DE CENTRO
GRAVITATIS
SOLIDORVM
LIBRITRES.
LVCÆ VALERII
in Gymna&longs;io Romano profe&longs;&longs;oris.
ROMÆ, Typis Bartholom ri Bonfadini.
MDC IIII.
SVPERIORVM PERMISSV.
Imprimatur
Si placet R. P.
Magi&longs;tro S. Palat
B.
Gyp&longs;ius Vice&longs;ger.
Io.
Maria Bra&longs;ichellen.
Sacri Pal.
Apostol.
Magi&longs;t.
SANCTISSIMO
DOMINO NOSTRO
CLEMENTI VIII
PONT. OPT. MAX.
Grata Principi munera,
P. B. ex Philo&longs;ophiæ late
bris deprompta, qua&longs;i aurum
&longs;oli expo&longs;itum illico &longs;plen
dent, & publicæ vtilitatis
&longs;pem o&longs;tendunt, magno or
nata præ&longs;idio in primos liuo
ris impetus illius approbatione, cuius officium e&longs;t
alia à rep. auertere, alia imperare.
Hinc por
rò factum e&longs;t, vt omnis ferè &longs;criptor exi&longs;ti matio
nis periculum aditurus, aliquem ex principibus
opus ab inuidorum mor&longs;ibus &longs;eruetur incolume. Hanc ergo con&longs;uetudinem amanti mihi &longs;anè feli
citer cecidit, vt tu &longs;ola tua propria benignitate
permotus in tuos me familiares vltro a&longs;criberes. Siue eniming enij mei debilis partus
troni de&longs;iderat autoritatem: tu principum orbis
terrarum princeps &longs;emper digni&longs;&longs;imam principa
tu &longs;apientiam præ&longs;titi&longs;ti. Seu tam elatæ dedica
tiones &longs;olent alienas à &longs;apientiæ &longs;tudio &longs;pes olere:
lux tanti patrocinij,
neficiorum, atram &longs;u&longs;picionem amouebit. Quòd
verò ad vitam ip&longs;ius operis attinet, quam nulla
per te velim temporum permutatione terminari:
vereor vt id &longs;ua luce multis alijs vitali a&longs;piciat
illa, quæ tua &longs;tudia, & res ge&longs;tas omnium lin
guis, & litteris celebrabit æternitas. quantum
enim tuam excel&longs;am &longs;u&longs;picio dignitatem, tantum
de&longs;picor i&longs;tius doni incredibilem cum illa com
parati humilitatem: neque id ni&longs;i diuinitus cre
diderim perpetuam in tuis laudibus famam ha
biturum. Quare illud non &longs;olum tibi diuini gre
gis anti&longs;titi cupio gratum accidere, cuius auto
ritate protectum in tanta nouarum rerum po&longs;t
tam graues autores contemptione, minimo meo
cum rubore in medium prodeat: &longs;ed ip&longs;i diuinita
ti ex voluntate donum expendenti, penes quam
e&longs;t æternitas, & cui primum dicata omnia e&longs;&longs;e
oportet: vt hi, quostuis luminibus dignaris, de
te&longs;tes libelli à mortis æmula me obliuione defen
dant. Stomacharis hic, arbitror, quòd tantum
&longs;pectem de nihilo; &longs;ed magis confe&longs;&longs;ionis impu
dentia. At verò non impetus animi ad gloriam,
cuius nullum mihi natura &longs;emen impartiuit (&longs;it
gloriæ loco ignauiæ fugi&longs;&longs;e dedecus) &longs;ed tua er
ga me voluntas, meisapta &longs;tudijs liberalitate te
&longs;tata hunc ardorem expre&longs;&longs;it. Tanta enim e&longs;t
venu&longs;tas tuæ virtutis ex mei meriti penuria, vt
putem &longs;ine me indice illam diminutum &longs;ui &longs;pecta
culum po&longs;teris præbituram. Nihil ergo minus
cogitans quàm quî tua beneficia cumulando per
turbatis iudicijs &longs;atisfacerem, &longs;cientia &longs;cilicet,
& virtute illa, qua maximè &longs;uperbit eneruata, &
are&longs;cens Mundiætas; nullum opulentiæ meæ, ar
tis alienæ &longs;pecimen pro munere gratiæ à te acce
pto partem tibi reddidi: &longs;ed ingenij mei partum,
qualis is cumque e&longs;t; quod & grati animi quæ&longs;i
tum monumentum crimine me audaciæ liberet,
&longs;i quodimpendeat, palam dedicaui. Alij tibi co
lumnas hone&longs;ti&longs;&longs;imis titulis ornatas erigant: &longs;ta
tuas in foris collocent: magnificas ædes extruant,
quarum in frontibus grandes marmoreæ tabulæ
flammantibus auro &longs;yderibus, & peregrinis lapi
dibus intextæ ea de te viuo referant &longs;axum impu
dens, quæ verecunda hæc pagina prætermittit. Ego incredibilis tuæ benignitatis non tam gra
uia te&longs;timonia, quæ loco moueri nequeant: &longs;ed
de&longs;tinaui. Quem quidem eo minus vereor ne
non tu, quamobrem Telchines forta&longs;&longs;e aliqui in
&longs;ectaturi, di&longs;pari &longs;is voluntate protecturus, quòd
in his tàm reconditis naturæ arcanis geometrica
demon&longs;tratione patefactis, tanquam in &longs;emine
multiplicem præ&longs;criptionem, ac normam e&longs;&longs;e in
telliges ip&longs;e pacis inter tuos greges autor, lupi
otomani terror, ciuili, & bellicæ architecturæ
maximè nece&longs;&longs;ariam. Quòd que, cum ad theologi
cam quandam veritatem chri&longs;tiano generi maxi
me &longs;alutarem illu&longs;trandam, per Philo&longs;ophi<17> etiam
campos &longs;apientium hominum corona decoratus,
nulla tantæ molis, quantam &longs;u&longs;tines negotiorum
iactura lati&longs;&longs;imè vageris; nempe illam cre&longs;cere,
atque illu&longs;trari indies magis ex optas, cuius con
&longs;uetudine tantopere delectaris. Quod denique
&longs;cientiæ ciuilis ip&longs;e periti&longs;&longs;imus omnium optimè
intelligis, quanti referat ad humanæ &longs;ocietatis for
mam & candorem, regum, atque optimatum a
mor in &longs;tudio&longs;os bonarum litterarum. contrà au
tem ex de&longs;pectione in hos cadente abijs, quorum
mores pro legibus haberi &longs;olent, no&longs;ti commu
nem ingeniorum veternum, mox tyrannidem gi
gni, magna cu&longs;tode adempta mode&longs;tiæ imperi
tantium crebra ciuium &longs;apientia, quæ prauis ti
morem efficit, melioribus pudorem, Quod &longs;i meæ
expectationi exitus re&longs;pondebit, vt te hoc munu
&longs;culo vel leuiter lætari &longs;entiam; alia non iniucun-
DINVS tuus nepos, domi fori&longs;que clari&longs;&longs;imus
Cardinalis, cuius inter familiares itidem,
ficijsque
manitatis te&longs;timonia ab inuidiæ &longs;atellite & mi
ni&longs;tra calumnia tueatur: quando duobus talibus
viris animi mei captum beneficentia &longs;ua pericli
tantibus, duplex periculum &longs;ubire &longs;um coactus. Sed iam verbo&longs;æ epi&longs;tolæ, & tuo fa&longs;tidio finem im
po&longs;iturus peto à te vnum; vt tibi per&longs;uadeas, me
inter tuos famulos, quos ære proprio, & victu quo
tidiano liberaliter &longs;u&longs;tentas, eorum, qui pro te
emori po&longs;&longs;unt, amore, con&longs;tantia, fidelitate nemini
planè concedere. Sic tua omnia præ&longs;tanti&longs;&longs;ima
facinora Princeps magnanime, & pietatis colu
men, Deus Opt. Max.
tibi fortunet, quem ad ma
iores in dies res gerendas in longum æuum inco
lumen, felicemque con&longs;eruet. Valet.
*e*i*s *t*a *a*u*t*o*u *k*e*n*t*r*a
*st
*b
*me/my
*be/ltion
*p
*d
*lu
LVC AE
VALER II
DE CENTRO
GRAVITATIS
SOLIDORVM
Propo&longs;itum e&longs;t mihi in hi&longs;ce tribus li
bris, ò Geometra, cuiu&longs;cumque figuræ
&longs;olidæ in geometria ratio haberi &longs;olet,
centrum grauitatis inuenire. Huius
autem prouinciæ mihi &longs;u&longs;cipiendæ oc
ca&longs;io fuit liber ille iam pridem editus
Federici Commandini Vrbinatis, in
quo cum ille corporum planis termi
nis definitorum; necnon cylindri, & coni, & fru&longs;ti conici,
& &longs;phæræ, & &longs;phæroidis centrum grauitatis o&longs;tendi&longs;&longs;et;
aliorum autem, quæ &longs;uperficie mixta continentur vno co
noide parabolico tentato &longs;yllogi&longs;mi iactura operam per
didi&longs;&longs;et, ego &longs;pe magis, ad quam vir ille exar&longs;erat incita
metriæ partem tamdiu de&longs;iderari cognitione digni&longs;&longs;imam;
cum ante exercitationis cau&longs;a omnium, quæ propo&longs;ui &longs;oli
dorum, excepto conoide parabolico, centra grauitatis aliis
viis indaga&longs;&longs;em; po&longs;tea non &longs;olum parabolici, &longs;ed ante me
tentata nemini, hyperbolici conoidis, & fru&longs;ti vtriu&longs;que, &
portionis vtriu&longs;que conoidis, & portionis fru&longs;ti, & hemi
&longs;phærij, & hemi&longs;phæroidis, & cuiu&longs;libet portionis &longs;phæ
ræ, & &longs;phæroidis vno, & duobus planis parallelis ab&longs;ci&longs;&longs;æ
dam triplici via. Taceo nunc alia eiu&longs;dem generis, quæ
cum vtilia, tum geometriæ &longs;tudio&longs;is non iniucunda, vt arbi
tror, futura in po&longs;teriores libros di&longs;tribuimus. Quòd autem
aliquot propo&longs;itiones, alias Archimedis lemmaticas, alias
Commandini meis rationibus attuli demon&longs;tratas; non tàm
idcirco id fcci, ne meæ lucubrationes
vel &longs;tylo Euclidis magis con&longs;onæ, vel ad percipiendum eo
minus laborio&longs;æ, quo ad inueniendum &longs;unt difficiliores,
vel meo propo&longs;ito aptiores viderentur. Earum propo&longs;itio
num, Archimedis duo &longs;unt in primo libro, decimaquarta,
& &longs;eptima, & &longs;ecunda pars vige&longs;imæ; in &longs;ecundo autem vna. Omne conoides parabolicum &longs;e&longs;quialterum e&longs;&longs;e coni ean
dem ba&longs;im, & eandem altitudinem habentis. Comman
dini autem omnes in primo libro nouem; vige&longs;ima tertia, &
quinta: trige&longs;ima &longs;ecunda, tertia, quarta, &longs;eptima, & nona:
quadrage&longs;ima prima, & &longs;ecunda. Sed multa hic noua inue
nies ita ad præ&longs;ens in&longs;titutum nece&longs;&longs;aria, vt per &longs;e
in geometria locum habere debeant, maxime verò tres pri
mæ &longs;ecundi libri propo&longs;itiones, quippe quibus magnam, ac
perdifficilem geometriæ partem demon&longs;tratione recta, &
generali ad viam regiam redactam e&longs;se intelliges. Ita Deus
Opt. Max.
cuius auxilio hæc feci, quibus prode&longs;se alicui
vehementer cupio, reliquis meis conatibus opem ferat. Sed
ad definitiones accedamus.
DEFINITIONES.
I.
Figuræ aliquæ planæ multilateræ centrum ha
bere dicuntur punctum illud, in quo omnes rectæ
lineæ vel angulos oppo&longs;itos iungentes bifariam
&longs;ecantur, vel ab angulis ductæ ad laterum op
po&longs;itorum bipartitas &longs;ectiones in ea&longs;dem ra
tiones.
II.
Circa diametrum e&longs;t figura plana, in qua re
cta quædam, quæ diameter figuræ dicitur, omnes
rectas alicui parallelas, à figura terminatas bi
fariam diuidit.
III.
Octaedrum communiter dictum, e&longs;t figura &longs;oli
da octo triangulis binis parallelis, æqualibus, &
&longs;imilibus comprehen&longs;a.
IIII.
Polyedri regularis centrum dicitur punctum,
in quo omnes rectæ lineæ, quæ ad angulos oppo
&longs;itos pertinent bifariam diuiduntur.
V.
Cuiu&longs;libet figuræ grauis centrum grauitatis
e&longs;t punctum illud, à quo &longs;u&longs;pen&longs;um graue per&longs;e
manet partibus quomodocumque circa con&longs;ti
tutis.
VI.
Axis pri&longs;matis, & pyramidis & eius fru&longs;ti di
citur recta linea, quæ in pyramide à vertice ad
ba&longs;is centrum figuræ vel grauitatis pertinet: in
reliquis autem, quæ ba&longs;ium oppo&longs;itarum figuræ
vel grauitatis centra iungit.
VII.
Si qua figura &longs;olida planis parallelis ita &longs;eca
ri po&longs;&longs;it, vt quæcumque &longs;ectiones centrum ha
beant, & &longs;int inter &longs;e &longs;imiles; aliqua autem recta
linea, &longs;iue ad centra ba&longs;ium oppo&longs;itarum prædi
ctis &longs;ectionibus parallelarum, & &longs;imilium, vt in
cylindro; &longs;iue ad verticem, & centrum ba&longs;is ter
minata, vt in cono, hemi&longs;phærio, & conoide, tran
&longs;eat per centra omnium prædictarum &longs;ectionum;
ea talis figuræ axis nominetur: ip&longs;a autem figura,
&longs;olidum circa axim. Quæ &longs;i vel vnam tantum ha
beat ba&longs;im, vel duas inæquales, & parallelas: dua
rum autem quarumlibet prædictarum &longs;ectionum
vertici, vel minori ba&longs;i propinquior &longs;it minor re-
ficiens nominetur: quo nomine &longs;ignificari etiam
volumus ea &longs;olida, quorum quælibet &longs;ectiones
ba&longs;i parallelæ quamuis ba&longs;i non &longs;int omnino &longs;imi
les, tamen ijs figuris deficiunt, quæ &longs;unt &longs;imiles
ha&longs;i, ac totis ijs, à quibus ip&longs;æ ablatæ intelli
guntur, ita vt tota figura & ablata habeant com
mune centrum in vna recta linea ad centrum ba
&longs;is terminata, quæ & ip&longs;a talis &longs;olidi axis nomi
netur.
Vt in figura, &longs;olidi ABDC deficientis &longs;olido CED
ba&longs;is e&longs;t circulus AB, terminus ba&longs;i oppo&longs;itus circum
ferentia circuli CMD. axis communis omnibus EF,
per cuius quodlibet punctum I plano ba&longs;i AB paralle
lo &longs;ecante &longs;olidum ABDC, & ablatum CED, & re
&longs;iduum, e&longs;t totius
&longs;ectio circulus G
H, ablati vero cir
culus KL, & re&longs;i
dui &longs;ectio reliquum
circuli GH dem
pto circulo KL.
quarum &longs;ectionum
omnium centrum
commune e&longs;t I. Quod &longs;i &longs;uper duos
circulos GH, KL circa axem communem EI cylin
dri de&longs;cribantur, (erunt autem eiu&longs;dem altitudinis) erit
reliquum cylindri GB, dempto cylindro cuius ba&longs;is
KL, axis EI, con&longs;titutum &longs;uper ba&longs;im G,
axim EI, quæ &longs;uo loco expectatur cogitatio.
POSTVLATA.
I.
Omnis figuræ grauis vnum e&longs;&longs;e centrum gra
uitatis.
II.
Omnium figurarum &longs;ibi mutuo congruentium
centra grauitatis mutuo &longs;ibi congruere.
III.
Omnis figuræ, cuius termini omnis cauitas
e&longs;t interior, intra terminum e&longs;&longs;e centrum graui
tatis.
IIII.
Similium triangulorum &longs;imiliter po&longs;ita e&longs;se
centra grauitatis. In triangulis autem &longs;imilibus
&longs;imiliter po&longs;ita puncta e&longs;&longs;e dicuntur, à quibus re
ctæ ad angulos æquales ductæ cum lateribus ho
mologis angulos æquales faciunt.
V.
Æqualia grauia ab æqualibus longitudinibus
&longs;ecundum centrum grauitatis &longs;u&longs;pen&longs;a æquipon
derare.
VI.
A quibus longitudinibus duo grauia æquipon
derant, ab ij&longs;dem alia duo quælibet illis æqualia
æquiponderare.
PROPOSITIO
PRIMA.
Si &longs;int quotcumque magnitu
dines inæquales deinceps
proportionales; exce&longs;&longs;us, qui
bus differunt deinceps pro
portionales erunt, in propor
tione totarum magnitudi
num.
Sint quotcumque inæquales magnitudines deinceps
proportionales AB, CD, EF, & G,
differentes exce&longs;&longs;ibus BH, DK, FL, mi
nima autem &longs;it G. Dico BH, DK, FL,
deinceps proportionales e&longs;se in proportio
ne, quæ e&longs;t AB, ad CD, &longs;eu CD, ad
EF. Quoniam enim e&longs;t vt AB, ad
CD, ita CD ad EF; hoc e&longs;t vt AB, ad
AH, ita CD, ad CK, permutando
erit, vt AB, ad CD, ita AH, ad CK:
vt igitur tota AB, ad totam CD, ita
reliqua BH, ad reliquam DK. Simili
ter o&longs;tenderemus e&longs;se vt CD ad EF,
ita DK ad FL; vt igitur BH ad DK,
ita erit DK ad FL, in proportione, quæ
e&longs;t AB ad CD, & CD ad EF. Quod demon&longs;tran
dum erat.
In omni triangulo vnum dumtaxat punctum
e&longs;t, in quo rectæ ab angulis ad latera incidentes
&longs;ecant &longs;e&longs;e in ea&longs;dem rationes. & &longs;egmenta, quæ
ad angulos, &longs;unt reliquorum dupla. & prædictæ
incidentes &longs;ecant trianguli latera bifariam.
Sit triangulum ABC, cuius duo quælibet latera AB,
AC, &longs;int bifariam &longs;ecta in punctis D, E, & ductæ rectæ
lineæ BE, CFD, AFG. Dico CF duplam e&longs;&longs;e ip&longs;ius
FD, & AF, ip&longs;ius FG, & BF, ip&longs;ius FE. Et in nullo alio
puncto à puncto F tres rectas ab angulis ad latera inciden
tes &longs;ecare &longs;e &longs;e in ea&longs;dem rationes. Et reliquum latus BC
&longs;ectum e&longs;&longs;e bifariam in puncto G. Quoniam enim e&longs;t vt BA
ad AD, ita CA ad AE: hoc e&longs;t, vt triangulum ABC ad
triangulum ADC, ita triangulum idem ABC ad trian
gulum AEB; æqualia
erunt triangula ADC,
AEB, & ablato trape
zio DE communi re
liquum triangulum BD
F reliquo triangulo C
EF æquale erit: &longs;ed
triangulum ADF e&longs;t
æquale triangulo BDF;
& triangulum AFE
triangulo EFC, pro
pter æquales ba&longs;es, &
communes altitudines; totum igitur triangulum AFB
toti AFC, triangulo æquale erit: &longs;ed vt triangulum AFB
triangulum AFC ad triangulum FCG; triangulum er
go FBG triangulo FCG æquale erit, & ba&longs;is BG ba
&longs;i GC æqualis. Quoniam igitur & AE e&longs;t æqualis
EC, &longs;imiliter vt ante, o&longs;tenderemus, triangulum BCF,
triangulo ACF, eademque ratione triangulum ABF,
triangulo BCF æquale e&longs;&longs;e: igitur vnumquodque trian
gulorum ABF, ACF, BCF, tertia pars e&longs;t trianguli
ABC: &longs;ed vt triangulum ABC, ad triangulum BCF,
ita e&longs;t AG, ad GF; tripla igitur e&longs;t AG ip&longs;ius GF,
ac proinde AF, ip&longs;ius FG dupla. Eadem ratione
BE, ip&longs;ius FE, & CF, ip&longs;ius FD, dupla concludetur.
Sed &longs;int &longs;i fieri pote&longs;t, trianguli ABC duo centra qua
lia diximus D, E: & ab ip&longs;is ad &longs;ingulos angulos du
cantur binæ rectæ lineæ:
& eadat D in aliquo trian
gulo BEC. Quoniam
igitur D e&longs;t centrum trian
guli ABC erit triangu
lum BDC tertia pars
trianguli ABC. Eadem
ratione triangulum BEC
tertia pars erit trianguli
ABC; triangulum ergo
DBC æquale erit trian
gulo BEC pars toti, quod
fieri non pote&longs;t, atqui
ab&longs;urdum &longs;equitur, &longs;i punctum D cadat in aliquo latere
triangulorum, quorum vertex E; Manife&longs;tum e&longs;t igitur
propo&longs;itum.
In &longs;imilibus triangulis rectæ lineæ, quæ inter
centra, & alia in ijs &longs;imiliter po&longs;ita puncta in
terijciuntur, proportionales &longs;unt in proportione
laterum homologorum.
Sint triangula &longs;imilia, & &longs;imiliter po&longs;ita ABC, DEF,
quorum &longs;int centra O, P, in ijs autem triangulis &longs;int pun
cta &longs;imiliter po&longs;ita K, L, quæ cadant primum in rectis
BG, EH, quæ ab angulis æqualibus B, E, ba&longs;es bifa
riam diuidunt. Dico e&longs;&longs;e OK ad PL, vt e&longs;t latus AB,
ad latus DE. iunctis enim AK, KC, DL, LF, quo
niam angulus KAC, æqualis e&longs;t angulo LDF, & angu
lus KCA, angulo LFD, ob &longs;imiliter po&longs;ita puncta K,
L, triangulum AKC, triangulo LDF &longs;imile erit, & vt
KA ad AC, ita LD ad DF: &longs;ed vt CA ad AG, ita
e&longs;t FD ad DH, expræcedenti; vt igitur KA, ad AG
ita erit LD, ad DH, circa æquales angulos: &longs;imilia igi
tur &longs;unt triangula AGK, DHL, & angulus AGK,
HD: &longs;ed vt GA, ad AC, ita e&longs;t HD ad DF: & vt
AC ad AB, ita DF ad DE, ex æquali igitur erit vt
KG ad AB, ita LH ad DE: &longs;ed vt AB ad BG, ita
e&longs;t DE ad EH, propter &longs;imilitudinem triangulorum
ABG, DEH: & vt BG ad GO ita e&longs;t EH ad HP,
propter triangulorum centra O, P; ex æquali igitur erit
vt KG ad GO, ita LH ad HP: & permutando vt
OG ad PH, ide&longs;t vt BG ad EH, ide&longs;t vt AB ad ED,
ita KG ad LH, & reliqua OK ad reliquam PL.
Sed &longs;int puncta &longs;imiliter po&longs;ita M, N, quæ cadant ex
tra lineas BG, EH, iunctæque OM, PN. Dico iti
dem e&longs;se vt AB ad ED, ita OM ad PN. Iungantur
enim rectæ MB, NE, quæ cum quibus lateribus homo
logis angulos æquales faciunt, ea &longs;int AB, DE, quod
propter i&longs;o&longs;celia triangula &longs;it dictum in &longs;imiliter po&longs;itis
triangulis. igitur etiam angulus BAM, æqualis erit an
gulo EDN; &longs;imilia igitur triangula ABM, DEN: &
vt MB ad BA, ita erit NE ad ED: &longs;ed vt AB ad
BG, ita e&longs;t DE ad EH, propter &longs;imilitudinem trian
gulorum, & vt BG ad BO, ita e&longs;t EH ad EP, ob
triangulorum &longs;imilium centra O, P: ex æquali igitur
erit vt MB, ad BO, ita NE ad EP. Rur&longs;us quo
niam angulus ABM, æqualis e&longs;t angulo DEN, quorum
angulus ABG, æqualis e&longs;t angulo DEH: erit reliquus
angulus OBM, æqualis reliquo angulo PEN: &longs;ed vt MB
ad BO, ita erat NE ad EP; triangulum igitur OBM
triangulo PEN, &longs;imile erit, & vt BO ad EP, hoc e&longs;t
BG ad EH, hoc e&longs;t AB ad DE, ita OM ad PN. Quod demon&longs;trandum erat.
Datis duobus triangulis &longs;calenis &longs;imilibus, &
dato puncto in altero eorum, vnum duntaxat pun
ctum in reliquo triangulo prædicto puncto &longs;imi
liter po&longs;itum pote&longs;t inueniri.
Sint data duo triangula &longs;calena &longs;imilia ABC, DEF,
& in triangulio ABC datum punctum G: &longs;int autem
hæc triangula &longs;imiliter po&longs;ita. Dico in triangulo DEF,
vnum duntaxat punctum puncto G &longs;imiliter po&longs;itum in
ueniri po&longs;se. Iunctis enim AG, BG, GC, ponatur
angulus EDH, æqualis angulo BAG, & angulus DEH,
æqualis angulo ABG, & HF iungatur. Manife&longs;tum
e&longs;t igitur ex præcedentis Theorematis demon&longs;tratione,
triangula EDH, HDF, FEH, &longs;imilia e&longs;se triangulis
BAG, GAC, CBG, prout inter &longs;e re&longs;pondent po&longs;i
tione, quorum &longs;ex triangulorum binis quibu&longs;que binæ ba
&longs;es homologæ re&longs;pondent: AB ED, AC DF, BC
ABC, DEF. Ex definitione igitur, duo puncta G, H,
in triangulis ABC, DEF, &longs;imiliter po&longs;ita erunt. At
enim &longs;i fieri pote&longs;t &longs;it aliud punctum K, in triangulo
DEF, &longs;imiliter po&longs;itum puncto G. Vel igitur punctum
K in aliquo triangulorum, quorum e&longs;t communis vertex
H, vel in aliquo eorundem latere cadet. cadat in latere
FH, & iungatur DK: triangulum ergo DFK, &longs;imile
erit triangulo ACG. Sed & triangulum EDF, &longs;imile
e&longs;t triangulo BAC; vtraque igitur horum ad illorum &longs;i
bi re&longs;pondens triangulorum duplicatam eorundem late
rum homologorum AC, DF, habebunt proportionem:
vt igitur e&longs;t triangulum EDF, ad triangulum BAC, ita
erit triangulum DFK, ad triangulum ACG: & per
mutando, vt triangulum ACG, ad triangulum ABC,
ita triangulum DFK, ad triangulum EDF: eadem ra
tione, vt triangulum ACG, ad triangulum ABC, ita
erit triangulum DFH, ad triangulum DEF: vt igitur
triangulum DFK, ad triangulum EDF; ita erit trian
gulum DFH, ad triangulum EDF; triangulum ergo
DFK, triangulo DFH, æquale erit, pars toti, quod e&longs;t
ab&longs;urdum: idem autem ab&longs;urdum &longs;equeretur, &longs;i punctum
triangulo DFH; Non igitur aliud punctum à puncto H,
in triangulo EDF, &longs;imiliter po&longs;itum erit puncto G. Quod demon&longs;trandum erat.
Cuilibet figuræ planæ rectangulum æquale
pote&longs;t e&longs;&longs;e.
Sit quælibet figura plana A.
Dico figuræ A, rectan
gulum æquale po&longs;se exi&longs;tere. Exponatur enim rectan
gulum BC, cuius latus BD, in infinitum producatur
ver&longs;us E. Quoniam igitur e&longs;t vt rectangulum BD, ad
planam figuram A, ita recta BD, ad aliquam lineam
rectam &longs;it vt BC, ad A, ita BD, ad DE, & comple
atur rectan
gulum EC. Quoniam igi
tur e&longs;t vt BD
ad DE, ita
rectangulum
BC, ad figu
ram A: &longs;ed
vt BD, ad
DE, ita e&longs;t
rectangulum BC, ad rectangulum CE; vt igitur re
ctangulum BC, ad figuram A, ita e&longs;t rectangulum
BC, ad rectangulum CE; rectangulum ergo CE, fi
guræ A, æquale erit. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omni figuræ circa diametrum in alte ram par
tem deficienti figura quædam ex parallelogram
mis æqualium altitudinum in&longs;cribi pote&longs;t, & al
tera circum&longs;cribi, ita vt circum&longs;cripta &longs;uperet in
&longs;criptam minori &longs;pacio quantacumque magnitu
dine propo&longs;ita. Semper autem in &longs;imilibus intelli
ge, eiu&longs;dem generis.
Sit figura plana ABC circa diametrum AD, ad par-
Dico fieri po&longs;se quod
proponitur: ducta enim per verticem figuræ A, ba&longs;i BC,
parallela, atque ideo figuram ip&longs;am contingente, ab&longs;ol
uatur parallelogrammum BL, &longs;ectaque diametro AD,
bifariam, & &longs;ingulis eius partibus &longs;emper bifariam, du
cantur per puncta &longs;ectionum rectæ lineæ ba&longs;i BC, & in
ter &longs;e parallelæ, atque ita multiplicatæ &longs;int &longs;ectiones,
vt &longs;ecti parallelogrammi in parallelogramma æqua
lia, & eiu&longs;dem altitudinis quælibet pars, vt paralle
logrammum BF, &longs;it minus &longs;uperficie propo&longs;ita, cu
ius parallelogram
mi latus EF, &longs;e
cet figuræ termi
num BAC, in
punctis GH, &
diametrum AD, in
puncto K. erit igi
tur GK, æqualis
KH: per omnia
igitur puncta &longs;e
ctionum termini
BAC, quæ à prædictis fiunt lineis parallelis, &longs;i ducan
tur diametro AD parallelæ, figura quædam ip&longs;i ABC,
in&longs;cribetur, & altera circum&longs;cribetur ex parallelogram
mis æqualium altitudinum. Dico harum figurarum
in&longs;criptam &longs;uperari à circum&longs;cripta minori &longs;pacio &longs;uper
ficie propo&longs;ita. Quoniam enim omnia parallelogramma,
quibus figura circum&longs;cripta &longs;uperat in&longs;criptam &longs;imul &longs;um
pta &longs;unt æqualia BF parallelogrammo: &longs;ed parallelo
grammum BF, e&longs;t minus &longs;uperficie propo&longs;ita: exce&longs;&longs;us
igitur quo figura circum&longs;cripta in&longs;criptam &longs;uperat, minor
erit &longs;uperficie propo&longs;ita. Fieri igitur pote&longs;t, quod propo
nebatur.
Pyramides &longs;imilibus, & æqualibus triangulis
comprehen&longs;æ inter &longs;e &longs;unt æquales.
Sint pyramides ABCD, EFGH, &longs;imilibus, & æqua
libus triangulis comprehen&longs;æ, & &longs;i &longs;int &longs;imiliter po&longs;itæ, qua
rum vertices A, E, ba&longs;es autem triangula BCD, FGH. Dico pyramidem ABCD, pyramidi EFGH, æqualem
e&longs;se. A punctis enim A, E, manantia latera inferius pro
ducantur, & prædictis lateribus maiores, inter &longs;e autem
æquales ab&longs;cindantur AK, AL, AM, EN, EO, EP,
& con&longs;truantur pyramides AKLM, ENOP: pyramides
igitur hæ æqualibus, & &longs;imilibus triangulis comprehenden
tur, vt colligitur ex ip&longs;a con&longs;tructione; triangulis igitur inter
&longs;e æquilateris, & æquiangulis KLM, NOP, inter &longs;e con
gruentibus non congruat, &longs;i fieri pote&longs;t, pyramis ENOP,
pyramidi AKLM, &longs;ed cadat vertex E, pyramidis ENOP,
extra verticem A, pyramidis AKLM, & ex puncto A,
&longs;it R, ducatur recta AR, & ER iungatur. Quoniam igi
tur æquales rectæ &longs;unt AK, AL, AM, quæ ex puncto
A, in &longs;ublimi pertinent ad &longs;ubiectum planum: & punctum
R, e&longs;t centrum circuli tran&longs;euntis per puncta N, O, P; cadet
recta AR ad &longs;ubiectum planum perpendicularis. Eadem
ratione recta ER ducta à vertice E, pyramidis ENOP,
ad centrum R, circuli tran&longs;euntis per puncta N, O, P, hoc
e&longs;t, per puncta K, L, M, illis congruentia, cadet ad idem
planum, ad quod linea AR, perpendicularis; ita que ab
eodem puncto R, ad idem planum, & ad ea&longs;dem partes duæ
perpendiculares erunt excitatæ, quod fieri non pote&longs;t:
punctum igitur E non cadet extra punctum A: quare la
tus EN, congruet lateri AK, quorum EF, e&longs;t æqualis
AK; igitur & EF, ip&longs;i AB, congruet. eadem ratione la
tus AG, congruet lateri AC, & latus EH, lateri AD, &
triangula triangulis, & pyramis EFGH, pyramidi ABC
D, & ip&longs;i æqualis erit. Quod demon&longs;trandum erat.
Hinc facile colligitur omnia &longs;olida, quæ in py
ramides æqualibus, & &longs;imilibus triangulis com
prehen&longs;as multitudine æquales diuidi po&longs;&longs;unt, e&longs;
&longs;e inter &longs;e æqualia. Quocirca omnia pri&longs;mata, &
pyramides, & octahedra, omnia denique corpora
regularia æqualibus, & &longs;imilibus planis compre
hen&longs;a inter &longs;e æqualia erunt.
Omnis pyramidis triangulam ba&longs;im habentis
quatuor axes &longs;ecant &longs;e in vno puncto in ea&longs;dem ra
rum, quæ ad oppo&longs;ita triangula, &longs;int tripla; ex quo
puncto tota pyramis diuiditur in quatuor pyrami
des æquales. Et in nullo alio puncto quatuor re
ctæ lineæ ductæ ab angulis ad triangula oppo&longs;ita
pyramidis &longs;ecant &longs;e&longs;e in ea&longs;dem rationes. Vocetur
autem punctum hoc centrum dictæ pyramidis.
Sit pyramis ABCD, cuius vertex A, ba&longs;is autem
triangulum BCD, axes AE, BM, CL, DN, vnde qua
tuor triangulorum, quæ &longs;unt circa pyramidem ABCD,
centra erunt grauitatis E, L, M, N. Dico quatuor li
neas AE, BM, CL, DN, &longs;ecare &longs;e &longs;e in vno puncto in
ea&longs;dem rationes, quas prædixi, & quæ &longs;equuntur. Nam ex
puncto A, ducatur recta ALH, quæ ob trianguli ABD,
centrum L, &longs;ecabit latus BD, bifariam in puncto H; iun
cta igitur CE, & producta conueniet cum ALH, vt in
puncto H. eadem ratione iunctæ AM, BE, & productæ
conuenient in medio lateris CD, conueniant in puncto K,
necnon AN, DE, in medio ip&longs;ius BC, vt in puncto G. Quoniam igitur ob triangulorum centra, e&longs;t vt CE ad EH,
ita AL ad LH, dupla enim e&longs;t vtraque vtriu&longs;que, &longs;eca
bunt &longs;e&longs;e rectæ AE, CL, inter ea&longs;dem parallelas; quare
vt AF ad FE, ita erit CF ad FL, circum æquales angu
los ad verticem: triangula igitur AFL, CFE; & reci
proca, & æqualia inter &longs;e erunt. Cum igitur &longs;it vt AL ad
LH, ita CE ad EH, hoc e&longs;t vt triangulum AFL ad
triangulum FLH, (&longs;i ducatur FH) ita triangulum CFE,
ad triangulum FEH, erunt inter &longs;e æqualia triangula
FEH, FLH. Quare vt triangulum AFH, ad triangu
lum FLH, hoc e&longs;t vt AH ad HL, ita erit triangulum
AFH ad triangulum FEH, hoc e&longs;t AF ad FE: &longs;ed re
cta AH, e&longs;t tripla ip&longs;ius LH; igitur & AF, erit ip&longs;ius FE,
tur erit CF, ip&longs;ius FL. Similiter o&longs;tenderemus rectas
AE, BM, &longs;ecare &longs;e &longs;e in ea&longs;dem rationes, ita vt &longs;egmen
ta, quæ ad angulos, &longs;int tripla eorum, quæ &longs;unt ad centra
E, M, quorum AF, e&longs;t tripla ip&longs;ius FE: in puncto igitur
F, &longs;ecant &longs;e rectæ lineæ AE, BM. Eadem ratione & re
ctæ AE, DN, &longs;ecent &longs;e in puncto F, nece&longs;se erit: quare
vt AF ad FE, ita erit DF ad FN. Quatuor igitur
axes pyramidis ABCD, &longs;ecant&longs;e &longs;e in puncto F, in ea&longs;
dem rationes, ita vt
&longs;egmenta ad angulos,
&longs;int Rur&longs;us, quia compo
nendo, & conuerten
do, e&longs;t vt FE ad EA,
ita FL ad LC: hoc
e&longs;t, vt pyramis BCD
F, ad pyramidem A
BCD, ita pyramis
ABDF, ad pyrami
dem CBDA, (pro
pter ba&longs;ium commu
nitatem, & vertices in
eadem recta linea) erit
pyramis ABDF, æqualis pyramidi BCDF. Eadem ra
tione tam pyramis ACDF, quàm pyramis ABCF, æqua
lis e&longs;t pyramidi BCDF. Quatuor igitur pyramides, qua
rum communis vertex punctum F, ba&longs;es autem triangula,
quæ &longs;unt circa pyramidem ABCD, inter &longs;e æquales
& vnaquæque pyramidis ABCD, pars quarta. Dico in
nullo alio puncto à puncto F, quatuor rectas, quæ ab an
gulis ad triangula oppo&longs;ita pyramidis ABCD, ducantur,
&longs;ecare &longs;e in ea&longs;dem rationes. Si enim fieri pote&longs;t &longs;ecent
&longs;e tales rectæ in ea&longs;dem rationes in alio puncto S. Simi
tuor pyramidum, quarum communis vertex S, ba&longs;es au
tem triangula, quæ &longs;unt circa pyramidem ABCD, e&longs;se
quartam partem pyramidis ABCD. Siue igitur pun
ctum S, cadat intra vnam priorum quatuor pyrami
dum, &longs;iue in earum aliquo latere, &longs;eu triangulo; nece&longs;
&longs;ario erit pars æquali toti; tam enim tota vna pyramis
quatuor priorum, quarum communis vertex F, quàm eius
pars, vna quatuor pyramidum po&longs;teriorum, quarum com
munis vertex S, erit eiu&longs;dem ABCD, pyramidis pars
quarta. Ex ab&longs;urdo igitur non in alio puncto à puncto F
&longs;ecabunt &longs;e in ea&longs;dem rationes quatuor rectæ, quæ ab angu
lis ad oppo&longs;ita triangula pyramidis ABCD, ducantur. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis pyramis ba&longs;im habens triangulam di
uiditur in quatuor pyra mides æquales, & &longs;imiles
inter &longs;e, & toti, & vnum octaedrum totius pyrami
dis dimidium, ip &longs;i que concentricum.
Sit pyramis ABCD, cuius ba&longs;is triangulum ABC,
&longs;ectisque omnibus lateribus bifariam, iungantur rectæ FG,
GH, HF, FK, KL, LM, MDico quatuor pyramides DKLM, LFBG, KHFA,
MHGC, æquales e&longs;se, & &longs;imiles inter &longs;e, & toti pyrami
di ABCD: octaedrum autem e&longs;se LFGM
midium pyramidis ABCD, ip&longs;ique concentricum. Du
cantur enim rectæ DNH, BQH, LN: & po&longs;ita BE, du
pla ip&longs;ius BH, iungatur DOC, in triangulo DBH, &
ponatur DP, ip&longs;ius PE, tripla, & connectantur rectæ LP,
PH. Quoniam igitur E, e&longs;t centrum trianguli ABC,
DP e&longs;t triplum ip&longs;ius PE: igitur P centrum erit pyra
midis ABCD. Et quoniam tres rectæ FK, KH, HF,
&longs;unt parallelæ tribus BD, DC, CB, pro vt inter &longs;e re&longs;pon
dent, vt KH, ip&longs;i LG, quoniam vtraque lateri DC, ob
latera triangulorum &longs;ecta proportionaliter in punctis K, H,
L, G: & &longs;ic de reliquis; erit pyramis A
pyramidi ABCD. Similiter vnaquæque trium aliarum
pyramidum ab&longs;ci&longs;&longs;arum, videlicet FLBG, GHMC,
KDLM, &longs;imilis erit pyramidi ABCD, atque ideo in
ter &longs;e &longs;imiles. Rur&longs;us,
quoniam pyramidum
&longs;imilium latus AD e&longs;t
duplum lateris AK, ho
mologi; pyramis AB
CD, octupla erit py
ramidis AKFH, ob
triplicatam laterum ho
mologorum proportio
nem. Similiter
qunæque
rum pyramidum ab&longs;ci&longs;
&longs;arum erit octaua pars
pyramidis ABCD;
quatuor igitur pyramides ab&longs;ci&longs;&longs;æ &longs;imul &longs;umptæ dimi
dium erit pyramidis ABCD: & reliquum igitur &longs;oli
dum demptis quatuor pyramidibus, dimidium pyramidis
ABCD. Dico reliquum &longs;olidum LKMGFH, e&longs;&longs;e
octaedrum. Nam octo triangulis ip&longs;um contineri mani
fe&longs;tum e&longs;t. bina autem oppo&longs;ita e&longs;&longs;e parallela, & æqualia,
& &longs;imilia, &longs;ic o&longs;tendimus. Quoniam enim triangulum
FGH, e&longs;t in plano trianguli ABC, plano trianguli KLM
parallelo; erit triangulum FGH, parallelum triangu-
ABC, & triangulum KLM, &longs;imile eidem triangulo
ABC;
&longs;ed & æquale propter æqualitatem laterum homologo
rum. Similiter o&longs;tenderemus reliquum &longs;olidum LKM
GFH continentia triangula bina oppo&longs;ita æqualia
inter &longs;e, & &longs;imilia, & parallela; octaedrum e&longs;t igitur
LKMGFH. Dico iam punctum P, quod e&longs;t cen
trum pyramidis ABCD, e&longs;se centrum octaedri L
MGFH. Quoniam enim DP, ponitur tripla ip&longs;ius PE,
& DO, e&longs;t æqualis
OE (&longs;iquidem planum
trianguli KLM, plano
lum &longs;ecat proportione
ex puncto D, in &longs;ubli
mi pertinent ad &longs;ubie
ctum planum trianguli
ABC) erit OP, ip&longs;i
PE, æqualis. Et quo
niam BH e&longs;t dupla
ip&longs;ius QH, quarum
BE e&longs;t dupla ip&longs;ius
EH, &longs;iquidem E e&longs;t centrum trianguli ABC; erit reli
qua EH reliquæ EQ dupla: & quia e&longs;t vt LD ad DB,
ita LN ad BH, propter &longs;imilitudinem triangulorum, &
e&longs;t LD, dimidia ip&longs;ius BD, erit & LN, dimidia ip&longs;ius
BH: &longs;ed QH e&longs;t dimidia ip&longs;ius BH; æqualis igitur LN
ip&longs;i QH. Iam igitur quia e&longs;t vt BE ad EH, ita
LO ad ON: &longs;ed BE, e&longs;t dupla ip&longs;ius EH; dupla igi
tur LO, erit ip&longs;ius ON: &longs;ed & QH erat dupla ip&longs;ius
QE; vt igitur LN ad NO, ita erit HQ ad QE: &
HE: & permutando, vt LN ad QH, ita LO ad EH:
&longs;ed LN, o&longs;ten&longs;a e&longs;t æqualis QH; æqualis igitur LO,
erit ip&longs;i EH; &longs;ed & OP, e&longs;t æqualis ip&longs;i PE, vt o&longs;ten
dimus: duæ igitur LO, OP, duabus HE, EP æqua
les erunt altera alteri, & angulos æquales continent LOP,
PEH, parallelis exi&longs;tentibus LN, BH &longs;ectionibus tri
anguli DBH, quæ fiunt à duobus planis parallelis; ba
&longs;is igitur LP, trianguli LOP, æqualis e&longs;t ba&longs;i PH,
trianguli PEH, & angulus OPL, angulo EPH in pla
no trianguli DBH, in quo DPE, e&longs;t vna recta linea;
igitur LPH, erit vna recta linea, quæ cum &longs;it axis octa
edri LKMGFH, & &longs;ectus &longs;it in puncto P, bifariam,
erit punctum P, centrum octaedri LKMGEH. &longs;ed &
centrum pyramidis ABCD. Manife&longs;tum e&longs;t igitur pro
po&longs;itum.
Omne fru&longs;tum pyramidis triangulam ba&longs;im
habentis, &longs;iue coni, ad pyramidem, vel conum, cu
ius ba&longs;is e&longs;t eadem, quæ maior ba&longs;is fru&longs;ti, & ea
dem altitudo, eam habet proportionem, quam duo
latera homologa, vel duæ diametri ba&longs;ium ip&longs;ius
fru&longs;ti, vnà cum tertia minori proportionali ad
prædicta duo latera, vel diametros; ad maioris ba
&longs;is latus, vel diametrum. Ad pri&longs;ma autem, vel
cylindrum, cuius eadem e&longs;t ba&longs;is, quæ maior ba&longs;is
fru&longs;ti, & eadem altitudo; vt tres prædictæ deìn
ceps proportionales &longs;imul, ad triplam lateris, vel
diametri maioris ba&longs;is.
Sit fru&longs;tum ABCFGH, pyramidis, vel coni ABCD,
cuius ba&longs;is triangulum, vel circulus ABC, axis autem
DE: & vt e&longs;t AC ad FH, ita &longs;it FH ad N, & fru
&longs;ti axis EK, nec non idem pyramidis, vel coni AB
CK, vt &longs;it eadem altitudo. Dico fru&longs;tum ABCF
GH, ad pyramidem, vel conum, ABCK, e&longs;se vt
tres lineas AC, FH, NO, &longs;imul ad ip&longs;ius AC, tri
plam: ad pri&longs;ma autem, vel cylindrum, cuius ba&longs;is ABC,
altitudo autem eadem cum fru&longs;to, vttres AC, FH, NO,
&longs;imul, ad ip&longs;ius AC, triplam. Nam vt e&longs;t AC ad FH,
& FH ad NO, ita &longs;it NO ad P: & exce&longs;&longs;us, quo hæ
quatuor lineæ differunt, &longs;int AL, FM,
vt AC ad FH, ita erit AL ad FM, & FM ad
Quoniam igitur e&longs;t vt AC ad P, ita pyramis, vel conus
ABCD, ad &longs;imilem ip&longs;i pyramidem, vel conum DFGH,
ob triplicatam laterum homologorum proportionem; erit
diuidendo, vt tres AL, FM, OQ, &longs;imul ad P, ita fru
&longs;tum ABCFGH, ad pyramidem, vel conum DFGH:
&longs;ed conuertendo e&longs;t vt P, ad AC, ita pyramis, vel conus
DFGH, ad pyramidem, vel conum ABCD: ex æquali
igitur, vt tres AL, FM, OQ, &longs;imul ad AC, ita fru&longs;tum Rur&longs;us quoniam axis DE, & latera pyramidis, vel coni
ABCD, &longs;ecantur plano trianguli, vel circuli FGH, ba&longs;i
ABC, parallelo; erit componendo, vt AD, ad DF, hoc
e&longs;t, vt AC ad FH, propter &longs;imilitudinem triangulorum,
hoc e&longs;t vt AC, ad CL, ita ED, ad DK; & per conuer
&longs;ionem rationis, vt AC, ad AL, ita DE, ad EK: &longs;ed vt
DE ad EK, ita e&longs;t pyramis, vel conus ABCD, ad py
ramidem, vel conum ABCK; vt igitur AC, ad AL,
ita e&longs;t pyramis, vel conus ABCD, ad pyramidem, vel
conum ABCK; &longs;ed vt tres lineæ AL, FM, OQ &longs;imul
ad AC, ita erat fru&longs;tum ABCFGH, ad pyramidem,
vel conum ABCD; ex æquali igitur, erit vt tres lineæ
AL, FM, OQ, &longs;imul ad AL, ita fru&longs;tum ABCFGH,
ad pyramidem, vel conum ABCK. Rur&longs;us, quoniam
tres exce&longs;&longs;us AL, FM, OQ, &longs;unt deinceps proportio
nales in proportione totidem terminorum AC, FH, NO,
erunt vt AL, FM, OQ, &longs;imul ad AL, ita AC, FH,
NO, &longs;imul ad AC: &longs;ed vt AL, FM, OQ, &longs;im ul ad
AL, ita erat fru&longs;tum ABCFGH, ad pyamidem, vel
conum ABCK; vt igitur tres lineæ AC, FH, NO, &longs;i
mul, ad AC, ita erit fru&longs;tum ABCFGH, ad pyrami
dem, vel conum ABCK. Sed vt AC, ad &longs;ui triplam, ita
e&longs;t pyramis, vel conus ABCK ad pri&longs;ma, vel cylindrum,
cuius e&longs;t eadem ba&longs;is ABC, & eadem altitudo cum py
ramide, vel cono ABCK; ex æquali igitur, erit vt tres
lineæ AC, FH, NO, &longs;imul ad ip&longs;ius AC, triplam, ita
fru&longs;tum ABCFGH, ad pri&longs;ma, vel cylindrum, cu
ius ba&longs;is ABC, & eadem altitudo pyramidi, vel cono
ABCK: ide&longs;t eadem, fru&longs;to ABCFGH. Manife&longs;tum
e&longs;t igitur propo&longs;itum.
Omni &longs;olido circa axim in alteram partem defi
cienti, cuius ba&longs;is &longs;it circulus, vel ellyp&longs;is, figura
quædam ex cylindris, vel cylindri portionibus
æqualium altitudinum in&longs;cribi poteft, & altera
circum&longs;cribi, ita vt circum&longs;cripta &longs;uperet in&longs;cri
ptam minori exce&longs;&longs;u quacumque magnitudine
propo&longs;ita.
Sit &longs;olidum ABC, circa axim AD, in alteram par
tem deficiens, cuius vertex A, ba&longs;is autem circulus, vel
ellyp&longs;is, cuius diameter BC. Igitur &longs;uper hanc ba&longs;im
circa axim AD,
intelligatur de&longs;eri
ptus cylindrus, vel
cylindri portio
BL, quæ &longs;olidum
ABC, compre
hendet: &longs;ectoque
cylindro, vel cylin
dri portione BL,
planis ba&longs;i paralle
lis in tot cylindros, vel cylindri portiones æqualium al
ritudinum, vt quilibet eorum &longs;it minor magnitudine
propo&longs;ita; e&longs;to &longs;olidum ABC, &longs;ectum prædictis planis:
erunt autem &longs;ectiones circuli, vel ellyp&longs;es fimiles inter
&longs;e & ba&longs;i BC, &longs;olidi ABC &longs;uper quas &longs;ectiones tam
quam ba&longs;es cylindris, vel cylindri portionibus æqua
lium altitudinum intra, atque extra figuram con&longs;titutis,
quorum bini inter eadem plana parallela inter &longs;e refe-
D
rum commune centrum K: &longs;upremus autem, qui ad A,
ad nullum refertur. Quoniam igitur ex con&longs;tructione,
cylindrus, vel cylindri portio BF, e&longs;t minor magnitudi
ne propo&longs;ita; exce&longs;sus autem omnes, quibus cylindri, ex
quibus con&longs;tat figura circum&longs;cripta, excedunt eos, ex qui
bus con&longs;tat figura in&longs;cripta, pro vt bini inter &longs;e referun
tur, vna cum &longs;upremo, qui ad nullum refertur, &longs;unt æqua
les cylindro, vel cylindri portioni BF, figura circum
&longs;cripta &longs;olido ABC, excedet in&longs;criptam minori exce&longs;
&longs;u magnitudine propo&longs;ita. Fieri igitur pote&longs;t quod pro
ponebamus.
