
figura rectilinea X &longs;it maior figura y: erit & pyramis X py
ramide y maior. &longs;ed erat minor; quod rur&longs;us fieri non po
te&longs;t. non e&longs;t igitur conus, uel coni portio x neque maior,
neque minor pyramide y. ergo ip&longs;i nece&longs;&longs;ario e&longs;t æqualis.
Itaque quoniam ut conus ad conum, uel coni portio ad co
ni portionem, ita e&longs;t cylindrus ad cylindrum, uel cylin
dri portio ad cylindri portionem: & ut pyramis ad pyra
midem, ita pri&longs;ma ad pri&longs;ma, cum eadem &longs;it ba&longs;is, & æqua
lis altitudo; crit cylindrus uel cylindri portio x pri&longs;ma
ti y æqualis.
drus, uel cylindri portio ce ad cylindrum, uel cylindri por
tionem x. Con&longs;tat igitur cylindrum uel cylindri
c e, ad pri&longs;ma y, quippe cuius ba&longs;is e&longs;t figura rectilinea in
&longs;pacio gh de&longs;cripta, eandem proportionem habere, quam
&longs;pacium gh habet ad &longs;pacium x, hoc e&longs;t ad dictam figuram.
quod demon&longs;trandum fuerat.
cimi.
THEOREMA IX. PROPOSITIO IX.
Si pyramis &longs;ecetur plano ba&longs;i æquidi&longs;tante; &longs;e
ctio erit figura &longs;imilis ei, quæ e&longs;t ba&longs;is, centrum
grauitatis in axe habens.
SIT pyramis, cuius ba&longs;is triangulum abc; axis dc: &
&longs;ecetur plano ba&longs;i æquidi&longs;tante; quod
abc &longs;imile; cuius grauitatis centrum e&longs;t K.
duo plana æquidi&longs;tantia abc, fgh &longs;ecantur à plano abd;
communes corum &longs;ectiones ab, fg æquidi&longs;tantes erunt: &
cadem ratione æquidi&longs;tantes ip&longs;æ bc, gh: & ca, hf. Quòd
cum duæ lineæ fg, gh, duabus ab, bc æquidi&longs;tent, nec
fintin eodem plano; angulus ad g æqualis e&longs;t angulo ad
b. & &longs;imiliter angulus ad h angulo ad c:
qui ad a e&longs;t æqualis. triangulum igitur fgh &longs;imile e&longs;t tri
angulo abc. Atuero punctum k centrum e&longs;&longs;e grauita
tis trianguli fgh hoc modo o&longs;tendemus. Ducantur pla
na per axem, & per lineas da, db, dc: erunt communes &longs;e
ctiones fK, ae æquidi&longs;tantes:
quare angulus kfh angulo eac; & angulus kfg ip&longs;i eab
e&longs;t æqualis. Eadem ratione
anguli ad g angulis ad b: &
anguli ad h iis, qui ad c æ
quales erunt. ergo puncta
eK in triangulis abc, fgh
&longs;imiliter &longs;unt po&longs;ita, per &longs;e
xtam po&longs;itionem Archime
dis in libro de centro graui
tatis planorum. Sed cum e
&longs;it centrum grauitatis trian
guli abc, erit ex undecima
propo&longs;itione eiu&longs;dem libri,
& K trianguli fgh grauita
tis centrum. id quod demon&longs;trare oportebat. Non aliter
in ceteris pyramidibus, quod propo&longs;itum e&longs;t demon&longs;tra
bitur.
PROBLEMA I. PROPOSITIO X.
DATA qualibet pyramide, fieri pote&longs;t, ut fi
gura &longs;olida in ip&longs;a in &longs;cribatur, & altera
batur
bentibus
magnitudine, quæ minor &longs;it
gnitudine propo&longs;ita.
Sit pyramis, cuius ba&longs;is
ba&longs;im habeat, & axem eun
dem. Itaque hoc pri&longs;ma
te continenter &longs;ecto bifa
riam, plano ba&longs;i
te, relinquetur
ma quoddam minus pro
po&longs;ita magnitudine: quod
quidem ba&longs;im eandem ha
beat, quam pyramis, & a
xem ef. diuidatur de in
partes æquales ip&longs;i ef in
punctis ghklmn: & per
diui&longs;iones plana
quæ ba&longs;ibus æquidi&longs;tent,
erunt &longs;ectiones, triangula
ip&longs;i abc &longs;imilia, ut proxi
me o&longs;tendimus. ab uno
quoque
gulorum duo pri&longs;mata
&longs;truantur; unum quidem
ad partes e; alterum ad in pyramide igitur in&longs;cripta erit quædam figura,
ex pri&longs;matibus æqualem altitudinem habentibus
ad partes e: & altera circum&longs;cripta ad partes d. Sed unum
quodque eorum pri&longs;matum, quæ in figura in&longs;cripta conti
nentur, æquale e&longs;t pri&longs;mati, quod ab eodem fit triangulo in
figura circumícripta: nam pri&longs;ma pq pri&longs;mati po e&longs;t æ
quale; pri&longs;ma st æquale pri&longs;mati sr; pri&longs;ma xy pri&longs;mati
xu; pri&longs;ma
ma re
linquitur ergo, ut circum&longs;cripta figura exuperet
pri&longs;mate, quod ba&longs;im habet abc triangulum, & axem ef.
Illud uero minus e&longs;t &longs;olida magnitudine propo&longs;ita.
ratione in&longs;cribetur, & circum&longs;cribetur &longs;olida figura in py
ramide, quæ quadrilateram, uel
PROBLEMA II. PROPOSITIO XI.
DATO cono, fieri pote&longs;t, ut figura &longs;olida in
&longs;cribatur, & altera circum&longs;cribatur ex cylindris
æqualem habentibus altitudinem, ita ut circum
&longs;cripta &longs;uperet in&longs;criptam, magnitudine, quæ &longs;o
lida magnitudine propo&longs;ita &longs;it minor.
SIT conus, cuius axis bd: & &longs;ecetur plano per axem
ducto, 'ut&longs;ectio &longs;it triangulum abc:
drus, qui ba&longs;im eandem, & eundem axem habeat. Hocigi
tur cylindro continenter bifariam &longs;ecto, relinquetur cylin
drus minor &longs;olida magnitudine propo&longs;ita. Sit autem is cy
lindrus, qui ba&longs;im habet circulum circa diametrum ac, &
axem de. Itaque diuidatur bd in partes æquales ip&longs;ide
in punctis fghKlm: & per ea ducantur plana conum &longs;e
cantia; quæ ba&longs;i æquidi&longs;tent. erunt &longs;ectiones circuli, cen
train axi habentes, ut in primo libro conicorum, propo&longs;i-Si igitur à &longs;ingu
lis horum circulorum, duo cylindri fiant; unus quidem ad
ba&longs;is partes; alter ad partes uerticis: in&longs;cripta erit in co
no &longs;olida quædam figura, & altera circum&longs;cripta ex cylin
dris æqualem altitudinem habentibus con&longs;tans; quorum
unu&longs;qui&longs;que, qui in
figura in&longs;cripta con
tinetur æqualis e&longs;t ei,
qui ab eodem fit cir
culo in figura
&longs;criptaItaque cylin
drus op æqualis e&longs;t
cylindro on; cylin
drus rs
cylindrus ux cylin
dro ut e&longs;t æqualis;
& alii aliis &longs;imiliter.
quare con&longs;tat
&longs;criptam
perare in&longs;criptam cy
lindro, cuius ba&longs;is e&longs;t
circulus circa diametrum ac, & axis de. atque hic e&longs;t mi
nor &longs;olida magnitudine propo&longs;ita.
PROBLEMA III. PROPOSITIO XII.
DATA coni portione, pote&longs;t &longs;olida quædam
figura in&longs;cribi, & altera circum&longs;cribi ex cylindri
portionibus æqualem altitudinem habentibus;
ita ut circum&longs;cripta in&longs;criptam exuperet, magni
tudine, quæ minor fit &longs;olida magnitudine pro
po&longs;ita.
Figuram ciu&longs;modi, & in&longs;cribemus, &
ut in cono dictum e&longs;t.
PROBLEMA IIII. PROPOSITIO XIII.
DATA &longs;phæræ portione, quæ dimidia &longs;phæ
ra maior non &longs;it, pote&longs;t &longs;olida quædam portio in
&longs;cribi & altera circum&longs;cribi ex cylindris æqualem
altitudinem habentibus, ita ut circum&longs;cripta in
&longs;criptam excedat magnitudine, quæ &longs;olida ma
gnitudine propo&longs;ita &longs;it minor.
HOC etiam codem pror&longs;us modo &longs;iet: atque ut ab
Archimedc traditum e&longs;t in conoidum, & &longs;phæroidum por
tionibus, propo&longs;itione ulge&longs;imaprima libri de conoidi
bus, & &longs;phæroidibus.
THEOREMA X. PROPOSITIO XIIII.
Cuiuslibet pyramidis, & cuiuslibet coni, uel
coni portionis, centrum grauitatis in axe
SIT pyramis, cuius ba&longs;is triangulum abc: & axis de.
Dico in linea de ip&longs;ius grauitatis centrum ine&longs;&longs;e. Si enim
fieri pote&longs;t, &longs;it centrum f: & ab f ducatur ad ba&longs;im pyrami
dis linea fg, axi æquidi&longs;tans:
guli abc producatur in h. quam uero proportionem ha
bet linea he ad eg, habeat pyramis ad aliud &longs;olidum, in
quo K:
cum&longs;cribatur ex pri&longs;matibus æqualem habentibus altitu
dinem, ita ut circum&longs;cripta in&longs;criptam exuperet magnitu
dine, quæ &longs;olido k &longs;it minor. Et quoniam in pyramide pla
num ba&longs;i æquidi&longs;tans ductum &longs;ectionem facit figuram &longs;i
milem ei, quæ e&longs;t ba&longs;is;
tem: erit pri&longs;matis st grauitatis
matis ux centrum in linea qp, pri&longs;inatis yz in linea po;
pri&longs;matis
matis quare to
ctaque
producta, à
puncto h du
catur linea a
xi pyramidis
æquidi&longs;tans,
quæ
in habebit
dem propor
tionem,
he ad eg.
Quoniam igi
tur exce&longs;&longs;us,
quo
pta figura in
&longs;criptam &longs;upe
rat, minor e&longs;t
&longs;olido
ramis ad eun
ioré propor
tioné habet,
quàm ad K &longs;o
lidum: uideli
cet maiorem,
quàm linea h
e ad eg; hoc
e&longs;t quàm
ad
ce&longs;&longs;us, quæ intra pyrimidem comprehenditur. Itaque ha
gura &longs;olida in&longs;cripta ad partem exce&longs;&longs;us, quæ e&longs;t intra pyra
midem. Cum ergo à pyramide, cuius grauitatis
punctum f, &longs;olida figura in&longs;cripta auferatur, cuius
e&longs;t intra pyramidem, centrum grauitatis erit in linea
producta, & in puncto quod fieri non pote&longs;t. Sequitur
igitur, ut centrum grauitatis pyramidis in linea de; hoc
e&longs;t in eius axe con&longs;i&longs;tat.
Sit conus, uel coni portio, cuius axis bd: & &longs;ecetur plano
per axem, ut &longs;ectio &longs;it triangulum abc. Dico centrum gra
uitatis ip&longs;ius e&longs;&longs;e in linea bd. Sit enim, &longs;i fieri pote&longs;t,
tionem habet cd ad df, habeat conus, uel coni portio ad
&longs;olidum g. in&longs;cribatur ergo in cono, uel coni portione &longs;oli
dri portionibus, &longs;icuti dictum e&longs;t, ita ut exce&longs;&longs;us, quo &longs;igu
ra circum&longs;cripta in&longs;criptam &longs;uperat, &longs;it &longs;olido g minor.
Itaque centrum grauitatis cylindri, uel cylindri portionit
qr e&longs;t in linea po; cylindri, uel cylindri portionis st cen
trum in linea on; centrum ux in linea nm; yz in mb;
in lk; ergo figu
ræ in&longs;criptæ centrum e&longs;t in linea pd. Sit autem
cta
rit axi æquidi&longs;tans, conueniat in
ad df: & conus, &longs;eu coni portio ad exce&longs;&longs;um, quo circum
&longs;cripta figura in&longs;criptam &longs;uperat, habebit maiorem pro
portionem, quàm ergo ad partem exce&longs;&longs;us, quæ
intra ip&longs;ius &longs;uperficiem comprehenditur, multo maiorem
proportionem habebit. habeat eam, quam erit
tem, ut
cuius grauitatis centrum e&longs;t e, au&longs;ertur figura in&longs;cripta,
cuius centrum
te exce&longs;&longs;us, quæ intra coni, uel coni portionis &longs;uper&longs;iciem
continetur, centrum grauitatis erit in linea e protracta,
atque in puncto t. quod c&longs;t ab&longs;urdum.
grauitatis coni, uel coni portionis, e&longs;&longs;e in axe bd: quod de
mon&longs;trandum propo&longs;uimus.