Dato parallelepipedo erecto circa datam re
ctam lineam tamquam axim, erectum parallele
pipedum æquale con&longs;tituere.
Sit datum parallelepipedum AB, erectum, cuius ba
&longs;is AC, altitudo autem latus BC: & data recta linea
finita ED. Oportet circa rectam ED, tamquam axim
parallelepipedo AB, æquale parallelepipedum erectum
con&longs;tituere. Per punctum igitur E, extendatur pla
num erectum ad lineam ED, & vt e&longs;t DE, ad BC, ita
fiat ba&longs;is AC, ad quadratum F: & ad punctum E, in
plano erecto ad lineam ED, quartæ parti quadrati F,
æquale GE, quadratum de&longs;cribatur, & compleatur
quadratum GH, quadruplum quadrati EG, &longs;eu qua
drato F, æquale: & ex puncto K, erecta KL, ip&longs;i EF,
æquali, & ad &longs;ubiectum planum perpendiculari &longs;uper ba
&longs;im GH, con&longs;tituatur parallelepipedum GK. Dico
& rectam DE, axim parallelepipedi GK. Iungantur
enim ba&longs;ium oppo&longs;itarum diametri GH, LK. Quo
niam igitur qua
drata &longs;unt EG,
GH, communem
que habent angu
lum, qui ad G,
con&longs;i&longs;tent circa di
ametrum GH; in
recta igitur GH,
erit punctum E. Et quoniam qua
dratum GH, e&longs;t
quadrati EG, qua
druplum; erit dia
meter GH, diametri EG, dupla; punctum igitur E,
erit in medio diametri GH. Rur&longs;us, quoniam ob pa
rallelepipedum GK, recta GL, æqualis e&longs;t, & paral
lela ip&longs;i KH, erit LH, parallelogrammum: & quia
vtraque DE, KH, e&longs;t ad &longs;ubiectum planum perpendi
cularis, parallelæ erunt, & in eodem plano parallelogram
mi LH; in quo cum LG, &longs;it parallela ip&longs;i KH; erit &
ED, ip&longs;i LG, parallela: e&longs;t autem, & æqualis vtrilibet
ip&longs;arum GL, GH, oppo&longs;itarum; punctum igitur D, e&longs;t
in recta LK, & tam KD, ip&longs;i EH, quàm LD, ip&longs;i
EG, æqualis erit, & inter &longs;e æquales LD, DK. pun
ctum igitur D, erit in medio diametri LK; &longs;ed & pun
ctum E, erat in medio diametri GH; recta igitur ED,
axis e&longs;t parallelepipedi GK, cuius parallelepipedi cum
altitudo DE, &longs;it ad BC, altitudinem parallelepipedi AB,
vt e&longs;t ba&longs;is AC, ad quadratum F, hoc e&longs;t ad ba&longs;im GH,
parallelepipedi GK; parallelepipedum GK, parallelepipe
do AB, æquale erit, Factum igitur e&longs;t quod oportebat.
Cuilibet figuræ &longs;olidæ
le pote&longs;t e&longs;&longs;e.
Sit quælibet figura &longs;olida A.
Dico &longs;olido A, parallele
pipedum æquale po&longs;se exi&longs;tere. Exponatur enim paral
lelepipedum BC, cuius ba&longs;is BG. Quoniam igitur e&longs;t vt
&longs;olidum BC, ad &longs;olidum A, ita recta linea, &longs;iue latus BD,
ad aliam rectam lineam; producto latere BD, &longs;it vt BC,
ad A, ita recta BD, ad rectam DE, & compleatur pa
rallelepipedum CE. Quoniam ita que e&longs;t vt BD, ad DE,
ita parallelogrammum &longs;iue ba&longs;is BG, ad parallelogram
mum, &longs;iue ba&longs;im EG; hoc e&longs;t parallelepipedum BC, ad
parallelepipedum CE: &longs;ed vt BD, ad DE, ita e&longs;t paral
lelepipedum BC, ad &longs;olidum A; vt igitur parallelepipe
dum BC, ad &longs;olidum A, ita erit parallelepipedum BC,
ad parallelepipedum CE; parallelepipedum igitur CE
æquale erit &longs;olido A. Quod fieri po&longs;se propo&longs;uimus.
Omnis parallelogtammi centrum grauitatis
diametrum bifariam diuidit.
Sit parallelogrammum ABCD, cuius duo latera AB,
BC, &longs;int primum in æqualia: &
mum habet &longs;altem duos angulos
recto, e&longs;to vterque angulorum B, D, non minor recto, &longs;it
que ducta diameter AC, &longs;ectaque in puncto G, bifariam. Dico G, e&longs;se centrum grauitatis parallelogrammi ABCD.
Trianguli enim ABC, &longs;it centrum grauitatis H; iuncta
que HG, & producta, ponatur GK, æqualis GH, & re
ctæ à punctis K, H, ad angulos ducantur. Quoniam igi
tur AG, e&longs;t æqualis GC, &
GH, ip&longs;i GK, & angulus
AGK, æqualis angulo CGH,
erit ba&longs;is AK, æqualis ba&longs;i
CH, & angulus GAK, æqua
lis angulo GCK: &longs;ed totus
angulus DAK, æqualis e&longs;t to
ti angulo BCA; reliquus igi
tur DAK, reliquo BCH,
æqualis erit, circa quos angu
los latus BC e&longs;t æquale lateri
AD, & CH, ip&longs;i AK; angu
lus igitur CBH, æqualis erit
angulo ADK. Similiter o&longs;tenderemus angulum CAH,
angulo ACK, & angulum BAH, angulo DCK, & an
gulum ABH, angulo CDK, æquales e&longs;se: &longs;ed latera
triangulorum, cum quibus rectæ ductæ à punctis K, H, ad
angulos triangulorum &longs;imilium ABC, CDA, &longs;unt ho-
&longs;imiliter po&longs;ita. Rur&longs;us quoniam angulus ABC, non
e&longs;t minor recto, acuti erunt reliqui ACB, BAC; igitur
latus AC, maximum erit: ponitur autem AB maius,
quàm BC; triangulum igitur ABC, &longs;calenum erit. Eadem ratione &longs;calenum e&longs;t triangulum ACD.
Quare
in triangulo ACD, vnum duntaxat punctum K, &longs;imili
ter po&longs;itum erit, ac punctum H, in triangulo ABC. Cum
igitur H &longs;it centrum grauitatis trianguli ABC, erit &
K, centrum grauitatis trianguli ACD. Sed longitudo
GK, æqualis e&longs;t longitudini GH; punctum igitur G erit
centrum grauitatis parallelogrammi ABCD, in quo ni
mirum &longs;ecta e&longs;t bifariam diameter AC: quare &longs;i ducatur
altera diameter BD, in medio etiam diametri BD, erit
idem centrum grauitatis G.
Sed &longs;int omnia latera æqualia
Sectisque duobus lateribus AD, BC, bifariam in E, F
iungantur EF, AE, ED,
AGC, & per punctum G,
ducatur ip&longs;i AD, vel BC,
parallela HGK. Quoniam
igitur EC, e&longs;t æqualis
AF, erit CG æqualis AG,
& EG, æqualis GF, pro
pter &longs;imilitudinem triangu
lorum: nec non EH, ip&longs;i
AH, & EK, ip&longs;i KD: tres
igitur diametri AC, AE,
ED, erunt &longs;ectæ bifariam
in punctis K, G, H: & quoniam ex æquali propter triangu
la &longs;imilia e&longs;t vt AF, ad FD, ita HG, ad GK, erit HG,
æqualis ip&longs;i GK: &longs;ed puncta K, H, &longs;unt centra grauitatis
parallelogrammorum BF, FC; igitur totius parallelo
grammi ABCD, centrum grauitatis erit G, in medio Quod e&longs;t propo&longs;itum.
Hinc manife&longs;tum e&longs;t, omnis parallelogrammi
centrum grauitatis e&longs;&longs;e in medio rectæ, quæ op
po&longs;itorum bipartitorum laterum &longs;ectiones iungit.
Si quodlibet parallelogrammum in duo paral
lelogramma diuidatur, & eorum
iungantur recta linea; totius diui&longs;i parallelogram
mi centrum grauitatis prædictam lineam ita di
uidit, vt eius &longs;egmenta è contrario re&longs;pondeant
prædictis partibus parallelogrammis.
Sit parallelogrammum ABCD, &longs;ectum in duo paral
lelogramma AE, ED, &
parallelogrammi AE, &longs;it
centrum grauitatis H, pa
rallelogrammi autem ED,
centrum grauitatis K: &
parallelogrammi ABCD,
&longs;it centrum grauitatis G:
& iungatur KH. Dico re
ctam KH, diuidi à puncto
G, ita vt &longs;it KG, ad G
H, vt e&longs;t parallelogrammum
AE, ad parallelogrammum
ED, Iungantur enim diametri AC, AE, ED. Igitur
ctis H, G, K. Quoniam igitur e&longs;t vt EH, ad HA, ita
EK ad KD, parallela erit KH, ip&longs;i AD; igitur & EC;
&longs;ed recta KH, &longs;ecat latus AE, trianguli AEC, bifariam
in puncto H, ergo & latus AC, bifariam &longs;ecabit; igitur
in puncto G. punctum igitur G, e&longs;t in linea KH. Rur&longs;us,
quoniam e&longs;t vt GA, ad AC, ita GH, ad EC, propter &longs;i
militudinem triangulorum; &longs;ed dimidia e&longs;t GA, ip&longs;ius
AC, igitur & GH, erit dimidia ip&longs;ius EC, hoc e&longs;t ip&longs;ius
FD. Similiter o&longs;tenderemus dimidiam e&longs;se KH ip&longs;ius
AD. vt igitur KH, ad AD, ita erit GH, ad FD: & per
mutando, vt AD, ad DF, ita KH, ad HG, & diui
dendo, vt AF, ad FD, hoc e&longs;t vt parallelogrammum AE,
ad parallelogrammum ED, ita KG, ad GH. Quod de
mon&longs;trandum erat.
Plana grauia æquiponderant à longitudini
bus ex contraria parte re&longs;pondentibus.
Sint plana grauia N, R, quorum centra grauitatis &longs;int
N, R, & longitudo aliqua AB: & vt e&longs;t N, ad R, ita &longs;it
BC, ad CA. Dico &longs;u&longs;pen&longs;is magnitudinibus &longs;ecundum
centra grauitatis N, in puncto A, & R, in puncto B, vtri
u&longs;que magnitudinis N, R, &longs;imul centrum grauitatis e&longs;se
C. Nam &longs;i N, R, magnitudines &longs;int æquales, manife&longs;tum
e&longs;t propo&longs;itum. Si autem inæquales, ab&longs;cindatur BD,
æqualis AC, vt &longs;it AD, ad DB, vt BC, ad CA. Et quo
niam &longs;pacio R, rectangulum æquale pote&longs;t e&longs;se; applice
tur ad lineam BD, rectangulum BDKE, æquale quar
tæ parti rectanguli æqualis ip&longs;i R, hoc e&longs;t quartæ parti
ip&longs;ius R; & po&longs;ita DG, æquali, & in directum ip&longs;i DK,
cta conueniant in punctis F, H: & fiant parallelogramma
FL, AK. Quoniam igitur e&longs;t vt N, ad R, ita BC, ad
CA, hoc e&longs;t AD, ad DB, hoc e&longs;t rectangulum AK, ad
rectangulum BK; erit permutando vt rectangulum AK,
ad N, ita rectangulum BK, ad R; &longs;ed rectangulum BK,
e&longs;t pars quarta ip&longs;ius R, ergo & rectangulum AK, erit
pars quarta ip&longs;ius N. Rur&longs;us quia e&longs;t vt GD, ad D
ita GA, ad AF, & GB, ad BH: &longs;ed GD e&longs;t æqualis
DK; ergo & GA, ip&longs;i AF, & GB, ip&longs;i BH, æquales
erunt & centra grauita
tis A, quidem rectangu
li MK, B, vero rectan
guli KL, & rectangulum
AK, pars quarta ip&longs;ius
M
& B
N, rectanguli AK, qua
druplum erat, quemad
modum & R ip&longs;ius BK;
igitur rectangulum MK,
&longs;pacio N, & rectangulum
KL, &longs;pacio R, æquale
erit. Sed vt BC, ad CA,
ita e&longs;t N, ad R; vt igi
tur BC, ad CA, ita
rectangulum MK, ad rectangulum KL; &longs;ed A e&longs;t cen
trum grauitatis rectanguli MK, & B, rectanguli KL; to
tius ergo rectanguli FL, hoc e&longs;t duorum rectangulorum
MK, KL, &longs;imul centrum grauitatis erit C. Sed rectan
gulo MK, æquale e&longs;t &longs;pacium N; & rectangulo KL, &longs;pa
cium R. Igitur &longs;i pro rectangulo MK, &longs;it &longs;u&longs;pen&longs;um N
&longs;pacium &longs;ecundum centrum grauitatis in puncto A, & pro
rectangulo KL, &longs;pacium R, &longs;ecundum centrum graui-
gitudinibus AC, CB; eritque vtriu&longs;que plani N, R, &longs;i
mul centrum grauitatis C. Quod demon&longs;trandum erat.
Hinc manife&longs;tum e&longs;t &longs;i cuiuslibet figuræ pla
næ vtcumque &longs;ectæ centra grauitatis partium
iungantur recta linea, talem lineam à centro gra
uitatis totius prædicti plani ita &longs;ecari, vt &longs;egmen
ta ex contrario re&longs;pondeant prædictis partibus.
Si totum quoduis planum, & pars aliqua non
habeant idem centrum grauitatis, & eorum cen
tra iungantur recta linea; in ea producta ad par
tes centri grauitatis totius, erit reliquæ partis cen
trum grauitatis.
Sit totum quoduis planum
ABC, cuius centrum graui
tatis E, & pars illius AB, cuius
aliud centrum D, & iuncta
DE, producatur ad partes E,
in infinitum v&longs;que in H. Dico
reliquæ partis BC, centrum
grauitatis, quod &longs;it G, e&longs;se in
linea EH. Quoniam enim D,
G, &longs;unt centra grauitatis par
tium AB, BC, cadet totius ABC, centrum grauitatis
cta D, E, G, &longs;unt in eadem recta linea. in qua igitur &longs;unt
puncta D, E, in eadem e&longs;t punctum G; &longs;ed puncta D, E, &longs;unt
in recta DH; igitur & punctum G, erit in recta DH: &longs;ed
extra ip&longs;am DE, vt modo o&longs;tendimus, in reliqua igitur
EH. Quod demon&longs;trandum erat.
Sit totum quoduis planum &longs;it vni parti concen
tricum &longs;ecundum centrum grauitatis, & reliquæ
erit concentricum. Et &longs;i partes inter &longs;e &longs;int con
centricæ, & toti erunt concentricæ.
Sit totum quoduis planum AB, quod cum vna parte
AC habeat commune centrum grauitatis E. Dico & re
liquæ partis CD, e&longs;se
idem centrum grauitatis
E. Si enim illud non
e&longs;t, erit aliud; e&longs;to F, &
EF iungatur. Quoniam
igitur partium AC, CD,
centra grauitatis &longs;unt E,
F; erit totius AB, in re
cta EF, centrum graui
tatis: &longs;ed & in puncto E,
vnius ergo magnitudinis
duo centra grauitatis e
runt. Quod e&longs;t ab&longs;urdum;
idem igitur E erit centrum grauitatis vtriuslibet partium
AC, CD. Sed vtriuslibet partium AC, CD, &longs;it cen
trum grauitatis E. Dico idem E totius AB, e&longs;se cen-
Si enim non e&longs;t, erit aliud, e&longs;to G: &
iunctatur EG, producatur ad partes G, in infinitum v&longs;
que ìn F. Quoniam igitur E, e&longs;t centrum grauitatis vnius
partis AC, & G, totius AB; erit reliquæ partis CD, in
linea GF centrum grauitatis: &longs;ed & in puncto E; eiu&longs;
dem igitur magnitudinis AB, duo centra grauitatis erunt. Quod fieri non pote&longs;t; totius igitur AB, erit centrum gra
uitatis idem E. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis trianguli rectilinei idem e&longs;t centrum
grauitatis, & figuræ.
Sit triangulum rectilineum ABC, cuius centrum G.
Dico G, e&longs;se centrum grauitatis trianguli ABC.
Si enim
fieri pote&longs;t, &longs;it aliud punctum N, centrum grauitatis trian
guli ABC, & per punctum G, ducantur rectæ AF, BD,
CE, & DHE, ERF, FKD, Quo
niam igitur quæ ab angulis A, B, C, ductæ &longs;unt rectæ
lineæ per G, &longs;ecant bifariam latera AB, BC, CA; erit
triangulum EDF, &longs;imile triangulo ABC, ob latera pa
rallela vt &longs;unt EF, AC. Et quoniam triangulum EDF,
dimidium e&longs;t cuius vis trium parallelogrammorum AF,
BD, CE, æqualia inter &longs;e erunt ea parallelogramma
omnifariam &longs;umpta, quorum centra grauitatis H, K, R;
intelligantur autem tria parallelogramma AF, BD, CE,
di&longs;tincta penitus, ita vt inter &longs;e congruant &longs;ecundum tria
triangula DEF, inter &longs;e congruentia: trium igitur trian
gulorum DEF, inter &longs;e congruentium & centra grauita
tis inter &longs;e congruent in puncto M. Quoniam igitur in
ter duas parallelas EF, KH, &longs;ecant &longs;e rectæ lineæ FH,
LR, in puncto G; erit vt FG, ad GH, ita RG, ad GL; Et quoniam in triangu
lo AGC, recta GD, &longs;ecat AC, bifariam in puncto D;
ip&longs;i AC, parallelam KH, bifariam &longs;ecabit in puncto L,
duorum igitur æqualium parallelogrammorum AF, EG;
&longs;imul, quorum centra grauitatis &longs;unt K, H, centrum gra
uitatis erit L. Sed duo parallelogramma AF, EC, &longs;i
mul &longs;unt paralle
logrammi BD, du
plum; trium igitur
parallelogrammo
rum AF, EC,
BD, &longs;imul: hoc
e&longs;t
vnà cum duobus
trium
inter &longs;e congruen
tium EDF, cen
trum grauitatis e
rit G. Sed triangu
li ABC, ponitur
centrum grauitatis N; producta igitur NG, occurret
centro M, reliquæ partis, ide&longs;t duorum triangulorum DEF;
quare vt triangulum ABC, ad duo triangula DEF, &longs;i
mul, ita erit MG, ad GN. Sed triangulum ABC, e&longs;t
duplum duorum triangulorum EDF: igitur & MG, erit
ip&longs;ius GN, dupla. Rur&longs;us quoniam vtriuslibet duorum
triangulorum EDF, centrum grauitatis erat M; erit &longs;i
militer po&longs;itum M, in triangulo EDF, ac centrum N, in
triangulo ABC, propter &longs;imilitudinem triangulorum:
Sed propter hæc &longs;imiliter po&longs;ita centra, quia homologo
rum laterum e&longs;t vt AB, ad DF, ita NG, ad GM: &
AB, e&longs;t dupla ip&longs;ius EB, erit & NG, dupla ip&longs;ius GM. Sed GM, erat dupla ip&longs;ius GN: igitur GN, erit &longs;ui ip&longs;ius
quadrupla. Quod e&longs;t ab&longs;urdum.
Non igitur centrum
ctum igitur G, erit centrum grauitatis trianguli ABC. Quod demon&longs;trandum erat.
Quod autem ex huius theorematis demon&longs;tratione li
quet centrum grauitatis trianguli e&longs;se in ea recta linea,
quæ ab angulo ad bipartiti lateris &longs;ectionem pertinet,
Archimedes per in&longs;criptionem figuræ ex parallelogram
mis demon&longs;trauit, aliter autem per diui&longs;ionem trianguli
in triangula nequaquam: qua enim ratione hoc ille tentat,
ea ex nono theoremate eiu&longs;dem prioris libri de æquipon
derantibus nece&longs;sario pendet. Cum igitur in illo ante ceden
ti &longs;it fallacia accipientis latenter &longs;peciem trianguli; &longs;cale
num &longs;cilicet pro genere triangulo, neque con&longs;equens erit
demon&longs;tratum. Quod autem dico manife&longs;tum e&longs;t: Datis
enim duobus triangulis &longs;imilibus, & in altero eorum dato
puncto, quod &longs;it trianguli centrum grauitatis, punctum in
altero triangulo modo &longs;imiliter po&longs;itum &longs;it prædicto pun
cto, nititur demon&longs;trare e&longs;se alterius trianguli centrum
grauitatis: cum autem nondum con&longs;tet centrum graui
tatis trianguli e&longs;se in recta, quæ ab angulo latus oppo&longs;i
tum bifariam &longs;ecat, &longs;ed ex nono theoremate &longs;it demon&longs;tran
dum medio decimo, non pote&longs;t illud accipi in nono theo
remate, quod ad demon&longs;trationem e&longs;set nece&longs;sarium. per
mittitur igitur aduer&longs;ario ponere centrum grauitatis trian
guli, vbicumque vult intra illius limites. atqui cum datis
duobus triangulis i&longs;o&longs;celiis &longs;imilibus, & in altero eorum
dato puncto, quod non &longs;it in prædicta recta linea, po&longs;sint
in altero duo puncta prædicto &longs;imiliter po&longs;ita inueniri, quo
rum vnum duntaxat concedet aduer&longs;arius e&longs;se alterius
trianguli centrum grauitatis, non autem non &longs;imiliter po
&longs;itum, ex quo ab&longs;urdum infertur partem anguli æqualem
e&longs;se toti: quid quod datis duobus triangulis æquilateris, &
in altero eorum dato puncto, quod non &longs;it centrum trian-
laterum &longs;ectiones cadunt, nece&longs;se e&longs;t in altero triangulo
tria puncta prædicto puncto e&longs;se &longs;imiliter po&longs;ita? quod &longs;i
etiam extra i&longs;tas lineas cadat vnius trianguli punctum, ne
ce&longs;se e&longs;t illi &longs;ex puncta in altero triangulo e&longs;se &longs;imiliter po
&longs;ita: &longs;ed &longs;i quod diximus de i&longs;o&longs;celiis &longs;imilibus, & æquila
teris triangulis demon&longs;trauerimus, rem velut ante oculos
expo&longs;uerimus.
Datis duobustriangulis i&longs;o&longs;celijs &longs;imilibus, &
in altero eorum dato puncto extra rectam, quæ à
vertice ad medium ba&longs;is cadit, duo puncta in re
liquo triangulo prædicto puncto &longs;imiliter po&longs;ita
inuenire.
Sint duo triangula i&longs;o&longs;celia, & &longs;imilia ABC, DEF:
quorum in altero ABC, à vertice A, ad ba&longs;im BC, bi
partitam in puncto G, cadat recta AG: atque extra hanc
in triangulo ABC, &longs;it quoduis punctum H: & iuncta AH,
fiat angulus EDK æqualis angulo BAH; & vt BA, ad
æqualis e&longs;t angulo EDF: quorum angulus EDK,
æqualis e&longs;t angulo BAH, erit reliquus angulus
æqualis reliquo angulo HAC; &longs;ed angulus HAC, e&longs;t
maior angulo BAH; ergo & angulus KDF, maior erit
angulo BAH; po&longs;ito igitur angulo FDL, æquali an
gulo BAH, ac proinde minori, quàm &longs;it angulus FD
fiat vt BA, ad AH, ita FD, ad DL. Dico, in triangu
lo EDF, duo puncta K, L, &longs;imiliter po&longs;ita e&longs;se ac pun
ctum H, in triangulo BAC. Iungantur enim rectæ AH,
BH, CH, EK, KF, FL, LE. Quoniam igitur an
gulus ED
homologis continentur; erit angulus DE
gulo ABH: &longs;ed totus angulus DEF, æqualis e&longs;t toti an
gulo ABC; reliquus igitur angulus KEF, æqualis erit
reliquo HBC: &longs;ed ex æquali e&longs;t vt CB, ad BH, ita
FE, ad EK; igitur vt antea erit angulus KFE, æqualis
angulo HCB, & angulus DFK, æqualis angulo ACH,
& angulus FDK, æqualis angulo CAH; punctum igi
tur K, &longs;imiliter po&longs;itum erit in triangulo EDF, ac pun
ctum H, in triangulo ABC. Rur&longs;us quoniam angulus
FDL, æqualis e&longs;t angulo BAH, & latus AB, homo
logum lateri DF, (e&longs;t enim vt BA, ad AC, ita FD, ad
DE) &longs;ed vt BA, ad AH, ita e&longs;t FD, ad DL, per con
&longs;tructionem; &longs;imiliter vt ante, o&longs;tenderemus, punctum L,
in triangulo EDF, &longs;imiliter po&longs;itum e&longs;se puncto H; in
uenta igitur &longs;unt duo puncta in triangulo DEF, &longs;imili
ter po&longs;ita ac punctum H, in triangulo BAC. Quod pro
po&longs;itum erat.
Omnis trapezij habentis duo latera parallela
centrum grauitatis e&longs;t in illa recta, quæ prædi
ctorum bipartitorum laterum &longs;ectiones iungit. atque in eo puncto, in quo tertia pars eius media
&longs;ic diuiditur, vt &longs;egmentum propinquius mino
ri parallelarum ad reliquum eam proportionem
habeat, quam maior parallelarum ad minorem. Talis autem rectæ lineæ &longs;ic diui&longs;æ, &longs;egmentum
minorem parallelarum attingens e&longs;t ad reliquum,
vt dupla maioris parallelarum vna cum minori,
ad duplam minoris vna cum maiori.
Sit trapezium ABCD, cuius duæ AD, BC, &longs;int pa
rallelæ: &longs;itque AD, maior. Secti&longs;que AD, BC, bifa
riam in punctis F, E,
iunctaque EF, & &longs;e
cta in tres partes æ
quales in punctis K,
H, fiat vt AD, ad
BC, ita HG, ad GK. Dico G, e&longs;se centrum
grauitatis trapezij A
BCD: & vt e&longs;t du
pla ip&longs;ius AD, vna
cum BC, ad duplam
ip&longs;ius BC, vna cum
AD, ita e&longs;se EG, ad
GF. Ducta enim per punctum H, ip&longs;is AD, BC, pa-
iungantur ANE, EOD. Quoniam igitur NO ip&longs;i AD,
parallela &longs;ecat omnes ip&longs;is AD, EC, interceptas in ea&longs;
dem rationes, & e&longs;t EH, pars tertia ip&longs;ius EF, erit & EN
ip&longs;ius EA, & EO, ip&longs;ius ED, pars tertia. E&longs;t autem NO,
parallela ba&longs;ibus BE, EC, duorum triangulorum ABE,
ECD; in ip&longs;a igitur NO, erunt centra grauitatis duo
rum triangulorum ABE, ECD: ergo & compo&longs;iti ex
vtroque in linea NO, erit centrum grauitatis. Quoniam
igitur K, centrum grauitatis trianguli AED, e&longs;t in EF, &
totius trapezij ABCD, centrum grauitatis in eadem linea
EF; erit & reliquæ partis, duorum &longs;cilicet triangulorum
ABE, ECD, &longs;imul in linea EF, centrum grauitatis: &longs;ed &
in linea NO; in puncto igitur H. Rur&longs;us quoniam triangula
AED, ABE, ECD, &longs;unt inter ea&longs;dem parallelas, erit
vt AD, ad BC, ita triangulum AED, ad duo triangu
la ABE, ECD, &longs;imul: &longs;ed vt AD, ad BC, ita e&longs;t HG,
ad GK; vt igitur triangulum AED, ad duo triangula
ABE, ECD, &longs;imul, ita erit HG, ad GK. &longs;ed K, e&longs;t
centrum grauitatis trianguli AED: & H, duorum trian
gulorum ABE, ECD, &longs;imul; totius igitur trapezij AB
CD, centrum grauitatis erit G. Rurius quoniam EL,
e&longs;t æqualis GK, æqualium EH, HK; erit reliqua LH,
æqualis reliquæ GH; tota igitur EG; erit bis GH, vna
cum GK: eadem ratione quoniam FM, e&longs;t æqualis GK,
& MK, æqualis GH, erit FG, bis GK, vna cum GH:
vt igitur HG, bis vna cum GK, ad GK, bis vna cum
GH, ita erit EG, ad GF. Sed vt HG, bis vna cum
GK, ad GK bis vna cum GH, ita e&longs;t AD, bis vna cum
BC, ad BC, bis vna cum AB, propterea quod e&longs;t vt
AD, ad BC, ita HG, ad GK; vt igitur e&longs;t AD, bis vna
cum BC, ad BC, bis vna cum AD, ita erit EG, ad GF. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis polygoni æquilateri, & æquianguli
idem e&longs;t centrum grauitatis, & figuræ.
Sit polygonum æquilaterum, & æquiangulum ABC
DEFG, cuius &longs;it primo laterum numerus impar, centrum
autem &longs;it L. Dico punctum L, e&longs;se centrum grauitatis
polygoni ABCDEFG; &longs;ectis enim duobus lateribus
DE, FG, bifariam in punctis K, H, ducantur ab angulis
oppo&longs;itis rectæ AH, CK. & rectæ BMG, CNF, CM,
MF, iungantur. Quoniam igitur ex decima tertia quar
ti Elem.
quemadmodum in pentagono, ita in omni præ
dicto polygono imparium multitudine laterum plane col
ligitur centrum po
lygoni e&longs;se in qua
libet recta, quæ ab
angulo ad medium
lateris oppo&longs;iti du
citur, quoniam ab
omnibus angulis &longs;ic
ductæ &longs;ecant &longs;e &longs;e
in eadem proportio
ne æqualitatis, ita
vt eadem &longs;it propor
tio &longs;egmentorum,
quæ ad angulos, ad
ea, quæ ad latera
illis angulis oppo&longs;ita; rectæ AH, CK, &longs;ecabunt &longs;e &longs;e in
puncto L. Rurfus quoniam ex eadem Euclidis angulus
BAL, æqualis e&longs;t angulo GAL, &longs;ed AB, e&longs;t æqualis
AG, & AM, communis, erit ba&longs;is BM, æqualis ba&longs;i
toti AGF, e&longs;t æqualis; reliquus igitur angulus CBG,
reliquo BGF, æqualis erit: &longs;ed circa hos æquales an
gulos recta BM, o&longs;ten&longs;a e&longs;t æqualis rectæ MG, & CB,
e&longs;t æqualis GF; ba&longs;is igitur CM, ba&longs;i GF, & angulus
CMB, angulo FMG, æqualis erit; &longs;ed totus BMN,
æqualis e&longs;t toti GMN; quia vterque rectus; reliquus
igitur CMN, reliquo NMF, æqualis erit, quos circa
recta CM, e&longs;t æqualis MF, & MN, communis; ba&longs;is
igitur CN, ba&longs;i NF, & anguli, qui ad N, æquales erunt,
atque ideo recti: &longs;ed & qui ad M, &longs;unt recti, & BM, e&longs;t
æqualis GM; parallelæ igitur &longs;unt BG, CF, & trape
zij CBGF, centrum grauitatis e&longs;t in linea MN: &longs;ed &
trianguli ABG, centrum grauitatis e&longs;t in linea AM; to
tius igitur figuræ ABCFG, centrum grauitatis e&longs;t in li
nea AN; hoc e&longs;t in linea AH. Rur&longs;us quoniam omnis
quadrilateri quatuor anguli &longs;unt æquales quatuor rectis:
& tres anguli ABM, BMN, MNC, &longs;unt æquales tri
bus angulis FGM, GMN, MNF, reliquus angulus
BCF, reliquo CFG, æqualis erit: &longs;ed totus angulus
BCD, e&longs;t æqualis toti angulo GFE; reliquus ergo
DCF, reliquo CFE, æqualis erit: &longs;ed linea CN, e&longs;t
æqualis NF, & anguli, qui ad N, &longs;unt recti; &longs;imiliter
ergo vt antea, centrum grauitatis trapezij CDEF, erit
in linea AH: &longs;ed & totius figuræ ABCFG, e&longs;t in li
nea AH; totius igitur polygoni ABCDEFG, in li
nea AH, e&longs;t centrum grauitatis, quod idem &longs;imiliter in
linea CK, e&longs;se oftenderemus; in communi igitur &longs;ectione
puncto L, e&longs;t centrum grauitatis polygoni ABCDEFG. Similiter quotcumque plurium laterum numero impa
rium e&longs;set polygonum æquilaterum, & æquiangulum,
&longs;emper deueniendo ab vno triangulo ad quotcumque eius
trapezia; propo&longs;itum concluderemus.
Sed e&longs;to polygonum æquilaterum, & æquiangulum,
ABCDEF, cuius laterum numerus &longs;it par, & centrum
e&longs;to G. Dico idem G, e&longs;se centrum grauitatis polygoni
ABCDEF. Iungantur enim angulorum oppo&longs;itorum
puncta rectis lineis AD, BE, CF. Ex quarto igitur
Elem.
&longs;ecabunt &longs;e&longs;e hæ rectæ omnes bifariam in vno pun
cto, quod talis figuræ centrum definiuimus: &longs;ed G poni
tur centrum; in puncto igitur G. Quoniam igitur duo
rum triangulorum CBG, GFE, anguli ad verticem
BGC, FGE, &longs;unt æquales; & vterlibet angulorum CBG,
GCB, æqualis e&longs;t vtrilibet ip&longs;orum EFG, GEF; ex
quarto Elem.
& circa æquales angulos latera proportio
nalia horum triangu
lorum &longs;unt æqualia;
&longs;imilia, & æqualia
erunt triangula BC
G, GFE: po&longs;itis
igitur centris graui
tatis K, H, duorum
triangulorum EFG,
GBC, iunctifque
KG, GH, erit v
terlibet angulorum
BGH, HGC, æ
qualis vtrilibet an
gulorum CGK, KGE, propter &longs;imilitudinem po&longs;itio
nis centrorum K, H, in i&longs;o&longs;celijs triangulis CBG,
GFE: (nam GH, &longs;i produceretur latus BC, bifariam
&longs;ecaret: &longs;imiliter GK, latus EF) &longs;ed CG, e&longs;t in directum
po&longs;ita ip&longs;i GF; igitur & GH ip&longs;i GK: & &longs;unt æquales,
vtpote lateribus triangulorum BCG, GFE, æqualibus
homologæ; cum igitur eorundem triangulorum centra
grauitatis &longs;int K, H; centrum grauitatis duorum triangu
lorum CBG, GFE, &longs;imul, erit punctum G. Eadem
duorum AFG, CDG, &longs;imul, centrum grauitatis erit G;
totius igitur polygoni ABCDEF; centrum grauitatis
erit idem G. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis figuræ circa diametrum in alteram par
tem deficientis, in diametro e&longs;t centrum graui
tatis.
Sit figura ABC, circa diametrum BD, in alteram par
tem deficiens ver&longs;us B. Dico centrum grauitatis figuræ
ABC, e&longs;se in linea BD. &longs;it enim punctum E, generali
ter extra lineam BD. Et per puncta E, C, ducantur ip&longs;i
BD, parallelæ EF,
CG, & vt e&longs;t CD,
ad DF, ita ponatur
figura ABC, ad ali
quod &longs;pacium M: &
figuræ ABC, in&longs;cri
batur figura ex paral
lelogrammis æqua
lium altitudinum de
ficiens à figura ABC,
minori defectu, quam
&longs;it &longs;pacium M, quan
tumcumque illud &longs;it:
minor igitur propor
tio erit figuræ ABC, ad &longs;pacium M, hoc e&longs;t minor pro
portio CD, ad DF, quàm figuræ ABC, ad &longs;ui reliquum,
dempta figura in&longs;cripta. Quoniam autem diameter BD,
ptorum ba&longs;i AC, parallela; erit in diametro BD, eorum
omnium parallelogrammorum centra grauitatis, atque
ideo totius figuræ in&longs;criptæ centrum grauitatis, quod &longs;it
H: & HEK, ducatur. Quoniam igitur EF, parallela
e&longs;t vtrique DH, CK; erit vt CD, ad DF, ita KH, ad
HE, &longs;ed minor e&longs;t proportio CD, ad DF, quàm figu
ræ ABC, ad re&longs;i
duum, dempta figu
ra in&longs;cripta; ergo &
KH, ad HE, minor
erit proportio, quàm
figuræ ABC, ad præ
dictum re&longs;iduum: ha
beat LKH, eandem
proportione
quàm figura ABC,
ad prædictum re&longs;i
duum. Quoniam
igitur punctum K,
cadit extra figuram
ABC; multo magis punctum L; non igitur punctum L,
erit prædicti re&longs;idui centrum grauitatis. Sed punctum
H, e&longs;t in&longs;criptæ figuræ centrum grauitatis: & vt figura
in&longs;cripta ad prædictum re&longs;iduum, diuidendo, ita e&longs;t LE,
ad EH; non igitur E, e&longs;t centrum grauitatis figuræ ABC:
&longs;ed ponitur E, generaliter punctum extra lineam BD;
Nullum igitur punctum extra lineam BD, e&longs;t centrum
grauitatis figuræ ABC; in linea igitur BD, erit figu
ræ ABC, centrum grauitatis. Quod demon&longs;trandum
erat.
Ex huius theorematis demon&longs;tratione con&longs;tat,
omnis figuræ planæ, &longs;iue &longs;olidæ, cuius termini
omnis cauitas &longs;it interior, atque ideo intra ter
minum centrum grauitatis; & cuius pars aliqua
e&longs;se po&longs;sit, quæ à tota figura deficiens minori
defectu quacumque magnitudine propo&longs;ita habe
at centrum grauitatis in aliqua certa linea recta
intra terminum figuræ con&longs;tituta, e&longs;&longs;e in ea recta
linea totius figuræ centrum grauitatis. Ac proin
de, cum per vndecimam huius, omni &longs;olido circa
axim in alteram partem deficienti, & ba&longs;im ha
benti circulum, vel ellyp&longs;im figura in&longs;cribi po&longs;&longs;it
ex cylindris, vel cylindri portionibus, à prædicto
&longs;olido deficiens minori &longs;pacio quacumque ma
gnitudine propo&longs;ita: talis autem figuræ in&longs;criptæ,
quemadmodum & circum&longs;criptæ centrum gra
uitatis &longs;it in axe, vt ex &longs;equentibus patebit, &
nunc cogitanti facilè patere pote&longs;t; manife&longs;tum
e&longs;t omnis &longs;olidi circa axim in alteram partem de
ficientis centrum grauitatis e&longs;&longs;e in axe.
Circuli, & Ellyp&longs;is idem e&longs;t centrum grauita
tis, & figuræ.
Sit circulus, vel ellyp&longs;is ABCD, cuius centrum E.
Dico centrum grauitatis figuræ ABCD, e&longs;se punctum E.
Ducantur enim duæ diametri ad rectos inter &longs;e angulos
AC, BD; in ellyp&longs;i autem &longs;int diametri coniugatæ. Quoniam igitur omnes rectæ lineæ, quæ in &longs;emicirculo,
vel dimidia ellyp&longs;i diametro ducantur parallelæ bifariam
&longs;ecantur à &longs;emidiametro, & quo à ba&longs;i remotiores, eo &longs;unt
minores; erit centrum grauitatis &longs;emicirculi, &longs;iue dimidiæ
ellyp&longs;is ABC, in linea BE; &longs;icut & &longs;emicirculi, &longs;iue di
midiæ ellyp&longs;is ADC, centrum grauitatis in linea DE.
e&longs;t autem BED, vna recta linea: in diametro igitur BD,
erit centrum grauitatis circuli, &longs;iue ellyp&longs;is ABCD. Eadem ratione o&longs;tenderemus idem centrum grauitatis e&longs;se
in altera diametro AC: in communi igitur vtriu&longs;que &longs;e
ctione puncto E. Quod demon&longs;trandum erat.
Si duarum pyramidum triangul as ba&longs;es haben
tium æqualium, & &longs;imilium inter &longs;e, tria latera
tribus lateribus homologis fuerint in directum
con&longs;tituta, in vertice communi erit vtriu&longs;que &longs;i
mul centrum grauitatis.
Sint duæ pyramides &longs;imiles, & æquales, quarum ver
tex communis G, ba&longs;es autem triangula ABC, DEF. Et &longs;int latera homologa pyramidum in directum inter &longs;e
con&longs;tituta: vt AG, GF: & BG, GD, & CG, GE. Dico compo&longs;iti ex duabus pyramidibus ABCG, GDEF,
ita con&longs;titut is centrum gra
uitatis e&longs;se in puncto G. E&longs;to enim H, centrum gra
uitatis pyramidis ABCG,
& ducta HGK, ponatur
G
gantur EK, KD, BH,
CH. Quoniam igitur e&longs;t
vt HG, ad GK, ita CG,
ad GE, & proportio e&longs;t
æqualitatis: & angulus
HGC, æqualis angulo EG
&longs;imile, & æquale triangulo EGK. Similiter triangulum
BGH, trian gulo DGK; & triangulum BGC, triangu
lo DGE: quare & triangulum BCH, triangulo DEK.
pyramis igitur BCGH, &longs;imilis, & æqualis e&longs;t pyramidi
EDGK. Congruentibus igitur inter &longs;e duobus triangu
BCGH, pyramidi GDEK congruet, & puncto K, pun
ctum H: & eadem ratione
pyramis ABCG, pyra
midi DEFG. congruente
igitur pyramide ABCG,
pyramidi DEFG, & pun
ctum K, congruet puncto
H. &longs;ed H, e&longs;t centrum gra
uitatis pyramidis ABCG:
igitur K, erit centrum gra
uitatis pyramidis DEFG:
&longs;ed e&longs;t GK, æqualis ip
&longs;i GH; vtriufque igitur
pyramidis ABCG, DE
FG, &longs;imul centrum grauitatis erit K; Quod demon&longs;tran
dum erat.
Omnis parallelepipedi centrum grauitatis e&longs;t in
medio axis.
Sit parallelepipedum ABCDEFGH, cuius axis
LM, isque &longs;ectus bifariam in puncto K. Dico K e&longs;se
centrum grauitatis parallelepipedi ABCDEFGH.
iungantur enim diametri AG, BH, CE, DF, quæ
omnes nece&longs;sario tran&longs;ibunt per punctum K, & in eo
puncto bifariam diuidentur. Iunctis igitur BD, FH:
quoniam triangulum EFK, &longs;imile e&longs;t, & æquale trian
gulo CDK, propter latera circa æquales angulos ad
triangulum E
triangulo BDK; erit pyramis KEFH, &longs;imilis, & æqua
lis pyramidi KBCD: habent autem tria latera tribus
lateribus homologis, ide&longs;t æ
qualibus, in directum, prout
inter &longs;e re&longs;pondent, con&longs;tituta;
duarum igitur pyramidum KE
FH, KBCD, &longs;imul centrum
grauitatis erit K: non aliter
duarum pyramidum
KBDA, &longs;imul centrum gra
uitatis erit K; totius igitur com
po&longs;iti ex quatuor pyramidibus;
ide&longs;t duabus oppo&longs;itis ABC
DK, EFGHK, centrum gra
uitatis erit idem K. Eadem
ratione tam duarum pyrami
dum AEHDK, BCGFK, &longs;imul, quàm duarum AB
FEK, CDHGK, &longs;imul centrum grauitatis erit K. To
tius igitur parallelepipedi ABCDEFG
grauitatis erit K. Quod demon&longs;trandum erat.
Si parallelepipedum in duo parallelepipeda
&longs;ecetur, &longs;egmenta axis à centris grauitatis totius
parallelepipedi, & partium terminata ex contra
rio parallelepipedi partibus re&longs;pondent.
Si parallelepipedum AB, cuius axis CD, &longs;ectum in
duo parallelepipeda AE, EN, quare & axis CD, in
axes CL, LD, parallelepipedorum AE, EN. Et &longs;int
centra grauitatis; F, parallelepipedi EN, & G, paral
lelepipedi AE, & H, parallelepipedi AB, in medio cu
iu&longs;que axis ex antecedenti. Dico e&longs;se FH, ad HG,
vt parallelepipedum AE, ad EN, parallelepipedum. Iungantur enim diametri ba&longs;ium oppo&longs;itarum, quæ per
puncta axium D, L, G, tran&longs;ibunt, ADM, KLE,
NCB; iamque parallelogramma
erunt AB, AE, EN, DB, DE,
EC, propter eas, quæ parallelas
iungunt, & æquales: quorum bi
na latera oppo&longs;ita &longs;ecta erunt bi
fariam in punctis C, L, D, per
definitionem axis: punctum igitur
F, in medio rectæ CL, oppo&longs;i
torum laterum bipartitorum &longs;ectio
nes coniungentis, erit parallelo
grammi EN, centrum grauitatis. Eadem ratione & parallelogram
mi AE, centrum grauitatis erit G, & H, parallelogram
mi AB. Vt igitur parallelogrammum AE, ad paralle
logrammum EN, hoc e&longs;t, vt ba&longs;is ME, ad ba&longs;im EB;
hoc e&longs;t, vt parallelogrammum MO, ad parallelogram
mum OB: hoc e&longs;t, vt parallelepipedum AE, ad paral
lelepipedum EN: ita erit FH, ad HG. Quod de
mon&longs;trandum erat.
Solida grauia æquiponderant à longitudini
bus ex contraria parte re&longs;pondentibus.
Sint &longs;olida grauia A, & B, quorum centra grauitatis
&longs;int A, B, &longs;ecundum quæ &longs;u&longs;pen&longs;a intelligantur A, in
puncto C, & B, in puncto D, cuiuslibet rectæ GH, quæ
&longs;it ita diui&longs;a in puncto E, vt &longs;it DE, ad EC, vt e&longs;t A,
ad B. Dico &longs;olida A, E, æquiponderare à longitudini
bus DE, EC; hoc e&longs;t vtriu&longs;que &longs;imul centrum grauita
tis e&longs;se E. Nam &longs;i A, B, &longs;int æqualia, manife&longs;tum e&longs;t
propo&longs;itum: &longs;i au
tem inæqualia, e&longs;to
maius A: maior igi
tur erit DE, quam
EC. ab&longs;cindatur
DF, æqualis EC:
erit igitur DE, æ
qualis GF: & CD,
vtrin que producta,
ponatur DH, æ
qualis DF: & CG,
ip&longs;i CF. & circa
axim, &
GH, e&longs;to paralle
lepipedum KL, æ
quale duobus &longs;o
lidis A, B, &longs;imul & parallelepipedum KL, &longs;ecetur plano
per punctum F, oppo&longs;itis planis parallelo, in duo paral
lelepipeda KN, ML. Quoniam igitur e&longs;t vt GF, ad
FH, ita parallelepipedum KN, ad parallelepipedum
EC, hoc e&longs;t &longs;olidum A, ad &longs;olidum B; erit vt parallelepipe
dum KN, ad parallelepipedum ML, ita &longs;olidum A, ad &longs;oli
dum B. componendo igitur, & permutando, vt parallelepi
pedum KL, ad duo &longs;olida A, B, &longs;imul, ita parallelepi
pedum ML, ad &longs;olidum B: & reliquum ad reliquum: &longs;ed
parallelepipedum KL, æquale e&longs;t duobus &longs;olidis A, B, &longs;i
mul: parallelepipedum igitur KN, &longs;olido A, & paralle
lepipedum ML, &longs;olido B, æquale erit. Rur&longs;us, quo
niam e&longs;t vt GF, ad
ad FH, ita CF, ad
FD; hoc e&longs;t DE,
ad EC: &longs;ed vt GF,
ad FH, ita e&longs;t
rallelepipedum
ad
ML; erit vt DE,
ad EC, ita paralle
lepipedum KN, ad
parallelepipedum
ML; &longs;ed C e&longs;t pa
rallelepipedi KN,
& D, parallelepipe
di ML, centrum
grauitatis; totius igi
tur parallelepipedi KL, centrum grauitatis erit E. Igi
tur &longs;olido A, po&longs;ito ad punctum G, &longs;ecundum centrum
grauitatis A, & &longs;olidum B, ad punctum D, &longs;ecundum
centrum grauitatis B, quorum A, e&longs;t æquale parallele
pipedo KN, & B, parallelepipedo ML; ab ij&longs;dem lon
gitudinibus DE, EC, æquiponderabunt; eritque com
po&longs;iti ex vtroque &longs;olido A, B, centrum grauitatis E. Quod
demon&longs;trandum erat.
Quod &longs;i quis à me quærat, cur non hic vtar quinta illa
&longs;ed illud idem propo&longs;itum vna demon&longs;tratione in planis,
altera præ&longs;enti in &longs;olidis demon&longs;trauerim. Re&longs;pondeo:
quia Propo&longs;itio quarta primi Archimedis, ex qua quinta
nece&longs;&longs;ario pendet, habet, &longs;i quis attendat, aliquas difficul
tates phy&longs;icas, quæ mathematicis rationibus non facile
di&longs;&longs;oluantur: quæ cau&longs;a igitur illum adduxit ad &longs;imile quid plicata in &longs;ecundo &longs;uo libro planorum æquiponderantium,
qua&longs;i qui quartæ, ac quintæ illi generali non &longs;atis acquie
&longs;ceret; eadem me compulit ad hoc propo&longs;itum duabus de
mon&longs;trationibus generalibus, altera de planis, altera de &longs;o
lidis grauibus &longs;ecurius demon&longs;trandum.
Quarumlibet trium magnitudinum eiu&longs;dem
generis centra grauitatis cum centro magnitudi
nis ex ijs compo&longs;itæ &longs;unt in eodem plano.
Sint quælibet tres ma
gnitudines eiu&longs;dem gene
ris A, B, C: quarum cen
tra grauitatis A, B, C. Ex
ijs autem compo&longs;itæ &longs;it
centrum grauitatis E. Di
co quatuor puncta A, B,
C, E, e&longs;&longs;e in eodem pla
no. Iungantur enim re
ctæ AB, BC, CA: & vt
e&longs;t A, ad C, ita &longs;it CD,
ad DA, & BD, iungatur: Rur&longs;us quoniam recta BD, coniungit duo centra gra
uitatis duarum magnitu
dinum B &longs;cilicet, & AC,
erit compo&longs;itæ ACB, in
recta BD, centrum graui
tatis: e&longs;t autem illud E. Quoniam igitur in quo
plano e&longs;t recta BD, in
eodem &longs;unt duo puncta
B, E, in quo autem pla
no e&longs;t recta BD, in eo
dem e&longs;t recta AC, &
puncta A, C; in quo igi
tur plano &longs;unt puncta A,
C, in eodem erunt pun
cta B, E; quatuor igitur puncta A, B, C, E, erunt in eodem
plano; Quod demon&longs;tr andum erat.
Si à cuiuslibet trianguli centro, & tribus an
gulis quatuor rectæ inter &longs;e parallelæ plano trian
guli in&longs;i&longs;tant: tres autem magnitudines æquales
habeant centra grauitatis in ijs tribus, quæ ad
angulos; trium magnitudinum &longs;imul centrum
grauitatis erit in ea, quæ ad trianguli centrum
terminatur.