THEOREMA XI. PROPOSITIO XV.
Cuiuslibet portionis &longs;phæræ uel &longs;phæroidis,
quæ dimidia maior non &longs;it:item&qacute;; cuiuslibet por
tionis conoidis, uel ab&longs;ci&longs;&longs;æ plano ad axem recto,
uel non recto, centrum grauitatis in axe con
&longs;i&longs;tit.
Demon&longs;tratio &longs;imilis erit ei, quam &longs;upra in cono, uel co
ni portione attulimus, ne toties eadem fru&longs;tra iterentur.
THEOREMA XII. PROPOSITIO XVI.
In &longs;phæra, & &longs;phæroidc idem e&longs;t grauitatis, &
figuræ centrum.
Secetur &longs;phæra, uel &longs;phæroides piano per axem ducto;
quod &longs;ectionem &longs;aciat circulum, uel cllip&longs;im abcd, cuius
diameter, & &longs;phæræ, uel &longs;phæroidis axis db; & centrum e.
Dico e grauitatis etiam centrum e&longs;&longs;e. &longs;ecetur enim altero
plano per e, ad planum &longs;ecans recto, cuius &longs;ectio &longs;it circu
lus cir ca diametrum a ç. erunt adc, abc dimidiæ portio
nes &longs;phæræ, uel &longs;phæroidis. & quoniam portionis adc gra
uitatis centrum e&longs;i in linea d, & centrum portionis abc in
ip&longs;a be; totius &longs;phæræ, uel &longs;phæroidis grauitatis centrum
in axe db con&longs;i&longs;iet, Quòd &longs;i portionis adc centrum
tatis ponatur e&longs;&longs;c f & fiatip&longs;i fe æqualis eg.
tionis abc centrum crit. &longs;olidis cnim figuris &longs;imilibus &
æqualibus inter &longs;e aptatis, & centra grauitatis ip&longs;arum in
ter se aptentur nece&longs;le e&longs;t. ex quo fit, ut magnitudinis, quæ
ex utilique
uitatis centrum &longs;it in medio lincæ fg uidelicet in e. Sphæ
ræ igitur, uel &longs;phæroidis grauitatis centrum e&longs;tidem, quod
centrum figuræ.
titionem
medis.
Ex demon&longs;tratis per&longs;picue apparet, portioni
&longs;phæræ uel &longs;phæroidis, quæ dimidia maier e&longs;t,
trum grauitatis in axe con&longs;i&longs;tere.
Data cnim
qualibet maio
ri
ræ, uel &longs;phæroi
dis grauitatis
centrum e&longs;t in
axe; e&longs;t autem
& in axe cen
trum portio
nis minoris:
reliquæ portionis uidelicet maioris centrum in axe nece&longs;
&longs;ario con&longs;i&longs;tet.
THEOREMA XIII. PROPOSITIO XVII.
Cuiuslibet pyramidis
gularem ba&longs;im
uitatis centrum e&longs;t in pun
cto, in quo ip&longs;ius axes con
ueniunt.
Sit pyramis, cuius ba&longs;is trian
gulum abc, axis de:
guli bdc grauitatis centrum f:
& iungatur a &longs;. crit & af axis eiu&longs;
dem pyramidis ex tertia diffini
tione huius. Itaque quoniam centrum grauitatis e&longs;t in
axe de; e&longs;t autem & in axe af; &qgrave;uod proxime demon&longs;traui
g. in quo &longs;cilicet ip&longs;i axes conueniunt.
THEOREMA XIIII. PROPOSITIO XVIII.
SI &longs;olidum parallelepipedum &longs;ecetur plano
ba&longs;ibus æquidi&longs;tante; erir &longs;olidum ad &longs;olidum,
&longs;icut altitudo ad altitudinem, uel &longs;icut axis ad
axem.
Sit &longs;olidum parallelepipe
dum abcdefgh, cuius axis
kl:
æquidi&longs;tante, quod faciat
&longs;ectionem mnop; & axi in
Dico
&longs;olidum gm ad &longs;olidum mc
eam proportionem habere,
quam altitudo &longs;olidi gm ha
bet ad &longs;olidi mc altitudi
nem; uel quam axis kq ad
axem ql. Si enim axis Kl ad
ba&longs;is planum &longs;it perpendicu
laris, & linea gc, quæ ex quin
ta huius ip&longs;i kl æquidi&longs;tat,
perpendicularis erit ad
planum, & &longs;olidi altitudi
nem dimetietur. Itaque &longs;o
lidum gm ad &longs;olidum mc
eam proportionem habet,
quam parallelogram
ad parallelogrammum nc,
hoce&longs;t quam linea go, quæ
axis kq ad ql axem. Si uero axis kl non &longs;it perpendicularis
ad planum ba&longs;is; ducatur a puncto k ad idem planum per
pendicularis kr, &longs;imiliter
mon&longs;trabimus
ad axem ql. Sed ut Kq ad ql, ita ks altitudo ad altitudi
nem sr; nam lineæ Kl, Kr à planis æquidi&longs;tantibus in ea&longs;
dem proportiones &longs;ecantur. ergo &longs;olidum gm ad &longs;olidum
mc
dinemquod
mi.
cimi
THEOREMA XV. PROPOSITIO XIX.
Solida parallelepipeda in eadem ba&longs;i, uel in
æqualibus ba&longs;ibus con&longs;tituta eam inter &longs;e propor
tionem habent, quam altitudines: & &longs;i axes ip&longs;o
rum cum ba&longs;ibus æquales angulos contineant,
eam quoque, quam axes proportionem
Sint &longs;olida parallelepipeda in
abef: & &longs;it &longs;olidi abcd altitudo minor: producatur au
tem planum cd adeo, ut&longs;olidum abef &longs;ecet; cuius &longs;ectio
&longs;it gh.
da abcd, abgh
in eadem ba&longs;i,
& æquali altitu
dine inter &longs;e æ
qualia.
igitur &longs;olidum
abef &longs;ecatur
plano ba&longs;ibus
&longs;olidum ghef
adip&longs;um abgh
que &longs;olidum abgh, hoc e&longs;t &longs;olidum abcd ip&longs;i æquale, ad
&longs;olidum abef, ut altitudo &longs;olidi abcd ad &longs;olidi abef al
titudinem.
cimi
Sint &longs;olida parallelcpipeda ab, cd in æqualibus ba&longs;ibus
con&longs;tituta:
d f; quæ quidem maior &longs;it, quàm be. Dico &longs;olidum ab ad
&longs;olidum cd eandem habere proportionem, quam be ad
d f. ab&longs;cindatur enim à linea df æqualis ip&longs;i be, quæ &longs;it gf:
& per g ducatur planum &longs;ecans &longs;olidum cd; quod ba&longs;ibus
æquidi&longs;tet, K. crunt &longs;olida ab, ck æque
alta inter
&longs;e æqualia
les ba&longs;es
habeant.
Sed
hd ad &longs;oli
dum cK
e&longs;t, ut alti
tudo dg
ad g
tudinem
catur enim &longs;olidum cd plano ba&longs;i
bus æquidi&longs;tante: & rur&longs;us
nende
ad &longs;olidum cd, ut gf ad fd. ergo
&longs;olidum ab, quod e&longs;t æquale ip&longs;i
ck ad &longs;olidum cd eam proportio
nem habet, quam altitudo gf, hoc
e&longs;t be ad df altitudinem.
cimi
Sint deinde &longs;olida parallelepipe
da ab, ac in eadem ba&longs;i; quorum
axes de, &longs; e cum ip&longs;a æquales angu Dico &longs;olidum ab ad &longs;olidum ace idem ha
bere proportionem, quam axis de ad axem ef. Si enini
axes in eadem recta linea fuerint con&longs;tituti, bæc dao lo'i
da, in unum, atque idem &longs;olidum conuenient. quare ex
iis, quæ proxime tradita &longs;unt, habebit &longs;olidum ab ad &longs;o
lidum ac eandem proportionem, quam axis de ad ef
axem. Si uero axes non &longs;int in eadem recta linea, demittan
tur a punctis d, &longs; perpendiculares ad ba&longs;is planum, dg, fh:
& jungantur eg, eh. Quoniam igitur axes cum ba&longs;ibus
æquales angulos continent, erit deg angulus æqualis an
gulo feh: & &longs;unt
anguli ad gh re
cti, quare & re
liquus edg æqua
lis erit reliquo
efh: & triangu
lum deg
loer
go gd ad de e&longs;t',
ut hf ad e: & per
mutando gd ad
hf, ut de ad cf.
Sed &longs;olidum ab
ad &longs;olidum ac
candem propor
tionem habet,
quam dg altitu
do ad
f h. ergo &
dem
axis de ad ef
Po&longs;tremo &longs;int
&longs;olidi paral le pi
peda ab, cd in
gulos faciant. Dico &longs;olidum ab
ef ad axem gh: nam &longs;i axes ad planum ba&longs;is recti &longs;int, il
lud per&longs;picue con&longs;tat: quoniam eadem linea, & axem & &longs;oli
di altitudinem determinabit. Si uero &longs;int inclinati, à pun
ctis eg ad &longs;ubiectum planum perpendiculares ducantur
ek, gl: & iungantur fk, hl. rur&longs;us quoniam axes cum ba
&longs;ibus æquales faciunt angulos, eodem modo demon&longs;trabi
tur, triangulum efK triangulo ghl &longs;imile e&longs;&longs;e: & ek ad gl,
ut ef ad gh. Solidum autem ab ad &longs;olidum cd e&longs;t, ut
eK ad gl. ergo & ut axis ef ad axem gh. quæ omnia de
mon&longs;trarc oportebat.
Ex iis quæ demon&longs;trata &longs;unt, facile con&longs;tare
pote&longs;t, pri&longs;mata omnia & pyramides, quæ trian
gulares ba&longs;es habent, &longs;iue in ei&longs;dem, &longs;iue in æqua
libus ba&longs;ibus con&longs;tituantur, eandem proportio
nem habere, quam altitudines: & &longs;i axes cum ba
&longs;ibus æquales angulos contineant, &longs;imiliter ean
dem, quam axes, habere proportionem: &longs;unt
enim &longs;olida parallelepipeda pri&longs;matum triangula
res ba&longs;es
cimi.
cimi.
THEOREMA XVI. PROPOSITIO XX.
Pri&longs;mata omnia & pyramides, quæ in ei&longs;dem,
uel æqualibus ba&longs;ibus con&longs;tituuntur, eam inter
&longs;e proportionem habent, quam altitudines: & &longs;i
axes cum ba&longs;ibus faciant angulos æquales, eam
etiam, quam axes habent proportionem.
Sint duo pri&longs;mata ae, af, quorum eadem ba&longs;is quadri
latera abcd:
af altitudo fh. Dico pri&longs;ma ae ad pri&longs;ma af eam habere
proportionem, quam eg ad fh. iungatur enim ac: & in
unoquoque pri&longs;mate duo pri&longs;mata intelligantur, quorum
ba&longs;es &longs;int triangu
la abc, acd. habe
bunt duo pri&longs;ma
te in eadem ba&longs;i
abc con&longs;tituta,
proportionem
dem, quam ip&longs;o
rum altitudines e
g, fh, ex iam de
mon&longs;tratis. & &longs;i
militer alia duo,
quæ &longs;unt in ba&longs;i a
c d. quare totum pri&longs;ma ae ad pri&longs;ma af eandem propor
tionem habebit, quam altitudo eg ad fh altitudinem.
Quòd cum pri&longs;mata &longs;int pyramidum tripla, & ip&longs;æ pyrami
des, quarum eadem e&longs;t ba&longs;is quadrilatera, & altitudo pri&longs;
matum altitudini æqualis, eam inter &longs;e proportionem ha
bebunt, quam altitudines.
Si uero pri&longs;mata ba&longs;es æquales habeant,
duo eiu&longs;modi pri&longs;mata ae, fl: & &longs;it ba&longs;is pri&longs;matis ae qua
drilaterum abcd; & pri&longs;matis fl quadrilaterum fghk.
Dico pri&longs;ma ae ad pri&longs;ma fl ita e&longs;&longs;e, ut altitudo illius ad
huius altitudinem. nam &longs;i altitudo &longs;it eadem,
duæ pyramides abcde, fghkl. quæ
cum æquales ba&longs;es, & altitudinem eandem habeant. quare
& pri&longs;mata ae, fl, quæ &longs;unt
lia &longs;int nece&longs;&longs;e e&longs;t. ex quibus per&longs;picue con&longs;tat
Si uero altitudo pri&longs;matis fl &longs;it maior, à pri&longs;mate fl ab
&longs;cindatur pri&longs;ma fm, quod æque altum &longs;it,
erunt eædem ra
tione pri&longs;mata a
e, fm inter &longs;e æ
qualia. quare &longs;i
militer demon
&longs;trabitur pri&longs;ma
fm ad pri&longs;ma fl
eandem habere
proportionem,
quam pri&longs;matis
fm altitudo ad
altitudinem ip
&longs;ius fl. ergo & pri&longs;ma ae ad pri&longs;ma fl eandem propor
tionem habebit, quam altitudo ad altitudinem. &longs;equitur
igitur ut & pyramides, quæ in æqualibus ba&longs;ibus
tur, eandem inter &longs;e &longs;e, quam altitudines, proportionem
habeant.
cimi
Sint deinde pri&longs;mata ae, af in eadem ba&longs;i abcd;
axes cum ba&longs;ibus æquales angulos contineant: & &longs;it pri&longs;Dico pri&longs;ma
ae ad pri&longs;ma af eam proportionem habere, quam gh ad
h l. ducantur à punctis gl perpendiculares ad ba&longs;is pla
num gK, lm: & iungantur kh,
h m. Itaque quoniam anguli gh
k, lhm &longs;unt æquales, &longs;imiliter ut
&longs;upra demon&longs;trabimus, triangu
la ghK, lhm &longs;imilia e&longs;&longs;e; & ut g
K ad lm, ita gh ad hl. habet au
tem pri&longs;ma ae ad pri&longs;ma af ean
dem proportionem, quam altitu
do gK ad altitudinem lm, &longs;icuti
demon&longs;tratum e&longs;t. ergo & can
dem habebit, quam gh, ad hl. py
ramis igitur abcdg ad pyrami
dem abcdl eandem proportio
nem habebit, quam axis gh ad hl axem.
Denique &longs;int pri&longs;mata ae, ko in æqualibus ba&longs;ibus ab
cd, klmn con&longs;tituta; quorum axes cum ba&longs;ibus æquales
faciant angulos:
pri&longs;matis autem ko axis pq, & altitudo pr. Dico pri&longs;ma
ae ad pri&longs;ma ko ita e&longs;&longs;e, ut fg ad pq. iunctis enim gh,
ad pr. &longs;ed pri&longs;ma ae ad ip&longs;um ko e&longs;t, ut fh ad pr. ergo
& ut fg axis ad axem pq. ex quibus &longs;it, ut pyramis abcdf
ad
dem
eandem ha
beat pro
portion&etilde;,
quod
Simili ra
tione in a
liis pri&longs;ma
tibus & py
ramidibus eadem demon&longs;trabuntur.
THEOREMA XVII. PROPOSITIO XXI.
Pri&longs;mata omnia, & pyramides inter &longs;e propor
tionem habent compo&longs;itam ex proportione ba
&longs;ium, & proportione altitudinum.
Sint duo pri&longs;mata ae, gm:
drilaterum abcd, & altitudo ef: pri&longs;matis uero gm ba
&longs;is quadrilaterum ghKl, & altitudo mn. Dico pri&longs;ma ae
ad pri&longs;ma gm proportionem habere compo&longs;itam ex pro
portione ba&longs;is abcd ad ba&longs;im ghkl, & ex proportione
altitudinis ef, ad altitudinem mn.
Sint enim primum ef, mn æquales: & ut ba&longs;is abcd
ad ba&longs;im ghkl, ita fiat linea, in qua o ad lineam, in qua p:
ut autem ef ad mn, ita linea p ad lineam q. erunt lineæ
pq inter &longs;e æquales. Itaque pri&longs;ma ae ad pri&longs;ma gm
&longs;i enim intelligantur duæ pyramides abcde, ghklm, ha
bebunt hæ inter &longs;e proportionem eandem, quam ip&longs;ar um
ba&longs;es ex &longs;exta duodecimi elementorum. Sed ut ba&longs;is abcd
ad ghKl ba&longs;im, ita linea o ad lineam p; hoc e&longs;t ad lineam q
ei æqualem. ergo pri&longs;ma ae ad pri&longs;ma gm e&longs;t, ut linea o
ad lineam q. proportio autem o ad q copo&longs;ita e&longs;t ex pro
portione o ad p, & ex proportione p ad q. quare pri&longs;ma
ae ad pri&longs;ma gm, & idcirco pyramis abcde, ad pyrami
dem ghKlm proportionem habet ex ei&longs;dem proportio
nibus compo&longs;itam, uidelicet ex proportione ba&longs;is abcd
ad ba&longs;im ghKl, & ex proportione altitudinis ef ad mn al
titudinem. Quòd &longs;i lineæ ef, mn inæquales ponantur, &longs;it
ef minor: & ut ef ad mn, ita fiat linea p ad lineam u: de
inde ab ip&longs;a mn ab&longs;cindatur rn æqualis ef: & per r duca
tur planum, quod oppo&longs;itis planis æquidi&longs;tans faciat &longs;e
ctionem st. erit pri&longs;ma ae, ad pri&longs;ma gt, ut ba&longs;is abcd
ad ba&longs;im ghkl; hoc e&longs;t ut o ad p: ut autem pri&longs;ma gt ad
pri&longs;ma gm, ita altitudo rn; hoc e&longs;t ef ad altitudine mn;
uidelicet linea p ad lineam u. ergo ex æquali pri&longs;ma ae ad
pri&longs;ma gm e&longs;t, ut linea o ad ip&longs;am u. Sed proportio o ad
u
ad ba&longs;im ghkl; & ex proportione p ad u, quæ e&longs;t altitudi
nis ef ad altitudinem mn. pri&longs;ma igitur ae ad pri&longs;ma gm
& proportione altitudinum. Quare & pyramis, cuius ba
&longs;is e&longs;t quadrilaterum abcd, & altitudo ef ad pyramidem,
cuius ba&longs;is quadrilaterum ghKl, & altitudo mn, compo&longs;i
tam habet proportionem ex proportione ba&longs;ium abcd,
ghkl, & ex proportione altitudinum ef, mn. quod qui
dem demon&longs;tra&longs;&longs;e oportebat.
Ex iam demon&longs;tratis per&longs;picuum e&longs;t, pri&longs;ma
ta omnia, & pyramides, in quibus axes cum ba&longs;i
bus æquales angulos continent, proportionem
habere compo&longs;itam ex ba&longs;ium proportione, &
proportione axium. demon&longs;ttatum e&longs;t enim, a
xes inter &longs;e eandem proportionem habere, quam
ip&longs;æ altitudines.
THEOREMA XVIII. PROPOSITIO XXII.
CVIVSLIBEt pyramidis, & cuiuslibet coni,
ditur, ut pars, quæ terminatur ad uerticem reli
quæ partis, quæ ad ba&longs;im, &longs;it tripla.
Sit pyramis, cuius ba&longs;is triangulum abc; axis de; & gra
uitatis centrum K. Dico lineam dk ip&longs;ius Ke triplam e&longs;&longs;e.
trianguli enim bdc centrum grauitatis &longs;it punctum f;
guli adc
b g, ch. Quoniam igitur
nient omnes in
Itaque animo concipiamus hanc pyramidem diui&longs;am in
quatuor pyramides, quarum ba&longs;es &longs;int ip&longs;a pyramidis
triangula; &
ctum k quæ quidem py
ramides inter &longs;e æquales
&longs;unt, ut
Ducatur
dc, de planum
&longs;it ip&longs;ius, & ba&longs;is abc
munis &longs;ectio recta linea
cel:
guli
l b: nam centrum graui
tatis trianguli con&longs;i&longs;tit
in linea, quæ ab angulo
ad dimidiam ba&longs;im per
ducitur, ex tertia deci
ma Archimedis.
quare
triangulum acl æquale
e&longs;t triangulo bcl: & propterea pyramis, cuius ba&longs;is trian
gulum acl, uer tex d, e&longs;t æqualis pyramidi, cuius ba&longs;is bcl
triangulum, & idem uertex. pyramides enim, quæ ab
ba&longs;es. eadem ratione pyramis aclk pyramidi bclk & py
ramis adlk ip&longs;i bdlk pyramidi æqualis erit. Itaque &longs;i apy
ramide acld auferantur pyramides aclk, adlk: & à pyra
mide bcld
quuntur erunt æqualia. æqualis igitur e&longs;t pyramis acdk
pyramidi bcdK. Rur&longs;us &longs;i per lineas ad, de ducatur pla
num quod pyramidem &longs;ccet:
&longs;ectio aem: &longs;imiliter o&longs;tendetur pyramis abdK æqualis
pyramidi acdk. ducto denique alio plano per lineas ca,
a f: ut eius, & trianguli cdb communis &longs;ectio &longs;it cfn, py
ramis abck pyramidi acdk æqualis demon&longs;trabitur.
ergo tres pyramides bcdk, abdk, abck uni, & eidem py
ramidi acdk &longs;int æquales, omnes inter &longs;e &longs;e æquales
Sed ut pyramis abcd ad pyramidem abck ita de axis ad
axem ke, ex uige&longs;ima propo&longs;itione huius: &longs;unt enim hæ
pyramides in eadem ba&longs;i, & axes cum ba&longs;ibus æquales con
tinent angulos, quòd in eadem recta linea con&longs;tituantur.
quare diuidendo, ut tres pyramides acdk, bcdK, abdK
ad pyramidem abcK, ita dk ad Ke. con&longs;tat igitur lineam
dK ip&longs;ius Ke triplam e&longs;&longs;e. &longs;ed & ak tripla e&longs;t Kf: itemque
bK ip&longs;ius kg: & ck ip&longs;ius kl tripla. quod eodem modo
demon&longs;trabimus.
cimi.
Sit pyramis, cuius ba&longs;is quadrilaterum abcd; axis ef:
& diuidatur ef in g, ita ut eg ip&longs;ius gf &longs;it tripla. Dico cen
trum grauitatis pyramidis e&longs;&longs;e punctum g. ducatur enim
linea bd diuidens ba&longs;im in duo triangula abd, bcd: ex
quibus
&longs;itque pyramidis abde axis eh; & pyramidis bcde axis
e K: & iungatur hK, quæ per f tran&longs;ibit: e&longs;t enim in ip&longs;a hK
centrum grauitatis magnitudinis compo&longs;itæ ex triangulis
abd, bcd, hoc e&longs;t ip&longs;ius quadrilateri. Itaque centrum gra
uitatis pyramidis abde &longs;it punctum l: & pyramidis bcde
&longs;it m. ducta igitur lm ip&longs;i hm lineæ æquidi&longs;tabit. nam el ad
cet triplam. quare linea lm ip&longs;am ef &longs;ecabit in puncto g:
etenim eg ad gf e&longs;t, ut el ad lh. præterea quoniam hk, lm
æquidi&longs;tant, erunt triangula hef, leg &longs;imilia:
&longs;e &longs;imilia fek gem: & ut ef ad eg, ita hf ad lg: & ita fK ad
gm. ergo ut hf ad lg, ita fk ad gm: & permutando ut hf
ad fK, ita lg ad gm. &longs;ed cum h &longs;it centrum trianguli abd;
& k
centrum: erit ex 8. Archimedis de centro grauitatis plano
rum hf ad fk ut triangulum bcd ad triangulum abd: ut,
autem bcd triangulum ad triangulum abd, ita pyramis
bcde ad pyramidem abde. ergo
linea lg ad gm erit, ut pyramis
bcde ad ex quo
&longs;equitur, ut totius pyramidis
abcde punctum g &longs;it grauitatis
centrum. Rur&longs;us &longs;it pyramis ba
&longs;im habens pentagonum abcde:
& axem fg:
cto h, ita ut fh ad hg triplam habe
at proportionem. Dico h grauita
tis
iungatur enim eb:
pyramis, cuius uertex f, & ba&longs;is
triangulum abe: & alia pyramis
intelligatur eundem uerticem ha
bens, & ba&longs;im bcde
&longs;it autem pyramidis abe faxis fk
& grauitatis centrum l: & pyrami
dis bcdef axis fm, & centrum gra
uitatis n:
quæ per puncta gh tran&longs;ibunt.