Sit triangulum ABC, cuius centrum N, à tribus au
tem angulis A, B, C, & centro N, in&longs;i&longs;tant plano trian-
CF, NM, tres autem magnitudines æquales habeant cen
tra grauitatis G, H, K, in tribus AD, BE, CF. Di
co trium magnitudinum &longs;imul, quarum centra grauitatis
G, H, K, e&longs;&longs;e in linea NM. Iungantur enim rectæ GH,
H
parallela, & iungatur LH. Quoniam igitur rectæ BP, LH,
iungunt duas parallelas LP, BH; erunt quatuor rectæ BH,
LP, BP, LH, in eodem plano. Et
guli PH, &longs;ecat planum trianguli ABC, à communi autem
&longs;ectione BP, &longs;urgunt
duæ parallelæ PL, MN;
quarum PL, e&longs;t in pla
no quadranguli PH,
erit etiam MN, in eo
dem plano quadranguli
PH: & &longs;ecabit LH. &longs;e
cet in puncto O: qùare
vt LO, ad OH, ita erit
PN, ad NB, propter
parallelas: &longs;ed PN, e&longs;t
dimidia ip&longs;ius NB; er
go & LO, e&longs;t dimidia ip
&longs;ius OH. Eadem ratio
ne, quoniam AP, æqua
lis e&longs;t PC, erit & GL, æqualis LK. Duarum igitur
magnitudinum G, K, &longs;imul centrum grauitatis erit L: &longs;ed
reliquæ magnitudinis, quæ ad H, e&longs;t centrum grauitatis
H; & vt compo&longs;itum ex duabus magnitudinibus G,
K, ad magnitudinem H, ita ex contraria parte e&longs;t HO,
ad OL; Trium igitur magnitudinum G, H, K, &longs;imul cen
trum grauitatis erit O, & in linea MN. Quod demon
&longs;trandum erat.
Omnis octaedri idem e&longs;t centrum grauitatis,
& figuræ.
E&longs;to octaedrum ABCDEF, cuius centrum G.
Di
co G, e&longs;se centrum grauitatis octaedri ABCDEF. Ductis enim axibus AC, BD, EF, communis eorum
&longs;ectio erit centrum G, in quo axes bifariam &longs;ecabuntur:
omnium autem angulorum, qui ad G, bini qui que ad
verticem &longs;unt æquales, qui æqualibus altera alteri rectis
continentur; &longs;imilia igi
tur, & æqualia erunt trian
gula, nimirum EBG,
GDF, & ECG, ip&longs;i
GFA, & BCG, ip&longs;i
GDA: igitur & BCE,
ip&longs;i ADF; pyramis igi
tur EBCG, &longs;imilis, &
æqualis e&longs;t pyramidi A
DFG, quarum latera ho
mologa &longs;unt indirectum
inter &longs;e con&longs;tituta; dua
rum igitur pyramidum
EBCG, ADFG, &longs;imul centrum grauitatis erit G. Eadem ratione &longs;ex reliquarum pyramidum binis quibu&longs;
que oppo&longs;itis &longs;imul &longs;umptis centrum grauitatis erit G. Totius igitur octaedri ABCDEF, centrum grauitatis
erit G. Quod demon&longs;trandum erat.
Omnis pyramidis triangulam ba&longs;im habentis
idem e&longs;t centrum grauitatis, & figuræ.
Sit pyramis ABCD, cuius ba&longs;is triangulum ABC,
centrum autem E. Dico E, e&longs;&longs;e centrum grauitatis pyra
midis ABCD. Secta enim ABCD, pyramide in quatuor
pyramides, &longs;imiles, & æquales inter &longs;e, & toti pyramidi
ABCD, & vnum octaedrum, &longs;int eæ pyramides DKLM,
MGCH, LBGF,
AKFH. Octaedrum
autem FGHKLM,
quod dimidium erit
pyramidis ABCD, &
&longs;int axes pyramidum
DSN, DS, KO, LP,
MQ: & ARG, iunga
tur. Quoniam igitur
FH, e&longs;t parallela ip&longs;i
BC, & &longs;ecta e&longs;t BC,
bifariam in puncto G,
centra
& N, ad quæ axes KO,
DN, terminantur; manife&longs;tum hoc e&longs;t ex &longs;uperioribus:
eritque dupla AO, ip&longs;ius OR, nec non AN, dupla ip&longs;ius
NG, componendo igitur erit vt AG, ad GN, ita AR,
ad RO, & permutando, vt AG, ad AR, ita GN, ad
RO: &longs;ed AG, e&longs;t dupla ip&longs;ius AR, quoniam & AB, ip
&longs;ius AF; igitur & GN, erit dupla ip&longs;ius RO: &longs;ed & GN,
e&longs;t dupla ip&longs;ius NR, nam N, e&longs;t centrum trianguli GFH;
æqualis e&longs;t igitur NR, ip&longs;i RO, atque hinc dupla NO,
igit
AO, ad ON: igitur in triangulo ADN, erit KO, ip&longs;i
DN, parallela. Eadem ratione &longs;i iungerentur rectæ BH,
CF o&longs;tenderemus & duos reliquos axes LP, MQ, e&longs;
&longs;e axi DN parallelos: quatuor autem prædicti axes in
&longs;i&longs;tunt plano trianguli KLM, ita vt DN tran&longs;eat per
centrum S: reliqui autem KO, LP, MQ, terminentur
ad angulorum vertices K, L, M, trianguli KLM; igi
tur &longs;i tres æquales magnitudines habeant centra grauita
tis in axibus KO, LP,
tribus magnitudinibus
in axe DN erit
grauitatis. Rur&longs;us
quoniam E ponitur
erit idem E centrum
octaedri FGHKLM,
idque in axe DN: e&longs;t
autem idem
uitatis octaedri, & figu
ræ: centrum igitur E
octaedri FCHKLM
erit in axe DN. Quod
&longs;i quatuor reliquæ pyramides dempto prædicto octaedro
&longs;imiliter diuidantur, ac pyramis ABCD diui&longs;a fuit, erunt
rur&longs;us in &longs;ingulis quatuor prædictarum pyramidum &longs;in
gula octaedra centrum grauitatis habentia vnumquodque
in axe &longs;uæ pyramidis: quæ pyramides cum &longs;int inter &longs;e
æquales, earum dimidia octaedr a ip&longs;is in&longs;cripta inter &longs;e
erunt æqualia: &longs;unt autem eorum centra grauitatis in axi
bus ab&longs;ci&longs;sarum pyramidum, DS, KO, LP, MQ
axis autem DS: e&longs;t in axe DN; per ea igitur, quæ de-
dibus AFHK, FBGL, GHOM &longs;imul, centrum gra
uitatis erit in axe D
LM, & octaedri FGHKLM centra grauitatis &longs;unt
in axe DN; omnium igitur quinque octaedrorum, quæ
&longs;unt in tota pyramide ABCD &longs;imul centrum grauitatis
e&longs;t in axe DN. Quod &longs;i rur&longs;us in &longs;ingulis quatuor præ
dictarum pyramidum modo dicta ratione quina octaedra
de&longs;cripta intelligantur, &longs;imiliter o&longs;ten&longs;um erit quina octa
edra in &longs;ingulis quatuor ab&longs;ci&longs;&longs;arum pyramidum, velut
quatuor magnitudines, centra grauitatis habere in axibus
quatuor prædictarum pyramidum: &longs;unt autem hæc qua
tuor compo&longs;ita ex quinis octaedris inter &longs;e æqualia, pro
pter æqualitatem octaedrorum multitudine æqualium,
quæ æqualibus &longs;unt pyramidibus ip&longs;orum duplis ord ine
diui&longs;ionis inter &longs;e re&longs;pondentibus in&longs;cripta; igitur vt ante,
quater quinorum octaedrorum &longs;imul in axe DN erit
centrum grauitatis: &longs;ed & octaedri FGHKLM centrum
grauitatis e&longs;t in axe DN; vnius igitur & viginti octae
drorum in pyramide ABCD exi&longs;tentium ex hac &longs;ecun
da diui&longs;ione, tanquàm vnius magnitudinis in axe DN erit
centrum grauitatis. Ab hoc igitur numero vnius & vi
ginti octaedrorum in pyramide ABCD exi&longs;tentium, &longs;i
mili diui&longs;ione illius reliquarum quatuor pyramidum primo
ab&longs;ci&longs;&longs;arum procedentes, & eundem &longs;emper gyrum, quem
fecimus à quinario repetentes, poterunt e&longs;se in tota AB
CD pyramide tot, quemadmodum diximus, de&longs;cripta,
octaedra, vt eorum numerus &longs;uperet quemcumque propo
&longs;itum numerum, & omnium tanquàm vnius magnitudinis
in axe DN, &longs;it centrum grauitatis. Sic autem facienti, &
reliquarum pyramidum demptis præcedentibus octaedris,
dimidia octaedra &longs;emper auferenti, tandem relinquen
tur pyramides minores &longs;imul &longs;umptæ quantacumque
magnitudine propo&longs;ita. Totius igitur pyramidis ABCD
Eadem ratione in
quolibet reliquorum trium axium, pyramidis ABCD, ip
&longs;ius centrum grauitatis e&longs;se o&longs;tenderemus; communis igi
tur &longs;ectio quatuor axium pyramidis ABCD, quod e&longs;t
ip&longs;ius centrum E, erit centrum grauitatis pyramidis AB
CD. Quod demon&longs;trandum erat.
Hinc manife&longs;tum e&longs;t centrum grauitatis pyra
midis triangulam ba&longs;im habentis e&longs;&longs;e in eopun
cto, in quo axis &longs;ic diuiditur, vt pars quæ ad ver
icem &longs;it reliquæ tripla.
Ominis pyramidis ba&longs;im plu&longs;quam trilate
ram habentis centrum grauitatis axim ita diui
dit, vt pars, quæ e&longs;t ad verticem &longs;it tripla re
liquæ.
Sit pyramis ABCDE, cui vertex E, ba&longs;is autem
quadrilatera ABCD, & e&longs;to axis EF, &longs;egmentum EM,
reliqui MF, triplum. Dico punctum M, e&longs;&longs;e centrum
grauitatis pyramidis ABCDE. Ducta enim AC, &longs;it
trianguli ABC, centrum grauitatis H, &longs;icut & K, trian
guli ACD: & iungantur KH, HE, EK: Factaque vt
EM, ad MF, ita EL ad LH, & EN ad N
gatur LN. Quoniam igitur EF e&longs;t axis pyramidis
ABCDE, erit ba&longs;is ABCD centrum grauitatis F.
lorum ABC, CDA, erunt EH, EK, axes pyramidum
ABCE, ACDA: quorum EL, e&longs;t tripla ip&longs;ius LH,
nec non EN, tripla ip&longs;ius EK; pyramidis igitur ABCE,
centrum grauitatis erit L, &longs;icut & K, pyramidis ACDE.
Rur&longs;us, quoniam totius quadrilateri ABCD, e&longs;t cen
trum grauitatis F, cuius magnitudinis partium triangu
lorum ABC, CDA, centra grauitatis &longs;unt K, H; recta
KH, à puncto F, &longs;ic
diuiditur, vt &longs;it HF, ad
FK, vt triangulum
ACD, ad triangulum
ABC, hoc e&longs;t, vt py
ramis ACDE, ad py
ramidem ABCE. &longs;ed
vt HF, ad FK, ita
e&longs;t LM, ad MN; vt
igitur e&longs;t pyramis AC
DE, ad pyramidem
ABCE, ita erit LM,
ad MN. Sed N, e&longs;t
centrum grauitatis py
ramidis ACDE, & L pyramidis ABCE; punctum
igitur M, erit centrum grauitatis pyramidis ABCDE. Quod &longs;i pyramis habeat ba&longs;im quinquelateram; po&longs;ito
rur&longs;us axe totius pyramidis, & ba&longs;i &longs;ecta in triangulum,
& quadrilaterum, po&longs;itis vtriu&longs;que proprijs centris graui
tatis, eadem demon&longs;tratione propo&longs;itum concludetur. Quemadmodum &longs;i ba&longs;is &longs;it &longs;ex laterum, &longs;ecta ea in quinque
laterum, & triangulum, & reliquis vt antea po&longs;itis: & &longs;ic &longs;em
per deinceps. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis pri&longs;matis triangulam ba&longs;im habentis
centrum grauitatis e&longs;t in medio axis.
Sit pri&longs;ma ABCDEF, cuius ba&longs;es oppo&longs;itæ trian
gula ABC, DEF, axis autem GH, &longs;ectus &longs;it bifariam
in puncto K. Dico punctum K, e&longs;se pri&longs;inatis ABCD
EF, centrum grauitatis. Ducantur enim rectæ FGO,
CHP, PO. Quoniam igitur GH, e&longs;t axis pri&longs;matis
ABCDEF, erit punctum G, centrum grauitatis trian
guli DEF: &longs;icut & H, trian
guli ABC; vtraque igitur
dupla e&longs;t AG, ip&longs;ius GO,
& CH, ip&longs;ius PH, &longs;ectæ
que erunt AB, DE, bifa
riam in punctis P, O: pa
rallela igitur, & æqualis e&longs;t
OP, ip&longs;i DA, iamque ip&longs;i
FC. quæ igitur illas con
iungunt CP, FO, æqua
les &longs;unt, & parallelæ, & pa
rallelogrammum FP. Nunc &longs;ecta OP, bifariam in
puncto N, iungantur GN,
NF, AF, FH, FB, & fa
cta FL, tripla ip&longs;ius LH,
à puncto L, per punctum K, ducatur recta LKMR. Quoniam igitur e&longs;t vt FG, ad GO, ita CH, ad HP,
& parallelogrammum e&longs;t FCPO; parallelogramma
etiam erunt CG, GP, angulus igitur FGH, æqualis
erit angulo NGO, quos circa æquales angulos latera
dupla e&longs;t FG, ip&longs;ius GO, & GH, ip&longs;ius ON; angulus
igitur OGN, æqualis erit angulo GFH; parallela igi
tur GN, ip&longs;i FH, & propter&longs;imilitudinem triangulorum
dupla erit FH, ip&longs;ius GN. Rur&longs;us, quoniam recta
OP, &longs;ecat latera oppo&longs;ita parallelogrammi BD, bifa
riam in punctis O, P, &longs;ecta, & ip&longs;a bifariam in puncto N,
erit punctum N, parallelogrammi BD, centrum graui
tatis, atque ideo axis FN, pyramidis ABDEF. qua
ratione erit quoque axis FH, pyramidis ABCF: &longs;ed
FL, e&longs;t tripla ip&longs;ius LH; pyramidis igitur ABCF, cen
trum grauitatis erit L. Rur&longs;us quia e&longs;t vt GK, ad KH,
ita GR, ad LH, propter &longs;imilitudinem triangulorum,
erit æqualis GR, ip&longs;i LH: &longs;ed e&longs;t FH, quadrupla ip-,
&longs;ius LH, quadrupla igitur FH, ip&longs;ius GR: &longs;ed FH
erat dupla ip&longs;ius GN; quadrupla igitur FH, reliquæ
NR, ac proinde GR, RN, æquales erunt: recta igitur
FL, tripla erit vtriu&longs;que ip&longs;arum GR, RN, &longs;ed vt FL,
ad NR, ita e&longs;t FM, ad MN, propter &longs;imilitudinem trian
gulorum; recta igitur FM, erit ip&longs;ius MN, tripla, &longs;icut
& LM, ip&longs;ius MR: &longs;ed quia KH, e&longs;t æqualis GK,
erit & LK, æqualis RK; propter &longs;imilitudinem trian
gulorum; cum igitur LK, &longs;it tripla ip&longs;ius MR, erit LK,
ip&longs;ius KM, dupla; vt igitur e&longs;t pyramis ABEDF, ad
pyramidem ABCF, ita erit LK, ad KM; e&longs;t autem M,
centrum grauitatis pyramidis ABED, &longs;icut & L, pyrami
dis ABCF; totius igitur pri&longs;matis ABCDEF, centrum
grauitatis erit K. Quod demon&longs;trandum erat.
Omnis pri&longs;matis ba&longs;im plu&longs;quam trilateram
habentis centrum grauitatis e&longs;t in medio axis.
Sit pri&longs;ma ABCDEFGH, ba&longs;im habens quadrila
teram ABCD: axis autem
cto M. Dico punctum M, e&longs;se centrum grauitatis pri&longs;
matis ABCDEFGH. Iungantur enim rectæ BD, FH,
vt parallelogrammum &longs;it BH, &longs;ectumque totum pri&longs;ma
in duo pri&longs;mata, quorum ba
&longs;es &longs;unt triangula, in quæ &longs;ecta
&longs;unt quadrilatera AC, EG,
&longs;int autem axes duorum pri&longs;
matum triangulas ba&longs;es ha
bentium NO,
igitur centra grauitatis O, tri
anguli ABD, & L, quadri
lateri AC, & Q, trianguli
BCD, itemque N, trianguli
EFH, & K, quadrilateri EG,
& P, trianguli FGH: iun
ctæ igitur OQ, NP, per pun
cta L, K, tran&longs;ibunt: cumque tres prædicti axes &longs;int
lateribus pri&longs;matis, atque ideo inter &longs;e quoque paralleli;
parallelogramma erunt OP, NL, LP. ducta igitur per
punctum M, ip&longs;i OQ, vel NP, parallela RS, erit vt
NK, ad KP, ita RM, ad MS: & vt KM, ad ML, ita
NR, ad RO, & PS, ad SQ: &longs;ed KM, e&longs;t æqualis ML;
igitur & KR, ip&longs;i RO, & PS, ip&longs;i SQ, æqualis erit: &longs;unt
autem hæ &longs;egmenta axium NO,
R, e&longs;t centrum grauitatis pri&longs;matis ABDEFH: & per Quoniam igitur
quadrilateri EG, e&longs;t centrum grauitatis K, cuius duorum
triangulorum centra grauitatis &longs;unt P, N; erit vt triangu
lum FGH, ad triangulum EFH, hoc e&longs;t vt pri&longs;ma BC
DFGH, ad pri&longs;ma ABDEFH, ita NK, ad KP, hoc
e&longs;t RM, ad MS; cum igitur &longs;it R, centrum grauitatis
pri&longs;matis ABDEFH: &longs;icut & S, pri&longs;matis BCDFGH;
totius pri&longs;matis ABCDEFGH, centrum grauitatis erit
M. Quod &longs;i pri&longs;ma ba&longs;im habeat quinquelateram; ab
&longs;ci&longs;so rur&longs;us pri&longs;mate vno triangulam ba&longs;im habente,
&longs;umpti&longs;que axibus pri&longs;inatum, quorum alterum habebit
ba&longs;im quadrilateram, eadem demon&longs;tratione propo&longs;itum
concluderemus, & &longs;ic deinceps in aliis. Manife&longs;tum e&longs;t
igitur propo&longs;itum.
Omnis fru&longs;ti pyramidis triangulam ba&longs;im
ha bentis centrum grauitatis e&longs;t in axe, primum
ita diui&longs;o, vt &longs;egmentum attingens minorem
ba&longs;im &longs;it ad reliquum, vt duplum vnius laterum
maioris ba&longs;is vna cum latere homologo mino
ris, ad duplum prædicti lateris minoris ba&longs;is,
vna cum latere homologo maioris. Deinde
à puncto &longs;ectionis ab&longs;ci&longs;sa quarta parte &longs;eg
menti, quod maiorem ba&longs;im attingit, & à pun
cto, in quo ad minorem ba&longs;im axis termina
tur &longs;umpta item quarta parte totius axis; in
eo puncto, in quo &longs;egmentum axis duabus po
&longs;terioribus &longs;ectionibus finitum &longs;ic diuiditur, vt
tum prædictum interiectum &longs;egmentum, vt tertia
proportionalis minor ad duo latera homologa ba
&longs;ium oppo&longs;itarum, ad compo&longs;itam ex his tribus
deinceps proportionalibus.
Sit pyramidis fru&longs;tum, cuius ba&longs;es oppo&longs;itæ, & parallelæ,
maior triangulum ABC, minor autem triangulum DEF,
axis autem GH. triangulorum autem ABC, DEF, quæ
inter &longs;e &longs;imilia e&longs;se nece&longs;se e&longs;t, &longs;int duo latera homologa
BC, EF: & vt e&longs;t BC, ad EF, ita &longs;it EF, ad X: vt autem e&longs;t
duplum lateris BC, vna cum latere EF, ad duplum lateris
EF, vna cum la
tere BC, ita &longs;it
HN, ad NG,
& NO, pars quar
ta ip&longs;ius NG, &
HS, pars quar
ta ip&longs;ius GH; ip
&longs;ius autem SO,
&longs;it VO, ad OS,
vt e&longs;t X, ad com
po&longs;itam ex tri
bus BC, EF, X. Dico punctum V
(quod cadet ne
ce&longs;sario infra
punctum N, quanquam hoc ad demon&longs;trationem nihil re
fert) e&longs;se centrum grauitatis fru&longs;ti ABCDEF. Ducta
enim recta AGL; quoniam GH, e&longs;t axis fru&longs;ti ABCD
EF, & punctum G, centrum grauitatis trianguli ABC,
erit punctum L, in medio ba&longs;is BC: &longs;ecto igitur etiam la
tere EF, bifariam in puncto K, iungantur LK,
gatur: & vt e&longs;t GO, ad ON, ita fiat GP, ad PM, & iun
gantur MN, OP, FG, GD, GE. Quoniam igitur re
cta KL, &longs;ecat trapezij BCFE, latera parallela bifariam
in punctis K,L, & e&longs;t vt HN, ad NG, hoc e&longs;t vt duplum
lateris BC, vna cum latere EF, ad duplum lateris EF, vna
cum latere BC, ita KM, ad ML; erit punctum M, cen
trum grauitatis trapezij BCFE, & pyramidis GBCFE,
axis GM. Et quoniam vt GO, ad ON, ita e&longs;t GP, ad
PM, atque ideo GP, tripla ip&longs;ius PM, erit punctum P,
centrum grauitatis pyramidis GBCFE, atque ideo in
linea OP. Rur&longs;us quoniam angulus ACB; æqualis e&longs;t
angulo DFK: & vt AC, ad CK, ita e&longs;t DF, ad FK:
e&longs;t autem DF, parallela ip&longs;i AC, & FK, ip&longs;i CL; erit
reliqua DK, reliquæ AL, parallela; vnum igitur planum
e&longs;t, ADKL, in quo iacet triangulum GMN; cum igitur
&longs;it parallela KH, ip&longs;i GL, vtque HN, ad NG, ita
parallela ip&longs;i MN; &longs;ecant enim latera trianguli GMN,
in ea&longs;dem rationes; igitur OP, erit LG, parallela. Simi
liter ex puncto O, ad axes duarum pyramidum GABED,
GACFD, duæ aliæ rectæ lineæ ducerentnr, quas & cen
tra grauitatis pyramidum habere, & parallelas rectis GQ,
GR, alteram alteri e&longs;se o&longs;tenderemus, &longs;icut o&longs;tendimus
OP, habentem centrum grauitatis pyramidis GBCFE,
ip&longs;i GL, parallelam; &longs;ed tres rectæ GL, GQ, GR, &longs;unt
in eodem plano trianguli nimirum ABC; tres igitur præ
dictæ parallelæ, quæ ex puncto O, atque ideo trium præ
dictarum pyramidum centra grauitatis erunt in eodem pla
no, per punctum O, & trianguli ABC, parallelo. Quo
niam igitur fru&longs;ti ABCDE, centrum grauitatis e&longs;t in axe
GH; (manife&longs;tum hoc autem ex duobus centris grauitatis
pyramidis, cuius e&longs;t prædictum fru&longs;tum, & ablatæ, quæ
centra grauitatis &longs;unt in axe, cuius &longs;egmentum e&longs;t axis
O. Rur&longs;us quoniam vt tres deinceps proportionales BC,
EF, X, &longs;imul ad BC, ita e&longs;t fru&longs;tum ABCDEF, ad py
ramidem; &longs;i de&longs;cribatur ABCH: &longs;ed vt triangulum ABC,
ad &longs;imile triangulum EDF, hoc e&longs;t vt BC, ad X, ita e&longs;t
pyramis ABCH, ad pyramidem GDEF; erit ex æqua
li, vt tres lineæ
BC, EF, X, &longs;i
mul ad X, ita fru
&longs;tum ABCDEF,
ad pyramidem
GDEF: & con
uertendo, vt X,
ad compo&longs;itam
ex BC, EF, X,
hoc e&longs;t vt VO,
ad OS, ita pyra
mis GDEF, ad
fru&longs;tum ABC
DEF; & diui
dendo, vt pyra
mis GDEF, ad reliquas tres pyramides fru&longs;ti, ita OV,
ad VS; &longs;ed S, e&longs;t centrum grauitatis pyramidis GDEF,
& O, trium reliquarum; fru&longs;ti igitur ABCDEF, cen
trum grauitatis erit V. Quod demon&longs;trandum erat.
Omnis fru&longs;ti pyramidis ba&longs;im plu&longs;quam trila
teram habentis centrum grauitatis e&longs;t punctum
illud, in quo axis &longs;ic diuiditur, vt axis fru&longs;ti pyra
midis triangulam ba&longs;im habentis diuiditur ab
ip&longs;ius centro grauitatis.
Sit pyramidis quadrilateram ba&longs;im habentis fru&longs;tum
ABCDEFGH, cuius axis KL, atque in ip&longs;o centrum
grauitatis O. Dico axim KL, &longs;ectum e&longs;se in puncto O,
vt propo&longs;uimus. Ductis enim AC, EG, quæ &longs;imilium
&longs;ectionum angulos æquales &longs;ubtendant B, F, qui late
ribus homologis continentur, fru&longs;ta erunt pyramidum
triangulas ba&longs;es habentium AFG, AGH: &longs;it autem fru
&longs;ti AFG, axis
TP, & in eo eiu&longs;
dem fru&longs;ti cen
trum grauitatis
M, & fru&longs;ti AG
H, axis VQ, &
in eo centrum
grauitatis N, &
iungantur TV,
MN,
niam igitur e&longs;t
pyramidis fru
&longs;tum, quod pro
ponitur; omnia
cius producta latera concurrent in vno puncto, qui e&longs;t pyra
midis vertex: fru&longs;ta igitur, in quæ diui&longs;um e&longs;t fru&longs;tum pro
po&longs;itum earum &longs;unt pyramidum, quæ verticem habent
communem cum pyramide, cuius e&longs;t fru&longs;tum propo&longs;itum:
tres igitur talium fru&longs;torum axes, vt pote &longs;egmenta axium
trium prædictarum pyramidum in communi illo vertice
concurrent: quilibet igitur duo trium prædictorum axium
KL, TP, VQ, erunt in eodem plano: TP, igitur, &
VQ, &longs;unt in eodem plano. Eadem autem ratione, qua
vtebamur de pri&longs;mate K, centrum grauitatis K, ba&longs;is
EH, e&longs;t in linea TV, & L, ba&longs;is BD, centrum grauita
tis e&longs;t in linea
dem plano trapezij PTVQ, &longs;eque mutuo &longs;ecabunt: cum
tium pri&longs;matum AFG, AGH, atque obid O, totius pri&longs;
matis AFGH, in linea MN, centrum grauitatis; per pun
ctum O, recta MN, tran&longs;ibit. Et quoniam planum tra
pezij PV, &longs;ecatur duobus planis parallelis, erunt TV, PQ,
fectiones parallelæ. His demon&longs;tratis, fiat rur&longs;us vt AB,
bis vna cum EF, ad EF, bis vna cum AB, ita TY, ad
YP: & &longs;umatur T
quarta ip&longs;ius PY, & ad axim KL, ducantur ip&longs;is TV,
PQ, parallelæ
quæ rectas TP,
KL, &longs;ecabunt in
vt igitur TY, ad
AB, bis vna cum
EF, ad EF bis
vna cum AB, ita
erit
eritque KS, pars
quarta ip&longs;ius K
L, qualis & R
X, ip&longs;ius RL. Et quoniam M, e&longs;t centrum grauitatis fru
&longs;ti AFG; manife&longs;tum e&longs;t ex tribus prædictis axis TP, &longs;e
ctionibus
vt e&longs;t 6 ad compo&longs;itam ex tribus deinceps proportionalibus
AB, EF, 6; Fru&longs;ti igitur ABCDEFGH, centrum gra
uitatis O, axim KL, ita diuidit, vt propo&longs;uimus. Quod
&longs;i fru&longs;tum propo&longs;itum &longs;it pyramidis ba&longs;im habentis quin
quelateram, & quotcumque plurium deinceps fuerit la
terum, eadem demon&longs;tratione &longs;emper deinceps, vt in pri&longs;
mate monuimus, propo&longs;itum concluderemus.
Dodecaedri, & ico&longs;aedri idem e&longs;t centrum gra
uitatis, & figuræ.
Nam huiu&longs;modi figuras habere axes, qui omnes &longs;e &longs;e
bifariam &longs;ecant; (tale autem &longs;ectionis punctum centrum e&longs;t)
con&longs;tat ex talium corporum in &longs;phæra in&longs;criptione in de
cimotertio Euclidis Elemento: nec non omnem pyrami
dem, cuius vertex e&longs;t dodecaedri, vel octaedri centrum
idem cum centro &longs;phæræ, vt con&longs;tat ex ij&longs;dem Euclidis in
&longs;criptionibus; ba&longs;is autem triangulum æquilaterum, vel
pentagonum, vna ex ba&longs;ibus corporum prædictorum, ha
bere pyramidem oppo&longs;itam &longs;imilem ip&longs;i, & æqualem, cuius
latera eius lateribus homologis &longs;unt in directum po&longs;ita,
ba&longs;is autem triangulum, vel pentagonum, quale diximus;
Eadem igitur ratione, qua v&longs;i &longs;umus ad demon&longs;trandum
centrum grauitatis, & parallelepipedi, & octaedri, propo
&longs;itum concluderemus.
Data qualibet figura, cuius termini omnis
cauitas &longs;it interior, &longs;i certum in ea punctum talis
cius partis centrum grauitatis e&longs;se po&longs;sit, quæ ab
ca deficiat minori &longs;pacio quantacumque magnitu
dine propo&longs;ita; illud erit totius figuræ centrum
grauitatis.
E&longs;to figura AB, cuius termini omnis cauitas &longs;it interior
& certum in ea punctum E, talis partis AB, figuræ qua
lem diximus centrum grauitatis e&longs;se po&longs;sit. Dico pun
ctum E, e&longs;se figuræ AB, centrum grauitatis. Si enim
E, non e&longs;t, erit aliud, e&longs;to F: & iuncta EF producatur,
& &longs;umatur in illa extra figuræ AB, terminum, quodlibet
punctum G; & vt e&longs;t FE, ad EG, ita &longs;it alia magnitudo
K, ad figuram AB, &
ex vi hypothe&longs;is &longs;it pars
quædam CD, figuræ
AB, cuius centrum gra
uitatis E, talis vt abla
ta relinquat AC, minus
magnitudine Mi
nor igitur proportio erit
AC, ad AB, quàm K,
ad AB, hoc e&longs;t quàm
FE, ad EG; fiat vt
AC, ad AB, ita EF,
ad FGH: &longs;ed F, e&longs;t cen
trum grauitatis totius
AB, & E, vnius par
tis CD; reliquæ igitur
partis AC, centrum grauitatis erit H, vltra punctum G: &longs;ed
G, cadit extra terminum figuræ AC; multo igitur magis H:
Quod e&longs;t ab&longs;urdum. Non igitur aliud punctum à puncto
E; punctum igitur E, figuræ AB, erit centrum grauitatis
Quod demon&longs;trandum erat.
Omnis coni centrum grauitatis axim ita diui
dit, vt &longs;egmentum ad verticem &longs;it reliqui triplum.
Sit conus ABC, cuius vertex B, axis autem BD, cu
ius BE, &longs;it tripla ip&longs;ius ED. Dico punctum E, e&longs;se co
ni ABC, centrum grauitatis. Si enim cono ABC, pyramis
in&longs;cribatur, cuius ba&longs;is in&longs;cripta circulo AC, æquilatera &longs;it,
& æquiangula, eius centrum grauitatis erit idem quod &
figuræ centrum, &longs;ed centrum
talis figuræ circulo in&longs;criptæ
idem e&longs;t, quod centrum cir
culi, vt colligitur ex demon
&longs;trationibus quarti Elemen
torum; in&longs;criptæ igitur pyra
midis erit axis BD, & cen
trum grauitatis E. talis au
tem ea pyramis in&longs;cribi po
te&longs;t, vt à cono deficiat mino
ri &longs;pacio quantacumque ma
gnitudine propo&longs;ita; igitur
ABC, coni centrum graui
tatis erit E. Quod demon&longs;trandum erat.
Omnis fru&longs;ti conici centrum grauitatis idem
e&longs;t in axe centro grauitatis fru&longs;ti pyramidis ba&longs;im
habentis æquilateram, & æquiangul am in &longs;criptæ
cono, ab &longs;ci&longs;&longs;i eodem plano, quo coni fru&longs;tum.
Sit coni fru&longs;tum ABCD, cuius axis EF, fru&longs;to autem
ABCD, intelligatur in&longs;criptum fru&longs;tum pyramidis in&longs;cri
ptæ cono AHD, à quo ab&longs;ci&longs;sum e&longs;t fru&longs;tum ABCD,
ba&longs;im habentis æquilateram, & æquiangulam in&longs;criptam
circulo AD: quare eius centrum grauitatis, & figuræ erit
punctum F, vt diximus in præcedenti, axis autem FH, &longs;i
cut etiam pyramidis ab&longs;ci&longs;sæ vna cum cono BHC, axis
EH, quare & reliqui fru&longs;ti pyramidis axis erit EF, igi
tur in EF, &longs;it fru&longs;ti in&longs;cripti fru&longs;to ABCD, centrum gra
uitatis G. Dico punctum G, e&longs;se centrum grauitatis fru
&longs;ti ABCD. Ponatur enim
FL, pars quarta ip&longs;ius FH,
necnon EK, pars quarta ip
&longs;ius EH: punctum igitur K,
e&longs;t centrum grauitatis pyra
midis, & coni BHC, &longs;icut
& punctum L, pyramidis, &
coni AHD. cum igitur fru
&longs;ti pyramidis fru&longs;to ABCD,
in&longs;cripti &longs;it centrum grauita
tis G; erit vt GL, ad LK,
ita pyramis BHC, ad pyra
midis fru&longs;tum fru&longs;to ABCD,
in&longs;criptum: &longs;ed vt pyramis
BHC, ad pyramidis fru&longs;tum
fru&longs;to ABCD, in&longs;criptum,
ita e&longs;t diuidendo, conus BHC, ad fru&longs;tum ABCD, pro
pter eandem triplicatam communium conis, & pyramidi
bus &longs;imilibus laterum homologorum proportionem; vt igi
tur GL, ad LK, ita erit conus BHC: ad fru&longs;tum ABCD:
&longs;ed coni BHC, centrum grauitatis erat K, & coni AHD,
centrum grauitatis L; fru&longs;ti igitur ABCD, centrum gra
nitatis erit G. Quod demon&longs;trandum erat.
Omnis cylindri centrum grauitatis axim bifa
riam diuidit.
Sit cylindrus ABCD, cuius axis EF, & &longs;it &longs;ectus bi
fariam in puncto G. Dico punctum G, e&longs;se centrum
grauitatis cylindri ABCD. Nam &longs;i cylindro AD, in
&longs;criptum intelligatur pri&longs;ma,
cuius ba&longs;es oppo&longs;itæ æquilate
ræ &longs;int, & æquiangulæ; erunt,
qua ratione &longs;upra diximus, ea
rum centra figuræ, & grauitatis
E, F; axis igitur in&longs;cripti pri&longs;
matis erit EF: & centrum gra
uitatis G. pote&longs;t autem tale
pri&longs;ma &longs;ic in&longs;cribi cylindro
ABCD, vt ab illo deficiat
minori &longs;pacio quantacumque
magnitudine propo&longs;ita; cylin
dri igitur ABCD, centrum
grauitatis erit G. Quod demon&longs;trandum erat.
Sphæræ, & &longs;phæroidis idem e&longs;t centrum gra
uitatis, & figuræ.
Sit &longs;phæra, vel &longs;phæroides ABCD, cuius centrum E,
e&longs;se E. Sint enim bini axes &longs;phæræ, vel &longs;phæroidis inter
&longs;e ad rectos angulos; & in &longs;phæroide &longs;it maior diameter
BD, minor AC, per binos autem hos axes plana tran
&longs;euntia ad eos axes erecta, &longs;ecent &longs;phæram, vel &longs;phæroidem. Qua ratione axes dimidij erunt axes hemi&longs;phærij, vel he
mi&longs;phæroidis: hemi&longs;phærium autem, & &longs;phæroidis e&longs;t fi
gura circa axim in alteram partem deficiens, qualium om
nium figurarum centrum grauitatis e&longs;t in axe; igitur hemi
&longs;phærij, vel hemi&longs;phæroidis ABCD, centrum grauitatis
e&longs;t in axi BE, &longs;icut & reliqui ADA, in axi ED; totius
igitur &longs;phæræ, vel &longs;phæroidis ABCD centrum grauitatis
e&longs;t in axi BD. Eadem ratione & in axi AC; in communi
igitur &longs;ectione centro E. Quod demon&longs;trandum erat.
PRIMI LIBRI FINIS.
LVC AE
VALERII
DE CENTRO
GRAVITATIS
SOLIDORVM
Si duæ magnitudines vnà maio
res, vel minores prima, & ter
tia minori exce&longs;&longs;u, vel defe
ctu
ne propo&longs;ita eiu&longs;dem generis
cum illa, ad quam refertur,
eandem
rint, maior vel minor prima ad &longs;ecundam, & vnà
maior, vel minor tertia ad quartam; erit vt prima
ad &longs;ecundam, ita tertia ad quartam.
Sint quatuor magnitudines A prima, B &longs;ecunda, C ter
tia, & D quarta: quantacumque autem magnitudine propo
&longs;ita, ex infinitìs quæ proponi po&longs;&longs;unt eiu&longs;dem generis cum
A, C, vel vna tantum, &longs;i AC &longs;int eiu&longs;dem generis: vel
vna, & altera; &longs;i vna vnius, altera &longs;it alterius generis; &longs;emper
aliæ duæ magnitudines vnà maiores, quàm AC, minori
exce&longs;su magnitudine propo&longs;ita; eandem habeant proportio
nem, maior quàm A ad B, & maior quàm C ad D. Dico
e&longs;se vt A ad B, ita C ad D. Po&longs;ita enim E ad D, vt
A ad B, & F maiori quàm C vtcumque, &longs;int aliæ duæ ma
gnitudines, G maior quàm A minori exce&longs;su magnitudine
eiu&longs;dem generis cum A, quam quis voluerit, & H maior
quàm C minori exce&longs;su quàm
quo F &longs;uperat C, ide&longs;t, quæ ma
ior &longs;it quàm C, & minor quàm
F: &longs;it autem vt G ad B, ita H
ad D. Quoniam igitur F maior
e&longs;t, <34>H, maior erit proportio
ip&longs;ius F quàm H ad D, hoc e&longs;t
quàm G ad B. Sed
&longs;it quàm A, maior e&longs;t proportio
G ad B, quàm A ad B, multo igitur erit maior proportio F
ad D, quàm A ad B. Sed F ponitur maior quàm C, vtcum
que; nulla igitur magnitudo maior quàm C e&longs;t ad D, vt
A ad B: &longs;ed E ad D, e&longs;t vt A ad B; non igitur e&longs;t E ma
ior quàm C; nec maior proportio E ad D, hoc e&longs;t A ad
B, quàm C ad D. Eadem autem ratione nec maior erit
proportio C ad D quàm A ad B, hoc e&longs;t non minor A
ad B, quàm C ad D; eadem igitur proportio A ad B,
quæ C ad D.
Sed aliæ duæ magnitudines vnà minores quàm A, C
minori defectu quantacumque magnitudine propo&longs;ita,
eandem habeant proportionem, minor quàm A ad B, &
minor quàm C, ad D. Dico e&longs;se vt A ad B, ita C ad D.
C vtcumque, &longs;it G minor quam A, minori defectu magni
tudine eiu&longs;dem generis cum A, quam quis voluerit, & H
minor quàm C, & maior quàm F: &longs;it autem vt G ad B, ita
H ad D. Quoniam igitur F minor e&longs;t quàm H, minor erit
proportio ip&longs;ius F
hoc e&longs;t <34>G ad B: &longs;ed cum G &longs;it
minor <34>A, minor e&longs;t propor
tio G ad B, quàm A ad B; mul
to ergo minor proportio F ad
D, quàm A ad B: &longs;ed F poni
tur minor quàm C vtcumque;
nulla igitur magnitudo minor
quàm C e&longs;t ad D, vt A ad B: &longs;ed E e&longs;t ad D, vt A ad B:
non igitur e&longs;t E minor quàm C, nec minor proportio E ad
D, hoc e&longs;t A ad B, quàm C ad D. eadem autem ratione
non minor erit proportio C ad D, quàm A ad B; hoc e&longs;t
non maior A ad B, quàm C ad D; vt igitur A ad B, ita
e&longs;t C ad D. Quod demon&longs;trandum erat.
Dico e&longs;se vt A ad B, ita C ad
D. Si enim fieri pote&longs;t, &longs;it minor
proportio A ad B quàm C ad D.
alia igitur aliqua magnitudo G
maior quàm A, eandem habebit
proportionem ad B, quam C ad
D. Sit autem F maior quam C
minori exce&longs;su magnitudine,
quis voluerit, & E maior quàm
A, & minor quàm G: vt autem
E ad B, ita F ad D. Quoniamigitur F maior e&longs;t quàm
C, maior erit proportio F ad D, quàm C ad D. Sed vt
F ad D, ità e&longs;t E ad B: & vt C ad D, ita G ad B; maior
maior erit quàm G minor maiori, quod fieri non pote&longs;t. Non igitur minor e&longs;t proportio A ad B quàm C ad D.
Eadem autem ratione non minor erit proportio C ad D,
quàm A ad B, hoc e&longs;t non maior A ad B, quàm C ad D;
eadem igitur proportio A ad B, quæ C ad D.
In &longs;ecunda autem hypothe&longs;is parte, quæ pertinet ad mi
norem
quàm C ad D. erit igitur, & &longs;it aliqua alia magnitudo G
minor quàm A ad B, vt C ad D. Sit autê F minor quàm
C minori defectu magnitudine,
quam quis voluerit, & E minor
quàm A, & maior quàm G, vt au
tem E ad B ita F ad D. Quoniam
igitur maior e&longs;t proportio C ad D,
quàm F ad D: &longs;ed vt C ad D, ita
e&longs;t G ad B: & vt F ad D, ita E ad
B: maior erit proportio G ad B
quàm E ad B; quamobrem erit
G maior quàm E, minor maiori,
quod fieri non pote&longs;t; non igitur ma
ior e&longs;t proportio A ad B, quàm C ad D. Eadem autem ra
tione non maior erit proportio C ad D, quàm A ad B, hoc
e&longs;t non minor A ad B, quàm C ad D. Eadem igitur erit
proportio A ad B, quæ C ad D. Quod
Si maior, vel minor prima ad vnà maiorem, vel
minorem &longs;ecunda, minori
fectu
rit vt tertia ad quartam; erit vt prima ad &longs;ecun
dam, ita tertia ad quartam.
Sint quatuor magnitudines, A prima, B &longs;ecunda, C ter
tia, & D quarta: & aliæ duæ magnitudines E
F vnà maiores quàm A, B minori exce&longs;su
quantacumque magnitudine propo&longs;ita eiu&longs;
dem generis cum ip&longs;is A, B. Sit autem E
maior quàm A, ad F maiorem quàm B, vt
C ad D. Dico e&longs;se A ad B, vt C ad
D. E&longs;to enim, quod fieri pote&longs;t, alia ma
gnitudo G eiu&longs;dem generis cum EF ad
aliam H, vt C ad D, vel E ad F. Quoniam
igitur e&longs;t permutando vt E ad G, ita F ad H,
& &longs;unt EF vnà maiores quàm AB minori ex
ce&longs;su quantacumque magnitudine propo&longs;i
ta; erit per antecedentem, vt A ad G, ita B
ad H: & permutando A ad B, vt G ad H,
hoc e&longs;t vt C ad D. Idem autem &longs;imiliter o&longs;ten
deremus po&longs;itis EF minoribus quàm AB, &
proportionalibus vt
Ij&longs;dem po&longs;itis, &longs;i non e&longs;t A ad
B, vt C ad D; vel igitur ma
ior vel minor erit proportio A
ad B quàm C ad D: &longs;it autem
maior: vt igitur A ad B, ita erit
eadem A ad E&longs;to illa E. &longs;intque aliæ duæ ma
gnitudines, G maior quàm A
minori exce&longs;su magnitudine eiu&longs;dem generis cum A,
quam quis voluerit, & F maior quàm B, & minor quàm
E. &longs;it autem G ad F vt C ad D. Quoniam igitur & vt
C ad D, ita e&longs;t A ad E; erit vt G ad F, ita A ad E; &
permutando vt G ad A, ita F ad E: &longs;ed G e&longs;t maior
E, minor maiori, quod e&longs;t ab
&longs;urdum. Non igitur maior e&longs;t
proportio A ad B quàm C ad
D: eadem autem ratione non
maior erit proportio B ad A
D ad C, hoc e&longs;t non minor A
ad B, quàm C ad D; e&longs;t igitur
A ad B, vt C ad D.
Rur&longs;us in &longs;ecunda parte hypothe&longs;is, quæ attinet ad mi
norem defectum: &longs;i non e&longs;t A ad B vt C ad D; e&longs;to, &longs;i fie
ri pote&longs;t, minor proportio A ad B quàm C ad D. igitur A
ad aliam quam B minorem eandem habebit
quam C ad D, e&longs;to illa E: &longs;intque
aliæ duæ magnitudines, G minor
quàm A minori defectu magnitudi
ne eiu&longs;dem generis cum A, quam
quis voluerit, & F minor quàm B,
& maior quàm E: &longs;it autem G ad
F, vt C ad D, hoc e&longs;t vt A ad E. Quoniam igitur permutando e&longs;t vt
G ad A, ita F ad E, & G e&longs;t mi
nor quàm A; erit & F minor quàm E, maior mino
ri, quod e&longs;t ab&longs;urdum; non igitur minor e&longs;t proportio
A ad B quàm C ad D: eadem autem ratione non minor
erit proportio B ad A, quàm D ad C, hoc e&longs;t non maior
A ad B, quàm C ad D; e&longs;t igitur A ad B vt C ad D. Quod demon&longs;trandum erat.
Si maior, vel minor prima ad vnà maiorem, vel
minorem &longs;ecunda, minori exce&longs;&longs;u, vel defectu
tam habuerit proportionem; prima ad &longs;ecundam
eandem nominatam habebit proportionem.
Sint duæ magnitudines A, B duarum autem aliarum
EF vnà maiorum, vel minorum quàm AB minori ex
ce&longs;su vel defectu quantacumque magnitudine propo
&longs;ita, habeat E maior vel minor quàm A ad F vnà
maiorem, vel minorem quàm B certam ali quam nomina
tam proportionem, verbi gratia, &longs;e&longs;quialteram. Dico A
ad B, eandem nominatam habere proportionem: vt A
ip&longs;ius B e&longs;se &longs;e&longs;quialteram. Quoniam
enim omnis proportio in aliquibus ma
gnitudinibus con&longs;i&longs;tit; &longs;it magnitudo C
ip&longs;ius D &longs;e&longs;quialtera: &longs;ed & E e&longs;t ip&longs;ius
F &longs;e&longs;quialtera; vtigitur C, tertia ad D
quartam, ita erit E maior, vel minor quàm
A prima, ad F vnà maiorem, vel minorem
&longs;ecunda, minori, vt ponitur, vtriu&longs;que ex
ce&longs;su, vel defectu magnitudine propo&longs;ita
eiu&longs;dem generis cum A, B, quæcumque
illa, & quantacumque &longs;it; erit per præ
cedentem eadem proportio A ad B,
quæ C ad D: &longs;ed proportio quam ha
bet C ad D, e&longs;t &longs;e&longs;quialtera; ergo & A
ip&longs;ius B erit &longs;e&longs;quialtera. Similiter quo
cumque alio nomine notatam proportio
nem habeat E ad F, eandem habere A
ad B, o&longs;tenderemus, vt duplam, &longs;e&longs;quitertiam, alicuius du
plicatam, vel triplicatam, & &longs;ic de &longs;ingulis. Manife&longs;tum
e&longs;t igitur propo&longs;itum.
Hæc autem propo&longs;itio in paucis exemplaribus, quæ do
no quibu&longs;dam
uiret ijs, in quibus certæ proportionis nomen,
tum terminum &longs;ubob&longs;curè indicat, vt in &longs;equenti XII iilud,
proportio dupla. Illo autem Lemmate, quod prima propofi
tio in&longs;cribebatur, nunc ita non egeo, vt primam, &
quæ &longs;ecunda, & tertia erant, & facilius demon&longs;trem, & ea
rum &longs;en&longs;um paucioribus comprehendam. priora ergo ita
non improbo vt hæc ijs anteponam.
Si &longs;int tres magnitudines &longs;e &longs;e æqualiter exce
dentes, minor erit proportio minimæ ad mediam
quàm mediæ ad maximam.
Sint tres magnitudines inæquales A, BC, DE, qua
rum BC æquè excedat ip&longs;am A, ac DE ip&longs;am BC
Dico minorem e&longs;se proportionem A, ad
BC, quàm BC, ad DE. Nam vt e&longs;t
A ad BC, ita &longs;it BC ad LH, & au
feratur BF æqualis A, & DG, & LK
æquales BC. Quoniam igitur e&longs;t vt A,
hoc e&longs;t FB ad BC, ita BC hoc e&longs;t KL
ad LH; erit diuidendo vt BF ad FC,
ita LK ad KH: & componendo, ac per
mutando vt BC ad LH, ita FC ad
KH. &longs;ed BC e&longs;t minor quàm LH; ergo
& FC hoc e&longs;t EG erit minor quàm KH. Sed DE, LH, &longs;uperant BC exce&longs;sibus
EG, KH; minor igitur erit DE quàm
LH, & minor proportio BC ad LH,
quàm BC ad DE. Sed vt BC ad LH,
ita e&longs;t A ad BC; minor igitur proportio erit A ad BC,
quàm BC ad DE. Quod demon&longs;trandum erat.
Si &longs;it minor proportio primæ ad &longs;ecundam,
quàm &longs;ecundæ ad tertiam, ab ip&longs;is autem æquales
auferantur; erit minor proportio reliquæ primæ
ad reliquam &longs;ecundæ, quam reliquæ &longs;ecundæ ad
reliquam tertiæ.
Sit minor proportio AB, ad CD, quam CD, ad EF.
Sitque AB, minima.
ablatæ autem æquales fint AG, CH,
EK. Dico reliquarum minorem e&longs;se proportionem BG,
ad DH, quam BH, ad FH. Ponatur enim CL, æqua
lis AB, & EM, æqualis CD. Quoniam igitur maior e&longs;t
proportio DL ad LH, quam DL, ad LC;
erit componendo maior proportio DH ad
HL, quam DC ad CL. hoc e&longs;t, maior
proportio DH, ad BG, quam DC,
ad AB: & conuertendo, minor proportio
BG ad DH, quam AB, ad CD: hoc e&longs;t
maior proportio AB, ad CD, quam BG,
ad DH. Rur&longs;us, quoniam maior e&longs;t pro
portio CD, ad EF, quam AB, ad CD:
hoc e&longs;t quam CL, ad EM; erit permutan
do, maior proportio CD, ad CL, quam
FE, ad EM: & diuidendo, maior DL, ad
LC, quam FM, ad ME: & permutando,
maior DL, ad FM, quam CL, ad EM: hoc e&longs;t quam
AB, ad CD. Sed maior erat proportio AB, ad CD,
quam BG ad DH; multo igitur maior proportio erit DL,
ad FM, quam BG, ad DH: hoc e&longs;t quam LH, ad MK:
& permutando, maior proportio DL, ad LH, quam FM,
ad MK: & componendo, maior DH, ad HL, quam FK,
M
tio BG ad DH, quam DH, ad FK. Quod demon
&longs;trandum erat.
Si &longs;int tres magnitudines inæquales, & aliæ il
lis multitudine æquales binæque in duplicata pri
marum proportione. Sit autem minor proportio
primæ ad &longs;ecundam, quam &longs;ecundæ ad tertiam in
primis; erit minor proportio primæ ad &longs;ecundam,
quam &longs;ecundæ ad tertiam in &longs;ecundis.