Rur&longs;us eodemmodo, quo &longs;up ra,
demon&longs;trabimus lineas Kgm, lhn &longs;ibiip&longs;is æquidi&longs;tare:
centrum, & ita in alils.
Sit conus, uel coni portio axem habens bd: &longs;eceturque
plano per axem, quod &longs;ectionem faciat triangulum abc:
& bd axis diuidatur in c, ita ut be ip&longs;ius ed &longs;it tripla.
Dico punctum e coni, uel coni portionis, grauitatis
e&longs;&longs;e centrum. Sienim fieri pote&longs;t, &longs;itcentrum f: & pro
ducatur ef extra figuram in g. quam uero proportionem
habet ge ad ef, habeat ba&longs;is coni, uelconi portionis, hoc
e&longs;t circulus, uel ellip&longs;is circa diametrum ac ad aliud &longs;pa
cium, in quo h. Itaque in circulo, uel ellip&longs;i plane de&longs;cri
batur rectilinea figura axlmcnop, ita ut quæ
tur
mis ba&longs;im habens rectilineam figuram aKlmcnop, &
axem bd; cuius quidem grauitatis centrum erit punctum
e, utiam demon&longs;trauimus. Et quoniam portiones &longs;unt
minores &longs;pacio h, circulus, uel ellip&longs;is ad portiones ma
iorem proportionem habet, quam ge ad ef. &longs;ed ut circu
lus, uel ellip&longs;is ad figuram rectilineam &longs;ibi in&longs;criptam, ita
conus, uel coni portio ad pyramidem, quæ figuram rectili
neam pro ba&longs;i habet; & altitudinem æqualem: etenim &longs;u
pra demon&longs;tratum e&longs;t, ita e&longs;&longs;e cylindrum, uel cylindri por
tionem ad pri&longs;ma, cuius ba&longs;is rectilinea figura, & æqua
lis altitudo. ergo per conuer&longs;ionem rationis, ut circulus,
uel ellip&longs;is ad portiones, ita conus, uel coni portio ad por
tiones &longs;olidas. quare conus uel coni portio ad portiones
&longs;olidas maiorem habet proportionem, quam ge ad ef: &
diuidendo, pyramis ad portiones &longs;olidas maiorem pro
portionem habet, quam gf ad fe. fiat igitur qf ad fe
ut pyramis ad dictas portiones. Itaque quoniam a cono
uel coni portione, cuius grauitatis centrum e&longs;t f, aufer
tur pyramis, cuius centrum e; reliquæ magnitudinis,
quæ ex &longs;olidis portionibus con&longs;tat, centrum grauitatis
erit in linea ef protracta, & in puncto q. quod fieri
non pote&longs;t: e&longs;t enim centrum grauitatis intra. Con&longs;tat
igitur coni, uel coni portionis grauitatis centrum e&longs;&longs;e pun
ctum e. quæ omnia demon&longs;trare oportebat.
THEOREMA XIX. PROPOSITIO XXIII.
QVODLIBET fru&longs;tum à pyramide, quæ
triangularem ba&longs;im habeat, ab&longs;ci&longs;&longs;um, diuiditur
in tres pyramides proportionales, in ea proportio
ne, quæ e&longs;t lateris maioris ba&longs;is ad latus minoris
ip&longs;i re&longs;pondens.
Hoc demon&longs;trauit Leonardus Pi&longs;anus in libro, qui de
praxi geometriæ in&longs;cribitur. Sed quoniam is adhuc im
pre&longs;&longs;us non e&longs;t, nos ip&longs;ius demon&longs;trationem breuiter
per&longs;tringemus, rem ip&longs;am &longs;ecuti, non uerba. Sit fru
&longs;tum pyramidis abcdef, cuius maior ba&longs;is triangulum
abc, minor def: & iunctis ae, cc, cd, per, line
as ae, ec ducatur planum &longs;ecans fru&longs;tum: itemque per
lineas ec, cd; & per cd, da alia plana ducantur, quæ
diuident fru&longs;tum in trcs pyramides abce, adce, defc.
teris ab adlatus de, itaut earum maior &longs;it abce, me
dia adce, & minor defc. Quoniam enim lineæ de,
ab æquidi&longs;tant; & interip&longs;as &longs;unt triangula abe, ade;
erit triangulum abe
ad triangulum abe,
utlinea ab ad lineam
de. ut autem triangu
lum abe ad triangu
lum abe, ita pyramis
abec ad pyramidem
adec: habent enim
altitudinem eandem,
quæ e&longs;tà puncto cad
planum, in quo qua
drilaterum abed. er
go ut ab ad de, ita pyramis abec ad pyramidem adec.
Rur&longs;us quoniam æquidi&longs;tantes &longs;unt ac, df; erit eadem
ratione pyramis adce ad pyramidem cdfe, ut ac ad
d f. Sed ut ac ad df, ita ab ad de, quoniam triangula
abc, def &longs;imilia &longs;unt, ex nona huius. quare ut pyramis
abce ad pyramidem abce, ita pyramis adce ad ip&longs;am
defc. fru&longs;tum igitur abcdef diuiditur in tres pyramides
proportionales in ea proportione, quæ e&longs;t lateris ab ad de
latus, & earum maior e&longs;t cabe, media adce, & minor
defc. quod demon&longs;trare oportebat.
mi.
PROBLEMA V. PROPOSITIO XXIIII.
QVODLIBET fru&longs;tum pyramidis, uel coni,
uel coni portionis, plano ba&longs;i æquidi&longs;tanti ita &longs;e
care, ut &longs;ectio &longs;it proportionalis inter maiorem,
& minorem ba&longs;im.
SIT fru&longs;tum pyramidis ae, cuius maior ba&longs;is triangu
lum abc, minor def: & oporteat ip&longs;um plano, quod ba&longs;i
æquidi&longs;tet, ita &longs;ecare, ut &longs;ectio &longs;it proportionalis inter
gula abc, def. Inueniatur inter lineas ab, de media pro
portionalis, quæ &longs;it bg: & à puncto g erigatur gh æquidi
&longs;tans be,
ba&longs;ibus æquidi&longs;tans, cuius &longs;ectio &longs;it triangulum hkl. Dico
triangulum hKl proportionale e&longs;&longs;e inter triangula abc,
def, hoc e&longs;t triangulum abc ad
triangulum hKl eandem habere
proportionem, quam
hKl ad ip&longs;um def.
lineæ ab, hK æquidi&longs;tantium pla
norum &longs;ectiones inter &longs;e æquidi
&longs;tant: atque æquidi&longs;tant bk, gh:
linea hk ip&longs;i gb e&longs;t æqualis: & pro
pterea proportionalis inter ab,
de. quare ut ab ad hK, ita e&longs;t hk
ad de. fiat ut hk ad de, ita de
ad aliam lineam, in qua &longs;it m. erit
ex æquali ut ab ad de, ita hk ad
m. Et quoniam triangula abc,
hKl, def &longs;imilia &longs;unt;
abc ad triangulum hkl e&longs;t, ut li
nea ab ad lineam de:
autem hkl adip&longs;um def e&longs;t, ut hk ad m. ergo triangulum
abc ad triangulum hkl eandem proportionem habet,
quam triangulum hKl ad ip&longs;um def. Eodem modo in a
liis fru&longs;tis pyramidis idem demon&longs;trabitur.
cimi
corol.
Sit fru&longs;tum coni, uel coni portionis ad: & &longs;ecetur plano
per axem, cuius &longs;ectio &longs;it abcd, ita ut maior ip&longs;ius ba&longs;is &longs;it
circulus, uel ellip&longs;is circa diametrum ab; minor circa cd.
Rur&longs;us inter lineas ab, cd inueniatur proportionalis be:
& ab e ducta ef æquidi&longs;tante bd, quæ lineam ca in f &longs;ecet,
culus, uel ellip&longs;is circa diametrum fg. Dico &longs;ectionem ab
ad &longs;ectionem fg eandem proportionem habere, quam fg
ad ip&longs;am cd. Simili enim ratione, qua &longs;upra, demon&longs;trabi
tur quadratum ab ad quadratum fg ita e&longs;&longs;e, ut
fg ad cd quadratum. Sed circuli inter &longs;e eandem propor
tionem habent, quam diametrorum quadrata. ellip&longs;es au
tem circa ab, fg, cd, quæ &longs;imiles &longs;unt, ut o&longs;tendimus in
mentariis
& &longs;phæroidibus, eam
ta diametrorum, quæ eiu&longs;dem rationis &longs;unt, ex corollaio
&longs;eptimæ propo&longs;itionis eiu&longs;dem li
bri. ellip&longs;es enim nunc appello ip
&longs;a &longs;pacia ellip&longs;ibus contenta. ergo
circulus, uel ellip&longs;is ab ad
uel ellip&longs;im fg eam proportionem
habet, quam circulus, uel ellip&longs;is
fg ad circulum uel ellip&longs;im cd.
quod quidem faciendum propo
&longs;uimus.
cimi
THEOREMA XX. PROPOSITIO XXV.
QVODLIBET fru&longs;tum pyramidis, uel coni,
uel coni portionis ad pyramidem, uel conum, uel
coni portionem, cuius ba&longs;is eadem e&longs;t, & æqualis
altitudo, eandem
que ba&longs;es, maior, & minor &longs;imul &longs;umptæ vnà
ca, quæ inter ip&longs;as &longs;it proportionalis, ad ba&longs;im ma
iorem.
SIT
cuius maior ba&longs;is ab, minor cd. & &longs;ecetur altero plano
ba&longs;i æquidi&longs;tante, ita ut &longs;ectio ef &longs;it proportionalis inter
ba&longs;es ab, cd. con&longs;tituatur
ni portio agb, cuius ba&longs;is &longs;it eadem, quæ ba&longs;is maior fru
&longs;ti, & altitudo æqualis. Di
co fru&longs;tum ad ad pyrami
dem, uel conum, uel coni
portionem agb eandem
utræque ba&longs;es, ab, cd unà
cum ef ad ba&longs;im ab. e&longs;t
enim fru&longs;tum ad æquale
pyramidi, uel cono, uel co
ni portioni, cuius ba&longs;is ex
tribus ba&longs;ibus ab, ef, cd
con&longs;tat; & altitudo ip&longs;ius
altitudini e&longs;t æqualis: quod mox o&longs;tendemus. Sed pyrami
des, coni, uel coni
quæ &longs;unt æquali altitudine,
proportionem habent, &longs;icu
ti demon&longs;tratum e&longs;t, partim
ab Euclide in duodecimo li
bro elementorum, partim à
nobis in
decimam
chimedis de conoidibus, &
&longs;phæroidibus. quare pyra
mis, uel conus, uel coni por
tio, cuius ba&longs;is e&longs;t tribus illis
ba&longs;ibus æqualis ad agbeam
habet proportionem, quam
ba&longs;es ab, ef, cd ad ab ba&longs;im. Fru&longs;tum igitur ad ad agb
portionem habet, quam ba&longs;es ab, cd unà cum ef ad ba
&longs;im ab. quod demon&longs;trare uolebamus.
decimi
Fru&longs;tum uero ad æquale e&longs;&longs;e pyramidi, uel co
no, uel coni portioni, cuius ba&longs;is con&longs;tat ex ba&longs;i
bus ab, cd, ef, & altitudo fru&longs;ti altitudini e&longs;t æ
qualis, hoc modo o&longs;tendemus.
Sit fru&longs;tum pyramidis abcdef, cuius mabcdior ba&longs;is trian
gulum abc; minor def: & &longs;ecetur plano ba&longs;ibus æquidi
&longs;tante, quod &longs;ectionem faciat triangulum ghk inter trian
gula abc, def proportionale. Iam ex iis, quæ demon&longs;trata
&longs;unt in 23. huius, patet fru&longs;tum abcdef diuidi in tres pyra
mides proportionales; & earum maiorem e&longs;&longs;e
abcd ergo pyramis à triangulo ghk
con&longs;tituta, quæ altitudinem habeat fru&longs;ti altitudini æqua
lem, proportionalis e&longs;t inter pyramides abcd, defb: &
idcirco fru&longs;tum abcdef tribus dictis pyramidibus æqua
le erit. Itaque &longs;i intelligatur alia pyra
mis æque alta, quæ ba&longs;im habeat ex tri
bus ba&longs;ibus abc, def, ghk con&longs;tan
tem; per&longs;picuum e&longs;t ip&longs;am ei&longs;dem py
ramidibus, & propterea ip&longs;i fru&longs;to æ
qualem e&longs;&longs;e.
Rur&longs;us &longs;it fru&longs;tum pyramidis ag, cu
ius maior ba&longs;is quadrilaterum abcd,
minor efgh: & &longs;ecetur plano ba&longs;i
bus æquidi&longs;tante, ita ut fiat &longs;ectio qua
drilaterum Klmn, quod &longs;it proportio
nale inter quadrilatera abcd, efgh. Dico pyramidem,
cuius ba&longs;is &longs;it æqualis tribus quadrilateris abcd, klmn,
efgh, & altitudo æqualis altitudini fru&longs;ti, ip&longs;i fru&longs;to ag
æqualem e&longs;&longs;e. Ducatur enim planum per lincas fb, hd,
bentia, uidelicet in fru&longs;tum abdefh, & in
erit triangulum kln proportionale inter triangula abd,
efh: & triangulum lmn proportionale inter bcd, fgh.
&longs;ed pyramis æque alta, cuius ba&longs;is con&longs;tat ex tribus trian
gulis abd, klz, efh, demon&longs;trata
e&longs;t fru&longs;to abdcfh æqualis. & &longs;i
militer pyramis, cuius ba&longs;is con
&longs;tat ex triangulis bcd, lmn, fgh
æqualis fru&longs;to bcdfgh: compo
nuntur autem tria quadrilatera a
bcd, klmn, efgh è &longs;ex triangu
lis iam dictis. pyramis igitur ba
&longs;im habens æqualem tribus qua
drilateris, & altitudinem eandem
ip&longs;i fru&longs;to ag e&longs;t æqualis. Eodem
modo illud
eiu&longs;modi fru&longs;tis.
Sit fru&longs;tum coni, uel coni portionis ad; cuius maior ba
&longs;is circulus, uel ellip&longs;is circa diametrum ab; minor circa
c d: & &longs;ecetur plano, quod ba&longs;ibus æquidi&longs;tet,
ctionem circulum, uel ellip&longs;im circa diametrum ef, ita ut
inter circulos, uel ellip&longs;es ab, cd &longs;it proportionalis. Dico
conum, uel coni portionem, cuius ba&longs;is e&longs;t æqualis tribus
circulis, uel tribus ellip&longs;ibus ab, ef, cd; & altitudo eadem,
quæ fru&longs;ti ad, ip&longs;i fru&longs;to æqualem e&longs;&longs;e. producatur enim
fru&longs;ti &longs;uperficies quou&longs;que coeat in unum punctum, quod
&longs;it g: & coni, uel coni portionis agb axis &longs;it gh, occurrens
planis ab, ef, cd in punctis hkl: circa circulum uero de
&longs;cribatur quadratum mnop, & circa ellip&longs;im
mnop, quod ex ip&longs;ius diametris con&longs;tat:
g n, go, gp, ex eodem uertice intelligatur pyramis ba&longs;im
habens dictum quadratum, uel rectangulum: & plana in
quibus &longs;unt circuli, uel ellip&longs;es ef, cd u&longs;que ad eius latera Quoniam igitur pyramis &longs;ecatur planis ba&longs;i
æquidi&longs;tantibus, &longs;ectiones &longs;imiles erunt: atque erunt qua
drata, uel rectangula circa circulos, uel ellip&longs;es de&longs;cripta,
quemadmodum & in ip&longs;a ba&longs;i. Sed cum circuli inter &longs;e
proportionem habeant, quam diametrorum quadrata:
con&longs;tantia: & &longs;it circulus, uel ellip&longs;is circa diametrum ef
proportionalis inter circulos, uel ellip&longs;es ab, cd; erit re
ctangulum ef etiam inter rectangula ab, cd proportio
nale: per rectangulum enim nunc breuitatis cau&longs;a
&longs;um quadratum intelligemus. quare ex iis, quæ proxime
dicta &longs;unt, pyramis ba&longs;im habens æqualem dictis rectangu
lis, & altitudinem eandem, quam fru&longs;tum ad, ip&longs;i fru&longs;to à
pyramide ab&longs;ci&longs;&longs;o æqualis probabitur. ut autem rectangu
lum cd ad
circulum, uel ellip&longs;im:
e f, ad ef rectangulum, ita circuli, uel ellip&longs;es ed, ef, ad ef:
& ut rectangulum ef ad rectangulum ab, ita circulus, uel
ellip&longs;is ef ad ab circulum, uel ellip&longs;im. ergo ex æquali, &
componendo, ut In
telligatur pyramis q ba&longs;im habens æqualem tribus rectan
gulis ab, ef, cd; & altitudinem
intelligatur ctiam conus, uel coni portio q, eadem altitudi
ne, cuius ba&longs;is &longs;it tribus circulis, uel tribus ellip&longs;ibus ab,
e f, cd æqualis. po&longs;tremo intelligatur pyramis alb, cuius.
ba&longs;is &longs;it rectangulum mnop, & altitudo eadem, quæ fru
&longs;ti: itemq, intelligatur conus, uel coni portio alb, cuius
ba&longs;is circulus, uel ellip&longs;is circa diametrum ab, & eadem al
titudo. utigitur rectangula ab, ef, cd ad rectangulum ab,
ita pyramis q ad pyramidem alb; & ut circuli, uel ellip
&longs;es ab, ef, cd ad ab circulum, uel ellip&longs;im, ita conus, uel co
ni portio q ad conum, uel coni portionem alb. conus
igitur, uel coni portio q ad conum, uel coni portionem
alb e&longs;t, ut pyramis q ad pyramidem alb. &longs;ed pyramis
alb ad pyramidem agb e&longs;t, ut altitudo ad altitudinem, ex
20. huius: & ita e&longs;t conus, uel coni portio alb ad conum,
uel coni portionem agb ex 14. duodccimi elementorum,
& ex iis, quæ nos demon&longs;trauimus in commentariis in un
decimam de conoidibus, & &longs;phæroidibus, propo&longs;itione
quarta. pyramis autem agb ad pyramidem cgd propor
tionem habet compo&longs;itam ex proportione ba&longs;ium & pro
portione altitudinum, ex uige&longs;ima prima huius: & &longs;imili
ter conus, uel coni portio agb ad conum, uel coni portio
nem cgd proportionem habet
portionibus, per ea, quæ in dictis commentariis demon
&longs;trauimus, propo&longs;itione quinta, & &longs;exta: altitudo enim in
utri&longs;que eadem e&longs;t, & ba&longs;es inter &longs;e &longs;e eandem habent pro
portionem. ergo ut pyramis agb ad pyramidem cgd, ita
e&longs;t conus, uel coni portio agb ad agd conum, uel coni
portionem: & per
ad
agb ad fru&longs;tum ad. ex æquali igitur, ut pyramis q ad fru
&longs;tum à pyramide ab&longs;ci&longs;&longs;um, ita conus uel coni portio q ad Sed pyramis q æqualis e&longs;t fru&longs;to à pyramide
ab&longs;ci&longs;&longs;o, ut demon&longs;trauimus. ergo & conus, uel coni por
tio q, cuius ba&longs;is ex tribus circulis, uel ellip&longs;ibus ab, ef, cd
con&longs;tat, & altitudo eadem, quæ fru&longs;ti: ip&longs;i fru&longs;to ad e&longs;t æ
qualis. atque illud e&longs;t, quod demon&longs;trare oportebat.
cnni.
noidibus
& &longs;phæ
roidibus
decimi
THEOREMA XXI. PROPOSITIO XXVI.
CVIVSLIBET fru&longs;ti à pyramide, uel cono,
uel coni portione ab&longs;cis&longs;i, centrum grauitatis e&longs;t
in axe, ita ut eo primum in duas portiones diui
&longs;o, portio &longs;uperior, quæ minorem ba&longs;im attingit
ad portionem reliquam eam habeat proportio
nem, quam duplum lateris, uel diametri maioris
ba&longs;is, vnà cum latere, uel diametro minoris, ip&longs;i
re&longs;pondente, habet ad duplum lateris, uel diame
tri minoris ba&longs;is vnà
ioris: deinde à puncto diui&longs;ionis quarta parte &longs;u
perioris portionis in ip&longs;a &longs;umpta: & rur&longs;us ab in
ferioris portionis termino, qui e&longs;t ad ba&longs;im maio
rem, &longs;umpta quarta parte totius axis: centrum &longs;it
in linea, quæ his finibus continetur, atque in eo li
neæ puncto, quo &longs;ic diuiditur, ut tota linea ad par
tem propinquiorem minori ba&longs;i,
tionem habeat, quam fru&longs;tum ad
conum, uel coni portionem, cuius ba&longs;is &longs;it ea
dem, quæ ba&longs;is maior, & altitudo fru&longs;ti altitudini
æqualis.
Sit fru&longs;tum ae a pyramide, quæ triangularem ba&longs;im ha
beat ab&longs;ci&longs;&longs;um: cuius maior ba&longs;is triangulum abc, minor
def; & axis gh. ducto autem plano per axem & per
d a, quod &longs;ectionem faciat dakl quadrilaterum; puncta
Kl lineas bc, ef bifariam &longs;ecabunt. nam cum gh &longs;it axis
fru&longs;ti: erit h centrum grauitatis trianguli abc: & g
centrum trianguli def: cen
trum uero cuiuslibet triangu
li e&longs;t in recta linea, quæ ab an
gulo ip&longs;ius ad
ducitur ex decimatertia primi
libri Archimedis de
uitatis planorum. quare
trum
e&longs;t in linea kl, quod &longs;it m: & à
puncto m ad axem ducta mn
ip&longs;i ak, uel dl æquidi&longs;tante;
erit axis gh diui&longs;us in portio
nes gn, nh, quas diximus: ean
dem enim proportionem ha
bet gn ad nh,
At lm ad mK habet eam,
duplum lateris maioris ba&longs;is
bc una cum latere minoris ef
ad duplum lateris ef unà cum
latere bc, ex ultima eiu&longs;dem
libri Archimedis. Itaque à li
nea ng ab&longs;cindatur, quarta
pars, quæ fit np: & ab axe hg ab&longs;cindatur itidem
quarta pars ho: & quam proportionem habet fru&longs;tum ad
pyramidem, cuius maior ba&longs;is e&longs;t triangulum abc, & alti
tudo ip&longs;i æqualis; habeat op ad pq. Dico centrum graui
tatis fru&longs;ti e&longs;&longs;e in linea po, & in puncto q. namque ip&longs;um
e&longs;&longs;e in linea gh manife&longs;te con&longs;tat. protractis enim fru&longs;ti pla
midis abcr, & pyramidis defr grauitatis centrum in li
nca rh. ergo & reliquæ magnitudinis, uidelicet fru&longs;ti cen
trum in eadem linea nece&longs;lario comperietur. Iungantur
db, dc, dh, dm: & per lineas db, dc ducto altero plano
intelligatur fru&longs;tum in duas pyramides diui&longs;um: in pyra
midem quidem, cuius ba&longs;is e&longs;t triangulum abc, uertex d:
& in eam, cuius idem uertex, & ba&longs;is trapezium bcfe. erit
igitur pyramidis abcd axis dh, & pyramidis bcfed axis
d m: atque erunt tres axes gh, dh, dm in eodem plano
daKl. ducatur præterea per o linea &longs;t ip&longs;i aK
quæ lineam dh in u &longs;ecet: per p uero ducatur xy æquidi
&longs;tans eidem, &longs;ecansque dm in
z: & iungatur zu, quæ &longs;ecet
gh in
erunt
punctum; ut inferius appare
bit. Quoniam igitur linea uo
æquidi&longs;tat ip&longs;i dg, erit du ad
uh, ut go ad oh. Sed go tri
pla e&longs;t oh. quare & du ip&longs;ius
uh e&longs;t tripla: & ideo pyrami
dis abcd centrum grauitatis
erit punctum u. Rur&longs;us quo
niam zy ip&longs;i dl æquidi&longs;tat, dz
ad zm e&longs;t, ut ly ad ym: e&longs;tque
ly ad ym, ut gp ad pn. ergo
dz ad zm e&longs;t, ut gp ad pn.