Sint tres magnitudines A, B, C, & aliæ illis multitudine
æquales D, E, F. quarum ip&longs;ius D ad E proportio &longs;it du
plicata eius, quæ e&longs;t A ad B: & E ad F, duplicata eius,
quæ e&longs;t B ad C. &longs;it autem mi
nor proportio A ad B, quam
B ad C. Dico minorem e&longs;se
proportionem D ad E, quam
E ad F. Sit enim vt C ad B,
ita B ad G: & vt B ad A, ita
A ad H. Igitur G ad C dupli
cata erit proportio ip&longs;ius G ad
B, hoc e&longs;t B ad C: &longs;imiliter
erit H ad B, duplicata propor
tio ip&longs;ius A ad B. Vt igitur
e&longs;t H ad B, ita erit D ad E: &
vt G ad C, ita E ad F. Rur
&longs;us, quia minor e&longs;t proportio
A ad B, quam B ad C, &longs;ed vt A ad B, ita e&longs;t H ad A
H ad B, quam G ad C, &longs;ed vt H ad B, ita erat D, ad
E: & vt G ad C, ita E ad F; minor igitur proportio erit
D ad E, quam E ad F. Quod demon&longs;trandum erat.
Si &longs;int octo magnitudines quaternæ propor
tionales: tertiæ autem vtriu&longs;que ordinis inter &longs;o
&longs;int vt primæ; erit vt compo&longs;ita ex primis ad com
po&longs;itam ex &longs;ecundis, ita compo&longs;ita ex tertiis ad
compo&longs;itam ex quartis.
Sint octo magnitudines quaternæ &longs;um
ptæ proportionales, vt A ad B, ita C ad
D. & vt E ad F, ita G ad H. &longs;it autem vt
A ad E, ita C ad G. Dico e&longs;se vt AE, ad
ABF, ita CG, ad DH. Quoniam enim
componendo e&longs;t vt AE, ad E, ita, CG,
ad G; &longs;ed vt E ad F, ita e&longs;t G, ad H; erit
ex æquali, vt AE, ad F, ita CG, ad H. Eadem ratione erit vt AE, ad B, ita CG,
ad D: & conuertendo, vt B ad AE, ita
D ad CG. &longs;ed vt AE, ad F, ita erat
CG ad H; ex æquali igitur erit vt B
ad F, ita D, ad H: & componendo, vt
BF ad F, ita DH ad H: & conuerten
do, vt F ad BF, ita H, ad DH. Sed vt
AE, ad F, ita erat CG ad H; ex æqua
li igitur erit vt AE ad BF, ita CG,
ad DH. Quod demon&longs;trandum erat.
Si &longs;int tres magnitudines &longs;e &longs;e æqualiter exce
dentes; & aliæ eiu&longs;dem generis illis multitudine
æquales, binæque &longs;umptæ in duplicata primarum
proportione; erit vtriu&longs;que ordinis minor pro
portio compo&longs;itæ ex primis ad compo&longs;itam ex &longs;e
cundis, quam compo&longs;itæ ex &longs;ecundis ad compo&longs;i
tam ex tertijs.
Sint tres magnitudines A, B, C, quarum C maxima
æque &longs;uperet B, atque
B, ip&longs;am A. & totidem
eiu&longs;dem generis D, E,
F, &longs;itque F ad E du
plicata proportio ip&longs;ius
C ad B: & E ad D,
duplicata ip&longs;ius B ad
A. Dico AD, &longs;imul
ad BE, &longs;imul mino
tem e&longs;&longs;e proportionem
quam BE, &longs;imul ad
CF, &longs;imul. E&longs;to enim
recta quæpiam GH,
ad aliam rectam &longs;ibi in
directum po&longs;itam HK,
vt magnitudo A ad ip
&longs;ius F duplam (hoc
enim fieri pote&longs;t) &
&longs;uper ba&longs;im GK; con&longs;tituatur triangulum GLK, atque
in eo de&longs;cribatur parallelogrammum GHMN: & vt e&longs;t
MP, & ip&longs;i GK, parallelæ TPR, VQS, ducantur. Quoniam igitur e&longs;t vt C, ad duplam ip&longs;ius F, ita GH, ad
HK; erit vt C ad F, ita e&longs;t par llelogrammum GM, ad
triangulum MHK: &longs;ed vt C, ad B, ita e&longs;t HM, ad
hoc e&longs;t parallelogrammum GM, ad parallelogrammum
MV: & vt F, ad E, ita triangulum MHK, ad triangu
lum MQS, ob duplicatam proportionem eius, quæ e&longs;t
HM ad
NK, ad NS trapezium, ita erit, per præcedentem, CF,
&longs;imul ad BE &longs;imul. Rur&longs;us quoniam e&longs;t conuertendo, vt
parallelogrammum MV, ad parallelogrammum GM, ita
B ad C. &longs;ed vt parallelogrammum GM, ad triangulum
KHM, ita erat C, ad F: & vt triangulum KHM, ad
triangulum QSM, ita F ad E; erit ex æquali, vt paral
lelogrammum MV, ad triangulum SQM, ita B, ad E. Similiter ergo vt ante erit vt trapezium NS, ad NR tra
pezium, ita EB, &longs;imul ad AD, &longs;imul. Rur&longs;us, quoniam
æque excedit LV, ip&longs;am LT, atque LG, ip&longs;am LV;
minor erit proportio LT ad LV, quam LV, ad LG: e&longs;t
autem trianguli LTR ad triangulum LVS, duplicata
proportio ip&longs;ius LT, ad LV, & trianguli LVS, ad trian
gulum LGK, duplicata ip&longs;ius LV, ad LG, propter &longs;i
militudinem triangulorum; minor igitur proportio erit
trianguli LTR, ad triangulum LVS, quam trianguli
LVS, ad triangulum LGK; dempto igitur triangulo
LNM, communi, minor erit proportio trapezij NR, ad
trapezium NS, quam trapezij NS, ad trapezium NK. Sed vt trapezium NR, ad trapezium NS, ita e&longs;t conuer
tendo AD &longs;imul ad BE, &longs;imul: & vt trapezium NS, ad
trapezium NK, ita BE, &longs;imul ad CF, &longs;imul; minor igi
tur proportio erit AD, &longs;imul ad BE &longs;imul, quam BE &longs;i
mul ad CF, &longs;imul. Quod demon&longs;trandum erat.
Si recta linea vtcumque &longs;ecta fuerit, cubus qui
fit à tota æqualis e&longs;t duobus &longs;olidis rectangulis,
quæ ex partibus, & totius quadrato fiunt.
Sit recta linea AB &longs;ecta in puncto C vtcumque.
Di
co cubum ex AB æqualem e&longs;se duobus &longs;olidis rectangu
lis, quæ fiunt ex AC CB, & quadrato AB. Quoniam
enim communi altitudine AB, e&longs;t vt rectangulum BAC
ad quadratum AB, ita &longs;olidum ex AB, & rectangulo
BAC ad cubum ex AB, eademque ratione vt rectangu
lum ABC, ad quadratum AB, ita &longs;olidum e&longs;t AB, &
rectangulo ABC ad cubum ex AB; erunt vt duo rectan
gula BAC, ABC ad quadratum AB, ita duo &longs;olida
ex AB, & rectangulis BAC, ABC ad cubum ex AB. Sed duo rectangula BAC, ABC &longs;unt æqualia quadrato
AC; duo igitur &longs;olida ex AB, & rectangulis BAC, CBA,
æqualia &longs;unt cubo ex AB. Sed &longs;olidum ex AB & rectan
gulo BAC e&longs;t id quod fit ex AC, & AC & quadrato
AB; duo igitur &longs;olida ex AC, CB, & quadrato AB &longs;i
mul &longs;umpta æqualia &longs;ua cubo ex AB. Si igitur recta linea
vtcumque &longs;ecta fuerit, &c. Quod demon&longs;trandum erat.
Si recta linea vtcumque &longs;ecta fuerit, cubus qui
fit à tota æqualis e&longs;t cubis partium, & duobus &longs;o
lidis rectangulis, quæ partium triplis, & earun
dem quadratis reciproce continentur.
Sit recta linea AB &longs;ecta vtcumque in puncto C.
Dico
cubum ex AB æqualem e&longs;se duobus cubis ex AC, CB,
& duobus &longs;olidis rectangulis, quorum alterum fit ex tripla
ip&longs;ius AC, & quadrato BC; alterum
&longs;ius BC, & quadrato AC. Quoniam enim quadratum
ex AB æquale e&longs;t duobus quadratis ex AC, CB, & ei
quod bis fit ex AC CB: & parallelepipeda elu&longs;dem al
titudinis inter &longs;e &longs;unt vt ba&longs;es; erit rectangulorum folido
rum id quod fit ex AC, & quadrato AB æquale cubo ex
AC, & ei, quod fit ex AC, & rectangulo ACB bis, &
ei, quod ex AC, & quadrato BC. Eadem ratione erit
quod fit ex BC, & quadrato AB æquale cubo ex BC, &
ei, quod fit ex BC, & rectangulo ACB, bis & ei, quod ex
BC, & quadrato AC. Sed cubus ex AB æqualis e&longs;t
duobus &longs;olidis ex AC CB. & quadrato AB; cubus igi
tur ex AB æqualis e&longs;t duobus cubis ex AC CB, & &longs;ex
&longs;olidis, quorum tres fiunt ex AC, & duobus rectangulis
ex AC CB, & quadrato BC: tria vero ex BC, & duo
bus rectangulis ex AC CB, & quadrato AC. Sed quod
fit ex AC, & rectangulo ACB, e&longs;t quod fit ex BC, &
e&longs;t quod fit ex AC, & quadrato BC; cubus igitur ex
AB æqualis e&longs;t duobus cubis ex AC CB, vna cum &longs;ex
&longs;olidis, quorum tria fiunt ex AC, & BC quadrato, tria
autem ex BC, & quadrato AC, hoc e&longs;t duobus &longs;olidis,
quorum alterum fit ex tripla ip&longs;ius AC, & quadrato BC,
alterum ex tripla ip&longs;ius BC & quadrato AC. Quod de
mon&longs;trandum erat.
Si recta linea vtcumque &longs;ecta fuerit, cubus qui
fit à tota æqualis e&longs;t cubis partium vna cum &longs;oli
do rectangulo, quod totius tripla, & partibus
continetur.
Sit recta linea AB &longs;ecta in puncto C vtcumque.
Di
co cubum ex AB æqualem e&longs;se duobus cubis ex AC,
CB, vna cum &longs;olido rectangulo ex AC CB, & tripla
ip&longs;ius AB. Quoniam enim quod fit ex AC, & rectan
gulo ACB, e&longs;t id quod fit ex BC, & quadrato AC: &
quod fit ex BC, & rectangulo ACB, e&longs;t id, quod fit ex
AC & quadrato BC. &longs;ed duo &longs;olida ex AC CB, & re
ctangulo ACB &longs;unt id, quod fit ex compo&longs;ita vtriu&longs;que
altitudine AB, et rectangulo ACB; duo igitur prædi
cta &longs;olida, quæ ex AC CB, & earum quadratis recipro
ce fiunt æqualia &longs;unt &longs;olido ex AB BC CA, & triplum
triplo, videlicet duo &longs;olida, quæ fiunt reciproce ex triplis
mul ei, quod ter fit ex AB, BC, CA, hoc e&longs;t ei, quod
partibus AC CB, & totius AB tripla continetur: additis
igitur communibus duobus cubis ex AC, CB, erit id, quod
&longs;it ex AC CB, & tripla ip&longs;ius AB, & duo cubi ex AC
CB, æqualia duobus &longs;olidis, quæ fiunt reciproce ex triplis
ip&longs;arum AC, CB, & earundem AC, CB, quadratis, &
duobus cubis ex AC, CB, hoc e&longs;t cubo ex AC. Si igi
tur recta linea vtcumque &longs;ecta fuerit, &c. Quod demon
&longs;trandum erat.
Hemi&longs;phærium duplum e&longs;t coni, cylindri au
tem &longs;ub&longs;e&longs;quialterum eandem ip&longs;i ba&longs;im, & ean
dem altitudinem habentium.
E&longs;to hemi&longs;phærium; cuius axis BD, ba&longs;is circulus, cu
ius diameter AC, &longs;uper quem cylindrus AE, & conus
ABC, quorum communis axis &longs;it BD, ac propterea
etiam eadem altitudo. Dico hemi&longs;phærium ABC, co
ni ABC e&longs;se duplum: cylindri autem AE &longs;uper ba&longs;im enim circulum RE, vertice D de&longs;cribatur
Sectoque axe BD primo bifariam, deinde
&longs;ingulis eius partibus rur&longs;us bifariam, tran&longs;eant per pun
cta &longs;ectionum plana ba&longs;i hemi&longs;phærij AC æquidi&longs;tantia,
quæ &longs;ecent hemi&longs;phærium, conum, & cylindrum. Se
ctus igitur erit AE cylindrus in cylindros æqualium alti
tudinum: &longs;uper &longs;ectiones autem coni, atque hemi&longs;phærij
nempe circulos, quorum centra in axe BD exi&longs;tunt cy
lindri con&longs;tituti intelligantur binis quibu&longs;que proximis
æquidi&longs;tantibus planis interiecti, quorum axes omnes
æquales in BD. Erit igitur cono EDR in&longs;cripta, & ABC
hemi&longs;phærio circum&longs;cripta figura quædam ex cylindris
æqualium altitudinum. Sint autem hæ figuræ ea ratione
hæc circum&longs;cripta illa in&longs;cripta, vt circum&longs;cripta excedat
hemi&longs;phærium, minori exce&longs;su, in&longs;cripta vero deficiat à
cono minori defectu quam &longs;it magnitudo propo&longs;ita, quan
tacumque illa &longs;it. His con&longs;titutis, manife&longs;tum e&longs;t, reliquo
cylindri AE dempto hemi&longs;phærio in&longs;criptam e&longs;se figu
ram ex re&longs;iduis cylindrorum, in quos cylindrus AE &longs;e
ctus fuerit, demptis cylindris hemi&longs;phærio circum&longs;criptis,
deficientem à reliquo cylindri AE dempto hemi&longs;phærio
minori defectu magnitudine propo&longs;ita, eodem &longs;cilicet,
quo figura hemi&longs;phærio circum&longs;cripta excedit hemi&longs;phæ
rium, excepto re&longs;iduo cylindri infimi AS, dempta he
mi&longs;phærij portione, quam comprehendit. Sit autem om-
FE, cuius axis BH, & communis &longs;ectio plani per pun
ctum H tran&longs;euntis ba&longs;i hemi&longs;phærij cum plano per axim
BD, &longs;it recta FGKHMNL. Quoniam igitur rectan
gulum DHB bis vna cum duobus quadratis DH, BH,
æquale e&longs;t BD quadrato: & rectangulum DHB bis
vna cum quadrato BH, e&longs;t rectangulum ex BD DH tan
quam vna, & BH; rectangulum ex BD, DH tanquam
vna & BH, vna cum quadrato DH æquale erit quadra
to BD, hoc e&longs;t quadrato FH: quorum quadratum KH
æquale e&longs;t rectangulo ex BD, DH, tanquam vna, & BH;
reliquum igitur quadrati FH dempto quadrato KH æ
quale erit reliquo quadrato DH, hoc e&longs;t quadrato GH:
& quadruplum quadruplo reliquum quadrati FL dempto
quadrato MK toti GN quadrato, hoc e&longs;t reliquum circu
li, FL dempto circulo MK, æquale circulo GN. Qua
re & GP, cylindrus reliquo cylindri FE dempto QK,
cylindro æqualis erit, propter æqualitatem altitudinum. Similiter o&longs;tenderemus &longs;ingula reliqua cylindrorum eiu&longs;
dem altitudinis, in quos totus cylindrus AE &longs;ectus fuit,
demptis cylindris hemi&longs;phærio circum&longs;criptis æqualia e&longs;
&longs;e &longs;ingulis cylindris cono EDR in&longs;criptis, quæ inter ea
dem plana interijciuntur. Tota igitur figura ex prædictis
cylindrorum re&longs;iduis reliquo cylindri AE, dempto he
mi&longs;phærio in&longs;cripta æqualis erit figuræ cono EDR in
&longs;criptæ: deficit autem vtraque harum figurarum hæc à co
no ADR, illa à re&longs;iduo cylindri AE dempto hemi&longs;phæ
rio minori exce&longs;su magnitudine vtcumque propo&longs;ita; re
liquum igitur cylindri AE dempto hemi&longs;phærio æquale
e&longs;t cono EDR, &longs;ed conus EDR; hoc e&longs;t conus ABC cylin
dri AE e&longs;t pars tertia; reliquum igitur cylindri AE dem
pto hemi&longs;phærio, cylindri AE e&longs;t pars tertia, hoc e&longs;t cylin
drus AE triplus dicti re&longs;idui:
quialter hemi&longs;phærij ABC: & Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omnis minor &longs;phæræ portio, ad cylindrum,
cuius ba&longs;is æqualis e&longs;t circulo maximo, altitudo
autem eadem portioni, eam habet proportionem,
quam exce&longs;&longs;us, quo tripla &longs;emidiametri &longs;phæræ
excedit tres deinceps proportionales, quarum ma
xima e&longs;t &longs;phæræ &longs;emidiameter, media vero quæ
inter centra &longs;phæræ & ba&longs;is portionis interijci
tur; ad &longs;emidiametri &longs;phæræ triplam.
Sit &longs;phæræ, cuius centrum D, &longs;emidiameter BD, mi
nor portio ABC, cuius axis BG &longs;egmentum &longs;emidiame
tri BD, ba&longs;is autem circulus, cuius diameter AC. Sitque
EF, cylindrus, cu
ius axis, &longs;iue alti
tudo eadem BG:
ba&longs;is autem æqua
lis circulo maxi
mo, cuius &longs;emidia
meter BD. Dico
portionem ABC,
ad cylindrum EF
eam habere pro
portionem, quam exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;upe
rat tres BD, DG; & minorem extremam ad ip&longs;as, quæ
&longs;it M; ad ip&longs;ius BD triplam. vertice enim D, ba&longs;i cylin
dri EF, cuius diameter FH de&longs;cribatur conus FDH, cu
ius intelligatur fru&longs;tum FHKL ab&longs;ci&longs;sum plano, quod ab-Quoniam igitur fru&longs;tum FH
re&longs;iduo, dempta ABC portione, quod ex præcedenti theo
remate per&longs;picuum e&longs;se debet: erit portio ABC æqualis
ei, quod relinquitur cylindri EF, &longs;i fru&longs;tum auferatur
FHKL: &longs;ed hoc reliquum e&longs;t ad cylindrum EF, vt exce&longs;
&longs;us, quo tripla lineæ FH, &longs;uperat tres deinceps proportio
nales FH, KL, & minorem extrema, ad triplam lineæ FH:
tur exce&longs;&longs;us, quo tripla ip&longs;ius BD, &longs;uperat tres BD, DG,
& M, &longs;imul, ad lineæ BD triplam, ita erit portio ABC ad
cylindrum EF. Quod demon&longs;trandum erat.
Omnis portio &longs;phæræ ab&longs;ci&longs;sa duobus planis
parallelis alteroper centrum acto ad cylindrum,
cuius ba&longs;is e&longs;t eadem ba&longs;i portionis, &longs;iue circu
lo maximo, & eadem altitudo, eam habet pro
portionem, quam exce&longs;&longs;us, quo maior extrema ad
&longs;phæræ &longs;emidiametrum, & axim portionis exce
dit tertiam partem axis portionis; ad maiorem ex
tremam antedictam.
Sit portio AB
CD, &longs;phæræ, cu
ius centrum F,
ab&longs;ci&longs;&longs;a duobus
planis parallelis
altero per
F tran&longs;eunte;
axis autem por
tionis fit FG: &
autem, cuius diameter BC: & cylindrus AE, cuius ba&longs;is
circulus AD, axis FG; & vt FG ad FA, ita &longs;it FA, ad
MN, à qua ab&longs;cindatur NO, pars tertia ip&longs;ius FG. Dico
ABCD Po&longs;ita enim G
H, æquali ip&longs;i
FG, de&longs;criba
tur circa axim
FG, cylindrus
L
HFK. Quoniam
igitur duo cylin
dri AE, LK,
&longs;unt eiu&longs;dem al
titudinis, erunt inter &longs;e vt ba&longs;es, AD, KH. hoc e&longs;t cy
lindrus AE ad cylindrum LK, duplicatam habebit pro
portionem diametri AD, ad diametrum KH, hoc e&longs;t eius,
quæ e&longs;t &longs;emidiametri AF ad &longs;emidiametrum GH. hoc e&longs;t
eam, quæ e&longs;t MN ad GH, &longs;iue FG. Sed vt FG ad tertiam
&longs;ui partem NO, ita e&longs;t cylindrus KL, ad conum KFH;
ex æquali igitur, erit vt MN ad NO, ita cylindrus AE
ad conum
pta ABCD portione: & per conuer&longs;ionem rationis, vt
NM, ad MO, ita cylindrus AE ad portionem ABCD:
& conuertendo, vt MO ad MN, ita portio ABCD ad
cylindrum AE. Quod e&longs;t propo&longs;itum.
Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis
parallelis neutro per centrum, nec centrum inter
cipientibus ad cylindrum, cuius ba&longs;is æqualis e&longs;t
eam
extrema ad triplas &longs;emidiametri &longs;phæræ, & eius
quæ inter
nis interijcitur, &longs;uperat tres deinceps
proportionales, quarum maxima e&longs;t
quæ inter centra &longs;phæræ, & minoris
ba&longs;is, media autem, quæ inter cen
træ &longs;phæræ, & maioris ba&longs;is portio
nis interijcitur; ad maiorem extre
mam antedictam.
Sit portio ABCD &longs;phæræ, cuius centrum
E, ab&longs;ci&longs;sa duobus planis parallelis, neutro
per E tran&longs;eunte, nec E
maior ba&longs;is &longs;it circulus, cui diameter AD.
minor autem cuius diameter BC, axis GH.
circa quem cylindrus OS, con&longs;i&longs;tat, cuius
ba&longs;is &longs;it circulus circa SR æqualis circulo
maximo: &longs;phæræ autem &longs;emidiater &longs;it EHG.
& vt GE ad EH, ita &longs;it HE ad V: & po
&longs;ita T tripla ip&longs;ius EF, & X itidem tripla ip&longs;ius EG, vt X
&longs;imul &longs;it æqualis. Dico ABCD portio
nem ad cylindrum SO e&longs;se vt Ab&longs;ci&longs;sa enim GK ip&longs;i EG æquali, cylin
drus PN circa axim GH, & conus KEN
con&longs;tituantur vt in præcedenti. planum igi
tur ab&longs;cindens portionem facit fru&longs;tum coni
KEN, quod &longs;it KLMN, cuius minor ba
&longs;is circulus, cui diameter LM; maior autem
cui diameter KN. Et vt e&longs;t GE ad EF, hoc
e&longs;t GK ad SH, ita &longs;it EF, vel SH, ad I.
vt igitur in præcedenti, o&longs;tenderemus cylin
drum SO ad cylindrum PN e&longs;se vt I ad
GK &longs;iue ad EG. Quoniam igitur &longs;unt ter
næ deinceps proportionales GE, EF, I, &
X, T, ZY, e&longs;tque vt FE ad EG ita T ad X;
erit vt I ad EG, hoc e&longs;t vt cylindrus SO ad
PN Et quoniam e&longs;t vt
GE ad EH, ita EH ad V: hoc e&longs;t, vt GK ad
LH. ita LH ad V: & ponitur X tripla ip&longs;ius
EG, hoc e&longs;t ip&longs;ius GK, vt autem e&longs;t triplaip&longs;ius GK ad
tres deinceps proportionales GK, LH, V, ita e&longs;t cylin
drus PN ad fru&longs;tum LKNM; erit vt X ad tres GE, EH,
V &longs;imul hoc e&longs;t ad lineam Sed vt ZY ad X, ita erat cylindrus SO
ad PN cylindrum; ex æquali igitur erit vt ZY ad Z
ita cylindrus SO ad fru&longs;tum KLMN: hoc e&longs;t, ad reli
quum cylindri SO dempta ABCD portione, & per con
uer&longs;ionem rationis, vt ZY, ad Y
tio ABCD ad SO cylindrum. Quod
Omnis maior &longs;phæræ portio ad cylindrum, cu
ius ba&longs;is æqualis e&longs;t circulo maximo, altitudo au
tem eadem portioni eam habet proportionem,
quam ad axim portionis habet exce&longs;&longs;us, quo &longs;eg
mentum axis portionis inter &longs;phæræ centrum, &
ba&longs;im portionis interiectum &longs;uperat tertiam par
tem minoris extremæ maiori po&longs;ita prædicto axis
&longs;egmento in proportione &longs;emidiametri &longs;phæræ
ad prædictum
cum &longs;ub&longs;e&longs;qui
altera reliqui
axis &longs;egmenti.
Sit &longs;phæræ, cu
ius
meter DGE ma
ior portio ABC,
axis autem por
tionis BGF, com
munis cylindro
KH, cuius ba&longs;is æqualis &longs;it circulo maximo; ba&longs;is autem
ita &longs;it GF ad S, & S ad FM, cuius &longs;it pars tertia FN, &
ponatur ip&longs;ius BG, &longs;ub&longs;e&longs;quialtera GL. Dico portio
nem ABC ad cylindrum KH e&longs;se vt LN ad BF. Nam
vt FG ad GE, &longs;iue ad BG, ita &longs;it EG ad PQ, à qua
ab&longs;cindatur QR, pars tertia ip&longs;ius FG. Et plano per G
tran&longs;eunte ba&longs;ibus cylindri KH, & ABC portionis pa
rallelo &longs;ecentur vna cylindrus KH in duos cylindros DH,
EK: & portio ABC, in portionem ECAD, & DBE
hemi&longs;phærium. Quoniam igitur e&longs;t conuertendo, vt PQ
ad EG, ita EG
ad GF, & e&longs;t ip
&longs;ius GF pars ter
tia QR, erit por
tio DACE ad
cylindrum EK,
vt PR ad
Rur&longs;us, quia e&longs;t
vt EG ad GF:
hoc e&longs;t vt PQ ad
EG, ita GF ad
S, & vt EG ad
GF, ita e&longs;t S ad
FM; erit ex æqua
li, vt PQ ad GF, ita GF ad FM. Sed vt GF ad
ita e&longs;t MF ad FN, tertiam ip&longs;ius MF partem, ex æquali
igitur erit vt PQ ad QR, ita GF ad FN, & per conuer
&longs;ionem rationis, & conuertendo, vt PR ad PQ, ita NG ad
GF. Sed vt PR ad PQ, ita erat portio ECAD ad cy
lindrum EK; vtigitur NG ad GF, ita erit portio EC
AD ad cylindrum EK. Sed vt GF ad FB, ita e&longs;t cy
lindrus EK ad cylindrum KH: ex æquali igitur vt NG
ad BF, ita portio ECAD, ad cylindrum KH. Similiter
o&longs;tenderemus e&longs;se, vt GL ad BF, ita DBE hemi&longs;phæ-
mi&longs;phærium DBE ad cylindrum DH. vt igitur prima
cum quinta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam;
videlicet, vt tota LN ad BF, ita portio ABC ad cylin
drum KH. Quod erat demon&longs;trandum.
Omnis portio &longs;phæræ ab&longs;ci&longs;&longs;a duobus planis
parallelis centrum intercipientibus ad cylin
drum, eiu&longs;dem altitudinis, cuius ba&longs;is æqualis e&longs;t
circulo maximo, eam habet proportionem, quam
ad axim portionis habet exce&longs;&longs;us, quo axis portio
nis &longs;uperat tertiam partem compo&longs;itæ ex duabus
minoribus extremis, maioribus po&longs;itis duobus
axis &longs;egmentis, quæ fiunt à centro &longs;phæræ in ra
tionibus, &longs;emidiametri &longs;phæræ ad prædicta &longs;eg
menta.
Sit portio AB
CD, &longs;phæræ, cu
ius centrum G,
ab&longs;ci&longs;sa duobus
planis parallelis
centrum G inter
cipientibus, quod
erit in axe portio
nis, qui &longs;it HK. Sectiones autem
factæ à prædictis planis &longs;int circuli, quorum diametri AD,
BC, qui circuli erunt ba&longs;es oppo&longs;itæ portionis. Sectaque
per punctum G, portione ABCD plano ad axim erecto,
quæ erit circulus maximus, cuius diameter LM, duo cylin
dri de&longs;cripti intelligantur, ad oppo&longs;ita portionis ba&longs;ium pla
na terminati ex illis autem totus cylindrus compo&longs;itus EF,
cuius ba&longs;is æqua
lis circulo maxi
mo LM. Deinde
in &longs;egmento GH
&longs;umpta OH, ter
tia parte minoris
extremæ maiori
GH in proportio
ne, quæ e&longs;t LG ad
GH; & in &longs;egmen
to GK, &longs;umatur
NK, tertia pars minoris extremæ maiori GK, in propor
tione, quæ e&longs;t LG ad GK. Dico portionem ABCD
ad cylindrum EF, e&longs;se vt NO ad KH. Sumptis enim
ij&longs;dem, quæ in præcedentis &longs;ump&longs;imus, demon&longs;trationem
&longs;imiliter o&longs;tenderemus tam portionem LBCM ad cy
lindrum EF, e&longs;se vt OG ad
DM ad eundem EF cylindrum, vt NG ad eundem axim
KH, vt igitur prima cum quinta ad &longs;ecundam, ita tertia
cum &longs;exta ad quartam: videlicet, vt NO ad KH, ita por
tio ABCD ad EF cylindrum. Quod demon&longs;trandum
crat.
Omne conoides parabolicum dimidium e&longs;t
cylindri, coni autem &longs;e&longs;quialterum eandem ip&longs;i
ba&longs;im, & eandem altitudinem habentium.
Sit conoides parabolicum ABC, & cylindrus AE, &
conus ABC, quorum omnium &longs;it eadem ba&longs;is circulus,
cuins diameter AC, axis autem BD, ac proinde vna om
nium altitudo. Dico conoidis ABC e&longs;se cylindri AE
dimidium, coni autem ABC &longs;e&longs;quialterum. Secto eni
axe BD in tot partes æquales, quarum infima ad ba&longs;im &longs;it
MD, vt figura ex cylindris æqualium altitudinum conoi
di ABC circum&longs;cripta, in&longs;criptam &longs;uperet minori &longs;pacio
quantacu nque magnitudine propo&longs;ita, & &longs;it hoc factum. Et quoniam quibus planis parallelis tran&longs;euntibus per præ
dictas &longs;ectiones axis BD &longs;ecatur conoides ABC, ij&longs;dem
&longs;ecatur triangulum per axim ABC, eruntque &longs;ectiones
parallelæ: &longs;it triangulo ABC circum&longs;cripta figura ex pa
rallelogrammis æqualium altitudinum, quæ triangulum &
ip&longs;a excedat minori &longs;pacio quantacumque magnitudine
propo&longs;ita. Cylindrorum autem qui &longs;unt circa conoides, &
parallelogrammorum multitudine æqualium, quæ &longs;unt cir
ca triangulum ABC, duo proximi ba&longs;i AC cylindri &longs;int
AF, HL, & totidem parallelogramma illis re&longs;pondentia
inter eadem plana parallela &longs;int AF, GK. Quoniam igi-
plicatis e&longs;t vt BM ad BD longitudine, ita MH ad AD
potentia: hoc e&longs;t, ita circulus, cuius diameter HMN, ad
circulum, cuius diameter ADC, hoc e&longs;t ita cylindrus HL,
ad cylindrum AF propter æqualitatem altitudinum: &longs;ed
vt BM ad BD, ita e&longs;t GM ad AD, propter &longs;imilitudinem
triangulorum, hoc e&longs;t ita
rallelogrammum; ergo vt parallelogrammum GK ad paral Similiter o&longs;tenderemus reliqua parallelogramma, quæ &longs;unt
circa
circa conoides ABC bina &longs;umpta prout inter &longs;e re&longs;pon
dent in eadem proportione; &longs;emper igitur componendo, &
ex æquali erit vt tota figura triangulo ABC circum&longs;cripta
ad parallelogrammum AF, ita figura conoidi circum&longs;cri
pta ad AF cylindrum: &longs;ed vt parallelogrammum AF, ad
parallelogrammum AE, ita e&longs;t cylindrus AF ad cylindrum
AE, propter æqualitatem omnifariam &longs;umptarum altitu
dinum; ex æquali igitur erit vt figura triangulo ABC cir
cum&longs;cripta ad parallelogrammum AE, ita figura conoidi
circum&longs;criptarum figurarum excedit &longs;ibi in&longs;criptam mino
ri &longs;pacio quantacumque magnitudine propo&longs;ita, vt igitur
triangulum ABC, ad parallelogrammum AE, ita erit co
noides ABC, ad cylindrum AE. Sed triangulum ABC
e&longs;t parallelogrammi AE dimidium; igitur conoides ABC
e&longs;t cylindro AE dimidium: &longs;ed cylindrus AE e&longs;t coni
ABC, triplum: igitur conoides ABC, erit coni ABC
&longs;e&longs;quialterum. Quod demon&longs;trandum erat.
Omnis pri&longs;matis triangulam ba&longs;im habentis
centrum grauitatis rectam lineam, quæ cuiu&longs;libet
trium laterum bipartiti &longs;ectionem, & oppo&longs;iti pa
rallelogrammi centrum iungit, ita diuidit, vt
pars, quæ attingit latus &longs;it dupla reliquæ.
Sit pri&longs;ma, quale diximus AB
CDEF, &longs;ectoque vno ip&longs;ius la
tere BF in puncto G, bifariam
parallelogrammi oppo&longs;iti &longs;it cen
trum H, & iuncta GH, cuius
pars GK &longs;it dupla reliquæ Dico pri&longs;matis ABCDEF, cen
trum grauitatis e&longs;&longs;e K. Per pun
ctum enim H ducatur NO ip
&longs;i AE, vel CD parallela, quæ
ip&longs;as AC, ED, &longs;ecabit
iunctisque BN, FO, ducatur per
punctum
parallela LM. Quoniam igitur e&longs;t vt HK ad KG, ita
NL ad LB, & OM ad MF, erit NL, ip&longs;ius LB, & OM
ba&longs;es AC, ED, bifariam &longs;e
cant; erunt igitur puncta L, M,
centra grauitatis triangulorum
ABC, DEF, oppo&longs;itorum. Pri&longs;matis igitur ABCDEF
axis erit LM: quare in eius bi
partiti &longs;ectione pri&longs;matis ABC
DEF centrum grauitatis: &longs;ectus
autem e&longs;t axis LM bifariam in
puncto K; nam ob parallelogram
ma e&longs;t vt NH ad HO, ita LK
ad KM; pri&longs;matis igitur ABC
DEF, centrum grauitatis erit
Quod demon&longs;trandum erat.
Omnis pri&longs;matis ba&longs;im habentis trapezium, cu
ius duo latera inter &longs;e &longs;int parallela centrum gra
uitatis rectam lineam, quæ æque inter &longs;e di&longs;tan
tium parallelogrammorum centra iungit, ita di
uidit, vt pars, quæ dictorum parallelogrammorum
minus attingit &longs;it ad reliquam, vt duorum ba&longs;is la
terum parallelorum dupla maioris vna cum mino
ri ad duplam minoris vna cum maiori.
Sit pri&longs;ma ABCDEFGH, cuius ba&longs;is trapezium
ABCD, habens duo latera AD, BC, inter &longs;e paralle
la, &longs;itque eorum AD maius: parallela igitur erunt inter &longs;e
duo parallelogramma BG, AH. Sit parallelogrammi AH
centrum K, & BG parallelogrammi centrum L, iuncta-
plam ip&longs;ius BC vna cum AD, ita LR ad RK. Dico
pri&longs;matis AG centrum grauitatis e&longs;se R. Ducantur enim
per puncta L, K lateribus pri&longs;matis, atque ideo inter &longs;e
parallelæ MN, OP, quæ
ob centra K, L, &longs;ecabunt
oppo&longs;ita parallelogrammo
rum latera bifariam, eas
&longs;ectiones connectant MO,
NP, ip&longs;ique MN, vel
OP, parallela ducatur Q
RS. Quoniam igitur e&longs;t
vt LR ad R
dupla ip&longs;ius AD vna cum
BC ad duplam ip&longs;ius BC
vna cum AD, ita OQ ad
QM, & recta MO bifa
riam &longs;ecat AC trapezij latera parallela, punctum Q, AC
trapezij centrum grauitatis; &longs;imiliter & punctum S erit EG,
trapezij centrum grauitatis: pri&longs;matis igitur AG axis erit
QS, & centrum grauitatis R, quod e&longs;t in medio axis. Omnis igitur pri&longs;matis ba&longs;im habentis trapezium, &c.
Quod demon&longs;trandum erat.
Si à quolibet prædicto pri&longs;mate duo pri&longs;mata
be&longs;es habentia triangulas &longs;int ita ab&longs;ci&longs;&longs;a, vt pa
rallelepipedum relinquant ba&longs;im habens minus
parallelogrammorum inter &longs;e parallelorum præ
dicti pri&longs;matis, maioris autem partes æqualia pa
rallelogramma ip&longs;um parallelepipedum relin
matis tamquam vnius magnitudinis rectam line
lam, quæ prædicti pri&longs;matis parallelorum paral
lelogrammorum centra iungit, ita diuidit, vt
pars, quæ minus parallelogrammum attingit &longs;it
dupla reliquæ.
Sit pri&longs;ma ABCDEFGH, cuius ba&longs;es oppo&longs;itæ tra
pezia ADHE, BCGF. Sint autem AD, EH, paral
lelæ, quarum maior EH. Oppo&longs;ita igitur parallelogram
ma AC, EG, inter &longs;e erunt parallela, quorum maius EG. At per rectas AB, CD, &longs;ectum &longs;it pri&longs;ma.
ABCDEF
GH, ita vt ab&longs;ci&longs;&longs;a pri&longs;mata ABSFER, CDVHGT,
relinquant parallelepipedum AT, ip&longs;um autem AT, re
linquat duo parallelogramma æqualia ES, TH. Po&longs;ito
autem centro K
parallelogrammi
AC, & L, paral
lelogrammi EG,
iunctaque KL,
ponatur KM, du
pla ip&longs;ius ML. Dico
matum BER,
CVH, &longs;imul cen
trum grauitatis
e&longs;se M. Sectis enim AB, CD, bifariam in punctis P, Q,
&longs;umpti&longs;que parallelogrammorum ES, VG, centris N, O,
iungantur PN, QO, & po&longs;ita PX dupla ip&longs;ius XN, & QZ
dupla ip&longs;ius ZO, iungantur rectæ PKQ, XZ, NO. Quoniam igitur in quadrilatero PQON, recta XZ, pa
rallela e&longs;t vtrilibet ip&longs;arum PQ, NO, &longs;ecat ijs parallelis
interceptas in ea&longs;dem rationes; recta igitur XT per pun-Sed quia PK e&longs;t æqualis KQ, & NL
ip&longs;i LO, etiam XM æqualis erit ip&longs;i MZ ob parallelas;
cum igitur pri&longs;matum BER, CVH centra grauitatis &longs;int
X, Z; erit vtriu&longs;que pri&longs;matis prædicti &longs;imul centrum gra
uitatis M. Quod e&longs;t propo&longs;itum.
Si &longs;int duæ pyramides æquales, & æque altæ,
ba&longs;es habentes in eodem plano, quarum vertices
recta linea connectens cum ea, quæ ba&longs;ium centra
grauitatis iungit &longs;it in eodem plano; earum cen
trum grauitatis tamquam vnius magnitudinis re
ctam lineam, quæ inter vertices, & centra ba&longs;ium
interiectas bifariam &longs;ecat, itadiuidit, vt pars &longs;u
perior &longs;it inferioris tripla.
Sint duæ
pyramides æ
quales, & æ
que altæ, qua
rum ba&longs;es in
eodem plano
AC, DB, ver
tices autem
G, H, & ba
&longs;ium
F, iunctæque
EF, GH, quas
bifariam &longs;ecet recta KL, huius autem pars quarta &longs;it LM. Dico vtriu&longs;que pyramidis GAC, HDB, &longs;imul centrum
grauitatis e&longs;&longs;e M. Iunctis enim GE, HF, &longs;umantur ea
Quoniam
igitur propter æqualitatem altitudinum, & quia EF, GH,
&longs;unt in eodem plano, &longs;unt EF, GH, inter &longs;e parallelæ, &
vt GN ad NE, ita e&longs;t HO ad OF; erit NO ip&longs;i E Fivel
GH, paralle
la, quas KL
bifariam &longs;ecat:
igitur & ip&longs;am
NO &longs;ecabit bi
fariam, iungit
autem recta
NO centra
grauitatis
ramidum
lium GAC,
HDB, vtriu&longs;
que ergo pyramidis &longs;imul centrum grauitatis erit in com
muni &longs;ectione duarum linearum KL, NO, &longs;ed recta NO,
&longs;ecans &longs;imiliter ip&longs;as GE, KL, HF, ip&longs;am KL, &longs;ecabit
in puncto M; punctum igitur M, erit prædictarum pyrami
dum centrum grauitatis. Quod demon&longs;trandum erat.
Omnis fru&longs;ti pyramidis ba&longs;im habentis paral
lelogrammum centrum grauitatis maiori ba&longs;i e&longs;t
propinquius, quam punctum illud, in quo axis &longs;ic
diuiditur, vt pars minorem ba&longs;im attingens &longs;it ad
reliquam vt dupla cuiu&longs;uis laterum maioris ba&longs;is
vna cum latere minoris &longs;ibi re&longs;pondente, ad
dicti lateris minoris ba&longs;is vna cum maioris &longs;ibi
re&longs;pondente.
Sit pyramidis, cuius ba&longs;is parallelogrammum EFGH,
fru&longs;tum ABCDEFGH,
cto
EH, dupla ip&longs;ius EH vna cum AD ad duplam ip&longs;ius
AD vna cum EH, & fru&longs;ti ABCDEFGH &longs;it centrum
grauitatis Dico punctum
infra punctum A punctis enim A,B,C,D, ducantur
ad maiorem ba&longs;im axi KL, parallelæ AN, BO, CR, DS,
& parallelepipedum ABCDNORS compleatur, &
productis ba&longs;is NO lateribus, de&longs;criptæ &longs;int quatuor py
ramides AEMNZ, BOPFY, CGXRQ, DHVST,
quarum ba&longs;es erunt parallelogramma circa diametrum
æqualia, atque &longs;imilia: & quatuor pri&longs;mata triangulas ba
&longs;es habentia, quorum binorum ex aduer&longs;o inter &longs;e re&longs;pon-
inter &longs;e æqualia, atque &longs;imilia, &longs;cilicet MS ip&longs;i OQ, &
ZO, ip&longs;is RV: &longs;itque axis KL pars tertia L
autem L Quoniam ìgitur ex &longs;upra demon&longs;tratis pri&longs;
matis ABCDTMPQ e&longs;t centrum grauitatis
rum autem pri&longs;matum oppo&longs;itorum ABYONZ, CDS
RXV, centrum grauitatis
CDEFGH demptis quatuor prædictis pyramidibus in Nam ex primo li
bro con&longs;tat punctum
pleatur trapezium ACGE, cuius diameter erit KL. Sed
earum quatuor pyramidum e&longs;t centrum grauitatis Si
enim ba&longs;ium, quibus binæ oppo&longs;itæ pyramides in&longs;i&longs;tunt
centra grauitatis, & bini oppo&longs;iti vertices &longs;ingulis rectis li-
ab axe
nife&longs;tat. Totius igitur fru&longs;ti ABCDEFGH, centrum
grauitatis
punctum
punctum Quod demon&longs;trandum erat.
Omnis fru&longs;ti conici centrum grauitatis pro
pinquius e&longs;t maiori ba&longs;i quam punctum illud, in
quo axis &longs;ic diuiditur, vt pars minorem ba&longs;im
attingens &longs;it ad reliquam, vt dupla diametri ma
ior is ba&longs;is vna cum minoris diametro ad duplam
diametri minoris ba&longs;is vna cum diametro ma
ioris.
Hoc eadem ratione deducetur ex antecedenti, qua cen
trum grauitatis fru&longs;ti conici in extremo primo libro demon
&longs;trauimus, quandoquidem &longs;imiliter vt ibi fecimus, omnis
pyramidis centro grauitatis idem probaremus accedere
quod prædictæ pyramidis in antecedente.
Si &longs;int quotcumque magnitudines, & aliæ illis
multitudine æquales, binæque &longs;umptæ in eadem
proportione, quæ commune habeant centrum gra
uitatis, centra autem grauitatis omnium &longs;int in
eadem recta linea; primæ & &longs;ecundæ tanquam
grauitatis.
Sit recta linea AB, & quotcumque magnitudines
FGH, & totidem KLM, binæ in eadem proportione:
nimirum vt F ad G ita K ad L: & vt G ad H ita L ad
M. in recta autem AB, &longs;int communia centra grauitatis,
C duarum FK, & D duarum GL: & E duarum HM. Om
nium autem primarum tamquam vnius magnitudinis &longs;it
centrum grauitatis O. Dico & omnium &longs;ecundarum &longs;i
mul centrum grauitatis e&longs;se O. Duarum enim FG &longs;i
mul &longs;it centrum grauitatis N. Vtigitur e&longs;t F ad G, hoc
e&longs;t, vt K ad L, ita erit DN, ad NC. punctum igitur N
e&longs;t centrum grauitatis duarum magnitudinum KL &longs;imul. Rur&longs;us, quia componendo, & ex æquali, e&longs;t vt FG &longs;imul
ad H, ita KL &longs;imul ad M: e&longs;t autem tam duarum FG,
quam duarum KL &longs;imul centrum grauitatis N, &longs;imiliter
vt ante o&longs;tenderemus duarum magnitudinum FGH,
KLM centrum grauitatis e&longs;se O. Quod e&longs;t propo&longs;itum.
Si &longs;int quotcumque magnitudines, & aliæ ip
&longs;is multitudine æquales primarum, ex quibus cen
tra grauitatis in eadem recta linea di&longs;po&longs;ita &longs;int
alternatim ad centra grauitatis &longs;ecundarum, qua-
mitur ab eodem prædictæ lineæ termino vnain
primis, & alterain &longs;ecundis inter &longs;e &longs;int æquales;
omnium primarum &longs;imul, ex quibus primæ cen
trum grauitatis propinquius e&longs;t prædicto lineæ
termino quàm primæ &longs;ecundarum, propinquius
erit prædicto lineæ termino quàm omnium &longs;ecun
darum &longs;imul centrum grauitatis.
Sint quotcumque magnitudines ABC primæ, & toti
dem &longs;ecundæ DEF, quarum centra grauitatis in recta
linea TV, primarum quidem G ip&longs;ius A proximum om
nium termino T, à quo &longs;umitur ordo. Deinde H ip&longs;ius B,
&
darum; videlicet vt centrum grauitatis L, ip&longs;ius D cadat
inter centra G, H, & M ip&longs;ius E inter centra H, K: & N
inter puncta
CF: & omnium ABC &longs;imul centrum grauitatis P, & om
nium DEF &longs;imul centrum grauitatis O. Dico punctum
P propinquius e&longs;&longs;e termino T, quàm punctum O. Duarum enim A, B &longs;it centrum grauitatis R: & S, dua
rum DB, & Q, duarum DE. Quoniam igitur Q e&longs;t
centrum grauitatis duarum magnitudinum DE &longs;imal; erit
vt D ad E, hoc e&longs;t ad B, ita MQ, ad QL: hoc e&longs;t HS,
ad SL. & componendo, vt ML, ad LQ, ita HL, ad
LS; & permutando, vt ML ad LH, ita LQ ad LS:
&longs;ed ML e&longs;t maior quàm LH; ergo & LQ erit maior
quàm LS. Eadem ratione quoniam S e&longs;t centrum gra
quales; erit RH maior quàm SH: &longs;ed quia LQ erat ma
ior quàm LS, e&longs;t & SH maior quàm QH; multo igitur
maior RH erit quàm QH: atque ideo punctum R pro
pinquius termino T, quàm punctum
niam tota magnitudo AB e&longs;t æqualis toti DE, & C æ
qualis F; erunt duæ primæ AB, & C, & totidem &longs;ecun
dæ DE, & F, quarum vnius po&longs;teriorum DE cen
trum grauitatis Q cadit inter R, K centra grauitatis
duarum priorum AB, & C, & reliquæ priorum C cen
trum grauitatis K cadit inter Q, N, duarum po&longs;terio
rum DE, & F centra grauitatis; erunt vt antea quatuor
magnitudines binæ proximæ æquales, &longs;cilicet AB, ip&longs;i
DE: & C ip&longs;i F, centra grauitatis habentes di&longs;pofita
alternatim in eadem recta TV. Cum igitur primæ prio
rum AB, centrum grauitatis R &longs;it termino T propin
quius quàm Q centrum grauitatis primæ po&longs;teriorum,
quæ e&longs;t tota DE; &longs;imiliter vt ante totius magnitudinis
ABC centrum grauitatis P erit termino T propinquius
quàm totius DEF centrum grauitatis O. Non aliter
o&longs;tenderemus, quotcumque plures magnitudines, quales
& quemadmodum diximus ad rectam TV, di&longs;po&longs;itæ
proponerentur, &longs;emper centrum grauitatis omnium prio
rum &longs;imul termino T propinquius cadere, quàm omnium
po&longs;teriorum &longs;imul centrum grauitatis. Manife&longs;tum e&longs;t
igitur propo&longs;itum.
Si &longs;int quotcumque magnitudines, & aliæ illis
multitudine æquales, quæ binæ commune habe
ant in eadem recta centrum grauitatis; &longs;umpto au
tem ordine ab vno eius lineæ termino, maior &longs;it
proportio primæ ad &longs;ecundam in primis, quàm
primæ ad &longs;ecundam in &longs;ecundis: & &longs;ecundæ ad
tertiam in primis maior quàm &longs;ecundæ ad ter
tiam in &longs;ecundis, & &longs;ic deinceps v&longs;que ad vltimas;
erit omnium primarum &longs;imul centrum grauitatis
propinquius prædicto lineæ termino, à quo &longs;umi
tur ordo, quàm omnium &longs;ecundarum.
Sint quotcumque magnitudines GHI, & totidem
LMN. Sitque maior proportio G ad H, quàm L ad M: &
H ad I, maior quàm M ad N: in recta autem AB &longs;int
communia centra grauitatis, C duarum magnitudinum
GL, & D duarum HM, & E duarum IN. omnium
autem primarum GHI &longs;imul &longs;it centrum grauitatis K: at
&longs;ecundarum omnium LMN centrum grauitatis R. Di
co centrum K cadere termino A propinquius quàm cen
trum R. Fiat enim vt G ad H, ita DP ad PC: & vt L
ad M, ita DQ ad QC. Maior igitur proportio erit DP
ad CP, quàm DC ad CQ: minor igitur CP erit quàm
CQ: quare DP maior quàm
ED, erit EP maior quàm
trum grauitatis omnium GHI &longs;imul, & ip&longs;ius GH e&longs;t cen
trum grauitatis P, & reliquæ magnitudinis I, centrum
grauitatis E; erit vt GH ad I, ita EK ad KP. eadem
ratione vt vtraque LM ad N, ita erit ER ad
&longs;us, quia maior e&longs;t proportio G ad H, quàm L ad M, erit
componendo, maior proportio GH ad H, quàm LM ad
M: &longs;ed maior e&longs;t proportio H ad K, quàm M ad N; ex
æquali igitur, maior erit proportio GH ad I, quàm LM
ad N, hoc e&longs;t EK ad KP, quàm ER ad
ergo maior proportio EK ad KP, quàm ER ad RP: &
componendo maior proportio EP ad PK quàm EP ad
PR; minor igitur PK erit quàm PR, at que ideo centrum
K propinquius termino A quàm centrum R. Quod de
mon&longs;trandum erat.
Si &longs;int quotcumque magnitudines, & aliæ ip&longs;is
multitudine æquales, quarum omnium centra
grauitatis &longs;int in eadem recta linea, & centra pri
marum ad centra &longs;ecundarum di&longs;po&longs;ita &longs;int alter
natim: &longs;it autem maior proportio primæ ad &longs;ecun-
dis: & &longs;ecundæ ad tertiam in primis, maior quàm
&longs;ecundæ ad tertiam in &longs;e cundis, & &longs;ic deinceps v&longs;
que ad vltimas; erit omnium primarum &longs;imul cen
trum grauitatis propinquius prædictæ lineæ ter
mino à quo &longs;umitur ordo omnium &longs;ecundarum
centrum grauitatis.
Sit quotcumque magnitudines GHI, & totidem LMN
primarum autem &longs;int centra grauitatis CDE cum &longs;ecun
darum centris OPQ in eadem recta AB di&longs;po&longs;ita alter
natim, vt O cadat inter puncta CD, & P inter puncta
DE, & E inter puncta
ad H, quàm L ad M, & H ad I maior quàm M ad N.
omnium autem primarum GHI &longs;imul &longs;it centrum gra
uitatis T; at omnium &longs;ecundarum LMN, &longs;imul, cen
trum grauitatis V. Dico punctum T e&longs;&longs;e termino A
propinquius quàm punctum V. E&longs;to enim F æqualis
L, & K æqualis M, & X æqualis N, &longs;it autem cen
trum grauitatis ip&longs;ius F in puncto C, & ip&longs;ius K in pun
cto D, & ip&longs;ius X in puncto E. In recta igitur AB om
nium FKX, &longs;imul centrum grauitatis erit termino A, pro
pinquius quàm omnium LMN &longs;imul centrum grauitatis. Sed & omnium GHI, &longs;imul centrum grauitatis in eadem
recta AB propinquius e&longs;t termino A quàm omnium
FKX, &longs;imul centrum grauitatis; multo igitur termino A
propinquius erit omnium GHI &longs;imul quàm omnium Quod demon&longs;tran
dum erat.
Po&longs;ito enim R centro grauitatis duarum
H, & S
linea EB, vel in linea AE, &longs;i in puncto E vel in linea EB,
cum igitur T &longs;it
&longs;imul, & E ip&longs;ius I, erit punctum T propinquius termino
A quàm punctum V. Sed punctum V in linea AE cadat.
Veligitur S centrum grauitatis duarum magnitudinum L,
M, &longs;imul cadit in puncto D, &longs;iue in linea DB, vel in li
nea AD. &longs;i in puncto D, vel in linea DB; centrum gra
uitatis R duarum magnitudinum GH erit termino A
propinquius quàm ip&longs;um S, & recta ER maior quàm ES,
Sed cadat punctum S in linea AD. Quoniam igitur ma
ior e&longs;t proportio G ad H, quàm L ad M: & vt G ad H,
ita e&longs;t DR ad RG, & vt L ad M, ita PS ad SO, ma
ior erit proportio DR ad RC, quàm PS ad SO; mul
to ergo maior DR ad RC, quàm DS ad SO, & multo
maior quàm DS ad SC, & componendo maior propor
tio DC ad CR, quàm DC ad CS; erit igitur CR mi
nor quàm CS, atque adeo RD maior DS, addita igitur
ED communi, erit ER maior quàm ES. Rur&longs;us quia
componendo, & ex æquali maior e&longs;t proportio totius GH
ad I quàm totius LM ad N, hoc e&longs;t maior longitudinis
ET ad TR, quàm QV ad VS, & multo maior quàm
RT quàm ES ad SV: & per conuer&longs;ionem rationis mi
nor proportio FR ad ET; quàm ES ad EV, & permu
tando minor proportio ER ad ES quàm ET ad EV: &longs;ed
ER maior erat quàm ES, ergo ET maior erit quàm EV:
& punctum T propinquius termino A, quàm punctum V. Quod demon&longs;trandum erat.
Datæ figuræ circa diametrum, vel axim in alte
ram partem deficienti, &longs;uper ba&longs;im rectam lineam
vel circulum, vel ellip&longs;im; cuius figuræ ba&longs;is, &
&longs;ectiones omnes parallelæ &longs;egmenta æqualia dia
metri vel axis intercipientes ita &longs;e habeant, vt
quarumlibet trium proximarum minor proportio
&longs;it minimæ ad mediam, quàm mediæ ad maxi
mam; figura quædam ex cylindris, vel cylindri
portionibus, vel parallelogrammis æqualium al
titudinum circum&longs;cribi pote&longs;t, cuius
uitatis &longs;it propinquius ba&longs;i quàm cuiu&longs;libet datæ
figuræ, qualem diximus quæ prædictæ figuræ cir
cadiametrum, vel axim circum&longs;cripta &longs;it.
Sit figura circa diametrum, vel axim in alteram
ficiens qualem diximus, cuius bafis circulus, vel ellip&longs;is vel
recta linea AC, axis autem vel diameter BD. Et data figu
ra ip&longs;i ABC figuræ circum&longs;cripta compo&longs;ita ex cylindris,
vel cylindri portionibus, vel parallelogrammis æqualium
altitudinum EF, GH, AK. Dico figuræ ABC alteram
figuram, qualem diximus po&longs;&longs;e circum&longs;cribi, cuius centrum
&longs;iue termino D, quàm prædictæ datæ figuræ circum&longs;criptæ
centrum grauitatis, Omnium enim cylindrorum, vel cy
lindri portionum, vel parallelogrammorum, ex quibus con
&longs;tat prædicta data figura circum&longs;cripta &longs;int axes, vel quæ
oppo&longs;ita latera coniungunt rectæ BL, LM, MD, qui
bus &longs;ectis bifariam in punctis N, O, P, ac planis per ea
&longs;iue rectis tran&longs;euntibus ba&longs;i AC parallelis, &longs;ecantibus
que dictos cylindros, vel cylindri portiones, vel pa
rallelogramma, compleatur & figuræ ABC circum&longs;cri
batur altera figura
vt prior, quæ ob &longs;e
ctiones factas com
ponetur ex duplis
multitudine cylin
dris, vel cylindri por
tionibus, vel paralle
logrammis <17>qualium
altitudinum, eorum
ex quibus con&longs;tat da
ta figura circum&longs;cri
pta &longs;in
lindri, aut reliqua,
quæ diximus QR,
ES, TV, GX, ZI, AY. Quoniam igitur cylindro
rum, vel cylindri portionum, vel parallelogrammorum quæ
&longs;unt circa figuram ABC, minor e&longs;t proportio QR ad ES,
quàm ES, ad TV, propter &longs;ectiones circulos, vel &longs;imiles
ellip&longs;es, vel rectas lineas, &
propo&longs;itæ Sed
ad TV, quàm TV, ad GX; multo ergo minor proportio erit
QR ad ES, quam TV ad GX: & componendo, minor
proportio QR, ES, &longs;imul ad ES, quàm TV, GX, &longs;imul
ad GX. &longs;ed vt GX ad GH, ita e&longs;t ES ad EF; ex æqua-
quàm TV, GX &longs;imul ad GH. & permutando, minor
proportio QR, ES &longs;imul ad TV, GX &longs;imul quàm EF
ad GH. & conuertendo, maior proportio GX, TV &longs;i
mul ad ES, QR &longs;imul, quàm GH ad EF. Similiter
o&longs;tenderemus duo ZI, AY, &longs;imul ad TV, GX, &longs;imul,
maiorem habere proportionem, quàm AK ad rectarum
GH. Rur&longs;us quoniam puncta N, O, in medio BL, LM,
&longs;unt, ip&longs;orum EF, GH, centra grauitatis: duorum autem
QR, ES &longs;imul centrum grauitatis e&longs;t in linea NL, pro
pterea quòd ES maius e&longs;t quàm QR, & æquales BN,
NL, quas centra grauitatis ip&longs;orum QR, ES bifariam
diuidunt, cadet ip&longs;orum QR, ES, &longs;imul centrum grauita
tis propius termino D, quàm ip&longs;ius EF centrum grauitatis,
& duobus centris N, O, interijcietur. Eademque ratio
ne duorum TV, GX, &longs;imul centrum grauitatis termino
D erit propinquius quàm ip&longs;ius GH centrum grauitatis,
& duobus centris O, P, duorum GH, AK interijcietur. Et duorum ZI, AY &longs;imul centrum grauitatis propin
quius erit D termino, quàm P ip&longs;ius AK. Quoniam
igitur omnia primarum magnitudinum, ex quibus con&longs;tat
figura &longs;ecundo circum&longs;cripta centra grauitatis in eadem re
cta linea BD, di&longs;po&longs;ita &longs;unt alternatim ad centra grauita
tis &longs;ecundarum primis multitudine æqualium, ex quibus
data figura con&longs;tat ip&longs;i ABC figuræ circum&longs;cripta, &longs;unt
termino D propinquiora, quàm centra grauitatis &longs;ecunda
rum, &longs;i bina, prout inter &longs;e re&longs;pondent comparentur: maior
autem proportio o&longs;ten&longs;a e&longs;t primæ ad &longs;ecundam in primis,
quàm primæ ad &longs;ecundam in &longs;ecundis: & &longs;ecundæ ad ter
tiam in primis, quàm &longs;ecundæ ad tertiam in &longs;ecundis,
&longs;umpto ordine à termino D, erit centrum grauitatis om
nium primarum &longs;imul, ide&longs;t figuræ ip&longs;i ABC figuræ
&longs;ecundo circum&longs;criptæ termino D propinquius, quàm
datæ figuræ eidem ABC figuræ primo circum&longs;criptæ cenQuod demon&longs;trandum erat.
Omnis prædictæ figuræ centrum grauitatis
e&longs;t propinquius ba&longs;i, quàm cuiu&longs;libet figuræ ex
cylindris, vel cylindri portionibus, vel parallelo
grammis æqualium altitudinum ip&longs;i circum&longs;cri
ptæ.
Sit prædicta figura ABC, cuius axis vel diameter BD,
& data intelligatur figura ex quotcumque cylindris, vel cy
lindri portionibus, vel parallelogrammis æqualium altitu
dinum figuræ ABC circum&longs;cripta, cuius &longs;it centrum gra
uitatis E, nempe in axe vel
diametro BD. Dico cen
trum grauitatis figuræ ABC
propinquius e&longs;&longs;e puncto D,
quàm punctum E. Si enim
fieri pote&longs;t, centrum grauita
tis figuræ ABC, quod &longs;it
F, non cadat infra punctum
E, &longs;ed vel &longs;upra, vel con
gruat puncto E: figuræ ita
que ABC circum&longs;cribatur
figura quædam ex cylindris,
vel cylindri portionibus, vel
parallelogrammis <17>qualium
altitudinum, cuius centrum
grauitatis, quod &longs;it G, &longs;it propinquius D puncto, quàm
punctum E, ac propterea propinquius, quàm punctum F,
centrum grauitatis figuræ primo circum&longs;criptæ. Rur&longs;us
multiplicatis cylindris, vel cylindri portionibus, vel paral-
gura, quemadmodum diximus in præcedenti, cuius cen
trum grauitatis H, in linea GD cadat & &longs;it minor pro
portio re&longs;idui huius tertiæ figuræ circum&longs;criptæ ip&longs;i ABC,
ad figuram ABC, quàm FG ad GD. Multo ergo mi
nor proportio erit dicti re&longs;idui ad figuram ABC quam F
H ad HD, fiat igitur vt prædictum re&longs;iduum ad figuram
ABC, ita ex contraria parte FH ad HDK; prædicti igi
tur re&longs;idui centrum grauitatis erit K, extra ip&longs;ius terminos,
quod fieri non pote&longs;t: Non igitur F centrum grauitatis fi
guræ ABC cadit in puncto E, nec &longs;upra; ergo infra pun
ctum E: & ponitur E centrum grauitatis cuiuslibet figuræ
ex cylindris, vel cylindri portionibus, vel parallelogrammis
æqualium altitudinum quo modo diximus ip&longs;i ABC cir
cum&longs;criptæ. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Omni prædictæ figuræ figura quædam ex cylin
dris, vel cylindri portionibus, vel parallelogram
mis æqualium altitudi
num circum&longs;cribi po
te&longs;t, cuius centri graui
tatis di&longs;tantia à prædi
ctæ figuræ centro gra
uitatis &longs;it minor quan
tacunque longitudine
propo&longs;ita.
Sit figura ABC in
partem
cuius centrum grauitatis F, propo&longs;ita autem quanDico figuræ ABC figu
mis
grauitatis &longs;it propinquius puncto F, quàm punctum G: figu
ræ enim ABC figura, qualem diximus circum&longs;cribatur, cu
ius re&longs;iduum dempta figura ABC, ad figuram ABC mi
norem habeat proportionem, quàm FG, ad GB, &longs;it autem
figuræ circum&longs;criptæ centrum grauitatis K, nempe in axe,
vel diDico
lineam FK minorem e&longs;&longs;e
quàm FG, atque adeo lon
gitudine propo&longs;ita. Quo
niam enim F e&longs;t centrum
grauitatis figuræ ABC,
erit centrum grauitatis
figuræ circum&longs;criptæ ip&longs;i
ABC propinquius termi
no B, quàm punctum F,
&longs;ed centrum grauitatis fi
guræ ABC quòd e&longs;t F, &
figuræ circum&longs;criptæ, quod
e&longs;t K & eius re&longs;idui dem
pta figura ABC &longs;unt in communi axe, vel diametro BD;
erit igitur dicti re&longs;idui in linea BK, centrum grauitatis,
quod &longs;it H. Minor autem proportio e&longs;t prædicti re&longs;idui
ad figuram ABC, hoc e&longs;t ip&longs;ius FK ad KH, quàm FG
ad GB, & multo minor, quàm FG ad GH; & compo
nendo minor proportio FH ad HK, quàm FH ad HG;
ergo KH maior erit, quàm GH; reliqua igitur F
nor, quàm FG atque adeo longitudine propo&longs;ita. Fieri
ergo pote&longs;t, quod proponebatur.
Si duarum prædictarum figurarum circa com
munem axim, vel diametrum, vel alterius diame
trum alterius axim, ba&longs;es, & quotcumque &longs;ectio
nes quales diximus, binæ in eodem plano fue
rint proportionales; idem punctum in diametro,
vel axe erit vtriu&longs;que centrum grauitatis.
Sint duæ prædictæ figuræ ABC, DBE, circa eandem
diametrum, vel axim BF. figuræ autem ABC &longs;it cen
trum grauitatis G, nempe in linea BF. Dico G e&longs;&longs;e
centrum grauitatis
figuræ DBE. &longs;i
enim non e&longs;t, &longs;it a
liud punctum H,
quod cadat primo
&longs;upra punctum G. Figuræ igitur AB
C, figura circum
&longs;cribatur qualem
diximus ex cylin
dris, vel cylindri
portionibus, vel pa
rallelogrammis æ
qualium
cuius centri graui
tatis
centro G, figuræ ABC &longs;it minor quàm recta GH: & figu
ræ DBE, figura circum&longs;cribatur ex cylindris, vel cylindri
portionibus vel parallelogrammis æqualium altitudinum,
multitudine æqualium ijs, ex quibus con&longs;tat ip&longs;i ABC,
C erunt bina &longs;umpto ordine à puncto B, in eadem propor
tione inter eadem plana parallela, vel rectas parallelas
&longs;tentia
binorum autem quorumque homologorum idem erit in li
nea BF, centrum grauitatis: punctum igitur K, centrum
grauitatis figuræ ip&longs;i ABC circum&longs;criptæ, idem erit fi
guræ ip&longs;i DBE, circum&longs;criptæ centrum grauitatis: cadi
grauitatis H figu
ræ DBE, quod e&longs;t
ab&longs;urdum. Non
igitur centrum gra
uitatis figuræ DB
E, cadit &longs;upra pun
ctum G. Sed ca
dat infra, vt in pun
cto L. Rur&longs;us igi
tur figuræ DBE fi
gura, qualem dixi
mus circum&longs;cripta,
cuius centrum gra
uitatis M, &longs;it pro
pinquius centro L,
quàm punctum G, figuræ ABC altera qualem diximus
figura circum&longs;cribatur, cuius centrum grauitatis &longs;it idem
punctum M, quod fieri po&longs;&longs;e con&longs;tat ex &longs;uperioribus. Sed
G ponitur centrum grauitatis figuræ ABC; ergo centrum
grauitatis figuræ ip&longs;i ABC, circum&longs;criptæ erit propinquius
ba&longs;i & puncto F, quàm figuræ ABC centrum grauitatis,
quod fieri non pote&longs;t. Non igitur figuræ DBE centrum gra
uitatis cadit infra punctum G. Sed neque &longs;upra; punctum
igitur G erit commune duarum figurarum ABC, DBE,
centrum grauitatis. Quod demon&longs;trandum erat.
Manife&longs;tum e&longs;t autem omnia proximis qua
tuor propo&longs;itionibus
vel diametrum in alteram partem deficienti, ea
dem ij&longs;dem rationibus o&longs;ten &longs;a remanere de com
po&longs;ito ex duabus figuris circa communem axim
vel diametrum in alteram partem deficientibus,
tam per &longs;e con&longs;iderato, quàm ad alteram figuram
circa eundem axim, vel diametrum cum prædi
cto compo&longs;ito, in alteram partem deficiens, ac &longs;i
e&longs;&longs;ent duæ tantummodo dictæ figuræ, quales in
præcedenti proxima inter &longs;e comparauimus; ma
nente &longs;emper illa conditione, quàm de &longs;ectioni
bus in vige&longs;ima huius diximus. Tantum aduer
tendum e&longs;t, vt pro &longs;ectionibus, dicamus compo&longs;ita
ex binis &longs;ectionibus (quæ &longs;cilicet fiunt ab codem
plano, vel eadem recta linea) cum de prædicto com
po&longs;ito &longs;it &longs;ermo: & in demon&longs;tratione, procylin
dris, vel cylindri portionibus, vel parallelogram
mis, compo&longs;ita ex binis cylindris, vel cylindri por
tionibus, vel parallelogrammis(quæ &longs;cilicet &longs;unt
inter eadem plana parallela, vel lineas parallelas,
& circa eundem axim, vel diametrum totius vel
diametri, vel axis partem) &longs;icut & pro figura com
po&longs;itum ex duabus dictis figuris: pro re&longs;iduo, com
po&longs;itum ex re&longs;iduis. Nam cum vtriu&longs;que re&longs;idui
componentibus, & circa eundem axim, vel diame
trum exi&longs;tentibus, qua ratione diximus, circum
&longs;criptarum, centra grauitatis &longs;int in diametro, vel
axe; etiam compo&longs;iti ex ijs duobus re&longs;iduis (vt in
priori libro generaliter demon&longs;trauimus, cen
trum grauitatis erit in eadem diametro, vel axe:
vnde vim habent proximæ quatuor anteceden
tes demon&longs;trationes, exemplum erit in demon
&longs;tratione trige&longs;imæ quartæ huius.
Hemi&longs;phærij centrum grauitatis e&longs;t punctum
illud in quo axis &longs;ic diuiditur, vt pars, quæ ad ver
ticem &longs;it ad reliquam vt quin que ad tria.
E&longs;to hemifphærium ABC cuius vertex B, axis BD:
&longs;it autem BD &longs;ectus in G puncto, ita vt pars BG ad GD
&longs;it vt quinque ad tria. Dico G e&longs;se centrum grauitatis
hemi&longs;phærij ABC. Ab&longs;cindatur enim BK ip&longs;ius BD
pars quarta: & &longs;uper ba&longs;im eandem hemi&longs;phærij eundem
que axim BD cylindrus AF con&longs;i&longs;tat, & conus intelli
gatur EDF, cuius vertex D, ba&longs;is autem circulus circu
lo AC oppo&longs;itus, cuius diameter EBF. Sectoque axe
BD bifariam in puncto H, & &longs;ingulis eius partibus rur
&longs;us bifariam, quoad BD &longs;ecta &longs;it in partes æquales cu
iu&longs;cumque libuerit numeri paris, tran&longs;eant per puncta &longs;e
ctionum plana quædam ba&longs;i AC parallela, & &longs;ecantia,
hemi&longs;phærium, conum, & cylindrum, quorum omnes &longs;e
ctiones erunt circuli, terni in codem plano ad aliam atque Quod &longs;i præ
terea factæ &longs;ectiones hemi&longs;phærij ABC à cylindri AF
&longs;ectionibus, circuli à circulis concentricis auferri intelli
gantur; reliquæ totidem erunt &longs;ectiones reliquæ figuræ &longs;o
lidæ, dempto ABC hemi&longs;phærio ex toto AF cylin
dro, circuli deficientes circulis concentricis, hoc e&longs;t prædi
ctis ABC hemi&longs;phærij &longs;ectionibus prout inter &longs;e re&longs;pon
dent. Nunc &longs;uper &longs;ectiones hemi&longs;phærij ABC, & co
ni EDF cylindris con&longs;titutis circa axes, quæ &longs;unt &longs;eg
menta æqualia axis BD, intelligantur duæ figuræ ex cy
lindris æqualium altitudinum, altera in&longs;cripta hemi&longs;phæ
rio ABC, altera cono EDF circum&longs;cripta. Si igitur
à toto AF cylindro auferatur figura, quæ in&longs;cripta e&longs;t
hemi&longs;phærio ABC, relinquetur figura quædam ex cylin
dris circa prædictos axes, vt &longs;unt BK, KH, HL, LD,
deficientibus ijs cylindris, ex quibus con&longs;tat figura in&longs;cri
pta hemi&longs;phærio ABC, & vno integro &longs;upiemo XF
cylindro, circum&longs;cripta re&longs;iduo AF cylindri dempto A
BC hemi&longs;phærio, circum&longs;criptione interna: talis autem
figuræ circum&longs;criptæ centrum grauitatis, per ea, quæ in
primo libro, erit in axe BD, quemadmodum & aliarum
duarum figurarum ex cylindris, quarum altera in&longs;cripta
e&longs;t hemi&longs;phærio ABC, altera cono EDF circum&longs;cripta.
perat ex cylindris figuram &longs;ibi in&longs;criptam, eodem figura
circum&longs;cripta reliquo cylindri AF, dempto ABC he
mi&longs;phærio, &longs;uperat ip&longs;um re&longs;iduum; figura autem in&longs;cripta
hemi&longs;phærio ABC pote&longs;t e&longs;&longs;e eiu&longs;modi, quæ ab hemi
&longs;phærio de&longs;iciat minori defectu quantacumque magnitu
dine propo&longs;ita; poterit figura, quæ prædicto re&longs;iduo cir
c
no i exce&longs;su quantacumque magnitudine propo&longs;ita. Ru &longs;us, quia quemadmodum cylindrus AN infimus de
ficiens cylindro SR, æqualis e&longs;t cylindro TP, ex &longs;upe
rioribus, ita vnu&longs;qui&longs;que aliorum cylindrorum deficien
tium cylindris, qui &longs;unt in hemi&longs;phærio, ex quibus cylin
dris deficientibus con&longs;tat dicto re&longs;iduo figura circum&longs;cri
pta, æqualis e&longs;t cylindrorum circa conum EDF, ei, qui
cum ip&longs;o e&longs;t inter eadem plena parallela, & circa eundem
axem; erunt omnes cylindri circa conum EDF, in ea
dem proportione cum prædictis cylindris deficientibus,
circa prædictum re&longs;iduum, &longs;i bini &longs;umantur inter eadem
plana parallela, & circa eundem axem. Quemadmodum
igitur omnium cylindrorum, qui circa conum EDF mi
nor e&longs;t proportio primi ad verticem D, ad &longs;ecundum,
quàm &longs;ecundi ad tertium, & &longs;ecundi ad tertium, quàm ter-
XF (duplicatæ enim &longs;unt talium cylindrorum rationes
earum, quas inter &longs;e habent diametri æqualibus exce&longs;sibus
differentes circulorum, qui &longs;unt &longs;ectiones coni, & ba&longs;es cy
lindrorum, ex quibus con&longs;tat figura cono EDF circum
&longs;cripta, &longs;umpta progre&longs;&longs;ione proportionum eodem ordine
gradatim à minima diametro v&longs;que ad maximam EF) ita
erit cylindrorum deficientium, ex quibus con&longs;tat figura
circum&longs;cripta reliquo cylindri AF, dempto ABC hemi
&longs;phærio, minimi, cuius axis DL ad &longs;ecundum minor pro
portio, quàm &longs;ecundi ad tertium, & &longs;ic deinceps, v&longs;que ad
re&longs;iduum pertinentem, &longs;icut & eorum ba&longs;es circuli deficien
tes, quæ &longs;unt dicti re&longs;idui &longs;ectiones. Cum igitur tam maxi
mi cylindri XF communis, quàm binorum quorumque reli
quorum cylindrorum circa conum EDF, & prædictum re&longs;i
duum inter eadem plana parallela con&longs;i&longs;tentium, quorum
axis communis in BD, commune centrum grauitatis in axe
BD exi&longs;tat, erit ex antecedenti punctum K, quod pono
centrum grauitatis coni EDF, idem re&longs;idui ex cylindro
AF, dempto ABC, hemi&longs;phærio centrum grauitatis. Quoniam igitur quarum partium e&longs;t octo axis BD talium
e&longs;t BG quinque, & BK duarum (ponimus enim nunc K
coni EDF centrum grauitatis) qualium e&longs;t BD octo, ta
lium erit GK trium: &longs;ed KH e&longs;t æqualis BK; qualium
igitur partium e&longs;t GK trium, talium erit KH duarum, ta
li&longs;que vna GH; dupla igitur KH ip&longs;ius GH: &longs;ed ABC
hemi&longs;phærium duplum e&longs;t prædicti re&longs;idui, cum &longs;it cylin
dri AF, &longs;ub&longs;e&longs;quialterum; vt igitur e&longs;t
ad prædictum re&longs;iduum, ita ex contraria parte erit
KH, adlongitudinem GH: &longs;ed H e&longs;t centrum grauitatis
totius cylindri AF & K, prædicti re&longs;idui dempto ABC
hemi&longs;phærio; ergo ABC hemi&longs;phærij centrum grauitatis
erit G. Quod demon&longs;trandum erat.
Omnis minoris portionis &longs;phæræ centrum gra
uitatis e&longs;t in axe primum bifariam &longs;ecto: deinde
&longs;ecundum centrum grauitatis fru&longs;ti circa eun
dem axim, ab&longs;ci&longs;&longs;i à cono verticem habente cen
trum &longs;phæræ; in eo puncto, in quo dimidius axis
portionis ba&longs;im attingens &longs;ic diuiditur, vt pars
duabus prædictis &longs;ectionibus intercepta &longs;it ad
eam, quæ inter &longs;ecundam, & tertiam &longs;ectionem
interijcitur, vt exce&longs;&longs;us, quo tripla &longs;emidiametri
&longs;phæræ, cuius e&longs;t prædicta portio, &longs;uperattres de
inceps proportionales, quarum maxima e&longs;t &longs;phæ
ræ &longs;emidiameter, media autem, quæ inter centra
&longs;phæræ, & ba&longs;is portionis interijcitur; ad &longs;emi
diametri &longs;phæræ triplam.
Sit minor portio ABC, &longs;phæræ, cuius centrum D,
&longs;emidiameter BD, in qua axis portionis &longs;it BG, ba&longs;is
autem circulus, cuius diameter AC: & circa axim BD
de&longs;criptus e&longs;to conus HDF, cuius ba&longs;is circulus FH
tangens portionem in B puncto &longs;it æqualis circulo ma
ximo, & fru&longs;tum coni HDF ab&longs;ci&longs;&longs;um vna cum portio
ne ABC &longs;it KHFL, & vt BD ad DG, ita fiat DG
ad P: &longs;ectoque axe BG bifariam in puncto N, fiat vt
exce&longs;&longs;us, quo tripla ip&longs;ius BD &longs;uperat tres BD, DG,
P, tanquam vnam, ita NM, ad MNO. Dico portio
nis ABC centrum grauitatis e&longs;se O. Nam circa axim
BG, &longs;uper ba&longs;im FH &longs;tet cylindrus EF, cuius cen-
ABC portione centrum grauitatis M commune fru&longs;to
KLFH, vt colligitur ex demon&longs;tratione antecedentis. Quoniam igitur e&longs;t vt exce&longs;sus, quo tripla ip&longs;ius BD &longs;u
perat tres BD, DG, P tanquam vnam, ad ip&longs;ius BD
triplam, hoc e&longs;t vt NM ad MO, ita portio ABC ad
EF cylindrum, & diuidendo vt MN ad NO, ita por
tio ABC ad reliquum cylindri EF; & N e&longs;t cylindri
EF, & M prædicti re&longs;idui centrum grauitatis; erit reli
quæ portionis ABC centrum grauitatis O. Quod de
mon&longs;trandum erat.
Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla
nis parallelis, altero per centrum acto, centrum
grauitatis e&longs;t in axe primum bifariam &longs;ecto: dein
de &longs;umpta ad minorem ba&longs;im quarta parte axis
portionis; in eo puncto, in quo dimidius axis mi
norem ba&longs;im attingens &longs;ic diuiditur, vt pars dua
bus prædictis &longs;ectionibus intercepta &longs;it ad eam,
ijcitur, vt exce&longs;&longs;us, quo maior extrema ad &longs;phæræ
&longs;emidiametrum, & axim portionis &longs;uperat ter
tiam partem axis portionis; ad maiorem extre
mam antedictam.
Sit portio ABCD &longs;phæræ, cuius centrum F: axis au
tem portionis &longs;it EF ab&longs;ci&longs;sæ duobus planis parallelis,
quorum alterum tran&longs;iens per punctum F faciat &longs;ectio
num circulum maximum, cuius diameter AD, reliquam
autem &longs;ectionem minorem circulum, quæ minor ba&longs;is di
citur, cuius di
ameter BC:
& vt e&longs;t EF
ad AD, ita
fiat AD ad
OP, cuius P
R, &longs;it æqua
lis tertiæ parti
axis EF. Et
&longs;ecta EF bi
fariam in puncto M, & po&longs;ita EN ip&longs;ius EF quarta
parte, fiat vt RO ad OP, ita MN ad NL. Dico L e&longs;&longs;e
centrum grauitatis portionis ABCD. Nam circa axim
EF &longs;uper circulum maximum AD de&longs;cribatur cylindrus
AG, cuius centrum grauitatis erit M: reliqui autem ex
cylindro AG dempta ABCD portione centrum graui
tatis N. Quoniam igitur e&longs;t vt RO ad OP, hoc e&longs;t vt
MN ad NL, ita portio ABCD ad reliquum cylindri
AG, & diuidendo vt NM ad ML, ita portio ABCD ad
reliquum cylindri AG: & cylindri AG e&longs;t N, prædicti au
tem re&longs;idui centrum grauitatis M; erit reliquæ portionis
ABCD centrum grauitatis L. Quod
Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla
nis parallelis neutro per centrum acto, nec cen
trum intercipientibus, centrum grauitatis e&longs;t in
axe primum bifariam &longs;ecto: deinde &longs;ecundum
centrum grauitatis fru&longs;ti circa eundem axim,
ab&longs;ci&longs;&longs;i à cono verticem habente centrum &longs;phæ
ræ; in eo puncto in quo dimidius axis maiorem
ba&longs;im attingens &longs;ic diuiditur, vt pars duabus præ
dictis &longs;ectionibus finita &longs;it ad eam, quæ inter &longs;e
cundam, & vltimam &longs;ectionem interijcitur, vt
exce&longs;&longs;us, quo maior extrema ad triplas & &longs;emidia
metri &longs;phæræ, & eius quæ inter centra &longs;phæræ,
& minorem ba&longs;im portionis interijcitur, &longs;uperat
tres deinceps proportionales, quarum maxima
e&longs;t, quæ inter centra &longs;phæræ, & minoris ba&longs;is,
media autem, quæ inter centra &longs;phæræ, & maio
ris ba&longs;is portionis interijcitur; ad maiorem extre
mam antedictam.
Sit portio ABCD, &longs;phæræ, cuius centrum E, ab
&longs;ci&longs;sa duobus planis parallelis, neutro per E tran&longs;eun
te, nec E intercipientibus: axis autem portionis &longs;it GH,
maior ba&longs;is circulus, cuius diameter AD, minor cuius
diameter BC: producta autem GH v&longs;que in E intel
ligatur coni KEN rectanguli, cuius axis EG, fru&longs;tum
tio, & &longs;phæræ &longs;emidiameter &longs;it EHGS: & po
&longs;ita T tripla ip&longs;ius ES, & V ip&longs;ius EG tri
pla, e&longs;to vt V ad T ita T ad XZ: & vt GE
ad EH ita EH ad
æqualis tribus GE, EH,
XY: & &longs;ecto axe GH bifariam in puncto I, in
linea GI, &longs;umatur O, centrum grauitatis fru
&longs;ti KLMN: Et vt
ad OIP. Dico portionis ABCD centrum
grauitatis e&longs;&longs;e P. Nam circa axim GH pla
nis ba&longs;ium portionis interceptus &longs;tet cylin
drus QR, cuius ba&longs;is &longs;it æqualis circulo ma
ximo. Quoniam igitur e&longs;t vt YX ad XZ,
hoc e&longs;t vt IO ad OP, ita portio ABCD
ad cylindrum QR, & diuidendo vt OI ad
IP, ita portio ABCD ad reliquum cylindri
QR: & I e&longs;t cylindri QR, & O prædicti
re&longs;idui centrum grauitatis; erit reliquæ por
tionis ABCD centrum grauitatis P. Quod demon
&longs;trandum erat.
dam R in ip&longs;a DO, ita vt VD ip&longs;ius PD, ad DT ip&longs;ius DO,
&longs;it vt PD, ad DO: &longs;it autem maior proportio PS ad SO, quàm
VR, ad RT. Dico OS, minorem e&longs;&longs;e quàm OR.
Fiat enim vt PS, ad SO, ita VZ ad ZT; ma
ìor igitur erit proportio VZ, ad ZT, quàm VR, ad
RT: & componendo maior proportio VT, ad TZ,
quàm VT, ad TR; minor igitur TZ, quàm TR, ide&longs;t
maior DZ, quàm DR. Rur&longs;us quia componendo e&longs;t
vt PO ad OS, ita VT ad TZ: &longs;ed vt DO ad OP, ita
e&longs;t DT ad TV; erit ex æquali, vt DO ad OS, ita DT,
ad TZ; & per conuer&longs;ionem rationis, vt OD ad DS,
ita TD ad DZ: & permutando, vt DO ad DT, ita DS
ad DZ: &longs;ed DO, e&longs;t maior quàm DT, ergo & DS, erit
maior quàm DZ: &longs;ed DZ maior erat quàm DR; multo
ergo DS maior quàm DR, vnde minor erit OS quàm
OR. Quod demon&longs;trandum erat.
Si datæ maiori &longs;phæræ portioni cylindrus cir
cum&longs;cribatur circa eundem axim portionis, cen
trum grauitatis reliquæ figuræ ex cylindro cir
cum&longs;cripto ablata portione, propinquius erit ver
tici portionis, quàm
Sit &longs;phæræ cuius centrum D maior portio ABC, cu
ius axis BE, ba&longs;is circulus cuius diameter AC, & por
tioni ABC, cylindro XH circa axim BE circum&longs;cripto
vt &longs;upra fecimus: quoniam tam portionis ABC, quàm
cylindri XH, centrum grauitatis e&longs;t in axe BE; erit reli
qui ex cylindro XH, in axe BE centrum grauitatis, &longs;int
in axe BE centra grauitatis Q portionis ABC & S præ
dicti re&longs;idui. Dico e&longs;&longs;e punctum S vertici B propinquius
quàm punctum
ad axim BE erectum &longs;ecet cylindrum XH, & portionem
ABC in duos cylindros
KBL, & portionem AKLC, &longs;ectio autem circulus ma
ximus e&longs;to ille cuius diameter KL: & duo coni rectan
guli circa axes BD, DE, vertice D communi de&longs;cri
bantur GDH, MDN, quorum alterius ba&longs;is GH com
munis erit cylindro XH: alterius autem MDN, minor
quàm eiu&longs;dem cylindri XH, ba&longs;is GH. Denique
DE, in puncto V, bifariam & &longs;umatur BO, ip&longs;ius BD,
pars quarta, necnon EP pars quarta ip&longs;ius DE, primum
itaque quoniam ER e&longs;t maior, quàm ED, erit punctum
R, in &longs;egmento BD. Quoniam igitur ex &longs;upra o&longs;ten&longs;is O
e&longs;t centrum grauitatis commune cono DGH, & reliquo
cylindri KH dempto ABC hemi&longs;phærio: & eadem ra
tione punctum P, cum &longs;it centrum grauitatis coni MDN,
erit idem centrum grauitatis reliqui ex cylindro XL dem
pta AKLC portione: e&longs;t autem reliquum cylindri KH
dempto KBL hemi&longs;phærio, æquale cono DGH, qua
ratione & reliquum cylindri XL, dempta AKLC por
tione æquale e&longs;t cono MDN; cum igitur S &longs;it centrum
grauitatis totius reliqui ex toto cylindro XH, dempta
ABC portione, erit idem S, centrum grauitatis compo
&longs;iti ex conis GDH, MDL: &longs;unt autem horum conorum
centra grauitatis O, P; vt igitur conus GDH, ad co
num MDN, ita erit PS, ad SO: &longs;ed coni GDH ad
&longs;imilem ip&longs;i conum MDN triplicata e&longs;t proportio axis
BD, ad axim BE, hoc e&longs;t cylindri KH ad cylindrum
XL; maior igitur proportio erit PS ad SO, quàm cy
lindri KH ad cylindrum XL, &longs;ed vt cylindrus KH, ad
cylindrum XL, ita e&longs;t VR ad RT, ob centra grauiratis
V, R, T, maior igitur proportio erit PS ad SO, quàm
VR ad RT: &longs;ed eiu&longs;dem PO e&longs;t vt PD ad DO, ita
VD ad DT, ob &longs;ectiones axium proportionales; pun
ctum igitur S propinquius e&longs;t puncto O, quàm punctum
R, per Lemma. Quare & Stermino B propinquius quàm
punctum R: &longs;ed R e&longs;t centrum grauitatis totius cylindri
XH: & S reliqui ex cylindro XH dempta ABC por
tione; igitur Q reliquæ portionis ABC, centrum graui
tatis erit in linea ER, atque ideo à puncto B remotius
quàm punctnm S. Quod e&longs;t propo&longs;itum.
Manife&longs;tum e&longs;t autem ex demon&longs;tratione the
rematis, omnis re&longs;idui ex cylindro datæ maiori
&longs;phæræ portioni circum&longs;cripto circa eundem
axim portionis, cuius ba&longs;is &longs;it æqualis circulo ma
ximo, centrum grauitatis e&longs;&longs;e in axe ab&longs;ci&longs;&longs;a pri
mum quarta parte ad verticem portionis termina
ta &longs;egmenti axis portionis, quod centro &longs;phæræ,
& vertice portionis, & quarta parte eius quod
centro &longs;phæræ, & ba&longs;i portionis terminatur; ad
ba&longs;im terminata in eo puncto, in quo &longs;egmentum
axis portionis duabus prædictis &longs;ectionibus fini
tum &longs;ic diuiditur, vt &longs;egmentum propinquius ba&longs;i
&longs;it ad reliquum, vt cubus &longs;egmenti axis portionis
centro &longs;phæræ, & vertice portionis terminati ad
cubum reliqui quod ba&longs;im portionis tangit, &longs;i
quidem cubi triplicatam inter &longs;e habent laterum
proportionem, &longs;imul illud manife&longs;tum e&longs;t, hoc
idem eadem ratione po&longs;&longs;e demon&longs;trari de centro
grauitatis reliqui ex cylindro dempta &longs;phæræ por
tione ab&longs;ci&longs;&longs;a duobus planis paralìelis centrum
&longs;phæræ intercipientibus, ita vt axis portionis à
centro &longs;phæræ in partes inæquales diuidatur, cu
ius cylindri circum&longs;cripti &longs;it idem axis, qui & por
tionis, ba&longs;is autem æqualis circulo maximo. Si
militer enim de&longs;criptis duobus conis rectanguli
ræ, ba&longs;es autem minores ba&longs;ibus oppo&longs;itis cylin
dri circum&longs;cripti: æqualibus circulo maximo, &longs;u
mentes pro vertice minorem ba&longs;im, pro ba&longs;i, ma
iorem ba&longs;im portionis immotis reliquis propo&longs;i
tum demon&longs;traremus.
Omnis maioris portionis &longs;phæræ centrum gra
uitatis e&longs;t in axe primum bifariam &longs;ecto: Deinde
&longs;umpta ad verticem quarta parte &longs;egmenti axis,
quod centro &longs;phæræ, & portionis vertice finitur:
itemque ad ba&longs;im quarta parte reliqui &longs;egmenti
inter centrum &longs;phæræ, & ba&longs;im portionis interie
cti. Deinde &longs;egmento axis, inter eas quartas par
tes interiecto, ita diui&longs;o, vt pats propinquior ba&longs;i
&longs;it ad reliquam vt cubus &longs;egmenti axis, quod
quod centris &longs;phæræ, & ba&longs;is portionis termina
tur; in eo puncto, in quo &longs;egmentum axis centro
&longs;phæræ, & &longs;ectione penultima finitum &longs;ic diuidi
tur, vt pars prima & penultima &longs;ectione termina
ta &longs;it ad totam vltima & penultima &longs;ectione termi
natam, vt exce&longs;&longs;us, quo &longs;egmentum axis portionis
inter centrum, & ba&longs;im portionis interiectum &longs;u
perat tertiam partem minoris extremæ maiori po
&longs;ita dicto axis &longs;egmento in proportione &longs;emidia-
&longs;ub&longs;e&longs;quialtera reliqui &longs;egmenti, ad axim por
tionis.
Sit maior portio ABC &longs;phæræ, cuius centrum D, dia
meter KH, axis autem portionis &longs;it BE, ba&longs;is circulus,
cuius diameter AC, & &longs;it axis BE primum bifariam &longs;e
ctus in puncto G: &longs;umptaque ip&longs;ius BD, quarta parte
BP, itemque ip&longs;ius DE quarta parte EN, &longs;ecetur inter
iecta PN, ita in puncto F, vt NF, ad FP, &longs;it vt cubus ex
BD ad cubum ex DE; punctum igitur F, ex præcedenti
corollario erit centrum grauitatis reliqui ex cylindro LM
portioni ABC, vt in antecedenti circum&longs;cripto. Quo
niam igitur & prædicti re&longs;idui, ex antecedenti, & cylindri
LM, centra grauitatis &longs;unt in axe BE, erit & portionis
ABC in axe BE centrum grauitatis, quod &longs;it S: manife
&longs;tum e&longs;t igitur punctum S, cadere &longs;upra centrum D, in li
nea BD, minori ablata &longs;phæræ portione, cuius ba&longs;is cir-
quàm centrum S, con&longs;tat ex præcedenti: quare centrum
G, totius cylindri LM inter puncta F, S cadet. Dico
GF ad FS e&longs;&longs;e vt exce&longs;&longs;us, quo recta DE &longs;uperat tertiam
partem minoris extremæ maiori po&longs;ita ip&longs;a DE in propor
tione continua ip&longs;ius DH ad DE vnà cum &longs;ub&longs;e&longs;quial
tera ip&longs;ius BD, ad axim BE, ita GF ad FS. Quoniam
enim portio ABC ad cylindrum LM e&longs;t vt prædictus ex
ce&longs;&longs;us vnà cum &longs;ub&longs;e&longs;quialtera ip&longs;ius BD ad axim BE:
& vt portio ABC ad LM cylindrum, ita e&longs;t GF ad FS,
ob centra grauitatis F, G; erit vt prædictus exce&longs;&longs;us vna
cum &longs;ub&longs;e&longs;quialtera ip&longs;ius BD ad axim BE, ita GF ad
FS. Quod demon&longs;trandum erat.
Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla
nis parallelis centrum intercipientibus, & à cen
tro æqualiter di&longs;tantibus, centrum grauitatis e&longs;t
in medio axis, vel idem, quod centrum &longs;phæræ.
Sit portio ABCD, &longs;phæræ, cuius centrum G, ab&longs;ci&longs;sa
duobus planis parallelis
centrum G intercipien
tibus, & æquè ab eo di
&longs;tantibus: &longs;ectiones
circuli minores, quorum
diametri &longs;int AD, BC
centra autem F,E, qui
bus axis portionis termi
nabitur, eritque ad pla
na vtriu&longs;que circuli per
pendicularis tran&longs;iens per centrum G: & quia illa plana Dico
portionis ABCD centrum grauitatis e&longs;&longs;e G. De&longs;cripta
enim figura, vt &longs;upra fecimus, intelligantur duo coni re
ctanguli GNO, GPQ, vertice G, communi, axibus
autem eorum EG, GF: & cylindrus LM, portioni cir
cum&longs;criptus circa eun
dem axim EF, cuius ba
&longs;is æqualis e&longs;t circulo
maximo: & &longs;umatur EH
ip&longs;ius EG, pars quar
ta, itemque FK, pars
quarta ip&longs;ius FG. Quo
niam igitur conorum G
NO, PGO, axes FG,
GH, &longs;unt æquales, re
liquæ KG, GH, æqua
les erunt; centra autem grauitatis conorum &longs;unt K, H; pun
ctum igitur G e&longs;t centrum grauitatis compo&longs;iti ex duobus
conis æqualibus GNO, GPQ, hoc e&longs;t reliqui ex cylin
dro LM, dempta ABCD, portione, ex ante demon&longs;tra
tis: &longs;ed idem G e&longs;t centrum grauitatis totius cylindri LM;
reliquæ igitur ABCD, portionis centrum grauitatis erit
G. Quod demon&longs;trandum erat.
Omnis portionis &longs;phæræ ab&longs;ci&longs;&longs;æ duobus pla
nis parallelis centrum intercipientibus, & à cen
tro non æqualiter di&longs;tantibus centrum grauitatis
e&longs;t in axe primum bifariam &longs;ecto: Deinde &longs;umpta
ad minorem ba&longs;im portionis quarta parte &longs;egmen
ti axis, quod minorem ba&longs;im attingit: & ad maio-
rum, quæ à centro &longs;phæræ fiunt: Deinde recta
inter has quartas partes interiecta ita diui&longs;a, vt
pars maiori ba&longs;i propinquior &longs;it ad reliquam vt
cubus &longs;egmenti axis inter &longs;phæræ centrum, & mi
norem ba&longs;im, ad cubum eius, quod inter &longs;phæræ
centrum, & maiorem ba&longs;im portionis interijci
tur; in eo puncto, in quo &longs;egmentum axis centro
&longs;phæræ, & penultima &longs;ectione terminatum &longs;ic di
uiditur, vt pars quæ penultima, & prima &longs;ectione
terminatur &longs;it ad totam vltima, & penultima &longs;e
ctione terminatam, vt ad axim portionis e&longs;t exce&longs;
&longs;us, quo idem axis portionis &longs;uperat
compo&longs;itæ ex duabus minoribus extremis, maio
ribus po&longs;itis duobus axis &longs;egmentis, quæ fiunt à
centro &longs;phæræ in rationibus &longs;emidiametri &longs;phæ
ræ ad prædicta &longs;egmenta.
Sit portio ABCD &longs;phæræ, cuius centrum G, abci&longs;&longs;a
duobus planis parallelis centrum G intercipien
qui per centrum G tran&longs;ibit, vtpote parallelorum circu
lorum centra iungens: cumque eorum vtrumque &longs;it à cen
tro non æqualiter di&longs;tantium perpendicularis, erunt eius
&longs;egmenta EG, GF, inæqualia. E&longs;to EG, maius: &longs;ectoque
axe EF bifariam in puncto P, &longs;umptisque ip&longs;arum EG,
GF, quartis partibus EH, FK, &longs;ecetur interiecta
in puncto Q, ita vt KQ, ad QH, &longs;it vt cubus ex EG,
ad cubum ex GF, & portionis ABCD, &longs;it centrum gra
uitatis R: quod quidem cum punctis P, Q, e&longs;&longs;e in axe
EF: & cylindro LM, &longs;uper ba&longs;im æqualem circulo ma
ximo circa axim EF, portioni circum&longs;cripto, reliqui eius
dempta ABCD, portione centrum grauitatis e&longs;se Q, &
propinquius E puncto, quàm centrum grauitatis R por
tionis ABCD, manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis de
maioris portionis &longs;phæræ centro grauitatis: portionis autem
ABCD centrum grauitatis R e&longs;se in &longs;egmento EG &longs;e
quitur ex antecedente. Dico PQ ad QR e&longs;se vt ad axim
EF exce&longs;sus, quo axis EF &longs;uperat tertiam partem com
po&longs;itæ
maiori extrema EG in proportione continua ip&longs;ius NG
tinua ip&longs;ius NG ad GF. Quoniam enim ob centra gra
uitatis QPR e&longs;t vt QP ad PR, ita portio ABCD ad
reliquum cylindri LM, erit componendo, & per conuer
&longs;ionem rationis, & conuertendo, vt PQ ad QR, ita por
tio ABCD ad LM cylindrum: &longs;ed portio ABCD ad
LM cylindrum e&longs;t vt prædictus exce&longs;&longs;us ad axim EF;
vtigitur prædictus exce&longs;&longs;us ad axim EF, ita e&longs;t PQ ad
QR. Quod demon&longs;trandum erat.
Omnis conoidis parabolici centrum grauita
tis e&longs;t punctum illud, in quo axis &longs;ic diuiditur vt
pars, quæ e&longs;t ad verticem &longs;it dupla reliquæ.
Sit conoides parabolicum ABC, cuius vertex B, axis
autem BD &longs;ectus in puncto E ita vt EB &longs;it ip&longs;ius ED
dupla. Dico E e&longs;se centrum grauitatis conoidis ABC.
Nam in &longs;ectione per
axim parabola ABC,
cuius diameter erit B
D, de&longs;cribatur rian
gulum ABC; &longs;um
ptisque ip&longs;ius BD æ
qualibus DH, HO,
per puncta H, O, &longs;e
centur vnà parabola
& triangulum ABC
duabus rectis FGH
KL, MNOPQ: & per eas rectas &longs;ecetur conoi
des ABC planis ba&longs;i parallelis, factæ autem &longs;e
ctiones erunt circuli circa FL, MQ, & in parabola
FH, MO. Quoniam igitur tres rectæ OB, BH, BD
&longs;e&longs;e qualiter excedunt, quarum minima BO, maxi
ma e&longs;t BD, minor erit proportio BO ad BH, quàm
BH ad BD; hoc e&longs;t NP ad GK, quàm GKad AC.
&longs;ed vt OB ad BH hoc e&longs;t NO ad GH, vel NP ad
GK ita e&longs;t quadra
tum MO ad quadra
tum FH, hoc e&longs;t eo
no dis &longs;ectionum cir
culus MQ ad circu
lum FL: eademque
ratione vt GK ad
AC ita circulus FL
ad circulum AC; mi
nor igitur proportio
erit circuli MQ ad
circulum FL quàm
circuli FL ad circulum AC. Similiter autem o&longs;tende
remus ternas quaslibet alias ita factas &longs;ectiones trianguli,
& parabolæ ABC inter &longs;e & ba&longs;i parallelas proportio
nales e&longs;se, & minorem proportionem vtrobique minimæ
ad mediam, quàm mediæ ad maximam. Sed E e&longs;t cen
trum grauitatis trianguli ABC, igitur per vige&longs;imamter
tiam huius centrum grauitatis conoidis ABC erit idem E. Quod demon&longs;trandum erat,
Omnis fru&longs;ti conoidis parabolici centrum gra
uitatis axim ita diuidit, vt pars, quæ minorem
ba&longs;im attingit &longs;it ad reliquam; vt duplum maioris
cum maiori.