Quòd cum gp &longs;it tripla pn;
erit etiam dz ip&longs;ius zm tri
pla. atque ob eandem cau&longs;
&longs;am punctuniz e&longs;t
uitatis pyramidis bcfed. iun
ctaigitur zu, in ea erit
&longs;tat; hoc e&longs;t ip&longs;ius fru&longs;ti. Sed fru&longs;ti centrum e&longs;t etiam in a
xe gh. ergo in puncto
Itaque u
bcfed ad pyramidem abcd. & componendo uz ad z
eam habet, quam fru&longs;tum ad pyramidem abcd. Vt uero
uz ad z
uo
abcd. &longs;ed ita erat op ad pq. æquales igitur &longs;unt p
qex quibus &longs;equitur lineam.
z u &longs;ecare op in q: & propterea
uitatis centrum e&longs;&longs;e.
hu
ius.
iu&longs;dem
Archime
dis.
libri Ar
chimedis
de
grauta
tis plano
rum
Sit fru&longs;tum ag à pyramide, quæ quadrangularem ba&longs;im
habeat ab&longs;ci&longs;&longs;um, cuius maior ba&longs;is abcd, minor efgh,
& axis kl. diuidatur autem
tionem habet duplum lateris ab unà cum latere ef ad du
plum lateris ef unà cum ab; habeat km ad ml. deinde à
& rur&longs;us ab l &longs;umatur quarta pars totius axis lk, quæ &longs;it
lo. po&longs;tremo fiat on ad np, ut fru&longs;tum ag ad
cuius ba&longs;is &longs;it eadem, quæ fru&longs;ti, & altitudo æqualis. Dico
punctum p fru&longs;ti ag grauitatis centrum e&longs;&longs;e. ducantur
enim ac, eg: & intelligantur duo fru&longs;ta triangulares ba
&longs;es habentia, quorum alterum lf ex ba&longs;ibus abc, efg
&longs;tet
qr; in quo grauitatis centrum s: fru&longs;ti uero lh axis tu, &
x grauitatis centrum: deinde iungantur ur, tq, xs. tran&longs;i
bit ur per l: quoniam l e&longs;t centrum grauitatis quadran
guli abcd: & puncta ru grauitatis centra triangulorum
abc, acd; in quæ quadrangulum ip&longs;um diuiditur. eadem
quoque ratione tq per punctum k tran&longs;ibit. At uero pro
portiones, ex quibus fru&longs;torum grauitatis centra inquiri
mus, eædem &longs;unt in toto fru&longs;to ag, & in fru&longs;tis lf, lh. Sunt
enim per octauam huius quadrilatera abcd, efgh &longs;imilia:
coque
portionem &longs;eruant. Vt igitur duplum lateris ab unà
cum latere ef ad duplum lateris ef unà cum ab, ita e&longs;t
duplum ad late
ris una cum late
re eh ad duplum
eh unà cum ad:
& ita in aliis.
Rur&longs;us fru&longs;tum
ag ad
cuius eadem e&longs;t
ba&longs;is, & æqualis
altitudo eandem
bet, quam
lf ad
quæ e&longs;t
&longs;i, & æquali alti
tudine: & &longs;imili
ter quam lh fru
&longs;tum ad pyrami
dem, quæ ex
dem
altitudine con
&longs;tat. nam &longs;i inter
ip&longs;as ba&longs;es me
diæ proportio
nales con&longs;tituan
tur, tres ba&longs;es &longs;imul &longs;umptæ ad maiorem ba&longs;im in om
nibus codem modo &longs;e habebunt. Vnde fit, ut axes Kl,
qr, tu à punctis psx in eandem proportionem &longs;ecen
tur. ergo linea xs per p tran&longs;ibit: & lineæ ru, sx, qt in
ter &longs;e æquidi&longs;tantes erunt. Itaque cum fru&longs;ti ag latera pro
gula uyl, xyp, tyk inter &longs;e &longs;imilia: & &longs;imilia etiam triangu
la lyr, pys, kyq quare ut in 19 huius, demon&longs;trabitur
xp, ad ps:
quam ul ad lr. Sed ut ul ad lr, ita e&longs;t triangulum abc ad
triangulum acd: & ut tk ad Kq, ita triangulum efg ad
triangulum egh. Vt autem triangulum abc ad triangu
lum acd, ita pyramis abcy ad pyramidem acdy. & ut
triangulum efg ad triangulum egh, ita pyramis efgy
ad pyramidem eghy; ergo ut pyramis abcy ad
a cdy, ita pyramis efgy ad pyramidem eghy. reliquum
igitur
ad pyramidem acdy, hoc e&longs;t ut ul ad r, & ut xp ad ps.
Quòd cum fru&longs;ti lf centrum grauitatis &longs;its: & fru&longs;ti lh &longs;it
centrum x: con&longs;tat punctum p totius fru&longs;ti ag grauitatis
e&longs;&longs;e centrum. Eodem modo fret demon&longs;tratio etiam in
aliis pyramidibus.
&longs;exti.
medis.
Sit fru&longs;tum ad à cono, uel coni portione ab&longs;ci&longs;&longs;um, eu
ius maior ba&longs;is circulus, uel ellip&longs;is circa diametrum ab;
minor circa diametrum cd: & axis ef. diuidatur
in g, ita ut eg ad gf eandem proportionem habeat, quam
duplum diametri ab unà cum diametro ed ad duplum cd
unà cum ab.
quarta pars totius fe axis. Rur&longs;us quam proportionem
habet fru&longs;tum ad ad conum, uel coni portionem, in
ba&longs;i, & æquali altitudine, habeat linea Kh ad hl. Dico pun
ctum l fru&longs;ti ad grauitatis centrum e&longs;&longs;e. Si enini fieri po
te&longs;t, &longs;it m centrum:
& ut nl ad lm, ita fiat circulus, uel ellip&longs;is circa
ab ad aliud &longs;pacium, in quo &longs;it o. Itaque in circulo, uel
ellip&longs;i circa diametrum ab rectilinea figura plane de&longs;cri
batur, ita ut quæ relinquuntur portiones &longs;int o &longs;pacio mi
nores: & intelligatur pyramis apb, ba&longs;im habens rectili
neam figuram in circulo, uel ellip&longs;i ab de&longs;criptam: à qua erit ex iis quæ proxime
tradidimus, fru&longs;ti pyramidis ad ceutrum grauitatis l. Quo
niam igitur portiones &longs;pacio o minores &longs;unt; habebit cir
culus, uel ellip&longs;is ab ad
portiones dictas
proportionem, quàm nl
ad lm. &longs;ed ut circulus, uel
ellip&longs;is ab ad portiones,
ita apb conus, uel coni
portio ad &longs;olidas portio
nes, id quod &longs;upra demon
&longs;tratum e&longs;t: & ut circulus
uel ellip&longs;is cd ad portio
nes, quæ ip &longs;i in&longs;unt, ita co
nus, uel coni portio cpd
ad &longs;olidas ip&longs;ius portio
nes. Quòd cum figuræ in
circulis, uel ellip&longs;ibus ab
cd de&longs;criptæ &longs;imiles &longs;int,
erit proportio circuli, uel
ellip&longs;is ab ad &longs;uas portio
nes,
ellip&longs;is cd ad &longs;uas. ergo
conus, uel coni portio ap
b ad portiones &longs;olidas
dem
quam conus, uel coni por
tio cpd ad &longs;olidas ip&longs;ius
portiones. reliquum igi
tur coni, uel coni portionis
portiones &longs;olidas in ip&longs;o contentas eandem
habet, quam conus, uel coni portio apb ad &longs;olidas portio
nes: hoc e&longs;t eandem, quam circulus, uel ellip&longs;is ab ad por
tiones planas. quare fru&longs;tum coni, uel coni portionis ad
nl ad lm: & diuidendo fru&longs;tum pyramidis ad dictas por
tiones maiorem proportionem habet, quàm nm ad ml.
fiat igitur ut fru&longs;tum pyramidis ad portiones, ita qm ad
m l. Itaque quoniam à fru&longs;to coni, uel coni portionis ad,
cuius grauitatis centrum e&longs;tm, au&longs;ertur fru&longs;tum pyrami
dis habens centrum l; erit reliquæ magnitudinis, quæ ex
portionibus &longs;olidis con&longs;tat; grauitatis
producta, atque in puncto q, extra figuram po&longs;ito: quod
fieri nullo modo pote&longs;t. relinquitur ergo, ut punctum l &longs;it
fru&longs;ti ad grauitatis centrum. quz omnia demon&longs;tranda
proponebantur.
THEOREMA XXII. PROPOSITIO XXVII.
OMNIVM &longs;olidorum in &longs;phæra de&longs;cripto
rum, quæ æqualibus, & &longs;imilibus ba&longs;ibus conti
nentur, centrum grauitatis e&longs;t idem, quod &longs;phæ
ræ centrum.
Solida eiu&longs;modi corpora regularia appellare &longs;olent, de
quibus agitur in tribus ultimis libris elementorum: &longs;unt
autem numero quinque, tetrahedrum, uel pyramis, hexa
hedrum, uel cubus, octahedrum, dodecahedrum, & ico&longs;a
hedrum.
Sit primo abcd pyramis
ræ centrum &longs;it e. Dico e pyramidis abcd grauitatis e&longs;&longs;e
centrum. Si enim iuncta dc producatur ad ba&longs;im abc in
f; exiis, quæ demon&longs;trauit Campanus in quartodecimo li
bro elementorum, propo&longs;itione decima quinta, & decima
feptima, erit f centrum circuli circa triangulum abc de
fcripti: atque erit ef &longs;exta pars ip&longs;ius &longs;phæræ axis. quare
ex prima huius con&longs;tat trianguli abc grauitatis centrum
e&longs;&longs;e punctum f: & idcirco lineam df e&longs;&longs;e pyramidis axem.
At cum ef &longs;it &longs;exta pars axis
&longs;phæræ, crit d tripla ef. ergo
punctum e e&longs;t grauitatis cen
trum ip&longs;ius pyramidis: quod
in uige&longs;ima &longs;ecunda huius de
mon&longs;tratum &longs;uit. Sed e e&longs;t cen
trum &longs;phæræ. Sequitur igitur,
ut centrum grauitatis pyrami
dis in &longs;phæra de&longs;criptæ idem
&longs;it, quod ip&longs;ius &longs;phæræ cen
trum.
Sit cubus in &longs;phæra de&longs;criptus ab, & oppo&longs;itorum pla
norum lateribus bifariam diui&longs;is, per puncta diui&longs;ionum
plana ducantur, ut communis ip&longs;orum &longs;ectio &longs;it rectali
nea cd. Itaque &longs;i ducatur ab, &longs;olidi &longs;cilicet diameter, lineæ
ab, cd ex trige&longs;iman onaun decimi&longs;e&longs;e bifariam &longs;ecabunt.
&longs;ecent autem in puncto e. erit,
e
id quod demon&longs;tratum e&longs;t in
octaua huius. Sed quoniam ab
e&longs;t &longs;phæræ diametro æqualis,
ut in decima quinta propo&longs;i
tione tertii decimilibri
torum o&longs;tenditur: punctum e
&longs;phæræ quoque centrum erit.
Cubi igitur in &longs;phæra de&longs;cri
pti grauitatis centrum idem
e&longs;t, quod centrum ip&longs;ius &longs;phæræ.
Sit octahedrum abcdef, in &longs;phæra de&longs;criptum, cuius
&longs;phæræ centrum &longs;itg. Dico punctum g ip&longs;ius octahedri
grauitatis centrum e&longs;&longs;e. Con&longs;tat enim ex iis, quæ demon
&longs;trata &longs;unt à Campano in quinto decimo libro elemento
rum, propo&longs;itione &longs;extadecima eiu&longs;modi &longs;olidum diuidi
in duas pyramides æquales, & &longs;imiles; uidelicet in pyrami
in pyramidem, cuius
gf &longs;emidiametri &longs;phæræ, & linea una.
ræ centrum, erit etiam centrum circuli, qui circa
abcd de&longs;cribitur: & propterea eiu&longs;dem quadrati grauita
tis centrum: quod in prima propo&longs;itione huius demon
&longs;tratum e&longs;t. quare pyramidis abcde axis erit eg: & pyra
midis abcdf axis fg. Itaque &longs;ith centrum grauitatis py
ramidis abcde, & pyramidis abcdf centrum &longs;it
&longs;picuum e&longs;t ex uige&longs;ima &longs;ecunda propo&longs;itione huius,
ch triplam e&longs;&longs;e hg:
h quadruplam. &
ratione fg
ip&longs;ius gk quod cum e
g, gf &longs;int æquales, & h
g, g
les erunt. ergo ex quar
ta propo&longs;itione primi
libri Archimedis de
tro
totius octahedri, quod
ex dictis pyramidibus
con&longs;tat, centrum graui
tatis erit punctum g idem, quod ip&longs;ius &longs;phæræ centrum.