Sit conoidis parabolici ABC, cuius axis BD fru&longs;tum
AEFC, eius maior ba&longs;is circulus, cuius diameter AC, mi
nor, cuius diameter EF: in eadem parabola per axem, axis Dico e&longs;&longs;e vt duplum circuli AC, vnà cum circulo EF, ad
duplum circuli EF vna cum circulo AC, ita GH, ad HD.
ctæ AKB, BLC. Quoniam igitur
qua ratione o&longs;ten
dimus conoides,
& triangulum A
BC, commune
habere in linea
BD centrum gra
uitatis,
&longs;us remanet de
mon&longs;tratum, fru&longs;ti
AEFC
FC; erit duarum parallelarum AG, KL vt dupla ip&longs;ius
AC, vnà cum KL, ad duplam ip&longs;ius KL, vnà cum AC
ita GH ad HD: &longs;ecat enim DG ip&longs;as AC, KL bifa
riam. Sed vt AC ad
lum EF, ex demon&longs;tratione antecedentis, hoc e&longs;t vt dupla
ip&longs;ius AC vnà cum KL ad duplam ip&longs;ius KL vnà cum
AC, ita duplum circuli AC vna cum circulo KL ad du
plum circuli KL vnà cum circulo AC; vt igitur e&longs;t du
plum circuli AC, vnà cum circulo EF, ad duplum circu
li EF, vnà cum circulo AC; ita erit GH ad HD. Quod demon&longs;trandum erat.
Omnis conoidis hyperbolici centrum grauita
tis e&longs;t punctum illud, in quo duodecima pars axis
ordine quarta ab ea, quæ ba&longs;im attingit, &longs;ic diui
ditur, vt pars ba&longs;i propinquior &longs;it ad reliquam, vt
&longs;e&longs;quialtera tran&longs;uer&longs;i lateris hyperboles, quæ
conoides de&longs;cribit ad axim conoidis.
Sit conoides hyperbolicum ABC, cuius vertex B, axis
autem BD, qui etiam erit diameter hyperboles, quæ co
noides de&longs;crip&longs;it, ad quam rectæ ordinatim applicantur:
eiu&longs;dem autem hyperboles tran&longs;uer&longs;um latus &longs;it EB, cu
ius &longs;it &longs;e&longs;quialtera BEI, & &longs;umpta DQ quarta parte
axis BD, & DG, eiu&longs;dem tertia, qua ratione erit FG
duodecima pars axis BD, & ordine quarta ab ea cuius
terminus D, fiat vt IB, ad BD, ita QH, ad HG. Dico conoidis ABC, centrum grauitatis e&longs;&longs;e H.
Sumpto
enim in linea AD quolibet puncto M, vt e&longs;t EB ad
BD longitudine, ita fiat MD, ad DK ip&longs;ius AD po
tentia: & ab&longs;cindatur DN, æqualis DM, & DL æqua
lis DK; &longs;iue autem &longs;it DK minor, quàm DM, &longs;iue ma
ior, &longs;iue eadem illi; omnibus ca&longs;ibus communis erit demon
&longs;tratio. At per puncta M, N, vertice B, circa diametrum
BD, de&longs;cribatur parabola MBN, & triangulum KBL. Manente igitur BD, & circumductis figuris MBN,
KBL, de&longs;cribantur conoides parabolicum MBN, &
conus KBL, quorum communis axis erit BD, ba&longs;es
autem circuli, quorum diametri KL, MN, in eodem
plano cum ba&longs;e conoidis ABC. Rur&longs;us &longs;ecto axe BD
bifariam, & &longs;ingulis eius partibus &longs;emper bifariam in qua-
ba&longs;i DF, FQ: & per puncta FQ duo plana ba&longs;ium pla
no parallela tres prædictas figuras &longs;olidas &longs;ecare intelli
gantur: &longs;ecabunt autem & tres figuras per axim, eruntque
&longs;ectiones rectæ lineæ ad diametrum figurarum ordinatim
applicatæ propter
plana &longs;ecantia pa
rallela: trium au
tem &longs;olidorum &longs;e
ctiones & ba&longs;es
omnes circuli, ter
ni in &longs;ingulis pla
nis: ac primi qui
dem ordinis &longs;int
ij, quorum diame
tri &longs;unt ba&longs;es
trianguli &longs;cilicet,
parabolæ, & hy
perboles, quæ præ
dictas figuras &longs;oli
das de&longs;cribunt, re
ctæ lineæ AC,
MN, KL. Se
cundi verò reten
to eodem ordine
be, gd. Tertij
denique ordinis
SZ, TY, VX.
Quoniam igitur e&longs;t vt EB, ad BD, ità quadratum MD,
ad quadratum DK, ide&longs;t conus MBN, &longs;i de&longs;cribatur eo
dem vertice B, ad conum KBL. Et vt IB, ad BE, ità e&longs;t
conoides MBN, ad conum MBN, in proportione &longs;cili-
des MBN ad conum KBL: Sed vt IB, ad BD, ità
ponitur QH ad HG; vt igitur conoides MBN, ad co
num KBL, ità e&longs;t QH ad HG. Sed Q e&longs;t centrum
grauitatis coni KBL, & G conoidis MBN; compo&longs;i
ti igitur ex conoi
de MBN, & co
no KBL
grauitatis erit H. Rur&longs;us quoniam
tres rectæ lineæ B
D, BF, BQ, æ
qualibus exce&longs;&longs;i
bus inter &longs;e diffe
runt, minor erit
proportio BQ, ad
BF, quàm BF,
ad BD, hoc e&longs;t
rectanguli EBQ,
ad rectangulum
EBF, quàm re
ctanguli EBF, ad
rectangulum EB
D. Sed quadrati
BQ, ad quadra
tum BF, dupli
cata e&longs;t proportio
lateris BQ ad l
tus BF: hoc e&longs;t
rectanguli EBQ
ad rectangulum EBF: & quadrati BF, ad quadratum
BD duplicata eius, quæ e&longs;t rectanguli EBF, ad rectan
gulum EBD; compo&longs;itis igitur primis cum &longs;ecundis, mi
nor erit proportio rectanguli BQE, ad rectangulum BFE, Sed vt
rectangulum BQE ad rectangulum BFE, ita e&longs;t quadra
tum SQ ad quadratum
ad rectangulum BDE, ita quadratum
tum AD; minor igitur proportio erit quadrati SQ, ad
quadratum Sed vt quadratum SQ ad quadratum
dratum SZ ad quadratum
quadratum AD ita quadratum
AC; minor igitur proportio erit quadrati SZ ad quadra
tum
circuli SZ ad circulum
culum AC; qui circuli &longs;unt &longs;ectiones conoidis ABC
po&longs;iti vt in propo&longs;itionibus lemmaticis dicebamus. Rur&longs;us
quoniam &longs;unt quatuor primæ proportionales; vt rectangu
lum DBE ad rectangulum FBE, ita MD quadratum
ad quadratum
BD, ad quadratum BF, ita quadratum DK, ad quadra
tum F
&longs;ed vt EB, ad BD, hoc e&longs;t rectangulum DBE prima in
primis ad quadratum BD primam in &longs;ecundis, ita e&longs;t
quadratum MD tertia in primis ad quadratum DK ter
tiam in &longs;ecundis; vt igitur compo&longs;ita ex primis ad com
po&longs;itam ex &longs;ecundis, ità erit compo&longs;ita ex tertijs ad com
po&longs;itam ex quartis; videlicet vt rectangulum DBE
vnà cum quadrato BD, hoc e&longs;t rectangulum BDE
ad rectangulum BFE, hoc e&longs;t vt quadratum AD, ad
quadratum
ad compo&longs;itum ex quadratis
rumque, vt quadratum AC, ad quadratum
po&longs;itum ex quadratis MN, KL, ad compo&longs;itum ex qua
dratis
nes &longs;olidorum, vt circulus AC, ad circulum
po&longs;itum ex circulis MN, KL, ad compo&longs;itum ex circuEadem ratione erit vt circulus AC, ad cir
culum SZ, ità compo&longs;itum ex circulis MN, KL, ad
compo&longs;itum ex circulis TY, VX: & conuertendo, & ex
æquali, vt circulus SZ, ad circulum
ex circulis TY, VX, ad compo&longs;itum ex circulis
& vt circulus
ad circulum AC,
ità
circulis
ad
circulis MN,
L. Sunt igitur tria
compo&longs;ita ex bi
nis &longs;ectionibus cir
culis, & totidem
alij circuli, quos
diximus in
proportione, &longs;i bi
na
gulis planis &longs;ecan
tibus: eorum au
tem minor erat
proportio circuli
SZ ad circulum
AC; minor igitur
proportio erit
po&longs;iti
T
po&longs;itum
lis
po&longs;itum ex circulis MN, KL. Hac eadem ratione ad verti
cem deinceps progredienti manife&longs;tum erit, omnium com-
ter ad conum KBL pertinet, alter ad conoides MBN, in
eodem plano &longs;ecante prædictorum inter &longs;e parallelorum
exi&longs;tentibus, minorem e&longs;&longs;e proportionem incipienti ab eo,
quod e&longs;t proximum vertici, primi ad &longs;ecundum, quàm &longs;e
cundi ad tertium, & &longs;ecundi ad tertium, quàm tertij ad
quartum, & &longs;ic &longs;emper deinceps v&longs;que ad maximum & vl
timum compo&longs;itum ex circulis MN, KL: & eandem di
ctas &longs;ectiones compo&longs;itas ex coni, & conoidis parabolici
&longs;ectionibus inter &longs;e habere proportionem, quàm habent in
ter &longs;e circuli &longs;ectiones conoidis ABC, pro vt illis in
ij&longs;dem planis &longs;ecantibus, & æqualia axis BD &longs;egmenta
intercipientibus re&longs;pondent: Igitur per trige&longs;imam &longs;ecun
dam huius, & &longs;equens eam Corollarium, conoides ABC,
& compo&longs;itum ex conoide MBN, & cono BKL, com
mune habebunt in axe BD centrum grauitatis. Sed H
erat huius compo&longs;iti centrum grauitatis; Igitur conoidis
ABC centrum grauitatis erit idem H. Quod demon
&longs;trandum erat.
Eadem demon&longs;tratione con&longs;tat &longs;i prædicta tria
&longs;olida ita vt diximus di&longs;po&longs;ita &longs;ecentur plano ba
&longs;ibus parallelo; &longs;ru&longs;tum conoidis hyperbolici, &
compo&longs;itum ex fru&longs;tis coni, & conoidis paraboli
ci, commune habere in communi axe centrum
grauitatis.
Si conus & conoides parabolicum circa eun
dem axim &longs;ecentur plano ba&longs;i parallelo; fru&longs;ti co
nici ab&longs;ci&longs;&longs;i maiori ba&longs;i propinquius erit quàm
parabolici centrum grauitatis.
Sint conus ABC, & conoides parabolicum EBF,
quorum communis
axis BD, cuius per
quoduis punctum M,
planum &longs;ecans ea cor
pora plano ba&longs;ium,
quarum diametri A
C, EF, parallelo ab
&longs;cindat fru&longs;ta AKL
C, cuius centrum gra
uitatis N, & EGH
F, cuius centrum gra
uitatis O, quorum vtrumque erit in communi axe DM. Dico punctum N, propinquius e&longs;se ip&longs;i D quàm punctum
O. Quoniam enim e&longs;t parabolicifru&longs;ti EGHF centrum
grauitatis O; erit vt duplum maioris ba&longs;is, ide&longs;t circuli
EF vna cum minori circulo GH, ad duplum circuli GH
vna cum circulo EF, hoc e&longs;t vt duplum quadrati ED vna
cum quadrato ED ita MO ad OD. Sed vt quadratum
ED ad quadratum GM in parabola quæ conoides de
&longs;cribit, cuius diameter BD, ita e&longs;t DB ad BM, hoc e&longs;t
AC ad KL; vt igitur e&longs;t dupla ip&longs;ius AC vna cum KL
ad duplam ip&longs;ius KL vna cum AC ita erit MO ad OD:
&longs;ed N e&longs;t fru&longs;ti conoici AKLC, centrum grauitatis; pun
ctum igitur N, erit maiori ba&longs;i AC propinquius quàm
tatis. Si igitur conus, & conoides parabolicum circa eun
dem axim, &c. Quod demon&longs;trandum erat.
Omnis fru&longs;ti conoidis hyperbolici centrum
grauitatis e&longs;t in axe primum &longs;ecto &longs;ecundum cen
trum grauitatis cuiu&longs;uis fru&longs;ti conici circa axem
conoidis communi vertice, ab&longs;ci&longs;&longs;i vnà cum fru
&longs;to conoidis: deinde ita vt pars minorem ba&longs;im
attingens &longs;it ad reliquam, vt dupla axis conoidis
vna cum reliqua dempto axe fru&longs;ti, ad duplam
eiu&longs;dem reliquæ vna cum axe conoidis: dein
de po&longs;itis quatuor rectis lineis binis propor
tionalibus, potentia primis, &longs;ecundis longitu
dine, in proportione, quæ e&longs;t inter axem conoi
dis, & reliquam dempto axe fru&longs;ti; ita vt ma
ior primarum &longs;it media proportionalis inter axem
conoidis, & tran&longs;uer&longs;um latus hyperboles, quæ fi
guram de&longs;cribit, minoris autem potentia &longs;e&longs;qui
altera minor &longs;ecundarum; in eo puncto, in quo
&longs;egmentum axis fru&longs;ti dictis duabus &longs;ectionibus
terminatum &longs;ic diuiditur, vt pars minori ba&longs;i pro
pinquior &longs;it ad reliquam vt cubus, qui fit ab axe
fru&longs;ti vnà cum &longs;olido rectangulo, quod axe co
noidis, & reliqua dempto axe fru&longs;ti, & tripla
axis conoidis continetur, ad &longs;olidum rectangu
lum ex eadem reliqua parte conoidis, & eo, quo
rum &longs;ecundarum.
Sit conoidis hyperbolici ABC, cuius axis BD; &
tran&longs;uer&longs;um latus hyperboles, quæ figuram de&longs;cribit EB,
fru&longs;tum ALMC ab&longs;ci&longs;&longs;um vnà cum axe FD: cuius
ba&longs;es oppo&longs;itæ, maior circulus circa AC, minor circa LM:
&longs;ecto autem axe FD primum &longs;ecundum G centrum gra
uitatis fru&longs;ti ab&longs;ci&longs;&longs;i vnà cum fru&longs;to ALMC à quouis co
no, cuius axis BD, & vertex B, deinde in puncto H ita
vt FH ad HD &longs;it vt dupla ip&longs;ius BD vnà cum BF ad
duplam ip&longs;ius BF vnà cum BD, quo facto cadet G
punctum infra punctum H, ponantur vt DB ad BF,
tem N media proportionalis inter EB, BD, at P ip&longs;ius
O potentia &longs;e&longs;quialtera: quo autem Q plus pote&longs;t quàm
P &longs;it quadratum ex R: & vt cubus ex FD vna cum &longs;oli
do rectangulo ex BF, FD, & tripla ip&longs;ius BD, ad &longs;oli
dum rectangulum ex BF, & quadrato R, ita &longs;it HK ad
KG. Dico fru&longs;ti ALMC centrum grauitatis e&longs;&longs;e K.
Producta enim quà opus e&longs;t diametro AC ip&longs;i BD æqua
les ab&longs;cindantur DS, DV: necnon ip&longs;i N æquales
DT, DX, vt &longs;it TD ad DS potentia, vt EB, ad
BD longitudine, & de&longs;cribantur conoides paraboli
cum TBX, & conus SBV, quorum vertex commu
nis B, axis BD: &longs;ectis autem his tribus &longs;olidis plano
per axim, &longs;int &longs;ectiones hyperbole ABC, & parabo
la TBX, & triangulum SBV, quæ figuras de&longs;cribunt;
quas planum ba&longs;is fru&longs;ti propo&longs;iti circa LM &longs;ecans vnà
cum tribus &longs;olidis faciat cum parabola TBX rectam I
& cum triangulo SBV rectam
& coni SBV &longs;ectiones circulos circa I
circa SV, TX parallelos; vt &longs;int conoidis TBX fru
&longs;tum TIRur
&longs;us producta I. M, ponatur <37>F, æqualis Q, & ab
&longs;cindatur F
que IB, B
erunt in dicto plano &longs;ecante tria &longs;olida per punctum F. Quoniam igitur circuli inter &longs;e &longs;unt vt quæ fiunt à diame
tris, vel à &longs;emidiametris quadrata, coni autem eiu&longs;dem al
titudinis inter &longs;e vt ba&longs;es; erit vt
conus
alterum e&longs;t coni IB
noidi IBEt quoniam in parabola TBX ordinatim
ad diametrum applicatarum DT e&longs;t ad FI hoc e&longs;t N
æqualis N; ergo & IF æqualis erit O: cum igitur &
P ip&longs;ius O, &
F
Q ad P, hoc e&longs;t DB ad BF, ita erit <37>F ad F
cata igitur proportio erit quadrati ex F<37> ad quadratum ex
E
quadratum ex F
circa
<37>B
DB ad BF: &longs;ed & conoidis TBX ad conoides IB
plicata e&longs;t proportio eius, quæ e&longs;t DB ad BF, vt mon
&longs;trant alij; eadem igitur proportio e&longs;t coni <37>B
num
ni IB
le erit conoidis TBX fru&longs;to TIRur&longs;us quia e&longs;t vt
cubus ex BD ad cubum ex BI ita conus SBV ad &longs;ui &longs;i
milem conum YBZ, in triplicata &longs;cilicet proportione la
terum, &longs;iue a
propter &longs;imilitudinem triangulorum, e&longs;t vt cubus ex BF ad
&longs;olidum ex BF & quadrato ex F
ad quadratum ex F
circa
igitur erit vt cubus ex BD ad &longs;olidum ex BF, & quadra
to F
ex BF, & quadrato F
F<37>, ita e&longs;t &longs;imiliter vt ante conus
æquali igitur erit vt cubus ex BD ad &longs;olidum ex BF, &
quadrato F<37>, ita conus SBV, ad conum <37>B
uertendo, & per conuer&longs;ionem rationis, e&longs;t vt &longs;olidum ex
BF, & quadrato F<37>, ad &longs;olidum ex BF, & quadrato,
quo plus pote&longs;t F<37> quàm F
quum dempto cono <35>B
BD ad &longs;olidum ex BF & quadrato, quo plus pote&longs;t F<37>,
quàm F
ex R, ita erit conus SBV, ad reliquum coni <37>B
pto cono
niam duo cubi ex BF, FD, & &longs;olidum ex BF, FD, &
tripla ip&longs;ius BD, &longs;unt æqualia cubo ex BD; erit id quo
plus pote&longs;t cubice recta BD quàm BF, cubus ex
FD, & &longs;olidum ex BF, FD, & tripla ip&longs;ius BD: cum
igitur &longs;it vt cubus ex BD ad cubum ex BF, ita conus
SBV ad conum YBZ; erit per conuer&longs;ionem rationis, &
conuertendo, vt cubus ex FD vna cum &longs;olido ex BF,
FD, & tripla ip&longs;ius BD ad cubum ex BD, ita fru&longs;tum
SYZV, ad conum SBV: &longs;ed cubus ex BD, ad &longs;oli-
&longs;tum TI
cum &longs;olido ex BF, FD, & tripla ip&longs;ius BD, ad &longs;olidum
ex BF, & quadrato R, hoc e&longs;t vt H
contraria parte fru&longs;tum SYZV, ad fru&longs;tum TI
fru&longs;ti SYZV e&longs;t centrum grauitatis G: fru&longs;ti autem TI
duobus fru&longs;tis centrum grauitatis erit K: commune autem
e&longs;t centrum grauitatis compo&longs;iti ex duobus fru&longs;tis SYZV
& TI
rollarium; fru&longs;ti igitur ALMC, centrum grauitatis erit K. Quod demon&longs;trandum erat.
Ex omnibus demon&longs;trationibus eorum, quæ in
hoc &longs;ecundo libro propo&longs;uimus, manife&longs;tum e&longs;t
omnium &longs;upra dictorum corporum centra grauita
tis inuenire: quæ cum que enim in modum theore
matis propo&longs;uimus, eadem tanquam problema
ta proponi, & ij&longs;dem demon&longs;trationibus ab&longs;olui
po&longs;&longs;unt.
Idem dico de ijs, quæ in primo, & tertio &longs;equenti libro
demon&longs;trauimus. Porro autem multa lemmata in&longs;tituto
præcipuo nece&longs;&longs;aria, & alia addita inuentio &longs;atis iucun
da centri grauitatis conoidis, & portionis conoidis parabo
lici, & hyperbolici, & fru&longs;ti vtriu&longs;que ne &longs;ecundus hic liber
nimis longus, & confu&longs;us exi&longs;teret, tertium requirebant. Quem quidem meorum &longs;tudiorum autumnalium fructum
Anni à partu Virginis MDCIII. cum SS. Clementis
Pont. Max.
auctoritate, & Petri eius Nepotis Cardinalis
ampli&longs;&longs;imi Aldobrandini iu&longs;&longs;u bene de me merentium Ma
thematicam &longs;cientiam, & Philo&longs;ophiam ciuilem in almo
Vrbis Gymna&longs;io profiterer, in eorum gratiam compo&longs;ui,
qui me centra grauitatis portionum &longs;phæroidis imperfe
cti operis crimine condemnandum omittere nolebant; cu
ius prouinciæ iuuante Deo, & mira Mathematicæ &longs;tudio
&longs;is &longs;atisfaciendi voluntate, multas difficultates ita &longs;upe
raui, vt vno men&longs;e Octobri plus præ&longs;titerim, quam à me
requi&longs;i&longs;&longs;ent. &longs;iquidem quæ de &longs;phæræ portionibus in hoc
libro proprijs eius figuræ rationibus, eadem in &longs;equen
ti aliis communibus cuilibet portioni &longs;phæræ, & &longs;phæroi
dis tum lati, tum oblongi ab&longs;ci&longs;&longs;æ vno, vel duobus planis
æque inter &longs;e di&longs;tantibus, & vtcumque in figuram in cideu-
tentione compen&longs;aui, quòd facere non potui&longs;sem ni&longs;i illi,
quos &longs;upra nominaui meos patronos tranquillum otium
mihi &longs;ua benignitate peperi&longs;&longs;ent; ego autem quo&longs;dam ad
uer&longs;os flatus vehementes in meam vtilitatem verte
re didici&longs;sem, cuius rei monumentum flammæ
vento agitatæ &longs;imulacrum cum illo Ver
gilij HOC ACRIOR in fronte
operis po&longs;ui, vt meus quali&longs;
cumque hic labor vel ab
inuitis in me collati
bencficij memo
riam præ&longs;e
ferret.
SECVNDI LIBRI FINIS.
L V C AE
VALER II
DE CENTRO
GRAVITATIS
SOLIDORVM
Si recta linea &longs;ecta fuerit bifa
riam, & non bifariam; rectan
gulum partibus in æqualibus
contentum æquale e&longs;t rectan
gulo, quod bis fit ex dimidiæ
&longs;ectæ &longs;egmentis, vna cum
quadrato non intermedij eo
rundem &longs;egmentorum.
Sit recta linea AB &longs;ecta in puncto C bi&longs;ariam, & non
bifariam in puncto D. Dico rectangulum ADB æqua
le e&longs;&longs;e rectangulo BDC bis vnà cum quadrato BD. Quoniam enim rectangulum ADB, æquale e&longs;t duobus
rectangulis, & ex BD, DC, & ex AC, BD, hoc e&longs;t ex
CB, BD: &longs;ed rectangulum ex CB, BD, e&longs;t rectangu
lum ex BD, DC, vnà cum quadrato BD; rectangulum
igitur ex AD, DB, æquale e&longs;t duobus rectangulis ex
BD, DC, vnà cum quadiato BD. Si igitur recta linea
&longs;ecta fuerit bifariam, & non bifariam, &c. Quod demon
&longs;trandum erat.
Si circulum, vel ellip&longs;im duæ rectæ lineæ tan
gentes in terminis coniugatarum diametrorum,
conueniant: & punctum i
centrum figuræ
que hanc vnà cum prædictæ figuræ termino al
terutri diametrorum parallela &longs;ecuerit recta li
nea, ita ip&longs;a &longs;ecabitur in duobus punctis, vt re
ctangulum bis contentum &longs;egmentis, quorum al
terum inter diametrum, & terminum figuræ, al
terum inter figuræ terminum & contingentem
interijcitur, vnà cum huius quadrato, &longs;it æquale
quadrato reliqui &longs;egmenti inter diametrum, &
guræ iungit interiecta.
Sit circulus, vel ellip&longs;is ABCD, cuius diametri con
iugatæ AC, BED, & figuram tangentes BF, GF, con
ueniant in puncto F; (parallelæ enim erunt vtraque alteri
coniugatorum diametrorum:) & recta FE iungatur, & ex
quolibet puncto G, in recta BE ducatur ip&longs;i AC paral
lela GLKH. Dico rectangulum GKH bis vnà cum
quadrato KH æquale e&longs;&longs;e quadrato GL. Quoniam
enim rectangulum BGD æquale e&longs;t rectangulo BGE
bis vnà cum quadrato BG: & rectangulum BED, e&longs;t
quadratum BE, erit vt rectangulum BED, ad re
ctangulum BGD, ita quadratum BE, ad rectangu
lum BGE bis, vnà cum quadrato BG: &longs;ed vt rectangu
lum BED, ad rectangulum BGD, ita e&longs;t quadratum EC,
hoc e&longs;t quadratum GH ad quadratum GK, ex primo
conicorum, vt igitur e&longs;t quadratum BE ad rectangulum
BGE bis, vnà cum quadrato BG, ita erit quadratum
GH ad quadratum GK. Rur&longs;us quia e&longs;t vt BE ad EG,
ita BF ad GL, propter &longs;imilitudinem triangulorum; erit
vt quadratum BE ad quadratum EG, ita quadratum
conuer&longs;ionem rationis, vt quadratum BE ad rectangu
lum BGE bis, vnà cum quadrato BG, ita quadratum
GH ad rectangulum GLH bis, vnà cum quadrato LH:
&longs;ed vt quadratum BE ad rectangulum EGB bis, vnà
cum quadrato BG, ita erat quadratum GH ad quadra
tum GK; vt igitur quadratum GH ad quadratum GK,
ita erit idem quadratum GH ad rectangulum GLH bis,
vnà cum quadrato LH: quadratum igitur GK æquale
erit rectangulo GLH bis, vnà cum quadrato LH; demptis
igitur ab eodem quadrato GH æqualibus quadrato GK,
& rectangulo GLH bis, vnà cum quadrato LH, erit
rectangulum GKH, bis vnà cum quadrato KH æquale
quadrato GL. Quod demon&longs;trandum erat.
Per data duo puncta in duabus rectis lineis da
tum angulum continentibus, in earum plano pa
rabola tran&longs;ibit, cuius vertex &longs;it a&longs;&longs;ignatum præ
dictorum punctorum, in quo altera linea parabo-
quidi&longs;tans.
Sint data duo puncta.
A, C, in duabus rectis lincis da
tum angulum ABC continentibus, &longs;it autem a&longs;&longs;ignatum
punctum C. Dico per puncta A, C, parabolam tran&longs;i
re, ita vt ip&longs;am linea AC contingat in C puncto, altera
autem AB &longs;ecet in puncto A, diametro parabolæ æqui
di&longs;tans. Completo enim parallelogrammo BD, ad re
ctam CD applicetur rectangulum æquale quadrato AD,
faciens latitudinem E. Quoniam igitur in plano BD
parabola inueniri pote&longs;t, cu
ius &longs;it vertex C, diameter
CD, ita vt quædam ex &longs;e
ctione ad diametrum CD
applicata in dato angulo A
BC, ide&longs;t ADC, qualis
e&longs;t recta AD, po&longs;&longs;it rectan
gulum ex CD, & E, ex
primo conicorum elemen. to; &longs;it ea &longs;ectio parabola
AC; a&longs;&longs;ignatum e&longs;t autem punctum C; per puncta igi
tur A, C parabola AC tran&longs;ibit, cuius vertex e&longs;t a&longs;&longs;i
gnatum punctum C. Et quoniam quæ ex vertice recta
CB e&longs;t applicatæ DA parallela, &longs;ectionem AC in pun
cto C continget: e&longs;t autem AB diametro CD æquidi
di&longs;tans, ac proinde parabolam &longs;ecabit in puncto A. Ma
nife&longs;tum e&longs;t igitur propo&longs;itum,
Si recta linea parabolam contingat, omnes re
ctælineæ ex &longs;ectione ad contingentem applicatæ
tudine, vt inter applicatas & contactum, vel ver
ticem interiectæ inter &longs;e potentia. Productis au
tem dictis applicatis, erunt inter &longs;ectionem & ba
&longs;im interiectæ inter &longs;e longitudine, vt in circulo,
vel ellip&longs;e ad diametrum ordinatim applicatæ, &longs;e
cantesque illam in ea&longs;dem rationes, in quas aliæ
prædictæ applicatæ &longs;ecant ba&longs;im parabolæ, inter
&longs;e potentia.
Sit &longs;ectio parabola ABC, cuius vertex B, diameter
BD: & recta quadam BE &longs;ectionem contingente in pun
cto B, &longs;int quotcumque rectæ lineæ ex &longs;ectione ordinatim
ad BE contingentem applicatæ diametro BD &longs;ectionis
parallelæ FG, KH, quibus productis &longs;int ad ba&longs;im &longs;e
ctionis applicatæ GN, KO. Et expo&longs;ito primum circu
lo, PQRS, cuius diametri ad rectos inter &longs;e angulos &longs;int
QS, PR; &longs;ecta autem QT in punctis V, X, in ea&longs;
dem rationes, in quas &longs;ecta e&longs;t AD in punctis N, O,
&longs;umpto or
applicentur ad &longs;emidiametrum QT rectæ ZV, XY dia
metro PR æquidi&longs;tantes. Dico e&longs;&longs;e HK ad FG lon
git
tudine, vt ZY ad YX potentia. Iungantur enim KL,
GM, ba&longs;i AC parallelæ. Quoniam igitur e&longs;t vt MB
ad BI. longitudine, ita GM ad KL potentia: &longs;ed MB
e&longs;t æqualis ip&longs;i FG, & BL ip&longs;i KH, & BF ip&longs;i GM, &
BH ip&longs;i KL in parallelogrammis BG, BK; vt igitur
FG ad KH longitudine, ita erit BH ad BF potentia:
&longs;imiliter quotcumque plures e&longs;&longs;ent applicatæ idem o&longs;ten
deremus. Rur&longs;us, quoniam e&longs;t vt EA, hoc e&longs;t FN ad FG,
ita quadratum EB ad BF quadratum, hoc e&longs;t quadra
tum AD ad quadratum DN, hoc e&longs;t ita quadratum QT,
hoc e&longs;t quadratum TY, hoc e&longs;t duo quadrata TX, XY,
ad quadratum TX; erit per conuer&longs;ionem rationis, vt FN,
hoc e&longs;t BD ad GN, ita duo quadrata TX, X
hoc e&longs;t quadratum TY, hoc e&longs;t quadratum TP, ad qua
dratum XY. Similiter o&longs;tenderemus e&longs;&longs;e vt BD ad
OK, ita quadratum PT ad quadratum VZ. Conuer
tendo igitur erit vt OK ad BD, ita quadratum XY ad
PT quadratum: & ex æquali vt OK ad GN, ita qua
dratum VZ ad quadratum XY. Suntigitur tres rectæ
lineæ BD, OK, GN, inter &longs;e longitudine, vt in circu
lo PQSR totidem PT, ZV, XY inter &longs;e potentia,
prout inter &longs;e re&longs;pondent. Idem autem &longs;imiliter o&longs;ten
deremus de quotcumque aliis in circulo, & &longs;ectione para
bola vt prædictæ applicatis multitudine æqualibus. In
ellip&longs;e autem, ductis diametris quibu&longs;uis coniugatis, &
totidem quot in circulo ad vnam &longs;emidiametrum rectis li
neis ordinatim applicatis &longs;ecundum puncta &longs;ectionum eiu&longs;
dem diametri in ea&longs;dem prædictas rationes, codemque or
dine; quoniam ex XXI primi conicorum &longs;tatim apparet re
ctarum linearum ita vt diximus in circulo, & ellip&longs;e appli-
prædictæ inter &longs;ectionem parabolam, & ba&longs;im interiectæ
inter &longs;e longitudine, vt in ellip&longs;e ad diametrum &longs;imiliter
vt diximus applicatæ inter &longs;e potentia. Manife&longs;tum e&longs;t
igitur propo&longs;itum.
Omnis figuræ circa axim in alteram partem
deficientis, cuius &longs;uperficies, excepta ba&longs;e &longs;it to
ta interius concaua ba&longs;im habentis circulum, vel
ellip&longs;im; quælibet tres &longs;ectiones ba&longs;i parallelæ
æqualia axis &longs;egmenta intercipientes, ita &longs;e ha
bent, vt minor &longs;it proportio minimæ ad mediam,
quam mediæ ad maximam.
Sit figura ABC circa axem BD in alteram partem de
ficiens, qualem diximus: & po&longs;itis in axe BD tribus qui
buslibet punctis
F, E, L, æqualia
axis &longs;egmenta in
tercipientibus, in
telligatur
ABC &longs;ectum per
ea puncta planis
culo, vel ellip&longs;i,
circa AC pa
rallelis: quare &longs;e
ctiones erunt cir
culi, vel ellip&longs;es &longs;imiles ba&longs;i, per definitionem, quarum dia
metri eiu&longs;dem rationis in eodem plano per axim &longs;int IK. Dico &longs;olidi ABC &longs;ectionum, minorem e&longs;&longs;e
proportionem, ip&longs;ius IK ad GH, quàm GH ad MN. Iunctis enim MRS, KSN; quoniam tres rectæ IK,
RS, MN, &longs;e&longs;e æqualiter excedunt in trapezio KM; mi
nor erit proportio IK ad RS, quàm RS ad MN: &longs;ed cir
culi, & &longs;imiles ellip&longs;es duplicatam habent inter &longs;e propor
tionem diametrorum eiu&longs;dem rationis; trium igitur præ
dictarum &longs;olidi ABC &longs;ectionum minor erit proportio IK
ad RS quàm RS ad MN: &longs;ed maior e&longs;t proportio circu
li, vel ellip&longs;is GH ad circulum, vel ellip&longs;im MN, quàm
circuli, vel ellip&longs;is RS, ad circulum, vel ellip&longs;im MN;
multo ergo minor proportio erit circuli, vel ellip&longs;is IK ad
circulum, vel ellip&longs;im RS, quàm circuli, vel ellip&longs;is GH ad
circulum, vel ellip&longs;im MN: &longs;ed minor e&longs;t proportio cir
culi vel ellip&longs;is I
eiu&longs;dem circuli, vel ellip&longs;is IK ad circulum, vel ellip&longs;im
RS; multo ergo minor proportio erit circuli, vel ellip&longs;is
IK ad circulum, vel ellip&longs;im GH quàm circuli, vel ellip
&longs;is GH ad circulum, vel ellip&longs;im MN. Quod demon
&longs;trandum erat.
Si &longs;phæroides &longs;ecetur plano vtcumque præter
quàm ad axem, circa quem &longs;phæroides de&longs;cribi
tur erecto nam tunc circulus fit. &longs;ectio ellip&longs;is erit:
&longs;imilis autem ip&longs;i alia quæcumque &longs;ectio &longs;phæ
roidis eidem parallela: earumque omnes diame
tri quæ eiu&longs;dem &longs;unt rationis erunt in eodem pla
no per axem.
Extant hæc demon&longs;trata ab Archimede in &longs;uo de &longs;phæ
roidibus, & conoidibus.
Si conoides parabolicum, vel hyperbolicum
&longs;ecetur plano vtcumque ad axim inclinato, &longs;ectio
ellip&longs;is erit: &longs;imilis autem ip&longs;i alia quæcumque
&longs;ectio conoidis eidem parallela: eruntque earum
omnes diametri, quæ eiu&longs;dem &longs;unt rationis in eo
dem plano per axem.
Manife&longs;ta &longs;unt hæc ex ijs, quæ Federicus Commandinus
demon&longs;trauit de &longs;ectionibus horum &longs;olidorum, in &longs;uis com
mentariis in eundem Archimedis librum de &longs;phæroidibus,
& conoidibus: quemadmodum & &longs;phæroidis, & conoi
dis vtriu&longs;que &longs;ectionem factam à plano ad axim erecto e&longs;
&longs;e circulum.
Super datam ellip&longs;im, circa datam rectam line
am ab eius centro eleuatam tanquam axem, coni,
& cylindri portionem inuenire. Datoque &longs;phæ
roidi, & conoidi, vel conoidis, &longs;phæroidi&longs;ve por
tioni circa datum axem &longs;phæroidis, vel cuiuslibet
dictarum portionum, cylindrus vel cylindri por
tio circum&longs;cripta e&longs;&longs;e pote&longs;t: vel comprehendere
inter eadem plana parallela, ita vt eius ba&longs;is &longs;it &longs;i
milis ba&longs;i, vel ba&longs;ibus comprehen&longs;æ portionis, vel
fru&longs;ti, &longs;i de conoidibus &longs;it &longs;ermo: & diametri, quæ
eiu&longs;dem &longs;unt rationis &longs;ectæ à centro bifariam &longs;int
in eadem recta linea.
Manife&longs;ta item &longs;unt hæc omnia, ex ijs, quæ in eodem li
bro de &longs;phæroidibus, & conoidibus demon&longs;trat Archi
medes.
Omnis fru&longs;ti pyramidis triangulam ba&longs;im ha
bentis ad pri&longs;tina, cuius ba&longs;is e&longs;t maior ba&longs;is fru
&longs;ti, & eadem altitudo, cam habet proportionem,
quàm rectangulum contentum duobus lateribus
homologis ba&longs;ium oppo&longs;itarum, vnà cum tertia
parte quadrati differentiæ dictorum laterum, ad
maioris lateris quadratum. Ad pyramidem autem,
cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti, & eadem altitu
do, vt prædictum rectangulum, vna cum prædicti
quadrati tertia parte, ad tertiam partem quadrati
maioris lateris.
Sit pyramidis triangulam ba&longs;im habentis fru&longs;tum AB
CD EF: laterum autem homo
logorum AB, DE, triangulorum
&longs;imilium oppo&longs;itorum ABC, D
EF, &longs;it differentia DG: & eiu&longs;
dem altitudinis fru&longs;to &longs;it pri&longs;ma
DEFCHK: & pyramis intelli
gatur ADEF. Dico fru&longs;tum
BDF ad pri&longs;ma HKF, e&longs;&longs;e vt
rectangulum DEG vna cum ter
tia parte quadrati DG. Ad qua
dratum DE: ad pyramidem au
tem ADEF, vt
gulum DEG, vnà cum tertia parte quadrati DG, ad terAb&longs;ci&longs;sis enim æqualibus EL
ip&longs;i BC, & FM ip&longs;i AC, & EG, ip&longs;i AB, con&longs;tituantur
pri&longs;mata ABCLEG, AGMFCL, ANHDGM, &
pyramis ADGM, & iungatur ML. Quoniam igitur ob pa
rallelas EF, GM, & DF, GL, &longs;imilia inter &longs;e &longs;unt trian
gula DEF, DGM, EGL, duplicatam inter &longs;e habebunt
laterum ho mologorum DE, DG, GE, proportionem,
hoc e&longs;t eandem, quæ totidem e&longs;t quadratorum ex ip&longs;is DE,
DG, GE, prout inter &longs;e re&longs;pondent: vt igitur DG qua
dratum ad quadratum DE, ita e&longs;t triangulum DGM
ad triangulum DEF: eademque ratione vt quadratum
GE ad DE quadratum, ita trian
gulum EGL ad triangulum D
EF: & vt prima cum quinta ad
&longs;ecundam, ita tertia cum &longs;exta ad
quartam: videlicet, vt duo qua
drata DG, GE, ad quadratum
DE, ita duo triangula DGM,
EGL, ad triangulum DEF. &
conuertendo, & per conuer&longs;ionem
rationis, vt quadratum DE ad
rectangulum DGE bis, ita trian
gulum DEF, ad parallelogram
mum GF: & conuertendo, vt rectangulum DGE bis, ad
quadratum DE, ita GF parallelogrammum ad triangu
lum DEF: & antecedentium dimidia, vt rectangulum
DGE ad quadratum DE, ita triangulum GML ad
triangulum DEF; hoc e&longs;t pri&longs;ma, cuius ba&longs;is triangulum
GLM, altitudo eadem pri&longs;mati H
Rur&longs;us, quoniam e&longs;t vt quadratum EG ad quadratum
ED, ita triangulum EGL ad triangulum DEF; erit &longs;i
militer vt quadratum EG ad quadratum ED, ita pri&longs;ma
BGL ad pri&longs;ma HKF: &longs;ed vt rectangulum DGE ad
quadratum DE, ita pri&longs;ma erat, cuius ba&longs;is triangulum G
ACGLFM illi æquale per vltimam XI. elem. ad pri&longs;ma
HKF: vt igitur prima cum quinta, rectangulum DGE
vna cum quadrato EG, hoc e&longs;t rectangulum DEG, ad
&longs;ecundam quadratum DE, ita erit tertia cum &longs;exta, duo
pri&longs;mata BGL, ACGLFM, ad quartam pri&longs;ma HKF. Præterea quoniam vt quadratum DG ad quadratum
DE, ita erat triangulum DGM ad triangulum DEF: &longs;ed
vt triangulum DGM ad triangulum DEF, ita e&longs;t pri&longs;ma,
HGM, ad pri&longs;ma HKF: & tertiæ antecedentium par
tes, videlicet, vt tertia pars quadrati DG, ad quadra
tum DE, ita pyramis ADGM ad pri&longs;ma HKF: &longs;ed
vt rectangulum DEG ad DE quadratum, ita erant duo
pri&longs;mata BGL, ACGLFM, ad pri&longs;ma HKF; vt igi
tur prima cum quinta, rectangulum DEG vna cum ter
tia parte DG quadrati, ad quadratum GD &longs;ecundam,
ita erit tertia cum &longs;exta, duo pri&longs;mata BGL, ACGLFM
vna cum pyramide ADGM, hoc e&longs;t integrum fru&longs;tum
ABCDEF ad pri&longs;ma HKF quartam. Ex hoc patet &longs;e
cunda pars propo&longs;iti. Quoniam enim e&longs;t vt rectangulum
DEG, vna cum tertia parte quadrati DG, ad quadra
tum DE, ita fru&longs;tum ABGDEF ad pri&longs;ma HKF: vt
autem quadratum DE, ad tertiam &longs;ui partem, ita e&longs;t pri&longs;
ma HKF ad pyramidem, cuius ba&longs;is triangulum DEF,
altitudo eadem pri&longs;mati HKF; erit ex æquali vt re
ctangulum DEG vna cum tertia parte quadrati DG
ad tertiam partem quadrati DE, ita fru&longs;tum ABCDEF,
ad pyramidem &longs;i compleatur ADEF. Manife&longs;tum e&longs;t
igitur propo&longs;itum.
Hinc manife&longs;tum e&longs;t eadem demon&longs;tratione,
qua vtimur ad propo&longs;itionem XXXVI. primili
bri; fru&longs;tum cuiuslibet pyramidis ba&longs;im habentis
pluribus quàm tribus lateribus contentam, ad pri&longs;
ma, &longs;eu pyramidem, cuius ba&longs;is e&longs;t eadem quæ ma
ior ba&longs;is fru&longs;ti, & eadem altitudo: & reliquum ip
&longs;ius pri&longs;matis dempto fru&longs;to, ad ip&longs;um pri&longs;ma, eas
habere rationes, quæ à ba&longs;ium fru&longs;ti oppo&longs;itarum
homologis lateribus eorumque differentia deri
uantur eo modo, quo in præcedenti theoremate
dicebamus.
Omne fru&longs;tum coni, vel portionis conicæ, ad cy
lindrum, vel cylindri portionem, cuius ba&longs;is e&longs;t ea
dem, quæ maior ba&longs;is fru&longs;ti, & eadem altitudo,
eam habet proportionem, quàm rectangulum con
tentum ba&longs;ium diametris eiu&longs;dem rationis, vnà
eum tertia parte quadrati differentiæ earumdem
diametrorum, ad maioris ba&longs;is quadratum. Ad
conum autem, vel coni portionem, cuius ba&longs;is e&longs;t
eadem, quæ maior ba&longs;is fru&longs;ti, & eadem altitudo;
vt prædictum rectangulum, vnà cum prædicti qua
drati tertia parte, ad tertiam partem quadrati ex
diametro maioris ba&longs;is. Prædicti autem cylindri,
ad totum cylindrum, vel cylindri portionem; vt
rectangulum contentum diametro minoris ba&longs;is
fru&longs;ti, & differentia diametri maioris, vnà cum
duabus tertiis quadrati differentiæ, ad quadra
tum diametri maioris ba&longs;is.
Sit coni, vel eius portionis fru&longs;tum ABCD, cuius ba&longs;es
oppo&longs;itæ, circuli vel &longs;imiles ellip&longs;es, quarum diametri mi
noris ba&longs;is AB cuius centrum E: maioris autem CD,
& &longs;uper ba&longs;im circulum, vel ellip&longs;im CD &longs;tet cylindrus,
vel portio cylindrica CG comprehendens fru&longs;tum AB
CD, eiu&longs;demque altitudinis cum ip&longs;o, & conus, vel co
ni portio ECD. quo autem AC diameter &longs;uperat dia
metrum AB, quæ differentia di
citur, &longs;it DF. Dico fru&longs;tum AD
ad cylindrum, vel portionem cy
lindricam CG, e&longs;&longs;e vt rectangu
lum DCF vnà cum tertia parte
quadrati DF, ad quadratum CD. Ad conum autem vel coni portio
nem ECD, vt rectangulum DCF,
vna cum tertia parte quadrati DF,
ad tertiam partem quadrati CD. Cylindri autem, vel cylindri por
tionis CG re&longs;iduum dempto fru
&longs;to AD, ad cylindrum, vel portionem cylindricam CG,
vt rectangulum CFD vna cum duabus tertiis quadrati
FD, ad quadratum CD. Cono enim, vel portioni coni
cæ, cuius fru&longs;tum AD, & cylindro, vel portioni cylindri
cæ, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is CD, altitudo au
tem eadem completo cono, vel portioni conicæ iam dictæ,
illi pyramis, huic pri&longs;ma in&longs;cripta intelligantur, quorum
æquilaterum, & æquiangulum; in ellip&longs;e autem, quod pro
Archimede de&longs;cribit Commandinus, ita vt & à cylindro,
vel cylindri portione pri&longs;ina, & à cono, vel coni portione
pyramis deficiat minori &longs;pacio quantacumque magnitudi
ne propo&longs;ita: quo modo autem in portione cylindrica, vel
conica hoc fieri po&longs;&longs;it, eadem quæ de cono atque cylindro
Euclides in duodecimo docuit manife&longs;tant. Ab&longs;ci&longs;&longs;ione
igitur facta fru&longs;ti AD, & cylindri, vel portionis cylindricæ
CG, ab&longs;ci&longs;&longs;a &longs;imul erunt fru&longs;tum pyramidis in&longs;criptum
fru&longs;to AD, & pri&longs;ma in&longs;criptum cylindro, vel portioni cy
lindricæ CG, eiu&longs;dem altitudinis inter &longs;e, & duobus præ
dictis &longs;olidis AD, CG, deficien
tia vnum à fru&longs;to, alterum à cy
lindro, vel portione cylindrica
multo minori &longs;pacio magnitudine
propo&longs;ita: &longs;ectiones autem pri&longs;ma
tis, & pyramidis erunt polygona
circulis, vel ellip&longs;ibus ip&longs;i CD op
po&longs;itis & &longs;imilibus in&longs;cripta in
ter &longs;e &longs;imilia, vt multi o&longs;tendunt. erunt etiam &longs;imilium polygono
rum circulis, vel ellip&longs;ibus &longs;imili
bus, quæ &longs;unt ba&longs;es oppo&longs;itæ fru
&longs;ti AD, in&longs;criptorum diametri eædem AB, CD. Quo
niam igitur &longs;imilium polygonorum circulis, & &longs;imilibus
ellip&longs;ibus in&longs;criptorum latera homologa inter &longs;e &longs;unt vt
diametri dictorum circulorum, vel ellip&longs;ium, eadem erit
proportio inter duas diametros AB, CD, hoc e&longs;t FC,
CD, quæ inter duo quælibet latera homologa polyga
norum circulis, vel ellip&longs;ibus &longs;imilibus AB, CD in
&longs;criptorum. Sed pyramidis fru&longs;tum fru&longs;to CB in&longs;cri
ptum ad pri&longs;ma, cuius ba&longs;is e&longs;t maior ba&longs;is fru&longs;ti pyrami
dis, & eadem altitudo, &longs;olido CG in&longs;criptum, e&longs;t vt re-
&longs;itarum, vna cum tertia parte quadrati differentiæ, ad ma
ioris lateris quadratum; idem igitur fru&longs;tum pyramidis
ad idem pri&longs;ma, erit vt rectangulum DCF, vna cum
tertia parte quadrati DF ad quadratum CD: deficit
autem vtrumque & pyramidis fru&longs;tum fru&longs;to CB in&longs;cri
ptum ab ip&longs;o CB fru&longs;to, & pri&longs;ma ip&longs;i CG in&longs;criptum
ab ìp&longs;o CG, minori &longs;pacio quantacumque propo&longs;ita ma
gnitudine; per tertiam igitur huius, erit vt rectangulum
DCF vna cum tertia parte quadrati DF, ad CD qua
dratum, ita fru&longs;tum CB ad cylindrum, vel portionem
cylindricam CG. Cum igitur conus, vel coni portio E
CD &longs;it pars tertia cylindri, vel portionis cylindricæ CG,
erit ex æquali, vt idem rectangulum DCF, vna cum ter
tia parte quadrati DF, ad tertiam partem quadrati CD,
ita fru&longs;tum BC, ad conum vel coni portionem ECD. P
terea, quia quadratum CD æquale e&longs;t duobus quadratis
ex CF, FD, vna cum rectangulo bis ex CF, FD: quorum
rectangulo CFD, vna cum quadrato CF æquale e&longs;t rectan
gulum DCF; erit quadratum CD æquale rectangulo
DCF vna cum quadrato DF; demptis igitur rectangu
lo DCF, & tertia parte quadrati DF; quod remanet
CD quadrati erit rectangulum CFD vna cum duabus
tertiis quadrati DF. quoniam igitur e&longs;t conuertendo vt
quadratum CD ad rectangulum DCF, vna cum tertia
parte quadrati DF, ita cylindris, vel portio cylindrica
CG ad fru&longs;tum CB, erit per conuer&longs;ionem rationis, &
conuertendo; vt rectangulum CFD vna cum duabus ter
tiis DF quadrati, ad quadratum CD, ita reliquum cy
lindri, vel portionis cylindricæ CG d
ad cylindrum, vel portionem cylindricam. Manife&longs;tum
e&longs;t igitur propo&longs;itum.
Si &longs;phæra, vel &longs;phæroides &longs;ecetur duobus pla
nis parallelis vtcumque, neutro per
quædam autem ex centro recta linea tran&longs;eat per
centrum alterutrius &longs;ectionum; per centrum re
liquæ tran&longs;ibit.
Sit &longs;phæra, vel &longs;phæroides &longs;ectum duobus planis pa
callelis vtcumque neutro per centrum ducto, quod &longs;it E:
per &longs;ectionum autem, quæ &longs;unt circuli, vel &longs;imiles el
lip&longs;es, alterutrius centrum F tran&longs;iens recta EFB oc
currat reliquæ &longs;ectionis plano in puncto G. Dico reli
quæ &longs;ectionis centrum e&longs;&longs;e G. Planum enim per OB &longs;e
cans &longs;phæram, vel &longs;phæroides, faciensque &longs;ectionem circu
lum, vel ellip&longs;im ABCD, &longs;ecabit, & &longs;ecet prædictas &longs;e
ctiones, circulos inquam, vel &longs;imiles ellip&longs;es parallelas, qua
rum alterius centrum ponitur F. Faciatque &longs;ectiones re
ctas parallelas AFC, KGH: &longs;imiliter aliud quodlibet
ctionem circulum, vel ellip&longs;im, & in ea parallelas LFM,
NGO, communes &longs;ectiones iam factæ &longs;ectionis &longs;phæræ
vel &longs;phæroidis cum circulis, vel ellip&longs;ibus inter &longs;e paral
lelis quarum diametri &longs;unt AC, KH. Quoniam igitur
E e&longs;t centrum &longs;phæræ, vel &longs;phæroidis; omnes in eo per
punctum E, tran&longs;euntes rectæ lineæ bifariam &longs;ecabuntur:
&longs;ed idem E e&longs;t in &longs;ectione &longs;phæræ, vel &longs;phæroidis, circu
lo, vel ellip&longs;e ABCD; omnes igitur in ip&longs;a rectas lineas
bifariam &longs;ecabit punctum E, & centrum erit circuli,
vel ellip&longs;is ABCD: quædam igitur ex centro recta EB
&longs;ecans parallelarum neutrius per centrum ductæ alteram
AC bifariam in circuli, vel ellip&longs;is ALCM centro F,
& reliquam in puncto G bifariam &longs;ecabit. Similiter
o&longs;tenderemus rectam NO &longs;ectam e&longs;se bifariam in pun
cto G: atque adeo circuli, vel ellip&longs;is KNHO centrum
e&longs;&longs;e G. Recta igitur E, tran&longs;iens per centrum &longs;ectionis
ALCM, tran&longs;ibit per centrum reliquæ KNHO ip&longs;i
ALCM parallelæ. Quod demon&longs;trandum erat.
Hinc manife&longs;tum e&longs;t, &longs;i &longs;phæra, vel &longs;phæroides
&longs;ecetur plano non per centrum: & recta linea &longs;phæ
ræ, vel &longs;phæroidis, & factæ &longs;ectionis centra iun
gens ad &longs;uperficiem vtrinque producatur; talis
axis &longs;egmenta e&longs;&longs;e
vertices extrema dicti axis, vt in figura theorema
tis &longs;unt puncta B, D.
Si hemi&longs;phærium, vel hemi&longs;phæroides vtcum
que ab &longs;ci&longs;&longs;um: & cylindrus, vel cylindri portio
illi circum&longs;cripta: & conus, vel coni portio, cu
ius ba&longs;is e&longs;t eadem &longs;olido circum&longs;cripto, hemi
&longs;phærium, vel hemi&longs;phæroides ad verticem
tingens
hemi&longs;phærij, vel hemi&longs;phæroidis parallelo: &longs;uper
&longs;ectiones autem prædicti coni, vel portionis coni
cæ, & hemi&longs;phærij, vel hemi&longs;phæroidis, circa hu
ius ab&longs;ci&longs;sæ portionis axem duo cylindri, vel por
tiones cylindricæ con&longs;titerint; reliquum cylindri
vel portionis cylindricæ prædicto plano ab&longs;ci&longs;sæ,
ne cylindrica, cuius ba&longs;is e&longs;t &longs;ectio hemi&longs;phærij,
vel hemi&longs;phæroidis, æquale erit reliquo cylindro,
vel portioni cylindricæ, cuius ba&longs;is e&longs;t &longs;ectio præ
dicti coni, vel portionis conicæ.
E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cuius
axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diameter AC. Et &longs;olido ABC circum&longs;criptus cylindrus, vel portio cy
lindrica, cuius ba&longs;es oppo&longs;itæ erunt circuli, vel &longs;imiles elli
p&longs;es, quarum diametri eiu&longs;dem rationis ADC, EF, la
tera oppo&longs;ita parallelogrammi per axem AFGC: & &longs;u
per ba&longs;im, cuius diameter EF, circa axim BD, de&longs;criptus
e&longs;to conus, vel coni portio EDF. Iam tria &longs;olida ABC,
EDF, AC, &longs;ecentur plano &longs;olidi ABC ba&longs;i parallelo,
quod &longs;ecabit, & &longs;ecet vnà figuras planas per axim BD
AF parallelogrammum, triangulum EDF, & &longs;emicir
culus, vel &longs;emi ellip&longs;is ABC: & &longs;int &longs;ectiones rectæ GO,
HN, KM: hæ igitnr erunt diametri eiu&longs;dem rationis trium
&longs;ectionum, &longs;cilicet circulorum, vel ellip&longs;ium &longs;irnilium, qui
bus erit commune centrum L, in quo nimirum axis BD
tres dictas lineas GO, HN, KM, bifariam &longs;ecat. Vt
igitur de &longs;olido AF diximus, &longs;int circa axem BL, & &longs;uper
ba&longs;es circulos, vel ellip&longs;es circa HN, KM cylindri, vel
portiones cylindricæ HP, KQ, qui vnà cum portione
cylindrica, vel cylindro GF ip&longs;a &longs;ectione facto, erunt inter
eadem plana paral
lela per EF, GO. Dico trium cylin
drorum, vel cylin
dri portionum GF,
HP, KQ,
ip&longs;ius GF dempto
HP, ip&longs;i KQ e&longs;se
æquale. Quoniam
enim cylindri, & cy
lindri portiones eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba
&longs;es, circuli autem, & &longs;imiles ellip&longs;es; inter &longs;e, vt quæ à
diametris eiu&longs;dem rationis fiunt quadrata; ex Archime
de, hoc e&longs;t vt earum quartæ partes, quæ à &longs;emidiame
tris quadrata de&longs;cribuntur; erit vt quadratum LO ad
quadratum LN, ita cylindrus, vel portio cylindrica
GF ad cylindrum, vel portionem cylindricam PH: &
diuidendo, vt rectangulum LNO bis vnà cum quadra
to NO, ad quadratum LN, ita reliquum cylindri, vel
portionis cylindricæ GF, dempto ip&longs;o PH, ad ip&longs;um
PH: &longs;ed vt quadratum LN ad quadratum LM, ita e&longs;t
vt &longs;upra, cylindrus, vel portio cylindrica HP ad cylin
drum, vel portionem cylindricam KQ, ex æquali igitur,
ad quadratum LM, ita reliquum cylindri, vel portionis
cylindricæ GF
pto
drum, vel
cylindricam KQ:
&longs;ed rectangulum L
NO bis vnà
drato NO æquale
e&longs;t quadrato LM;
reliquum igitur cy
lindri, vel portionis
cylindricæ GF,
pto
Quod erat demon&longs;trandum.
Cylindri, vel portionis cylindricæ hemi&longs;phæ
rio, vel hemi&longs;phæroidi circum&longs;criptæ reliquum
dempto hemi&longs;phærio, vel hemi&longs;phæroide, æqua
le e&longs;t cono, vel portioni conicæ eandem ba&longs;im he
mi&longs;phærio, vel hemi&longs;phæroidi, & eandem altitu
dinem habenti.
E&longs;to hemi&longs;phærio, vel hemi&longs;phæroidi ABC, cu
ius axis BD, ba&longs;is circulus, vel ellip&longs;is circa diametrum
ADC, circum&longs;criptus cylindrus, vel cylindrica portio
AE, circa communem &longs;cilicet axim BD. conus autem,
vel coni portio circa axim BD, ba&longs;im habens commu
nem &longs;olido ABC, intelligatur. Dico reliquum &longs;olidi
AE, dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC æ-Nam circa axim
BD, & &longs;uper ba&longs;im circulum, vel ellip&longs;im, cuius diame
ter RE, &longs;imilem & oppo&longs;itam ei, quæ circa AC, de&longs;cri
batur conus, vel coni portio RDE. Deinde axe BD bi
fariam &longs;ecto, & &longs;ingulis eius partibus rur&longs;us bifariam, vt
partes axis BD omnes &longs;int æquales, per puncta &longs;ectio
num, quotquot erunt, totidem plana parallela &longs;ecent vnà
cum &longs;olido AE duas ip&longs;ius partes, &longs;olida ABC, RDE. Omnes igitur factæ &longs;ectiones, vel erunt circuli, vel &longs;imiles
ellip&longs;es ei, quæ e&longs;t circa AC, atque adeo inter &longs;e &longs;imiles:
talium autem &longs;ectiones communes cum AE parallelo,
grammo per axim, erunt rectæ lineæ, ternæ in &longs;ingu
lis planis &longs;ecantibus, & in eadem recta linea; vt in proxi
ma ip&longs;i RE, &longs;unt FL, GN, KM, quæ quidem erunt
trium circulorum, vel &longs;imilium ellip&longs;ium diametri eiu&longs;dem
rationis ba&longs;ium trium &longs;olidorum, cylindri &longs;cilicet, vel por
tionis cylindricæ FL, fru&longs;ti GL, & portionis KBM, he
mi&longs;phærij, vel hemi&longs;phæroidis ABC. Itaque circa axem
BH cylindri, vel portionis cylindricæ FE, & &longs;uper ba
&longs;es circulos, vel ellip&longs;es circa GN, KM, de&longs;cribantur
cylindri, vel cylindri portiones GP, KQ, qui pat
tes erunt totius cylindri, vel portionis cylindricæ FE. Idem fiat circa reliquas axis partes BD tamquam axes,
&longs;ecantibus. Hac ratione habebimus iam duas figuras
compo&longs;itas ex cylindris, vel cylindri portionibus altitudi
ne, & multitudine æqualibus, alteram cono, vel portioni
conicæ RDE in&longs;criptam, alteram hemilphærio, vel he
mi&longs;phæroidi ABC circum&longs;criptam: quod ita factum e&longs;
&longs;e intelligatur, quemadmodum in primo libro fieri po&longs;se
demon&longs;trauimus, vt figura cono RDE in&longs;cripta ab eo
deficiat, hemi&longs;phærio autem, vel hemi&longs;phæroidi ABC
circum&longs;cripta ip&longs;um excedat minori &longs;pacio magnitudine
propo&longs;ita quantacumque illa &longs;it. Reliquo itaque cylin
dri, vel portionis cylindricæ AE dempto hemi&longs;phærio, vel
hemi&longs;phæroide ABC figura quædam in&longs;cripta relinque
tur ex cylindris, vel portionis cylindricæ re&longs;iduis æqualium
altitudinum, demptis ijs, ex quibus con&longs;tat figura hemi
&longs;phærio, vel hemi&longs;phæroidi ABC circum&longs;cripta, excepto
infimo cylindro, vel portione cylindrica AS. Et quo
niam (excepto exce&longs;su, quo &longs;olidum AS excedit &longs;ui par
tem portionem quandam hemi&longs;phærij, vel hemi&longs;phæroidis
ABC) quo &longs;pacio figura hemi&longs;phærio, vel hemi&longs;phæroidi
ABC circum&longs;cripta &longs;uperat ip&longs;um hemi&longs;phærium, vel he
præcedenti, reliquum cylindri, vel portionis cylindricæ
FE dempto cylindro, vel portione cylindrica KQ, æ
quale e&longs;t cylindro, vel portioni cylindricæ GP: eadem
que ratione &longs;ingula cylindrorum, vel cylindri portionum
re&longs;idua, quæ &longs;unt in reliqua figura cylindri, vel portionis
cylindricæ AE, dempto hemi&longs;phærio, vel hemi&longs;phæroi
de ABC, æqualia erunt &longs;ingulis cylindris, vel cylindri
portionibus, quæ &longs;unt in cono, vel portione conica RDE,
&longs;i bina &longs;umantur inter eadem plana parallela, vel circa
cundem axem; tota igitur figura in&longs;cripta prædicto re&longs;iduo,
toti figuræ in&longs;criptæ cono, vel portioni conicæ RDE æ
qualis erit: deficit autem vtraque figura in&longs;cripta à &longs;ibi
circum&longs;cripta minori &longs;pacio quantacumque magnitudine
propo&longs;ita; per tertiam igitur huius, reliquum cylindri, vel
portionis cylindricæ AE, dempto hemi&longs;phærin, vel he
mi&longs;phæroide ABC, æquale e&longs;t cono, vel portioni coni
cæ RDE, hoc e&longs;t ip&longs;i ABC. Quod erat demon&longs;trandum.
Si hemi&longs;phærium, vel hemi&longs;phæroides, & cylin
drus, vel portio cylindrica ip&longs;i circum&longs;cripta, &
conus, vel coni portio, cuius e&longs;t
ba&longs;is autem qu<17> opponitur communi ba&longs;i duorum
prædictorum &longs;olidorum, vnà &longs;ecentur duobus
planis ba&longs;i parallelis; portiones reliquæ figuræ
ex cylindro, vel cylindri portione hemi&longs;phærio,
vel hemi&longs;phæroidi circum&longs;cripta dempto hemi
&longs;phærio, vel hemi&longs;phæroide, quæ à duobus præ
dictis planis &longs;ecantibus fiunt, æquales &longs;unt &longs;in
partibus &longs;iue fru&longs;tis inter eadem plana parallela
re&longs;pondentibus.
E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cu
ius axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diame
ter ADC. &longs;olido autem ABC circum&longs;criptus cylindrus,
vel portio cylindrica AXEC: & conus, vel coni portio
&longs;it XDE, cuius vertex D, ba&longs;is circulus, vel ellip&longs;is cir
ca XBE ba&longs;i &longs;olidi AE, vel ABC, prædictæ oppo&longs;ita,
&longs;ecto autem &longs;olido AE, atque vnà cum ip&longs;o eius partibus,
&longs;olidis ABC, XD
E, duobus planis ba
&longs;i &longs;olidi AE, vel
ABC, atque ideo
inter &longs;e quoque pa
rallelis, intelligan
tur trium &longs;olidorum
portiones ternæ in
ter eadem plana pa
rallela: videlicet in
ter duo per XE,
FN, hemi&longs;phærij, vel hemi&longs;phæroidis minor portio HBL:
& reliquum cylindri, vel portionis cylindricæ FE dem
pta portione HBL: & coni, vel conicæ portionis fru&longs;tum
XGME. &longs;imiliter inter duo plana per FN, OV &longs;olidi
ABC portio PHLT, eaque ablata reliquum &longs;olidi ON,
& fru&longs;tum GQSM. Denique &longs;olidi ABC portio AP
TC, eaque ablata, reliquum &longs;olidi AV, & conus, vel
coni portio QDS. Dico reliquum &longs;olidi FE, dempto
HBL e&longs;&longs;e æquale fru&longs;to XGME: & reliquum &longs;olidi ON
dempto PHLT, æquale fru&longs;to GQSM: & reliquum
&longs;olidi AV dempto &longs;olido APTC æquale &longs;olido QDS.
dempto &longs;olido ABC æquale e&longs;se &longs;olido XDE, &longs;imili
ter o&longs;ten&longs;um remanet, tam reliquum &longs;olidi AN, dempto
&longs;olido AHLC, æquale e&longs;se &longs;olido GDM, quam reli
quum &longs;olidi AV dempto &longs;olido APTC æquale &longs;olido
QDS; erit demptis æqualibus, tam reliquum &longs;olidi FE,
dempto &longs;olido HBL, æquale &longs;olido XGME; quam
reliquum &longs;olidi ON, dempto &longs;olido PHLT æquale &longs;o
lido GQSM. At reliquum &longs;olidi AV dempto &longs;oli
do APTC &longs;olido QDS æquale erit. Manife&longs;tum e&longs;t
igitur propo&longs;itum.
Hemi&longs;phærium, vel hemi&longs;phæroides &longs;ub&longs;e&longs;qui
alterum e&longs;t cylindri; vel portionis cylindricæ ip&longs;i
circum&longs;criptæ.
E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC,
ip&longs;ique circum&longs;criptus cylindrus, vel portio cylindri
ca AE, circa eundem &longs;cilicet axem BD, & &longs;uper can
dem ba&longs;im circulum,
vel ellip&longs;im, circa AC:
nam hac ratione ba&longs;is
oppo&longs;ita &longs;olidum ABC
tanget ad verticem B. Dico
hemi&longs;phæroides ABC
e&longs;se cylindri, vel portio
nis cylindricæ AE &longs;ub
&longs;e&longs;quialterum. Nam
circa axem BD, &longs;uper prædictam ba&longs;em circa AC, e&longs;to
de&longs;criptus conus, vel coni portio ABC. Quoniam igitur
hemi&longs;phærio, vel hemi&longs;phæroide ABC æquale e&longs;t cono,
vel portioni conicæ ABC: & cylindrus, vel portio cylin
drica AE tripla e&longs;t co
ni, vel portionis conicæ
ABC; triplus itidem
erit cylindrus, vel cylin
drica portio AE dicti
re&longs;idui dempto hemi
&longs;phærio, vel hemi&longs;phæ
roide ABC; ac propte
rea hemi&longs;phærij, vel he
mi&longs;phæroidis ABC
&longs;e&longs;quialter, hoc e&longs;t hemi&longs;phærium, vel hemi&longs;phæroides
ABC cylindri, vel portionis cylindricæ AE &longs;ub&longs;e&longs;quial
terum. Quod erat demon&longs;trandum.
Omnis minor portio &longs;phæræ, vel &longs;phæroidis ad
cylindrum, vel cylindri portionem, cuius ba&longs;is
æqualis e&longs;t circulo maximo, vel æqualis, & &longs;imi
lis ellip&longs;i per centrum ba&longs;i portionis parallelæ,
& eadem altitudo portioni; eam habet proportio
nem, quam rectangulum contentum &longs;phæræ, vel
&longs;phæroidis dimidij axis axi portionis congruen
tis ijs, quæ à centro ba&longs;is portionis fiunt
vnà cum duobus tertiis quadrati axis portionis; ad
&longs;phæræ, vel &longs;phæroidis dimidij axis quadratum.
Sit minor portio ABC, &longs;phæræ, vel &longs;phæroidis, cuius
centrum D, axis autem axi portionis congruens BEDR:
portione ABC ex cylindro, vel portione cylindrica NO
circum&longs;cripta hemi&longs;phærio, vel hemi&longs;phæroidi NBO,
cuius ba&longs;is circa diametrum NO, &longs;it ba&longs;i portionis ABC
parallela: qua ratione ba&longs;is prædicti &longs;olidi FG, erit vel cir
culus, vel ellip&longs;is æqualis circulo maximo, vel &longs;imilis, &
æqualis ellip&longs;i circa NO, portionis ABC ba&longs;i paralle
læ. Dico portionem ABC ad cylindrum, vel portio
nem cylindricam FG, e&longs;se vt rectangulum BED, vnà
cum duabus tertiis qua
drati EB ad quadratum
BD. E&longs;to enim conus,
vel coni portio HDG,
cuius fru&longs;tum HKLG
prædicto plano ab&longs;ci&longs;&longs;um:
& omnino &longs;int
vel ellip&longs;ium &longs;imilium dia
metri eiu&longs;dem rationis
NO, vt ad XII huius, in
AC, KL, &longs;ectæ omnes bi
fariam in
& HBG, in eodem plano per axem. Quoniam igitur ex &longs;u
perioribus, reliquum &longs;olidi FG, dempto ABC, æquale e&longs;t
fru&longs;to HKLG; erit eiu&longs;dem &longs;olidi FG reliquum ABC
æquale reliquo &longs;olidi FG, dempto HKLG: &longs;ed hoc reli
quum dempto HKLG, &longs;upra o&longs;tendimus e&longs;se ad &longs;olidum
FG, vt rectangulum ex KL, & differentia HG, vnà
cum duabus tertiis quadrati differentiæ, ad quadratum
GH: & vt HG ad KL, ita e&longs;t BD ad DE, propter &longs;imi
litudinem triangulorum; vt igitur e&longs;t rectangulum BED,
vnà cum duabus tertiis quadrati BE, ad quadratum BD,
ita erit portio ABC, ad cylindrum, vel portionem cylin
dricam FG. Quod demon&longs;trandum erat.
Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;&longs;a
duobus planis parallelis, alteroper centrum du
cto, ad cy lindrum, vel cylindri portionem, cuius
ba&longs;is e&longs;t eadem, quæ maior ba&longs;is portionis, &
altitudo; eam habet proportionem, quam rectan
gulum contentum ijs, quæ à centro minoris ba&longs;is
fiunt axis &longs;phæræ, vel &longs;phæroidis &longs;egmentis, vnà
cum duabus tertiis quadrati axis portionis; ad
&longs;phæræ, vel &longs;phæroidis dimidij axis quadratum.
Sit portio NACO &longs;phæræ, vel &longs;phærodij, cuius cen
trum D, axis autem axi portionis congruens BEDR,
ab&longs;ci&longs;sa duobus planis parallelis altero per centrum D, &longs;e
ctionem faciente circulum
maximum, vel ellip&longs;im,
cuius diameter NO, & &longs;u
per dictam &longs;ectionem, cir
ca axem ED, &longs;tet cylin
drus, vel portio cylindrica
NM, ab&longs;ci&longs;sa ij&longs;dem pla
nis, quibus portio NAC
O, à cylindro, vel portio
ne cylindrica NG, &longs;it cir
cum&longs;cripta hemi&longs;phærio,
vel hemi&longs;phæroidi NBO:
qua ratione erit cylindri,
vel portionis cylindricæ NM ba&longs;is eadem, quæ maior
ba&longs;is portionis NACO, circulus &longs;cilicet, vel ellip&longs;is cir
ca NO, & eadem altitudo portioni. Dico portionem
e&longs;se vt rectangulum BER, vnà cum duabus tertiis ED
quadrati, ad quadratum BD. Ij&longs;dem enim quæ in præce
denti con&longs;tructis, & notatis, &longs;it præterea cylindrus, vel por
tio cylindrica PL, circa axim ED circum&longs;cripta cono,
vel portioni conicæ KDL, Quoniam igitur reliquum
cylindri, vel portionis cylindricæ NM, dempta portione
NACO æquale e&longs;t cono, vel portioni conicæ
erit reliqua portio NACO æqualis reliquo eiu&longs;dem NM,
dempto cono, vel portione conica KDL. Et quoniam cir
culi, & &longs;imiles ellip&longs;es inter &longs;e &longs;unt vt quadrata diametro
rum, vel
& portiones cylindricæ
erit vt quadratum EM, hoc e&longs;t quadratum BG, ad qua
dratum EL, hoc e&longs;t vt quadratum BD ad quadratum
DE, propter &longs;imilitudinem triangulorum, ita &longs;olidum NM
ad &longs;olidum PL: & per conuer&longs;ionem rationis, vt quadra
tum BD ad rectangulum BED bis, vnà cum quadrato
BE, ita &longs;olidum MN, ad &longs;ui reliquum dempto &longs;olido
PL: & conuertendo, vt rectangulum BED bis, vnà cum
quadrato BE, hoc e&longs;t rectangulum BER, ad quadratum
BD, ita reliquum &longs;olidi NM dempto &longs;olido PL ad &longs;o
lidum NM. Rur&longs;us, quoniam e&longs;t vt quadratum EL ad
quadratum EM, &longs;iue BG, hoc e&longs;t vt quadratum ED ad
quadratum BD, ita &longs;olidum PL ad &longs;olidum NM, ob
&longs;imilem rationem &longs;upradictæ: & duæ tertiæ partes &longs;olidi
PL e&longs;t &longs;olidum KDL; erit ex æquali, vt duæ tertiæ qua
drati ED ad quadratum BD, ita reliquum &longs;olidi PL
dempto &longs;olido KDL, ad &longs;olidum NM: &longs;ed vt rectangu
lum BER ad quadratum BD, ita erat &longs;olidi NM reli
quum dempto &longs;olido PL, ad &longs;olidum NM; vt igitur pri
ma cum quinta ad &longs;ecundam, ita erit tertia cum &longs;exta ad
quartam; videlicet, vt rectangulum BED, vnà cum dua
bus tertiis ED quadrati ad quadratum BD, ita reliquum
portione conica KDL, hoc e&longs;t portio NACO ip&longs;i æqua
lis, ad cylindrum, vel portionem cylindricam NM. Quod erat demon&longs;trandum.
Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;&longs;a
duobus planis parallelis, neutro per centrum du
cto, nec centrum intercipientibus, ad cylindrum,
vel cylindri portionem, cuius ba&longs;is æqualis e&longs;t
circulo maximo, vel ellip&longs;i per centrum ba&longs;ibus
portionis parallelæ &longs;imilis, & æqualis, eam ha
bet proportionem, quam duo rectangula; & quod
&longs;phæræ, vel &longs;phæroidis axis axi portionis
tis ijs, quæ à centro minoris ba&longs;is portionis fiunt
& &longs;phæræ, vel &longs;phæroidis centra iungit, & axe por
tionis continetur, vnà cum duabus tertijs quadra
ti axis portionis; ad &longs;phæræ, vel &longs;phæroidis dimi
dij axis quadratum.
Sit portio AQTC &longs;phæræ, vel &longs;phæroidis, cuius cen
trum D, axis autem axi portionis congruens BSEDR,
ab&longs;ci&longs;&longs;um duobus planis parallelis, neutro per centrum
D acto, nec ip&longs;um intercipientibus: & circa portionis
axim SE &longs;tet cylindrus, vel portio cylindrica FX ab
&longs;ci&longs;sa vnà cum portione AQTC ex toto cylindro, vel
portione cylindrica NG, hemi&longs;phærio, vel hemi&longs;phæroi
di NBO circum&longs;cripta, cuius ba&longs;is circulus maximus
qua ratione cylindrus, vel portionis cylindricæ FX eiu&longs;
dem altitudinis portioni AQTC, ba&longs;is erit circulus
æqualis circulo maximo, vel ellip&longs;is &longs;imilis, & æqualis ei,
cuius diameter NDO, ba&longs;ibus AQTC portionis paral
lelæ. Dico portionem AQTC ad cylindrum, vel por
tionem cylindricam FX, e&longs;&longs;e vt duo rectangula BSR,
DES, vnà cum duabus tertiis quadrati ES, ad quadra
tum BD. Ij&longs;dem enim con&longs;tructis, & notatis, quæ in an
circa axim ED &longs;teterat:
planum præterea minoris
ba&longs;is QT portionis AQ
TC extendatur: & &longs;e
cans tria &longs;olida, & figuras
planas per axim po&longs;itas in
eodem plano, faciat ternas
&longs;ectiones, circulos, vel elli
p&longs;es &longs;imiles ei, quæ e&longs;t cir
ca NO: & earum diame
tros IX, PV, QT, in
cadem recta linea commu
ni &longs;ectione exten&longs;i plani, &
eius, quod per axem: quæ quidem diametri &longs;ectæ eruntom
nes bifariam in centro S communi trium prædictarum pla
narum Denique coni, vel portionis conicæ HDG
fru&longs;to PKIV ab&longs;ci&longs;&longs;o vnà cum portione AQTC, &longs;it
circa axim SE circum&longs;criptus cylindrus vel portio cylin
drica ZV. Quoniam igitur per XIIII huius, reliquum
&longs;olidi FX, dempta portione AQTC, æquale e&longs;t fru&longs;to
PKLV; erit reliqua portio AQTC, reliquo eiu&longs;dem
&longs;olidi FX, dempto fru&longs;to PKLV æqualis. Et quoniam
e&longs;t vt PV ad KL, ita SD, DE, propter &longs;imilitudinem
triangulorum: & vt rectangulum ex KL, & differentia
ferentiæ, ad quadratum PV, ita e&longs;t reliquum &longs;olidi ZV
dempto fru&longs;to PKLV ad &longs;olidum ZV; erit vt rectangu
lum DES, vnà cum duabus tertiis quadrati ES, ad DS
quadratum, ita &longs;olidi ZV reliquum dempto fru&longs;to PK
LV ad &longs;olidum ZV: &longs;ed vt quadratum DS ad quadra
tum DB, hoc e&longs;t vt quadratum SV ad quadratum BG,
ide&longs;t ad quadratum SX, ita e&longs;t &longs;olidum ZV, ad &longs;olidum
FX; ex æquali igitur, vt rectangulum DES, vnà cum
duabus tertiis ES quadrati, ad quadratum BD, ita e&longs;t
reliquum &longs;olidi ZV, dem
pto &longs;olido PKLV ad &longs;o
lidum FX: &longs;ed vt rectan
gulum BSR ad quadra
tum BD, ita e&longs;t, eadem
ratione, qua in præcedenti
theoremate vtebamur, re
liquum &longs;olidi FX dem
pto &longs;olido ZV, ad &longs;oli
dum FX; vt igitur prima
cum quinta ad &longs;ecundam,
ita tertia cum &longs;exta ad
quartam; videlicet, vt duo
rectangula BSR, DES, vnà cum duabus tertiis quadra
ti ES ad quadratum BD, ita erit totum reliquum cylin
dri, vel portionis cylindricæ FX dempto fru&longs;to PKLV:
hoc e&longs;t &longs;phæræ, vel &longs;phæroidis portio AQTC ad cylin
drum, vel portionem cylindricam FX. Quod demon
&longs;trandum erat.
Omnis maior portio &longs;phæræ, vel &longs;phæroidis,
ad cylindrum, vel portionem cylindricam, cuius
ba&longs;is æqualis e&longs;t circulo maximo, vel æqualis, &
&longs;imilis ellip&longs;i per centrum ba&longs;i portionis paralle
læ, altitudo autem eadem portioni, eam habet
proportionem, quam &longs;olidum rectangulum con
tentum axe portionis, & reliquo axis &longs;phæræ, vel
&longs;phæroidis &longs;egmento, & eo, quod ba&longs;is portionis,
& &longs;phæræ, vel &longs;phæroidis centraiungit, vnà cum
binis tertiis partibus duorum cuborum: & eius
qui à &longs;phæræ, vel &longs;phæroidis axis dimidio; &
cius qui ab eo, quod &longs;phæræ, vel &longs;phæroidis, &
ba&longs;is portionis centra iungit &longs;it &longs;egmento; ad &longs;o
lidum rectangulum, quod axe portionis, & duo
bus &longs;phæræ, vel &longs;phæroidis axis fit dimidijs.
Sit maior portio AB
C, &longs;phæræ, vel &longs;phæroi
dis ABCF, cuius cen
trum D: ba&longs;is
tionis, circulus, vel elli
p&longs;is, cuius diameter A
C: Et &longs;ecta portione
ABC per centrum D
plano ba&longs;i AC paral
lelo, qua ratione &longs;ectio
erit circulus maximus,
vel ellip&longs;is &longs;imilis ba&longs;i
DE &longs;phæræ, vel &longs;phæroidis, & ba&longs;is portionis centra DE,
atque producta incidat in &longs;phæræ, vel &longs;phæroidis &longs;uperfi
ciem ad partes E in puncto F, & ad partes oppo&longs;itas in
puncto B: &longs;phæræ igitur, vel &longs;phæroidis axis axi portionis
BE congruens crit BDEF, nam vertex portionis erit B:
& hemi&longs;phærio, vel hemi&longs;phæroidi KBL &longs;it circum&longs;cri
ptas cylindrus, vel cylindrica portio KH, cuius &longs;cilicet
axis BD, & circa axim DE, alter cylindrus, vel portio
cylindrica GL portioni KACL circum&longs;cripta: quorum
circum&longs;criptorum &longs;olido
rum vtriulque communis
ba&longs;is erit circulus, vel
ellip&longs;is circa KL. Ita
que ex his compo&longs;itus to
tus cylindrus, vel cylin
dri portio GH erit por
tioni ABC circum&longs;cri
pta, habens axim BE, at
que ideo eandem altitu
dinem ABC portioni,
ba&longs;im autem, cuius dia
meter &longs;it GM &longs;imilem
& æqualem ei, quæ e&longs;t circa KL. Dico portionem ABC
ad cylindrum, vel portionem cylindricam GH, e&longs;se vt &longs;o
lidum rectangulum contentum ip&longs;is BE, EF, ED, vnà
cum binis tertiis duorum cuborum, duabus &longs;cilicet cubi
BD, & totidem cubi ED, ad &longs;olidum rectangulum con
tentum ip&longs;is EB, BD, DF. Quoniam enim parall ele
pipeda eiu&longs;dem altitudinis inter &longs;e &longs;unt vt ba&longs;es, erit vt re
ctangulum BEF vnà cum duabus tertiis ED quadrati ad
rectangulum BDF, ide&longs;t ad quadratum BD, &longs;iue DF,
ita &longs;olidum ex BE, EF, ED, communi altitudine DE,
vnà cum duabus tertiis cubi ED, ad &longs;olidum ex DE,
DE quadrati, ad quadratum DF, ita o&longs;tendimus e&longs;&longs;e
portionem AKLC ad &longs;olidum GL; vt igitur e&longs;t &longs;olidum
ex BE, EF, ED, vnà cum duabus tertiis cubi ED, com
muni altitudine DE, ad &longs;olidum ex ED, BD, DF, ita
erit portio AKLC ad &longs;olidum GL: &longs;ed vt &longs;olidum ex
ED, DB, DF, hoc e&longs;t id, cuius altitudo ED, ba&longs;is BD
quadratum, ad &longs;olidum ex EB, BD, DF, hoc e&longs;t ad id,
cuius altitudo BE, ba&longs;is quadratum BD, ita e&longs;t altitudo,
vel latus ED, ad altitudinem vel latum BE: hoc e&longs;t &longs;oli
dum GL ad &longs;olidum GH; quippe quorum dictæ lineæ
ED, BE &longs;unt axes; ex æquali igitur, vt &longs;olidum ex BE,
EF, ED, vnà cum duabus tertiis cubi DE, ad &longs;olidum
ex EB, BD, DE, cuius altitudo EB, ba&longs;is quadratum
BD, ita erit portio AKLC ad &longs;olidum GH. Rur&longs;us,
quoniam &longs;olidum HK e&longs;t hemi&longs;phærij, vel hemi&longs;phæroi
dis KBL &longs;e&longs;quialterum; erit vt duæ tertiæ partes cubi BD
ad cubum BD, ita hemi&longs;phærium, vel hemi&longs;phæroides
KBL ad &longs;olidum KH: &longs;ed vt cubus BD ad &longs;olidum ex
BD, DF, & altitudine BE, hoc e&longs;t vt altitudo BD ad
altitudinem BE, ita e&longs;t &longs;olidum KH ad &longs;olidum GH, quo
rum dictæ altitudines BD, BE &longs;unt axes, ex æquali igitur
erit vt duæ tertiæ partes cubi BD ad &longs;olidum ex EB, BD,
DF, ita hemi&longs;phærium, vel hemi&longs;phæroides KBL, ad &longs;oli
dum GH: &longs;ed vt
tertiis cubi ED ad &longs;olidum ex EB, BD, DF, erat por
tio AKLC ad cylindrum GH; vt igitur prima cum quin
ta ad &longs;ecundam, ita tertia cum &longs;exta ad quartam, videlicet,
vt duæ tertiæ cubi BD, vna cum duabus tertiis cubi BE,
& &longs;olido ex BE, EF, ED ad &longs;olidum ex EB, BD, DF,
ita erit &longs;phæræ, vel &longs;phæroidis maior portio ABC ad &longs;oli
dum, cylindrum &longs;cilicet, vel portionem cylindricam GH. Quod erat demon&longs;trandum.
Omnis portio &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;sa
duobus planis parallelis centrum intercipienti
bus, ad cylindrum, vel cylindri portionem, cuius
ba&longs;is æqualis e&longs;t circulo maximo, vel &longs;imilis, &
æqualis ellip&longs;i per centrum ba&longs;ibus portionis pa
rallelæ, & eadem altitudo portioni, eam habet
proportionem, quam duo &longs;olida rectangula ex ter
norum &longs;phæræ, vel &longs;phæroidis axis &longs;egmentorum
eundem terminum habentium alterutrius ba
&longs;ium portionis centrum, binis &longs;phæræ, vel &longs;phæ
roidis axem complentibus, & &longs;ingulis axis por
tionis itidem à centro &longs;phæræ, vel &longs;phæroidis fa
ctis, vnà cum binis tertijs partibus duorum cubo
rum ex &longs;egmentis axis portionis à centro &longs;phæræ,
vel &longs;phæroidis factis; ad &longs;olidum rectangulum,
quod duobus &longs;phæræ, vel &longs;phæroidis axis dimi
diis, & axe portionis continetur.
Sit portio ABCD &longs;phæræ, vel &longs;phæroidis, cuius cen
trum E, axis portionis KEH: ip&longs;i autem portioni cir
cum&longs;criptus cylindrus, vel cylindrica portio NO, vt in
antecedenti, cuius communis &longs;ectio cum &longs;phæra, vel &longs;phæ
roide AFDG, &longs;it circulus maximus, vel ellip&longs;is circa dia
metrum LEM; quamobrem ba&longs;is &longs;olidi NO, eiu&longs;dem
altitudinis portioni ABCD circulus erit æqualis circu
lo maximo, vel ellip&longs;is æqualis, & &longs;imilis ellip&longs;i circa LM
ba&longs;ibus portionis parallelæ. Dico portionem ABCD
&longs;olida ad rectangula, alterum ex FH, HG, EH: alterum
ex GK, KF, EK, vnà cum binis tertiis duorum cubo
rum ex EK, EH, ad &longs;olidum rectangulum ex GE,
EF KH, axe enim KH producto vt incidat in &longs;uper
ficiem in punctis F, G, &longs;it &longs;phæræ, vel &longs;phæroidis, ex
demon&longs;tratis, axis FK, EHG. Intelliganturque vt in
antecedenti duo cylindri, vel cylindri portiones NM,
LO, totius prædicti &longs;olidi NO: itemque duæ portiones
&longs;phæræ, vel &longs;phæroidis ALMD, LBCM, quorum qua
tuor &longs;olidorum commu
nis ba&longs;is e&longs;t circulus, vel
ellip&longs;is circa LEM. Quoniam igitur vt in
antecedenti o&longs;tendere
mus portionem ALM
D ad &longs;olidum NM e&longs;
&longs;e vt &longs;olidum ex FH,
HG, EH, vnà cum
duabus tertiis cubi EH
ad &longs;olidum ex FE, EG,
EH, communi altitu
dine EH: &longs;ed vt &longs;oli
dum ex FE, EG, EH,
altitudine EH, ad &longs;olidum ex FE, EG, KH altitudi
ne KH, ita e&longs;t altitudo EH ad altitudinem KH, hoc
e&longs;t &longs;olidum NM ad &longs;olidum NO, quippe quorum &longs;unt
axes EH, KH; ex æquali igitur erit vt &longs;olidum ex FH,
HG, EH, vnà cum duabus tertiis cubi EH, ad &longs;oli
dum ex FE, EG, KH, ita portio ALMD, ad &longs;oli
dum NO. Eadem ratione o&longs;tenderemus e&longs;&longs;e, vt &longs;olidum
ex GK, KF, EK, vnà cum duabus tertiis cubi EK, ad
&longs;olidum ex FE, EG, KH, ita portionem LBCM, ad
&longs;olidum NO; vt igitur prima cum quinta ad &longs;ecundam,
da, & quod &longs;it ex FH,
HG, EH, & quod
ex GK, KF, EK, vnà
cum duabus tertiis &
cubi ex EH, & cu
bi ex EK, ad &longs;olidum
ex FE, EG, KH, ita
erit tota &longs;phæræ, vel
&longs;phæroidis portio AB
CD, ad cylindrum, vel
portionem cylindricam
NO. Quod demon
&longs;trandum erat.
Omnis trianguli comprehen&longs;i &longs;ectione para
bola, ex duabus rectis lineis, quarum altera &longs;e
ctionem tangat, altera in eam incidat diametro
&longs;ectionis ex contactu æquidi&longs;tans, centrum graui
tatis e&longs;t punctum illud, in quo recta linea ex con
tactu diuidens incidentem ita vt pars, quæ &longs;ectio
nem attingit &longs;it &longs;e&longs;quialtera reliquæ, &longs;ic diui
ditur, vt pars quæ e&longs;t ad contactum &longs;it tripla
reliquæ.
Sit triangulum ABC comprehen&longs;um &longs;ectione parabo
la ADB, & duabus rectis lineis, quarum altera AC tan
gat &longs;ectionem in puncto A, reliqua autem BC, in eam
incidens in puncto B, &longs;ectionis diametro ex puncto A,
æquidi&longs;tans intelligatur: & per centrum grauitatis trian-Dico AF
e&longs;&longs;e ip&longs;ius FE triplam: at BE ip&longs;ius EC &longs;e&longs;quialteram. Completo enim triangulo rectilineo ABC, &longs;ectis que re
ctis lineis bifariam AB in puncto H, & AC in puncto K
ducatur HDK, quæ parallela erit ba&longs;i BC: parabolæ igi
tur &longs;egmenti BDA dia meter erit DH; in qua parabolæ
ADB, cuius vertex D &longs;it centrum grauitatis M: trian
guli autem rectilinei ABC centrum grauitatis N, & iun
gatur MN: producta igitur MN occurret trianguli ABC
mixti centro grauitatis F. &longs;int igitur centra M, N, F, in
eadem recta linea:
& ducta recta AN
G &longs;ecet ba&longs;im BC
bifariam in G pun
cto, nece&longs;&longs;e e&longs;t e
nim: & ex puncto
F ad rectam AG,
ducatur recta FO
ip&longs;is BC, KH pa
rallela, & BD, DA
iungantur.
igitur AG &longs;ecat
BC, KH paral
lelas in rectolineo
triangulo ABC,
in ea&longs;dem rationes; &longs;ecta erit HK bifariam à linea AG:
cumque HD diameter parabolæ ADC, cuius vertex D,
&longs;it parallela diametro parabolæ, cuius vertex A, atque
ideo etiam BC incidenti parallela, erit DH pars ip&longs;ius
KH: quoniam igitur in triangulo mixto ABC recta KD
applicata parallela e&longs;t ip&longs;i BC, quæ itidem e&longs;t parallela
diametro parabolæ, cuius vertex A; erit vt AC ad AK
potentia, ita BC ad DK longitudine, quod &longs;upra demon
&longs;trauimus: &longs;ed AC quadrupla e&longs;t potentia ip&longs;ius AK;
dupla ip&longs;ius KH, erit DK dimidia eiu&longs;dem KH, & &longs;ecta
bifariam KH in puncto D: &longs;ed recta AG &longs;ecabat eandem
KH bi fariam; per punctum igitur D tran&longs;ibit AG. Quo
niam igitur parabola ADC, cuius vertex D, &longs;e&longs;quiter
tia e&longs;t per Archimedem trianguli ADB, cuius duplum
e&longs;t triangulum ABG, &longs;icut & huius triangulum ABC;
triangulum ABC quadruplum erit trianguli ADB: qua
lium igitur partium æqualium e&longs;t triangulum ABC duo
decim, talium erit triangulum ADB trium, & parabola
ADB, cuius ver
tex D quatuor: du
plum igitur erit tri
angulum ABC
mixtum parabolæ
ADB, cuius ver
tex D, & cen
trum grauitatis M:
&longs;ed trianguli ABC
rectilinei e&longs;t cen
trum grauitatis N,
& F
mixti; dupla igitur
erit MN ip&longs;ius N
F, & MD ip&longs;ius
OF, & DN ip&longs;ius NO, propter &longs;imilitudinem triangulo
rum: &longs;ed & tota AN dupla e&longs;t totius NG, ob centrum
grauitatis N rectilinei trianguli ABC; reliqua igitur AD
dupla e&longs;t reliquæ GO. cum igitur AG &longs;it dupla ip&longs;ius
AD, quadrupla erit AG ip&longs;iu&longs;que GO. quare & quadru
pla AE ip&longs;ius FE ob parallelas: tripla igitur AF ip&longs;ius FE. Rur&longs;us quoniam ex Archimede &longs;e&longs;quialtera e&longs;t DM ip&longs;ius
MH, erit tota DH ad DM vt quinque ad tria, hoc e&longs;t
vt decem ad &longs;ex: &longs;ed MD erat dupla ip&longs;ius OF; tota igi-
e&longs;t ip&longs;ius DH; igitur GC ad FO vt viginti ad tria: &longs;ed
quia tripla exi&longs;tente AO ip&longs;ius OG, e&longs;t tota AG ip&longs;ius
AO &longs;e&longs;quitertia, erit quoque GE, ip&longs;ius OF &longs;e&longs;quiter
tia, propter &longs;imilitudinem triangulorum AGE, AOF,
hoc e&longs;t qualium partium æqualium OF trium, talium GE
quatuor; qualium e&longs;t GC hoc e&longs;t BG viginti, talium
erit EG quatuor, & EC &longs;exdecim: dempta igitur EG
ex GC, & addita ip&longs;i BG, qualium e&longs;t EC &longs;exdecim:
talium erit BE vigintiquatuor: &longs;ed vt vigintiquatuor ad
&longs;exdecim, ita &longs;unt tria ad duo, quæ proportio e&longs;t &longs;e&longs;qui
altera, &longs;e&longs;quialtera igitur erit BE ip&longs;ius EC, o&longs;ten&longs;a e&longs;t
autem AF ip&longs;i FE tripla. Manife&longs;tum e&longs;t igitur pro
po&longs;itum.
Si duo triangula mixta prædicti generis verti
cem communem habeant, qui e&longs;t contactus, &
ba&longs;es æquales in eadem recta linea, vel continuas,
vel &longs;egmento interiecto, tota extra &longs;iguram ver&longs;a
cauitate; centrum grauitatis compo&longs;iti ex vtro
que e&longs;t pun ctum illud, in quo recta linea à vertice
ad bipartitæ rectæ prædictis &longs;ectionibus interce
ptæ, in qua &longs;unt ba&longs;es dictorum triangulorum &longs;e
ctionis punctum pertinens &longs;ic diuiditur; vt pars,
quæ e&longs;t ad verticem &longs;it tripla reliquæ.
Sint duo prædicti generis triangula ABC, ADE ha
bentia verticem A communem, qui e&longs;t contactus recta. rum cum parabolis, tangente AB parabolam AC, &
rallelas parabolarum diametres per A, & in vna recta li
nea CE &longs;egmento BD interiecto: vtriu&longs;que autem &longs;e
ctionis AC, AE concauitas &longs;pectet extra figuram ACE:
&longs;ecta autem CE bifariam in F, iunctaque AF, ponatur
AG tripla ip&longs;ius GF. Dico compo&longs;iti ex triangulis A
BC, ADE centrum grauitatis e&longs;&longs;e G. Po&longs;ita enimvtra
que &longs;e&longs;quialtera, CH ip&longs;ius HB, & EK ip&longs;ius KD,
iunctisque AH, AK, ducatur per punctum G ip&longs;i CE
parallela &longs;ecans AH, AK in punctis L, M. Quoniam
igitur LM ip&longs;i CE parallela &longs;ecat eas quæ ex puncto A
ad rectam CD du
cuntur rectas lineas
in ea&longs;dem rationes, &
e&longs;t AG tripla ip&longs;ius
GF; tripla erit vtra
que AL ip&longs;ius LH,
& AM ip&longs;ius MK:
&longs;e&longs;quialtera autem e&longs;t
CH ip&longs;ius HB, &
EK ip&longs;ius KD; erit
igitur L centrum gra
uitatis trianguli AB
C, & M trianguli A
DE per præceden
tem. Rur&longs;us quoniam ab&longs;oluantur triangula rectiline
ACB, AEK, & æqualia erunt propter æquales ba&longs;es,
po&longs;ita inter ea&longs;dem parallelas, & vtrumque &longs;e&longs;quialterum
eius trianguli mixti, quod comprehendit, ex demon&longs;tra
tione antecedentis; æqualia igitur erunt triangula mixta
ABC, ADE, &longs;iquidem &longs;unt æqualium &longs;ub&longs;e&longs;quialtera. Et quoniam componendo, & permutando e&longs;t vt CB ad
DE ita BH ad DK, æqualis erit BH ip&longs;i DK: &longs;ed &longs;i ab
æqualibus po&longs;itis CF, FE ip&longs;as CB, DE æquales au-
ti FK æqualis e&longs;t: in triangulo autem AHK recta AF
&longs;ecat LM, HK parallelas in ea&longs;dem rationes; erit igitur
LG æqualis ip&longs;i GM; cum igitur æqualium triangulo
rum ABC, ADE centra grauitatis &longs;int L, M; erit com
po&longs;iti ex vtroque centrum grauitatis G. Idem o&longs;tendere
mus, quod proponitur, & &longs;i ba&longs;es prædictorum triangulo
rum &longs;int continuæ. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Si duæ parabolæ in eodem plano circa æqua
les diamet ros in directum inter &longs;e con&longs;titutas, ita
vt vertices &longs;int extrema ex diametris compo&longs;itæ,
communem habuerint aliquam ordinatim ad dia
metrum applicatarum, & vertices cum puncto con
uenientiæ iungantur rectis lineis: centrum gra
uitatis v triu&longs;que portionis ijs rectis lineis ab &longs;ci&longs;
&longs;æ, rectam lineam, quæ terminum communem
diamctrorum, & concur&longs;um parabolarum iungit
bifariam diuidit.
Circa æquales
diametros AD,
DC indirectum
inter &longs;e con&longs;titutas,
verticibus A, C,
duæ parabolæ in
eodem plano
munem
quam BD ordi
que AB, BC, &longs;it &longs;ecta BD bifariam in puncto G. Dico G e&longs;se centrum grauita tis duarum portionum AEB,
BFE &longs;imul. Si enim hoc non e&longs;t, &longs;it aliud punctum L. &
compleantur parallelogramma ANBD, DBRC, hoc
e&longs;t totum AR parallelogrammum: & &longs;ecta BG bifariam
in puncto H, ponatur DK ip&longs;ius BD pars tertia, vt pun
ctum K &longs;it trianguli ABC centrum grauitatis. Po&longs;ita au
tem &longs;e&longs;quialtera BP ip&longs;ius PN, & BQ ip&longs;ius QR, iun
ctisque AP, CQ, duoatur per punctum H ip&longs;i AC, vel
NR parallela, cum ip&longs;is AP, CQ conueniens in punctis
ST: & iuncta LG,
&longs;i punctum L non
&longs;it in linea BD,
e&longs;to LM quintu
pla ip&longs;ius MG. Quoniam igitur ob
parallelas AC, P
Q, ST in trape
zio APQC, e&longs;t
vt DH ad HB, ita
AS ad SP, & CT
ad TQ, erit AS ip&longs;ius SP, & CT ip&longs;ius TQ tripla:
&longs;ed e&longs;t BP &longs;e&longs;quialtera ip&longs;ius PN, & BQ ip&longs;ius QR;
mixti igitur trianguli ANB centrum grauitatis erit S, &
trianguli mixti CRB centrum grauitatis T. cum igitur
BP, BQ proportionales æqualibus NB, BR inter &longs;e
&longs;int æquales, & &longs;ecta AC bifariam in puncto D; etiam
ijs parallela ST &longs;ecta erit bifariam in puncto H: iungit
autem ST centra grauitatis mixtorum triangulorum AN
B, BRC; compo&longs;iti igitur ex vtroque centrum grauita
tis erit H. Rur&longs;us quoniam ex quadratura parabolæ, &longs;e
miparabola ABD &longs;e&longs;quitertia e&longs;t trianguli BDA, erit
triangulum BDA &longs;e&longs;quialterum mixti trianguli ANB:
erit &longs;e&longs;quialterum: totum igitur triangulum ABC &longs;e&longs;qui
alterum e&longs;t compo&longs;iti ex triangulis mixtis ANB, CRB. Et quoniam quarta pars e&longs;t GH ip&longs;ius BD, & DK ter
tia, DG verò dimidia; qualium duodecim partium æqua
lium e&longs;t BD, talium erit DK quatuor, & GH trium, &
DG &longs;ex, & reliqua KG duarum; &longs;e&longs;quialtera igitur e&longs;t
GH ip&longs;ius GK: quare vt triangulum ABC ad compo
&longs;itum ex prædictis triangulis mixtis, ita ex contraria parte
e&longs;t HG ad G
grauitatis H, trianguli autem ABC centrum grauitatis
K; erit dicti compo&longs;iti, & trianguli ABC &longs;imul centrum
grauitatis G. Rur&longs;us, quoniam triangulum ABC &longs;e&longs;
quialterum e&longs;t compo&longs;iti ex triangulis mixtis &longs;upra dictis,
& compo&longs;itum ex duabus &longs;emiparabolis ABD, CBD
&longs;e&longs;quitertium trianguli ABC; crit compo&longs;itum ex trian
gulis mixtis vnà cum triangulo ABC, quintuplum com
po&longs;iti ex portionibus AEB, BFC; hoc e&longs;t vt ex contra
ria parte LM ad MG: cum igitur G &longs;it centrum graui
tatis compo&longs;iti ex triangulis mixtis, & triangulo ABC, &
compo&longs;iti ex portionibus AEB, BFC centrum grauita
tis L; erit vtriu&longs;que dicti compo&longs;iti, hoc e&longs;t totius AR
parallelogrammi centrum grauitatis L: &longs;ed & punctum G
ex primo libro e&longs;t centrum grauitatis parallelogrammi
AR; eiu&longs;dem igitur parallelogrammi AR erunt duo cen
tra grauitatis G, L. Quod fieri non pote&longs;t: duarum igitur
portionum AEB, BFC &longs;imul centrum grauitatis erit G. Quod e&longs;t propo&longs;itum.