Sit ico&longs;ahedrum ad de&longs;criptum in &longs;phæra, cuius
&longs;it g. Dico g ip&longs;ius ico&longs;ahedri grauitatis e&longs;&longs;e centrum. Si
enim ab angulo a per g ducatur recta linea u&longs;que ad &longs;phæ
ræ &longs;uperficiem; con&longs;tat ex &longs;exta decima propo&longs;itione libri
tertii decimi elementorum, cadere eam in angulum ip&longs;i a
oppo&longs;itum. cadat in d:
angulum abc: & iunctæ bg,producantur, & cadant in
angulos ef, ip&longs;is bc oppo&longs;itos. Itaque per triangula
abc, def ducantur plana &longs;phæram &longs;ecantia. erunt hæ &longs;e-
do&longs;ii: unus quidem circa triangulum abc de&longs;criptus: al
ter uero circa def: & quoniam triangula abc, def æqua
lia &longs;unt, & &longs;imilia; erunt ex prima, & &longs;ecunda propo&longs;itione
duodecimi libri clementorum, circuli quoque inter &longs;e &longs;e
æquales. po&longs;tremo a centro g ad circulum abc perpendi
cularis ducatur gh; & alia perpendicularis ducatur ad cir
culum def, quæ &longs;it gk; & iungantur ah, dk per&longs;picuum
e&longs;t ex corollario primæ &longs;phæricorum Theodo&longs;ii, punctum
h centrum e&longs;&longs;e circuli abc, & k centrum circuli def. Quo
niam igitur triangulorum gah, gdK latus ag e&longs;t æquale la
teri gd; &longs;unt enim à centro &longs;phæræ ad &longs;uperficiem: atque
e&longs;t ah æquale dk: & ex &longs;exta propo&longs;itione libri primi &longs;phæ
ricorum Theodo&longs;ii gh ip&longs;igK: triangulum gah æquale
erit, & &longs;imile gdk triangulo: & angulus agh æqualis an
gulo dg
ctis. crgo & ip&longs;i hgd, dgk duobus rectis æquales erunt.
& idcirco hg, g cum autem
h &longs;it
anguli abc grauitatis cen
in prima propo&longs;itione hu
ius tradita &longs;unt. quare gh
erit pyramidis abcg axis.
& ob eandem cau&longs;&longs;am gk
axis pyramidis defg. lta
que centrum grauitatls py
ramidis abcg &longs;it
l, & pyramidis defg &longs;it m.
Simillter ut &longs;upra demon
&longs;trabimus mg, gl inter &longs;e æquales e&longs;&longs;e, & punctum g graui
tatis centrum magnitudinis, quæ ex utri&longs;que pyramidibus
con&longs;tat. eodem modo demon&longs;trabitur, quarumcunque
duarum pyramidum, quæ opponuntur, grauitatis Sequitur ergo utico&longs;ahedri centrum gra
uitatis &longs;it idem, quod ip&longs;ius &longs;phæræ centrum.
Sit dodecahedrum af in &longs;phæra de&longs;ignatum, &longs;itque &longs;phæ
ræ centrum m. Dico m centrum e&longs;&longs;e grauitatis ip&longs;ius do
decahedri. Sit enim pentagonum abcde una ex duode
cim ba&longs;ibus &longs;olidi af: & iuncta am producatur ad &longs;phæræ
&longs;uperficiem. cadet in angulum ip&longs;i a oppo&longs;itum; quod col
ligitur ex decima &longs;eptima propo&longs;iticne tertiidecimi libri
elementorum. cadat in f. at &longs;i ab aliis angulis bcde per
trum itidem lineæ ducantur ad &longs;uperficiem &longs;phæræ in pun
cta ghkl; cadent hæ in alios angulos ba&longs;is, quæ ip&longs;i abcd
ba&longs;i opponitur. tran&longs;eant ergo per pentagona abcde,
fghKl plana &longs;phæram &longs;ecantia, quæ facient &longs;ectiones cir
culos æquales inter &longs;e &longs;e: po&longs;tea ducantur ex centro &longs;phæræ
m perpendiculares ad pla
na dictorum
circulum quidem abcde
perpendicularis mn: & ad
circulum fghKl ip&longs;a mo,
erunt puncta no
centra: & lineæ mn, mo in
ter &longs;e æquales: quòd circu
li æquales &longs;int. Eodem mo
do, quo &longs;upra, demon&longs;trabi
mus lineas mn, mo in
atque eandem lineam con
uenire. ergo cum puncta no &longs;int centra circulorum, con
&longs;tat ex prima huius &
ponatur abcdem pyramidis grauitatis centrum p: & py
ramidis fghklm ip&longs;um q centrum. erunt pm, mq æqua
les, & punctum m grauitatis centrum magnitudinis, quæ
ex ip&longs;is pyramidibus con&longs;tat.
rumlibet pyramidum, quæ è regione opponuntur, patetigitur totius dodecahe
dri, centrum grauitatis
prehendentis centrum. quæ quidem omnia demon&longs;tra&longs;&longs;e
oportebat.
pri
mæ &longs;phæ
ricorum
Theod.
phærico
rum.
PROBLEMA VI. PROPOSITIO XXVIII.
DATA qualibet portione conoidis rectangu
li, ab&longs;ci&longs;&longs;a plano ad axem recto, uel non recto; fie
ri pote&longs;t, ut portio &longs;olida in&longs;cribatur, uel circum
&longs;cribatur ex cylindris, uel cylindri portionibus,
æqualem habentibus altitudinem, ita ut recta li
nea, quæ inter centrum grauitatis portionis, &
figuræ in&longs;criptæ, uel circum&longs;criptæ interiicitur,
&longs;it minor qualibet recta linea propo&longs;ita.
Sit portio conoidis rectanguli abc, cuius axis bd,
uitatisquequam ue
ro proportionem habet linea be ad lineam g, eandem ha
beat portio conoidis ad &longs;olidum h: & circum&longs;cribatur por
tioni figura, &longs;icuti dictum e&longs;t, ita ut portiones reliquæ &longs;int
&longs;olido h minores: cuius quidem figuræ centrum grauitatis
&longs;it punctum k. Dico
&longs;ita. ni&longs;i enim &longs;it minor, uel æqualis, uel maior erit. & quo
niam figura circum&longs;cripta ad reliquas portiones maiorem
proportionem habet, quàm portio conoidis ad &longs;olidum h;
hoc e&longs;t maiorem, quàm bc ad g: & be ad g non minorem
habet proportionem, quàm ad ke, propterea quod ke non
ponitur minor ip&longs;a g: habcbit figura circum&longs;cripta ad por
tiones reliquas maiorem proportionem quàm be ad ek:
& diuidendo portio conoidis ad reliquas portiones habe
bit maiorem, quàm bk ad Ke. quare &longs;i fiat ut portio co
ke: erit lk maior, quam bk: & ideo punctum l extra por
tionem cadet.
igitur à figura circum
&longs;cripta, cuius grauitatis
centrum e&longs;t k, aufertur
portio conoidis, cuius
centrum e.
ad Ke eam proportio
nem, quam portio co
noidis ad reliquas por
tiones; erit punctum l
extra portionem
centrum magnitudinis
ex reliquis portionibus compo&longs;itæ. illud autem fieri nullo
modo pote&longs;t. quare con&longs;tat lineam ke ip&longs;a g linea propo&longs;i
ta minorem e&longs;&longs;e.
ex tradi
tione
l
Rur&longs;us in&longs;cribatur portioni figura, uidelicet cylindr us
mn, ut &longs;it ip&longs;ius altitudo
æqualis dimidio axis bd:
& quam proportionem
habet be ad g, habeat mn
cylindrus ad &longs;olidum o.
in&longs;cribatur deinde eidem
alia figura, ita ut portio
nes reliquæ &longs;int &longs;olido o
minores: & centrum gra
uitatis figuræ &longs;it p. Dico
lineam pe ip&longs;a g
e&longs;&longs;e. &longs;i enim non &longs;it mi
nor, codem, quo &longs;upra modo demon&longs;trabimus figuram in
&longs;criptam ad reliquas portiones maiorem proportionem
habere, quàm be ad ep. & &longs;i fiat alia linea le ad ep, ut e&longs;t
figura in&longs;cripta ad reliquas portiones,
uitatis centrum e auferatur in&longs;cripta figura, centrum ha
bens p: & &longs;it le ad ep, ut figura in&longs;cripta ad portiones reli
quas: erit magnitudinis, quæ ex reliquis portionibus con
&longs;tat, centrum grauitatis punctum l, extra portionem ca
dens. quod ficrinequit. ergo linca pe minor e&longs;tip&longs;a g li
nea propo&longs;ita.
Ex quibus per&longs;picuum e&longs;t centrum grauitatis
figuræ in&longs;criptæ, & circum&longs;criptæ eo magis acce
dere ad portionis centrum, quo pluribus cylin
dris, uel cylindri portionibus con&longs;tet:
rain&longs;cripta maior, & circum&longs;cripta minor. &
quanquam continenter ad portionis
pius admoueatur: nunquam tamen ad ip&longs;um per
ueniet. &longs;equeretur enim figuram in&longs;criptam,
&longs;olum portioni, &longs;ed etiam circum&longs;criptæ figuræ
æqualem e&longs;&longs;e. quod e&longs;t ab&longs;urdum.
THEOREMA XXIII. PROPOSITIO XXIX.
CVIVSLIBET portionis conoidis rectangu
li axis à
terminatur ad uerticem, reliquæ partis, quæ ad ba
&longs;im &longs;it dupla.
SIT portio conoidis rectanguli uel ab&longs;ci&longs;&longs;a plano ad
axem recto, uel non recto: & &longs;ecta ip&longs;a altero plano per
&longs;it &longs;uperficici &longs;ectio ab crectanguli coni &longs;ectio, uel parabo
le; plani ab&longs;cindentis portionem &longs;ectio &longs;it recta linea ac:
axis portionis, & &longs;ectionis diameter bd. Sumatur autem
in linea bd punctum e, ita ut be &longs;itip&longs;ius ed dupla. Dico
e portionis ab
c grauitatis e&longs;&longs;e
centrum. Diui
datur enim bd
bifariam in m:
& rur&longs;us dm, m
b bifariam diui
dantur in pun
ctis n, o:
baturque
ni figura &longs;olida,
& altera circum
&longs;cribatur ex cy
lindris æqualem
altitudinem ha
bentibus, ut&longs;u
perius
Sit autem pri
mum figura in
&longs;cripta
f g: &
pta
ah, Kl con&longs;tet.
punctum n erit
centrum graui
tatis figuræ in
fcriptæ,
&longs;cilicet ip&longs;ius d
m axis:
erit centrum cy
lindri ah: & cy
lindri kl
o, axis bm me
dium. quare &longs;i li
neam on ita di
ui&longs;erimus in p,
ut
tionem
lindrus ah ad
cylindrum kl,
habeat linea op
ad pn: centrum
grauitatis toti
us figuræ
&longs;criptæ
ctum p. Sed cy
lindri, qui &longs;unt
æquali altitudi
ne, eandem in
ter &longs;e &longs;e, quam
ba&longs;es propor—
tionem habent:
ad bm, ita
dratum
ad
&longs;ius Km, ex uige
&longs;ima primi libri
quadratum ac
ad
g: hoc e&longs;t circu
lus circa diame
trum ac ad cir
culum circa dia
metrum kg. du
pla e&longs;t autem li
nea db lineæ crgo circulus ac circuli kg: & idcirco cylindrus
ah cylindri k. l duplus erit. quare & linea op dupla
ip&longs;ius pn. Deinde in&longs;cripta & circum&longs;cripta portioni
alia figura, ita ut in&longs;cripta con&longs;tituatur ex tribus cylin
dris qr, sg, tu: circum&longs;cripta uero ex quatuor ax, yz,
K
ita ut
ad cylindrum
dratum
magnitudinis compo&longs;itæ ex cylindris
mb &longs;it dupla bo, erit & præterea quo
niam cylindri yz centrum grauitatis e&longs;t
&longs;ain
drus yz ad duos cylindros K
dinis, quæ ex dictis tribus cylindris con&longs;tat. cylindrus
tem
ad 1: & ad cylindrum k
quare yz &
propterea linea
centrum grauitatis e&longs;t punctum
in
lindros yz, k
totius figuræ Sed cylindrus ax ad ip&longs;um yz
e&longs;t ut linea db ad bn: hoc e&longs;t ut 4 ad 3: & duo cylindri k
qlcylindrus igitur ax ad tres
iam dictos cylindros e&longs;t ut 2 ad 3. Sed
rum partium, &
tium quatuor:
trium. quare &longs;equitur ut punctum
&longs;criptæ &longs;it centrum. Itaque fiat
bifariam diuidatur in
ra o&longs;tendetur centrum magnitudinis compo&longs;itæ ex cylin-
dris sg, tu e&longs;&longs;e
punctum
totius figuræ in
&longs;criptæ, quæ
&longs;tat
qr, &longs; g, tu e&longs;&longs;e
centrum. Sunt
enim hi cylindri
æquales & &longs;imi
les cylindris yz,
K
circum&longs;criptæ.
ut be ad ed, ita
e&longs;t op ad pn;
triu&longs;que e&longs;t du
pla: erit compo
nendo, ut bd ad
de, ita on ad n
p; & permutan
do, ut bd ad o
n, ita de ad np.
Sed bd dupla
e&longs;t on. ergo &
ed ip&longs;ius np du
pla erit. quòd &longs;i
ed bifariam di
uidatur
qualis np: &
&longs;ublata en, quæ
e&longs;t
trique e cum autem be &longs;it dupla
ed, & op dupla pn, hoc e&longs;tip&longs;ius e
cet bo unà cum pe ip&longs;ius reliqui e&longs;tque
bo dupla ergo pe, hoc e&longs;t n&longs;ed dn
dupla e&longs;t n&longs;unt au
tem dqua
re con&longs;tat np ip&longs;ius pe duplam e&longs;&longs;e. & idcirco pe ip&longs;i en
æqualem. Rur&longs;us cum &longs;it
etiam reliqua Eadem quoque ratione
ergo ut
mquare & opræ
terea reliquaigitur
liquæ matque erat ergo m
quales &longs;unt: & ita æquales m
m
ter &longs;e &longs;int æquales. Sed ut
ad d
les ergo dSed etiam æ
quales nreliqua igitur
nquare dempta p
ne, relinquitur pe æqualis e
&longs;ecundo loco de&longs;criptarum a primis centris pn æquali in
teruallo recedunt. quòd &longs;i rur&longs;us aliæ figuræ de&longs;cribantur,
codem modo demon&longs;trabimus earum centra æqualiter ab
his recedere, & ad portionis conoidis centrum propius ad
moueri. Ex quibus con&longs;tat lineam
portionis diuidi in partes æquales. Si enim fieri pote&longs;t, non
&longs;it centrum in puncto e, quod e&longs;t lineæ
quædam figura in&longs;cribi pos&longs;it, ita utlinea, quæ inter cen
trum grauitatis portionis, & in&longs;criptæ figuræ interiicitur,
qualibet linea propo&longs;ita &longs;it minor, quod proxime demon
&longs;trauimus: perueniet tandem
figura æquali interuallo ad portionis centrum accedit, ubi
primum
portionis centrum &longs;e applicabit. quod fieri nullo modo
po&longs;&longs;e per&longs;picuum e&longs;t. non aliter idem ab&longs;urdum &longs;equetur,
fi ponamus centrum portionis recedere à medio ad par
tes
quod portionis ergo punctum e centrum erit gra
uitatis portionis abc. quod demon&longs;trare oportebat.
libri Ar
chimedis
decimi.
cimi
quinti
Quod autem &longs;upra
dis recta per figuras, quæ ex cylindris æqualem altitudi
dinem habentibus con&longs;tant, idem &longs;imiliter demon&longs;trabi
mus per figuras ex cylindri portionibus con&longs;antes in ea
portione, quæ plano non ad axem recto ab&longs;cinditur. ut
enim tradidimus in commentariis in undecimam propo&longs;i
tionem libri Archimedis de conoidibus & &longs;phæroidibus.
portiones cylindri, quæ æquali &longs;unt altitudine eam inter &longs;e
&longs;e proportionem habent, quam ip&longs;arum ba&longs;es: ba&longs;es
quæ &longs;unt ellip&longs;es &longs;imiles eandem proportionem habere,
quam quadrata diametrorum eiu&longs;dem rationis, ex corol
lario &longs;eptlmæ propo&longs;itionis libri de conoidibus, & &longs;phæ
roidibus, manife&longs;te apparet.
de conoi
dibus &
&longs;phæroi
dibus.
THEOREMA XXIIII. PROPOSITIO XXX.
Si à portione conoidis rectanguli alia portio
ab&longs;cindatur, plano ba&longs;i æquidi&longs;tante; habebit
portio tota ad eam, quæ ab&longs;ci&longs;&longs;a e&longs;t, duplam pro
portion em eius, quæ e&longs;t ba&longs;is maioris portionis
ad ba&longs;i m minoris, uel quæ axis maioris ad axem
minoris.
ABSCINDATVR à portione conoidis rectanguli
abc alia portio ebf, plano ba&longs;i æquidi&longs;tante: & eadem
portio &longs;ecetur alio plano per axem; ut &longs;uperficiei &longs;ectio &longs;it
parabole abc:
lincæ ac, ef: axis autem portionis, & &longs;ectionis diameter
bd; quam linea ef in puncto g &longs;ecet. Dico portionem co
noidis abc ad portionem ebf duplam proportionem ha
bere eius, quæ e&longs;t ba&longs;is ac ad ba&longs;im ef; uel axis db ad bg
axem. Intelligantur enim duo coni, &longs;eu coni portiones
abc, ebf,
lem habentes altitudinem. & quoniam abc portio conoi
dis fe&longs;quialtera e&longs;t coni, &longs;eu portionis coni abc; & portio
ebf coni feu portionis coni bf e&longs;t &longs;e&longs;quialtera, quod de
mon&longs;trauit Archimedes in propo&longs;itionibus 23, & 24 libri
de conoidibus, & &longs;phæroidibus: erit conoidis portio ad
conoidis portionem, ut conus ad conum, uel ut coni por
tio ad coni portionem. Sed conus, nel coni portio abc ad
conum, uel coni portionem ebf compo&longs;itam proportio
nem habet ex proportione ba&longs;is ac ad ba&longs;im ef, & ex pro
portione altitudinis coni, uel coni portionis abc ad alti
tudinem ipfius ebf, ut nos demon&longs;trauimus in com men
tariis in undecimam propo&longs;itionem eiu&longs;dem libri Archi
medis: altitudo autem ad altitudinem c&longs;t, ut axis ad axem.
quod quidem in conis rectis per&longs;picuum e&longs;t, in &longs;calenis ue Ducatur à puncto bad planum ba
&longs;is ac perpendicularis linea bh, quæ ip&longs;am ef in K &longs;ecet.
erit bh altitudo coni, uel coni portionis abc: & bK altitu
do efg. Quod cum lineæ ac, ef inter &longs;e æquidi&longs;tent, &longs;unt
enim planorum æquidi&longs;tantium &longs;ectiones: habebit db ad
bg proportionem eandem, quam hb ad bk quare por
tio conoidis abc ad portionem efg proportionem habet
compo&longs;itam ex proportione ba&longs;is ac ad ba&longs;im ef; & ex
proportione db axis ad axem bg. Sed circulus, uel
ellip&longs;is circa diametrum ac ad circulum, uel ellip&longs;im
circa ef, e&longs;t ut quadratum ac ad quadratum ef; hoc e&longs;t ut
tum eg e&longs;t, ut linea db ad lineam bg. circulus igitur, uel el
lip&longs;is circa diametrum ac ad
hoc e&longs;t ba&longs;is ad ba&longs;im eandem proportionem habet,
db axis ad axem bg. ex quibus &longs;equitur portionem abc
ad portionem ebf habere proportionem duplam eius,
quæ e&longs;t ba&longs;is ac ad ba&longs;im ef: uel axis db ad bg axem. quod
demon&longs;trandum proponebatur.
cimi.
cimi
noidibus
& &longs;phæ
roidibus
quinti
THEOREMA XXV. PROPOSITIO XXXI.
Cuiuslibet fru&longs;ti à portione rectanguli conoi
dis ab&longs;cis&longs;i, centrum grauitatis e&longs;t in axe, ita ut
demptis primum à quadrato, quod fit ex diame
tro maioris ba&longs;is, tertia ip&longs;ius parte, & duabus
tertiis quadrati, quod fit ex diamerro ba&longs;is mino
ris: deinde à tertia parte quadrati maioris ba&longs;is
rur&longs;us dempta portione, ad quam reliquum qua
drati ba&longs;is maioris unà cum dicta portione
proportionem habeat eius, quæ e&longs;t quadrati ma
eo axis puncto, quo ita diuiditur ut pars, quæ mi
norem ba&longs;im attingit ad alteram partem eandem
proportionem habeat, quam dempto quadrato
minoris ba&longs;is à duabus tertiis quadrati maioris,
habet id, quod reliquum e&longs;t unà cum portione à
tertia quadrati maioris parte dempta, ad
eiu&longs;dem tertiæ portionem.
SIT fru&longs;tum à portione rectanguli conoidis ab&longs;ci&longs;&longs;um
abcd, cuius maior ba&longs;is circulus, uel ellip&longs;is circa diame
trum bc, minor circa diametrum ad; & axis ef. de&longs;criba
tur autem portio conoidis, à quo illud ab&longs;ci&longs;&longs;um e&longs;t, & pla
no per axem ducto &longs;ecetur; ut &longs;uperficiei &longs;ectio &longs;it parabo
le bgc, cuius diameter, & axis portionis gf: deinde gf diui
datur in puncto h, ita ut gh &longs;it dupla hf: & rur&longs;us ge in ean
dem proportionem diuidatur:
ex iis, quæ proxime demon&longs;trauimus, con&longs;tat centrum gra
uitatis portionis bgc e&longs;&longs;e h punctum: & portionis agc
punctum k. &longs;umpto igitur infra h puncto l, ita ut kh ad hl
tionem agd; erit punctum l eius fru&longs;ti grauitatis
quam portio conoidis bgc ad agd portionem.
niam quadratum bf ad quadratum ae, hoc e&longs;t quadratum
bc ad quadratum ad e&longs;t, ut linea fg ad ge: erunt duæ ter
tiæ quadrati bc ad duas tertias quadrati ad, ut hg ad gk:
& &longs;i à duabus tertiis quadrati bc demptæ fuerint duæ ter
tiæ quadrati ad: erit
tertias quadrati ad, ut hk ad kg. Rur&longs;us duæ tertiæ quadra
ti ad ad duas tertias quadrati bc &longs;unt, ut kg ad gh: & duæ
tertiæ quadrati bc ad ergo
ex æquali id, quod relinquitur ex duabus tertiis quadrati
bc, demptis ab ip&longs;is quadrati ad duabus tertiis, ad
partem quadrati bc, ut kh ad hf: & ad portionem
tertiæ partis, ad quam unà cum ip&longs;a portione, duplam pro
portionem habeat eius, quæ e&longs;t quadrati bc ad
ad, ut Kl ad lh. habet enim Kl ad lh eandem proportio
nem, quam conoidis portio bgc ad portionem agd: por
tio autem bgc ad portionem agd duplam proportionem
habet eius, quæ e&longs;t ba&longs;is bc ad ba&longs;im ad: hoc e&longs;t quadrati
bc ad quadratum ad; ut proxime demon&longs;tratum e&longs;t. quare
dempto ad quadrato à duabus tertiis quadrati bc, erit id,
quod relinquitur unà cum dicta portione tertiæ partis ad
reliquam eiu&longs;dem portionem, ut el ad lf. Cum igitur cen
trum grauitatis fru&longs;ti abcd &longs;it l, à quo axis ef in eam,
diximus, proportionem diuidatur; con&longs;tat
quod demon&longs;trandum propo&longs;uimus.
corum.
FINIS LIBRI DE CENTROGRAVITATIS SOLIDORVM.
Impre&longs;&longs;. Bononiæ cum licentia Superiorum,