Omnis figuræ circa axim in alteram partem de
ficientis, cuius ba&longs;is e&longs;t circulus, vel ellip&longs;is, &longs;iue-
perficies tota interius concaua, centrum grauitatis
e&longs;t in dimidio axis &longs;egmento, quod ba&longs;im, vel ma
iorem ba&longs;im attingit.
Sit figura circa axim in alteram partem deficiens ABC,
cuius axis BD, ba&longs;is, vel maior ba&longs;is circulus, vel ellip&longs;is
circa diametrum AC, reliqua autem &longs;uperficies tota inte
rius concaua: &longs;ecto autem axe BD bifariam in puncto G,
&longs;it &longs;olidi ABC centrum grauitatis F nempe in axe BD. Dico punctum F e&longs;&longs;e in &longs;egmento ED.
Secto enim &longs;oli
do ABC, & figu
ra per axem pla
no per
ba&longs;i, vel ba&longs;ibus
parallelo, fiat &longs;e
ctio circulus, vel
ellip&longs;is &longs;imilis
ba&longs;i, per diffini
tionem, & &longs;ectio
nis diameter K
N: deinde figu
ra quædam ex
duobus cylindris, vel cylindri portionibus KL, AM cir
ca axes BE, ED, eiu&longs;dem altitudinis circum&longs;cribatur
&longs;olido ABC: &longs;ecanturque bifariam BE in puncto G, &
ED in puncto H. totius autem figuræ circum&longs;criptæ &longs;it
centrum grauitatis O, nempe in axe BD. Quoniam igi
tur propter bipartitorum axium &longs;ectiones G, H, e&longs;t &longs;olidi
KL centrum grauitatis G: &longs;olidi autem AM centrum
grauitatis H, erit in linea GH totius &longs;olidi AL centrum
grauitatis O, & vt &longs;olidum AM ad &longs;olidum KL, ita GO
ad OH: &longs;ed maior e&longs;t proportio &longs;olidi AM ad &longs;olidum KL
OH, quàm GE ad EH: & componendo, maior pro
portio GH ad HO, quàm eiu&longs;dem GH ad HE; mi
nor igitur OH erit quàm EH, & punctum O propin
quius puncto D quàm punctum E; verum quoniam ex
ijs, quæ in præcedenti libro demon&longs;trauimus, propo&longs;itæ
figuræ &longs;olidæ ABC centrum grauitatis e&longs;t puncto D
propinquius, quàm cuiuslibet figuræ ex cylindris, vel cy
lindri portionibus æqualium altitudinum ip&longs;i circum&longs;cri
ptæ, erit punctum F propinquius puncto D quàm pun
ctum O; multo igitur puncto D erit propinquius pun
ctum F quàm punctum E; ergo infra punctum E, & in
linea ED cadet &longs;olidi ABC centrum grauitatis F. Quod demon&longs;trandum erat.
Omnis fru&longs;ti coni, vel portionis conicæ cen
trum grauitatis e&longs;t punctum illud, in quo eius
axis &longs;ic diuiditur, vt pars quæ minorem ba&longs;im at
tingit a&longs;&longs;umens quartam partem axis ablati coni,
vel portionis conicæ, &longs;it ad eam, quæ inter po&longs;tre
mam &longs;ectionem, & quartæ partis ab&longs;ci&longs;&longs;<17> ad ba&longs;im
axis totius coni terminum interijcitur, vt cubus,
qui fit ab axe totius, ad cubum qui fit ab axe abla
ti coni.
Sit coni, vel portionis conicæ ABC fru&longs;tum BDEC,
cuius axis FG: conus autem, vel coni portio ablata AD
E: &longs;int centra grauitatis H &longs;olidi ABC, & K &longs;olidi
ADE, & L fru&longs;ti DC: quæ centra præterquam quod
centrum H, ex ijs, quæ in primo libro demon&longs;traui
mus. Dico e&longs;&longs;e KL ad LH vt cubum ex AG ad cu
bum ex AF. Quoniam enim
ob centra grauitatis
e&longs;t vt fru&longs;tum DC ad &longs;olidum
ADE, ita ex contraria parte
KH ad HL; erit componen
do, vt &longs;olidum ABC ad &longs;oli
dum ADE, ita KL ad LH:
&longs;ed vt
ADE, ita e&longs;t cubus ex AG
ad cubum ex AF: triplieata
enim e&longs;t vtraque proportio eiu&longs;
dem, quæ e&longs;t ip&longs;ius AG ad ip
&longs;am AF, propter &longs;imilitudi
nem &longs;olidorum; vt igitur e&longs;t cu
bus ex AG ad cubum ex AF,
ita erit KL ad LH. Quod demon&longs;trandum erat.
Re&longs;idui &longs;olidi ex cylindro, vel portione cylin
drica hemi&longs;phærio, vel hemi&longs;phæroidi circum
&longs;cripta, dempto hemi&longs;phærio, vel hemi&longs;phæroide,
centrum grauitatis e&longs;t punctum illud, in quo axis
&longs;ic diuiditur, vt pars ba&longs;im attingens hemi&longs;phæ
rij, vel hemi&longs;phæroidis &longs;it tripla reliquæ.
E&longs;to hemi&longs;phærio, vel hem&longs;phæroidi ABC, cuius axis
BD, circum&longs;criptus cylindrus, vel portio cylindrica AF:
& ponatur DDico reliqui ex &longs;oli- Nam
&longs;uper ba&longs;im circulum, vel ellip&longs;im, cuius diameter EF &longs;i
milem, & oppo&longs;itam &longs;olidi ABC, vel AF ba&longs;i, cuius dia
meter AC, &longs;tet cylindrus, vel portio cylindrica EDF: vt
&longs;itaxis BD communis quatuor &longs;olidis ABC, EDF,
AF, & reliquæ figuræ dempto &longs;olido ABC compre
hen&longs;æ &longs;uperficie cylindrica, & circulo, vel ellip&longs;e circa EF,
& dimidia &longs;uper&longs;icie &longs;phærica interiori, cuius figuræ &longs;oli
dæ ponimus centrum grauitatis Secto igitur axe
BD bifariam, & &longs;ingulis eius partibus rur&longs;us bifariam,
ducti&longs;que per puncta &longs;ectionum planis quibu&longs;dam planis
prædictarum ba&longs;ium oppofitarum parallelis, &longs;ecta &longs;int qua
tuor prædicta &longs;olida, quorum, excepto propo&longs;ito re&longs;iduo,
&longs;ectiones omnes erunt circuli
les, & in &longs;olido AF etiam æquales, quarum omnium
diametri eiu&longs;dem rationis erunt in eodem plano, in quo
&longs;it parallelogrammum per axim AEFC: &longs;olidiautem dicti
re&longs;idui &longs;ectiones, re&longs;idua &longs;ectionum &longs;olidi ABC. At circa
rum ABC, EDF, cylindri, vel portiones cylindricæ con
&longs;i&longs;tant altitudine, & multitudine æquales; ita vt duarum fi
gurarum ex ijs compofitarum altera fit cirdum&longs;cripta &longs;olihac igitur abla
ta ex &longs;olido AF, figura relinquetur ex re&longs;iduis cylindro
rum, vel cylindri portionum altitudine, & multitudine
æqualibus ijs cylindris, vel cylindri portionibus, ex quibus
con&longs;tat alterutra figurarum &longs;olidis ABC, DEF circum
&longs;criptarum: eruntque ex &longs;uperius demon&longs;tratis dicta re&longs;i
dua, & cylindri vel cylindri portiones, quæ circa &longs;olidum
EDF, inter &longs;e æqualia proutinter &longs;e re&longs;pondent inter ea
dem plana parallela, vt e&longs;t exempli gratia reliquum &longs;oli
di AN dempto &longs;olido SR, æquale &longs;olido TP: & &longs;ic de
inceps: &longs;ummus autem XF cylindrus, vel portio cylindrica
e&longs;t communis: Atqui bina hæc iam dicta &longs;olida centrum
grauitatis habent commune communis bipartiti axis &longs;ectio
nem in eadem recta linea BD, in qua e&longs;t etiam &longs;olidi XF
communis centrum grauitatis. duarum igitur dictarum figu
rarum &longs;olido EDF, & prædicto re&longs;iduo circum&longs;criptarum
idem aliquod punctum in axe BD erit commune centrum
grauitatis: &longs;ieri autem pote&longs;ts quod in &longs;ecundo libro demon
&longs;trauimus, vt duæ dictæ figuræ &longs;uperent vnaquæ que &longs;ibi in
&longs;criptam minori &longs;pacio quantacumque magnitudine pro
po&longs;ita. ex demon&longs;tratis igitur in primo libro; duo &longs;olida cir
ca axem BD in alteram partem deficientia commune ha
bebunt in axe BD centrum grauitatis: &longs;ed &longs;olidi, ide&longs;t co-
reliqui igitur ex cylindro, vel portione cylindrica AF dem
pto hemi&longs;phærio, vel hemi&longs;phæroide ABC centrum graui
tatis erit idem K. Quod erat demon&longs;trandum.
Si hemi&longs;phærium, vel hemi&longs;phæroides vna cum
cylindro, vel cylindri portione ip&longs;i circum&longs;cripta
&longs;ecetur plano ba&longs;i parallelo; reliqui ex cylindro,
vel portione cylindrica ab&longs;ci&longs;&longs;a ad partes verti
cis, dempta illa quæ ab&longs;ci&longs;&longs;a e&longs;t &longs;imul minori,
& &longs;phæræ, vel &longs;phæroidis portione, centrum gra
uitatis e&longs;t punctum illud, in quo eius axis &longs;ic diui
ditur, vt quæ inter hanc po&longs;tremam &longs;ectionem, &
centrum ba&longs;is vnà ab&longs;ci&longs;&longs;æ portionis interijci
tur, a&longs;&longs;umens quartam partem &longs;egmenti, quod di
ctæ ba&longs;is, & &longs;phæræ, vel &longs;phæroidis centra iungit,
&longs;it ad &longs;ui &longs;egmentum, quod inter po&longs;tremam &longs;e
ctionem, & quartæ partis axis hemi&longs;phærij, vel
hemi&longs;phæroidis ad verticem ab&longs;ci&longs;&longs;æ terminum
interijcitur, vt cubus axis hemi&longs;phærij, vel hemi
&longs;phæroidis, ad cubum eius, quæ ba&longs;is portionis &
hemi&longs;phærij, vel hemi&longs;phæroidis centra iungit. Reliqui autem ex cylindro, vel portione cylindri
ca vnà ab&longs;ci&longs;&longs;a
&longs;phæroidis portione, quæ e&longs;t ad ba&longs;im, dempta hac
portione centrum, grauitatis e&longs;t punctum illud,
quod quartam partem ab&longs;cindit axis portionis ad
E&longs;to hemi&longs;phærio, vel hemi&longs;phæroidi ABC, cuius axis
BD, ba&longs;is circulus vel ellip&longs;is, cuius diameter AC cir
cum&longs;criptus cylindrus, vel cylindri portio AF, cuius in
num per AC, BD, faciat &longs;ectiones &longs;emicirculum, vel &longs;e
miellip&longs;im ABC, & parallelogrammum per axem AE
FC; & per quodlibet punctum L axis BD, planum ba&longs;ibus
AC, EF &longs;olidi AF
AF, faciat &longs;ectiones circulos, vel ellip&longs;es &longs;imiles, & in &longs;olido
AF etiam æquales ijs, quæ circa AC, EF: earum autem dia
metros, &longs;ectiones cum
& cum &longs;emicirculo, vel &longs;emielliple ABC, ip&longs;am HN. Ita
que habebimus figuram quandam &longs;olidam GHBNO re&longs;i
duum cylindri, vel portionis cylindricæ GF dempta mino
ri &longs;phæræ, vel &longs;phæroidis portione HBN, cuius axis erit BL. Sumpta igitur BQ quarta parte axis BD, & LP quarta par
te ip&longs;ius DL fiat vt cu
bus ex BD ad cubum ex
DL, ita PR ad
Dico re&longs;idui GHBNO
centrum grauitatis e&longs;&longs;e
R. Reliqui autem ex
cylindro, vel portione
cylindrica AO dempta
portione AHNC, cen
trum grauitatis e&longs;&longs;e P.
Nam &longs;uper ba&longs;im circulum, vel ellip&longs;im EF, &longs;tet conus, vel
portio conica EDF: &longs;itque prædicto plano per L ab&longs;ci&longs;
&longs;us conus, vel coni portio KDM, cuius axis DL, quæ pro
pter planum &longs;ecans ba&longs;i EF parallelum, &longs;imilis erit toti
cono, vel portioni conicæ EDF. Quoniam igitur BQ
e&longs;t axis BD pars quarta, & LP pars quarta ip&longs;ius DL;
&longs;ius DKM. Et quoniam &longs;olidum DEF ad &longs;olidum D
KM e&longs;t vt cubus ex BD ad cubum ex DL, hoc e&longs;t vt
&longs;olidum EDF ad &longs;olidum KLM, & vt PR ad
erit diuidendo, vt fru&longs;tum EKMF ad ablatum KDM,
ita ex contraria parte PQ ad QR: cum igitur &longs;int
centra grauitatis P &longs;olidi DKM, & Q &longs;olidi DET;
erit reliqui fru&longs;ti EKMF centrum grauitatis R: &longs;ed
qua ratione in præcedenti con&longs;tat, reliqui ex &longs;olido AF,
dempto &longs;olido ABC centrum grauitatis e&longs;&longs;e Q, eadem
concluditur idem e&longs;&longs;e centrum grauitatis reliqui ex &longs;olido
GF, dempta portione HBN, quod & fru&longs;ti EKMF,
nempe punctum R: Et quoniam P e&longs;t centrum grauita
tis coni, vel portionis conicæ KDM, crit idem P centrum
grauitatis ieliqui ex cylindro, vel portione cylindrica
AO dempta portione AHNC. Manife&longs;tnm e&longs;t igitur
propo&longs;ituro.
Ij&longs;dem po&longs;itis &longs;olidis, vt in antecedenti, &longs;ectis
que per duo quælibet puncta axis duplici plano
ba&longs;i parallelo, reliqui ex cylindro, vel portione
cylindrica dictis duobus planis intercepta dem
pta &longs;phæræ, vel &longs;phæ roidis portione ip&longs;i inter ea
dem plana re&longs;pondente, centrum grauitatis e&longs;t
punctum illud, in quo eius axis &longs;ic diuiditur, vt
quæ inter hanc po&longs;tremam &longs;ectionem, & centrum
maioris ba&longs;is vnà ab&longs;ci&longs;sæ portionis interijcitur,
a&longs;&longs;umens quartam partem &longs;egmenti, quod prædi
ctæ ba&longs;is, & &longs;phæræ vel &longs;phæroidis centra iungit,
nem, & quartæ partis eius, quæ &longs;phæræ, vel hemi
&longs;phærij, & minoris ba&longs;is portionis centra iungit
ad minorem ba&longs;im ab&longs;ci&longs;sæ terminum interijci
tur, vt cubus eius, quæ minoris ba&longs;is, & &longs;phæræ,
vel &longs;phæroidis, ad
roidis, & maioris ba&longs;is portionis centra iungit.
Ij&longs;dem po&longs;itis &longs;olidis, vtque in antecedenti ponebantur
ABC, AF; per duo quælibet puncta RQ axis BD &longs;e
centur po&longs;ita &longs;olida duobus planis ba&longs;i, quæ circa AC, cir
culo &longs;cilicet, vel ellip&longs;i parallelis: quibus planis intercepta
hemi&longs;phærij, vel hemi&longs;phæroidis portio &longs;it MOPN, vnà
cum cylindro, vel portione cylindrica GL parte ip&longs;ius AF,
nis axis vnà ab&longs;ci&longs;&longs;us
ab axe BD &longs;olidi AB
C, &longs;it RQ: & &longs;umptis
quartis partibus RI ip
&longs;ius DR, & QZ ip&longs;ius
DQ, fiat vt cubus ex
DQ ad cubum ex D
R, ita IY ad YZ. Dico reliqui ex cylin
dro, vel portione cylindrica GL dempta portione MOP
N, centrum grauitatis e&longs;&longs;e Y. Facta enim con&longs;tructione
coni, vel portionis conicæ EDF, vt in &longs;uperioribus, erunt
&longs;imilium conorum, vel coni portionum SDT, VDX, ea
dem ordine axes DQ, DR: propter igitur factas diui&longs;io
nes, erunt
& demon&longs;tratio &longs;imilis antecedenti. dicti igitur re&longs;idui
GMOPMH centrum grauitatis Y. Quod e&longs;t propo
&longs;itum.
Si &longs;phæra, vel &longs;phæroides vnà cum cylindro,
vel portione cylindrica ip&longs;i circum&longs;cripta &longs;ecetur
plano, haud per centrum, ba&longs;ibus &longs;olidi circum
&longs;cripti parallelo; reliqui ex cylindro, vel portio
ne cylindrica ad maioris portionis &longs;phæræ, vel
&longs;phæroidis partes ab&longs;ci&longs;&longs;a, dempta &longs;phæræ, vel
&longs;phæroidis maiori portione, centrum grauita
tis e&longs;t punctum illud, in quo dicti reliqui &longs;olidi
axis &longs;egmentum inter duas quartas partes extre
mas &longs;egmentorum eiu&longs;dem axis, quæ à centro
&longs;phæræ, vel &longs;phæroidis fiunt interiectum, &longs;ic diui
ditur, vt pars propinquior ba&longs;i &longs;it ad reliquam, vt
prædictorum, quæ à centro fiunt axis &longs;egmento
rum maioris cubus ad cubum minoris.
Sit &longs;phæræ, vel &longs;phæ
roidi ABCD cuius cen
trum E, circum&longs;criptus
cylindrus, vel portio cy
lindrica FGHK, cum
quibus planum per axim
communem BED, fa
ciat &longs;ectiones, parallelo
grammum per axim FG
HK, & circulum, vel el
lip&longs;im ABCD: quas fi
guras vnà cum dictis &longs;o
que &longs;ectiones circulos, vel ellip&longs;es &longs;imiles &longs;cilicet ba
&longs;ibus oppo&longs;itis &longs;olidi FH, & &longs;ectionum diametros LM,
TV, ab&longs;cindat &longs;olidi ABCD maiorem portionem
LBM, & &longs;olidi FH cylindrum, vel portionem cy
lindricam TH, cuius axis BES: duorum autem &longs;egmen
corum BE, ES &longs;umptis duabus quartis partibus extre
mis BQ PS, fiat vt cubus ex BE ad cubum ex ES, ita
PR ad
tione cylindrica TH, portioni LBM circumfcripta, dem
pta portione LBM, centrum grauitatis e&longs;&longs;e R. Se
ctis enim parallelogrammo TH, & &longs;olidis LBM, TH,
plano per centrum E, ba&longs;ibus &longs;olidi TH parallelo, &longs;it &longs;e
ctio, (vna enim communis erit vtrique &longs;olido) circulus,
vel ellip&longs;is, cuius diameter AEC in parallelogrammo T
H diametris TV, GH
oppo&longs;itarum ba&longs;ium pa
rallela. Tum &longs;uper ba
&longs;es oppo&longs;itas circulos, vel
ellip&longs;es circa GH, FK
&longs;tent coni, vel portiones
conicæ GEH, FEK:
& planum per TV ba&longs;i
circa FK parallelum ab
&longs;cindat à &longs;olido FEK
conum, vel coni portio
nem NEO &longs;imilem vti
que ip&longs;i FEK, hoc e&longs;t
ip&longs;i GEH, propter &longs;imiles ba&longs;es, & &longs;imilia triangula per
axim in eodem parallelogrammo FH. Solidi itaque
NEO, ex ijs, quæ in primo libro demon&longs;trauimus, cen
trum grauitatis erit P; quemadmodum & Q &longs;olidi
NEO. Quoniam igitur tàm &longs;olidi GEH ad &longs;oli
dum NEO propter &longs;imilitudinem, quàm cubi ex BE
ad cubum ex ES, ita &longs;olidum GEH ad &longs;olidum NEO,
hoc e&longs;t in eadem proportione, quæ e&longs;t ex contraria parte ip
&longs;ius PR ad
&longs;olidi NEO, & Q &longs;olidi GEH; erit compo&longs;iti ex vtro
que centrum grauitatis R. Rur&longs;us, quoniam reliquum &longs;o
lidi AH dempto hemi&longs;phærio, vel hemi&longs;phæroide ABC,
æquale e&longs;t &longs;olido GEH: & reliquum &longs;olidi TC dempto
&longs;olido ALMC æquale &longs;olido NEO; erit vt &longs;olidum
GEH ad &longs;olidum NEO, ide&longs;t ex contraria parte, vt PR
ad RQ, ita reliquum &longs;olidi AH dempto ABC, ad re
liquum &longs;olidi TC, dempto ALMC: &longs;ed reliqui ex &longs;oli
do AH dempto ABC e&longs;t centrum grauitatis Q: & reli
qui ex &longs;olido TC dempto ALMC, centrum grauitatis
P, ex &longs;uperius demon&longs;tratis; totius igitur reliqui ex cy
lindro, vel portione cylindrica TH dempta &longs;phæræ, vel
&longs;phæroidis maiori portione LBM centrum grauitatis e&longs;t
R. Quod demon&longs;trandum erat.
Si &longs;phæra, vel &longs;phæroides vnà cum cylindro,
vel portione cylindrica ip&longs;i circum&longs;cripta, &longs;ece
tur duobus planis ba&longs;i &longs;olidi circum&longs;cripti pa
rallelis, centrum intercipientibus, & ab eo non
æqualiter di&longs;tantibus; reliqui ex cylindro, vel
portione cylindrica dictis planis intercepta, dem
pta portione &longs;phæræ, vel &longs;phæroidis ip&longs;i re&longs;pon
dente, centrum grauitatis e&longs;t punctum illud, in
quo prædicti reliqui &longs;olidi axis &longs;egmentum in
mentorum, quæ à centro &longs;phæræ, vel &longs;phæroi
dis fiunt interiectum &longs;ic diuiditur, vt pars ma
iori ba&longs;i propinquior &longs;it ad reliquam, vt prædi
ctorum axis &longs;egmentorum cubus maioris ad cu
bum minoris.
Ij&longs;dem po&longs;itis, & con&longs;tructis, quæ in antecedenti, rur
&longs;us per quodlibet axis BE punctum X, ductum planum
ba&longs;ibus &longs;olidi FH parallelum, &longs;ecansque vnà cylindrum,
vel portionem cylindricam FH, & &longs;phæram, vel &longs;phæroi
des ABCD: e&longs;to duobus planis per TV, ZY, inter &longs;e pa
rallelis, & centrum E intercipientibus abci&longs;&longs;a &longs;phæræ, vel
&longs;phæroidis portio L
cylindrica TY: & &longs;umatur ip&longs;ius EX pars quarta XQ,
qualis e&longs;t & PS ip&longs;ius E
S: & vt e&longs;t cubus ex EX
ad cubum ex ES, ita fiat
PR ad
qui ex cylindro, vel por
tione cylindrica TY dem
pta &longs;phæræ, vel &longs;phæroi
dis portione L
trum grauitatis e&longs;&longs;e R. E&longs;to
enim conus, vel coni por
tio
cto plano per ZY, & com
munibus axibus ES, EX,
&longs;imili igitur demon&longs;tratio
ne antecedentis manife&longs;tum e&longs;t quod proponebatur.
Hemi&longs;phærij, vel hemi&longs;phæroidis centrum
grauitatis e&longs;t punctum illud, in quo axis &longs;it diui
ditur, vt pars ad verticem &longs;it ad reliquam vt quin
que ad tria.
E&longs;to hemi&longs;phærium, vel hemi&longs;phæroides ABC, cuius
axis BD, ba&longs;is circulus, vel ellip&longs;is, cuius diameter AD
C: &longs;itque &longs;olidi ABC centrum grauitatis G, nempe
in axe BD. Dico BG ad GD e&longs;&longs;e vt quinque ad tria.
Nam circa axim BD &longs;uper ba&longs;im circulum, vel ellip&longs;im cir
ca AC, &longs;tet circum&longs;cri
ptus &longs;olido ABC cy
lindrus, vel portio cy
lindrica AE, & &longs;ecta
BD bifariam in F, rur
&longs;us FB bifariam &longs;ece
tur in puncto H. Quo
niam igitur &longs;olidum A
BC e&longs;t &longs;olidi AE, &longs;ub
&longs;e&longs;quialterum, erit di
uidendo &longs;olidum ABC reliqui ex &longs;olido AE duplum
cum igitur &longs;int centra grauitatis, G &longs;olidi ABC, & H
prædicti reliqui, & F totius AE; quo fit vt ex con
traria parte &longs;it vt &longs;olidum ABC ad prædictum re&longs;iduum,
ita HF ad FG, erit HF dupla ip&longs;ius FG; quadrupla
igitur BF ip&longs;ius FG: &longs;ed talium quatuor partium e&longs;t BF,
qualium BD e&longs;t octo, cum &longs;it BF dimidia ip&longs;ius BD;
qualium igitur octo e&longs;t BD, talium erit BG quinque, &
GD trium. Quod demon&longs;trandum erat.
Dico hemi&longs;phærij, vel hemi&longs;phæroidis ABC cen
trum grauitatis e&longs;&longs;e G. In plano enim &longs;emicirculi, vel &longs;e
miellip&longs;is per axem BD de&longs;criptæ intelligantur duæ pa
rabolæ, quarum diametri AD, DC, & communiter
ad vtranque ordinatim applicata &longs;it BD: & connectun
tur rectæ AB, BC: &longs;umptis autem in BD tribus qui
buslibet punctis, æqualia axis &longs;egmenta XF, FY interci
pientibus, &longs;ecent per ea puncta tres figuras hemi&longs;phærium,
vel hemi&longs;phæroides ABC, & &longs;emicirculum, vel &longs;emielli
p&longs;im per axem, & figuram planam ARBSC, quæ lineis pa
rabolicis ARB, BSC, & recta AC continetur, pla
na quædam ba&longs;i hemi&longs;phærij, vel hemi&longs;phæroidis paralle
la. Erunt igitur &longs;ectiones hemi&longs;phærij, vel hemi&longs;phæroidis
circuli, vel ellip&longs;es &longs;imiles ba&longs;i,
LFM, N
lineæ PXQ, RFS, TYV. Quoniamigitur per IV hu
ius e&longs;t vt KH ad LM potentia, ita KQ ad FS hoc
e&longs;t in earum duplis PQ ad RS longitudine; erit vt PQ
ad RS, ita circulus, vel ellip&longs;is KH ad circulum vel &longs;i
milem ellip&longs;im LM. Eadem ratione erit vt RS ad
TV, ita circulus, vel ellip&longs;is LM ad circulum, vel
quàm RS ad TV circuli igitur, vel ellip&longs;is KH ad
vel ellip&longs;im LM, minor erit proportio <34> circuli, vel ellip&longs;
LM ad circulum, vel ellip&longs;im NO: & duæ figuræ hemi
&longs;phærium, vel hemi&longs;phæroides ABC, & plana ARBSC,
&longs;unt circa axim, vel diametrum BD in alteram parte m
deficientes, quales definiuimus; vtriu&longs;que igitur dictæ fi
guræ vnum erit commune centrum grauitatis. Rur&longs;us
po&longs;ito puncto F in medio axis BD, & FG ip&longs;ius GE
tripla, quoniam ponitur BG ad GD vt quinque ad tria;
qualium partium æqualium ip&longs;i EG e&longs;t FG trium, ta
lium erit BG quindecim, & GD nouem, & talis EG
vna: dempta igitur GE ab ip&longs;a DG, & addita ip&longs;i BG,
qualium partium e&longs;t BE &longs;exdecim, talium erit ED octo;
dupla igitur BE ip&longs;ius ED, & trianguli ABC centrum
grauitatis E. Rur&longs;us quoniam ex quadratura parabolæ,
duarum portionum ARB, BSC triangulum ABC e&longs;t
triplum; hoe e&longs;t vt FG ad GE, ita ex contraria parte
triangulum ABC ad duas portiones ARB, BSC: Sed
trianguli ABC e&longs;t centrum grauitatis E, & duarum por
tionum ARB, BSC &longs;imul per XXIII huius, centrum
grauitatis F, totius igitur figuræ ARBSC centrum gra
uitatis erit G, commune autem hoc centrum grauitatis
e&longs;t hemi&longs;phærio, vel hemi&longs;phæroidi ABC. Manife&longs;tum
e&longs;t igitur propo&longs;itum.
Omnis minoris portionis &longs;phæræ, vel &longs;phæroi
dis centrum grauitatis e&longs;t in axe primum bifa
riam &longs;ecto: deinde &longs;ecundum centrum grauitatis
reliqui &longs;olidi dempta portione ex cylindro, vel
portione, ex cylindro, vel portione cylindrica,
&longs;phær<17>, vel &longs;phæroidis circa axim axi portionis
gruentem
dius axis portionis ba&longs;im
pars prima, & &longs;ecunda &longs;ectione terminata, &longs;it ad
totam &longs;ecunda, & po&longs;trema &longs;ectione terminatam,
vt rectangulum contentum axe portionis, & reli
quo &longs;phæræ, vel &longs;phæroidis dimidij axis &longs;egmen
to, vnà cum duabus tertijs quadrati axis portio
nis, ad &longs;phæræ, vel &longs;phæroidis dimidij axis axi
portionis congruentis quadratum.
Sit &longs;phæræ, vel &longs;phæroidis minor portio ABC, cuius
axis BD: & in eo centrum grauitatis F: &longs;ecto autem axe
BD primum bifariam
in puncto G, & rur
&longs;us BG in puncto
H centro grauitatis
reliqui dempta por
tione ex cylindro, vel
portione cylindrica
KL circa axim BD,
ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;
&longs;a codem plano cum
portione ABC, & cylindro, vel portione cylindri
ca, quæ circum&longs;criberetur &longs;phæræ, vel &longs;phæroidi, cu
ius e&longs;t portio ABC, circa axim, cuius dimidium BDE. Dico GH ad HF, (nam cadet centrum F infra biparti
ti axis BD &longs;ectionem G, ex XXIII huius) e&longs;&longs;e vt rectan
gulum BDE vnà cum duabus tertijs BD quadrati ad
quadratum BE. Quoniam enim totius &longs;olidi KL cen-
ex KL dempta ABC portione; erit vt portio ABC ad
prædictum re&longs;iduum, ita ex contraria parte HG ad GF:
& componendo, vt &longs;olidum KL ad prædictum re&longs;iduum,
ita HF ad FG: & per conuer&longs;ionem rationis, vt &longs;olidum
KL ad portionem ABC, ita FH ad HG: & conuerten
do, vt portio ABC ad &longs;olidum KL, ita GH ad HE:
&longs;ed vt portio ABC ad &longs;olidum KL, ita e&longs;t rectangulum
BDE vnà cum duabus tertiis quadrati BD ad quadra
tum EB; vt igitur rectangulum BDE, vnà cum duabus
tertiis quadrati BD, ad quadratum EB, ita erit GH ad
HF. Quod demonftrandum erat.
Omnis portionis &longs;phæræ, vel &longs;phæroidis ab&longs;ci&longs;
&longs;æ duobus planis parallelis, altero per centrum
acto, centrum grauitatis e&longs;t in axe primum bifa
riam &longs;ecto: deinde &longs;umpta eius quarta parte ad
minorem ba&longs;im; in eo puncto, in quo dimidius
axis maiorem ba&longs;im attingens &longs;ic diuiditur, vt
pars axis prima, & &longs;ecunda &longs;ectione terminata,
&longs;it ad eam, quæ prima, & po&longs;trema &longs;ectione ter
minatur, vt rectangulum contentum &longs;phæræ, vel
&longs;phæroidis axis axi portionis congruentis ijs &longs;eg
mentis, quæ fiunt à centro minoris ba&longs;is portio
nis, vnà cum duabus tertiis quadrati axis portio
nis; ad&longs;phæræ, vel &longs;phæroidis dimidij axis qua
dratum.
Sit &longs;phæræ, vel &longs;phæroidis cuius centrum E portio
per E, & &longs;ectionem faciente circulum maximum, vel
ellip&longs;im per centrum, cuius diameter AED: axis autem
portionis &longs;it EF, cui congruens &longs;phæræ, vel &longs;phæroidis axis
GFER: &longs;it autem FE bifariam &longs;ectus in puncto H: &
FH bifariam in puncto K, &longs;itque in EH, &longs;ic enim erit,
portionis ABCD centrum grauitatis L. Dico e&longs;&longs;e HK
ad KL, vt rectangulum GFR, vnà cum duabus tertiis
quadrati EF ad quadratum EG. Sit enim cylindrus, vel
portio cylindrica AM circa axim FE ab&longs;ci&longs;&longs;a ij&longs;dem pla
nis cum portione AB
CD, ex cylindro, vel
portione cylindrica cir
ca axim GR &longs;phæ
ræ, vel &longs;phæroidi AG
DR circum&longs;cripta. Quoniam igitur &longs;olidi
AM e&longs;t centrum gra
uitatis H: reliqui au
tem dempta ABCD
portione centrum gra
uitatis K: & portionis
ABCD ponitur cen
trum grauitatis L; erit
vt portio ABCD ad reliquum &longs;olidi AM, ita ex con
traria parte KH ad HL. componendo igitur vt in antece
denti, & per conuer&longs;ionem rationis, & conuertendo, erit
vt portio ABCD ad &longs;olidum AM; hoc e&longs;t vt rectangu
lum GFR, vnà cum duabus tertiis quadrati EF ad qua
dratum EG, ita HK ad KL. Quod demon&longs;trandum
erat.
Omnis portionis &longs;phæræ, vel &longs;phæroidis ab
&longs;ci&longs;&longs;æ duobusplanis parallelis, neutro per cen
trum acto, nec centrum intercipientibus, centrum
grauitatis e&longs;t in axe, primum bifariam &longs;ecto: de
inde &longs;ecundum centrum grauitatis reliqui dem
pta portione ex cylindro, vel portione cylindrica,
ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum portione à cylin
dro, vel portione cylindrica &longs;phæræ, vel &longs;phæroi
di circa eius axem axi portionis congruentem cir
cum&longs;cripta; in eo puncto, in quo dimidius axis
portionis maiorem ba&longs;im attingens &longs;ic diuiditur,
vt pars prima & &longs;ecunda &longs;ectione terminata &longs;it ad
eam, quæ prima, & po&longs;trema &longs;ectione terminatur,
vt duo rectangula, alterum contentum duobus
&longs;phæræ, vel &longs;phæroidis axis axi portionis
tis ijs &longs;egmentis, quæ fiunt à centro minoris ba&longs;is
portionis: alterum axe portionis, & &longs;egmento,
quod &longs;phæræ, vel &longs;phæroidis, & maioris ba&longs;is por
tionis centra iungit, vnà cum duabus tertiis qua
drati axis portionis, ad &longs;phæræ vel &longs;phæroidis di
midij axis quadratum.
Sit &longs;phæræ, vel &longs;phæroidis, cuius centrum E portio
ABCD, ab&longs;ci&longs;&longs;a duobus planis parallelis, neutro per E
tran&longs;eunte, nec E intercipientibus: portionis autem axis
&longs;it FS: maior ba&longs;is circulus, vel ellip&longs;is, cuius diame
drica MN ab&longs;ci&longs;&longs;a ij&longs;dem planis cum portione ABCD
ex cylindro, vel portione cylindrica, &longs;phæræ, vel &longs;phæroidi
BCR circa eius axim CFSR circum&longs;cripta, cuius &longs;it cen
trum grauitatis H, ac propterea &longs;ecta FS bifariam in pun
cto H. reliqui autem
dempta portione AB
CD ex &longs;olido MN &longs;it
centrum grauitatis K,
quod cadet in FH, &
portionis ABCD cen
trum grauitatis in ip&longs;a
HS cadet, quod &longs;it L. Dico e&longs;&longs;e HK ad KL,
vt duo rectangula GF
R, FSE, vnà cum
duabus tertiis quadra
ti FS, ad quadratum
EG. Quoniam enim
&longs;imiliter vt ante o&longs;tenderemus e&longs;&longs;e HK ad KL, vt e&longs;t
portio ABCD ad &longs;olidum MN: &longs;ed portio ABCD
ad &longs;olidum MN, e&longs;t vt duo rectaugula GFR, ESF, vnà
cum duabus tertiis quadrati FS, ad quadratum EG; vt
igitur duo prædicta rectangula, vnà cum duabus tertiis
quadrati FS ad quadratum EG, ita erit HK ad KL. Quod erat demon&longs;trandum.
Omnis maioris portionis &longs;phæræ, vel &longs;phæroi
dis centrum grauitatis e&longs;t in axe, primum bifa
riam &longs;ecto: deinde &longs;ecundum centrum grauitatis
reliqui dempta portione ex cylindro, vel portione
ne, à cylindro, vel portione cylindrica, &longs;phæræ, vel
&longs;phæroidi circa eius axim axi portionis
tem
nis &longs;ic diuiditur, vt pars prima, & &longs;ecunda &longs;ectione
terminata &longs;it ad eam, quæ prima & po&longs;trema &longs;e
ctione terminatur, vt &longs;olidum rectangulum ex axe
portionis, & reliquo &longs;egmento axis &longs;phæræ, vel
&longs;phæroidis axi portionis congruentis, & eo, quod
&longs;phæræ, vel &longs;phæroidis, & ba&longs;is portionis centra
iungit, vnà cum binis tertijs duorum cuborum; &
eius, qui à &longs;phæræ, vel &longs;phæroidis axis fit dimi
dio: & eius, qui ab ea, quæ &longs;phæræ, vel &longs;phæroidis,
& ba&longs;is portionis centra iungit; ad &longs;olidum rectan
gulum, quod duobus &longs;phæræ, vel &longs;phæroidis præ
dicti axis dimidijs, & axe portionis continetur.
Sit &longs;phæræ, vel &longs;phæ
roidis, cuius centrum
E maior portio ABC,
cuius axis BD, ba&longs;is
circulus, vel ellip&longs;is, cu
ius diameter AC: &
circa axem BD &longs;tet
cylindrus, vel portio
cylindrica KL, ab&longs;ci&longs;
&longs;a eodem plano cum
portione ABC, ex cy
lindro, vel portione cy
lindrica, &longs;phæræ, vel
&longs;phæroidi ABCR circa eius axim BDR circum&longs;cripta,
in ip&longs;a BH, centrum grauitatis reliqui dempta portione ex
&longs;olido KL, &longs;it portionis ABC in ip&longs;a DH centrum gra
uitatis F, per vim XXXVII &longs;ecundi. Dico e&longs;&longs;e HG ad GF,
vt &longs;olidum rectangulum ex BD, DR, DE vnà cum bini
tertiis duorum
ex BE, ED, ad &longs;oli
dum rectangulum ex
BD, BE, ER. Simi
liter enim vt &longs;upra de
mon&longs;trato e&longs;&longs;e vt HG
ad GF, ita portionem
ABC ad
quoniamportio ABC
ad &longs;olidum KL e&longs;t vt
&longs;olidum ex BD, DR,
DE, vnà cum binis ter
tiis duorum
BE, & ED, ad &longs;oli
dum ex BD, BE, ER; erit vt modo dicta antecedens
magnitudo ad dictam con&longs;equentem, ita HG, ad GF. Quod demon&longs;trandum erat.
Omnis portionis &longs;phæræ, vel &longs;phæroidis ab
&longs;ci&longs;&longs;æ duobus planis parallelis centrum interci
pientibus, & ab eo non æqualiter di&longs;tantibus, cen
trum grauitatis e&longs;t in axe, primum bifariam &longs;ecto:
deinde &longs;ecundum
pta portione ex cylindro, vel portione cylindrica,
ab&longs;ci&longs;&longs;o, vel ab&longs;ci&longs;&longs;a vnà cum portione, à cylin-
di circa eius axim axi portionis congruentem cir
cum&longs;cripta; in eopuncto, in quo maius &longs;egmen
tum axis portionis corum, quæ à centro fiunt &longs;ic
diuiditur, vt pars prima & &longs;ecunda &longs;ectione termi
nata &longs;it ad eam, quæ prima, & po&longs;trema &longs;ectione
terminatur, vt duo &longs;olida rectangula; & quod fit
ex duobus &longs;phæræ, vel &longs;phæroidis axis axi portio
nis congruentis ijs &longs;egmentis, quæ fiunt à centro
maioris ba&longs;is portionis, & ea, quæ maioris ba&longs;is
& &longs;phæræ, vel &longs;phæroidis centra iungit: & quod
ex &longs;phæræ, vel &longs;phæroidis eiu&longs;dem axis &longs;egmentis
à centro minoris ba&longs;is factis, & ea, quæ minoris ba
&longs;is, & &longs;phæræ, vel &longs;phæroidis centra iungit, vnà
cum binis tertiis partibus duorum cuborum exijs
&longs;egmentis axis portionis, quæ à centro &longs;phæræ,
vel &longs;phæroidis fiunt; ad &longs;olidum
duobus &longs;phæræ, vel &longs;phæroidis prædicti axis dimi
dijs, & axe portio
nis continetur.
Sit &longs;phæræ, vel &longs;phæ
roidis, cuius centrum
E, portio ABCD, ab
&longs;ci&longs;&longs;a duobus planis pa
rallelis centrum E in
tercipientibus, & ab eo
non æqualiter di&longs;tan
tibus: axis autem por
tionis &longs;it GH: maior
cuius diameter ABC: & circa axim GH, &longs;tet cylindrus,
vel portio cylindrica NO, ab&longs;ci&longs;&longs;a ij&longs;dem planis cum por
tione ABCD, ex cylindro, vel portione cylindrica &longs;phæ
ræ, vel &longs;phæroidi BCR circa axim FGHR circum&longs;cri
pta, cuius &longs;it centrum grauitatis K, &longs;ectio &longs;cilicet bipartiti
axis GH: reliqui autem ex &longs;olido NO dempta portione,
&longs;it centrum grauitatis L, nempe in axis GH &longs;egmento
GK, quod minorem
portionis ba&longs;im attln
git: portionis autem
ABCD &longs;it centrum
grauitatis M: quod qui
dem in reliquo &longs;eg
mento KH cadet. Dico e&longs;&longs;e KL ad LM,
vt duo &longs;olida rectan
gula ex FH, HR, EH,
& ex RG, GF, GK,
vnà cum binis tertiis
duorum cuborum ex
EG, EH; ad &longs;olidum
rectangulum ex GH, EF, ER. Similiter enim vt &longs;upra
demon&longs;trato e&longs;&longs;e vt KL ad LM, ita portionem ABCD
ad &longs;olidum NO; quoniam portio ABCD ad &longs;olidum
NO, e&longs;t vt duo &longs;olida rectangula ex GH, HR, EH, &
ex RG, GF, EG, vnà cum binis tertiis duorum cubo
rum ex EH, EG ad &longs;olidum ex GH, EF, ER, erit
vt totum iam dictum antecedens ad dictum con&longs;equens,
ita KL ad LM. Quod demon&longs;trandum erat.
Omnis portionis conoidis parabolici centrum
grauitatis e&longs;t punctum illud, in quo axis &longs;ic diui
ditur, vt pars quæ ad verticem &longs;it eius, quæ ad ba
&longs;im dupla.
Omnis fru&longs;ti portionis conoidis parabolici cen
trum grauitatis e&longs;t punctum illud, in quo axis &longs;ic
diuiditur, vt pars minorem ba&longs;im attingens &longs;it ad
reliquam, vt duplum maioris ba&longs;is vnà cum mino
ri, ad duplum minoris, vnà cum maiori.
Harum proportionum vtriu&longs;que non alia demon&longs;tratio
e&longs;t ab ea, quam in &longs;ecundo &longs;crip&longs;imus de centro grauitatis
conoidis parabolici, & eius fru&longs;ti: propterea quod omnis por
tionis conoidis parabolici, &longs;icut & hyperbolici &longs;ectio ba&longs;i
parallela ellip&longs;is e&longs;t &longs;imilis ba&longs;i. Ex corollario xv.
de conoi
dibus, & &longs;phæroidibus Archimedis.
Omnis conoidis hyperbolici, vel portionis hy
perbolici conoidis centrum grauitatis, e&longs;t pun
ctum illud, in quo duodecima pars axis ordine
quarta ab ea, quæ ba&longs;im attingit, &longs;ic diuiditur, vt
pars propinquior ba&longs;i &longs;it ad reliquam vt &longs;e&longs;quial
axem cono
Sit conoides hyperbolicum, vel portio conoidis hyper
bolici ABC, cuius axis BD, qui in portione non erit ad ba
&longs;im perpendicularij: ba&longs;is autem dicti conoidis, vel portio
nis &longs;it circulus, vel ellip&longs;is, cuius diameter ADC: & hyper
boles ABC, quæ vel conoides de&longs;cribit, vel e&longs;t &longs;ectio tan
tummodo per axem, cuius tran&longs;uer&longs;um latus &longs;it BE, &
te DF. & tertia DG: qua ratione erit FG duode cima
pars axis BD quarta ab ea, cuius terminus D; f
IB ad BD, ita FH ad HG. Dico conoidis, vel portio
nis ABC centrum grauitatis e&longs;&longs;e H. Nam vt e&longs;t EB
ad BD ita fiat DK ad KA: & ponatur KDY &longs;e&longs;qui
altera ip&longs;ius DK, & ex AK ab&longs;cindatur KM &longs;ub&longs;e&longs;
quialtera ip&longs;ius AK: & ip&longs;is DK DM, DA, æquales
eodem ordine ab&longs;cindantur DL, DN, DC: & de&longs;cri
bantur triangula, KBL, MBN: & per puncta ABC
vertice communi B, tran&longs;eant duæ &longs;ectiones parabolæ
AOB, & BPC, ita vt contingat recta BK parabolam
AOB, recta autem BL parabolam BPC; &longs;it autem
AKLC, parabolarum diametris parallela,. Deinde
&longs;ecto axe BD bifariam, & &longs;ingulis eius partibus rur&longs;us bi
fariam in quotlibet partes æquales, &longs;int ex illis duæ
partes DQ, QF: & per puncta QF planis quibu&longs;dam
ba&longs;i parallelis &longs;ecentur vnà &longs;olidum & hyperbole ABC:
&longs;intque hyperboles &longs;ectiones, quæ continent &longs;ectiones trian
gulorum ABC mixti, & rectilinei KBL, rectæ RTX
ZVS: &longs;olidi autem ABC &longs;ectiones erunt cir
culi, vel ellip&longs;es &longs;imiles ba&longs;i circa diametros RS, Quoniam igitur e&longs;t vt
vtrobique enim e&longs;t proportio &longs;e&longs;quialtera: erit permutan
do vt YK ad A
HG, ita D
triangulum BKM, hoc e&longs;t ad æquale huic ex demon
&longs;tratis triangulum A
triangulum BKL ad duo mixta rriangula AKB, BLC
&longs;imul. &longs;ed duorum triangulorum AKB, BLC &longs;imul e&longs;t
centrum grauitatis F, vt in hoc tertio libro demon&longs;tra
uimus: trianguli autem BKL, vt in primo, centrum gra
uitatis G; totius igitur trianguli ABC centrum graui
tatis erit H. Rur&longs;us quoniam e&longs;t vt BD ad BQ hoc
DK ad QX: & vt quadratum BK ad quadratum BX,
hoc e&longs;t vt quadratum BD ad quadratum BQ, ita e&longs;t
A
tionales; &longs;ed & earum primæ, & tertiæ &longs;unt proportiona
les; nam e&longs;t vt EB ad BD, hoc e&longs;t vt rectangulum EBD
prima in primis ad quadratum BD primam in &longs;ecundis,
&longs;itam ex &longs;ecundis, ita erit compo&longs;ita ex tertiis ad com
po&longs;itam ex quartis; videlicet vt rectangulum BDE, quod
æquale e&longs;t rectangulo EBD vna cum quadrato BD, ad
rectangulum BQE, quod æquale e&longs;t rectangulo EBQ
vnà cum quadrato BQ, ita erit tota AD ad totam
Sed vt rectangulum BDE ad rectangulum BQE ita e&longs;t
AD quadratum, ad quadratum RQ, hoc e&longs;t ita circu
lus, vel ellip&longs;is circa AC, ad circulum, vel &longs;imilem illi
ellip&longs;em circa RS; vtigitur AD ad TQ, hoc e&longs;t in ea
rum duplis vt AC ad TV, ita erit circulus, vel ellip&longs;is
circa AC ad circulum, vel ellip&longs;em circa RS. Similiter
o&longs;tenderemus e&longs;&longs;e vt AC ad
p&longs;im circa AC, ad circulum, vel ellip&longs;em, circa
uertendo igitur, & ex æquali erunt binæ in eadem propor
tione, vt
ad circulum, vel ellip&longs;im circa RS: & vt TV ad AC, ita
circulus, vel ellip&longs;is circa RS ad circulum, vel ellip&longs;im
circa AC. Rur&longs;us, quoniam tres rectæ lineæ incipienti
à minima
nales quadratis ex B
F
&longs;ius F
D
vtpote proportionales ip&longs;is BF, BQ, BD, propter &longs;i
militudinem triangulorum; minor igitur proportio erit
lium minor erit proportio circuli, vel ellip&longs;is circa
circulum, vel cllip&longs;im circa RS, quàm circuli, vel elli
p&longs;is circa RS, ad circulum, vel ellip&longs;im, circa AC. Similiter quæcumque &longs;ectiones per prædicta axis, vel dia
metri BD puncta &longs;ectionum fierent vt dictum e&longs;t ad ver
ticem retrocedenti o&longs;tenderentur quælibet ternæ inter &longs;e
proximæ, binæque &longs;umptæ vtriu&longs;que ordinis proportio-
diam quàm mediæ ad maximam; per XXXII igitur &longs;e
cundi, triangulum mixtum, & &longs;olidum ABC, in huius
axe illius autem diametro BD commune habebunt cen
trum grauitatis. &longs;ed demon&longs;trauimus H centrum grauita
tis trianguli ABC; conoidis igitur vel portionis ABC
centrum grauitatis erit idem H. Quod demon&longs;trandum
erat.
Et hic huius tertij Libri finis e&longs;&longs;et; ni&longs;i &longs;ecundo iam im
pre&longs;&longs;o, alia quædam via magis naturalis me ad conoidis hy
perbolici centrum grauitatis reduxi&longs;&longs;et. Ea igitur in &longs;ecun
dum librum aliàs in&longs;erenda, nunc in &longs;equenti appendice
&longs;eptem propo&longs;itionibus expo&longs;ita, per &longs;ectionem prædicti
conoidis in conoides parabolicum eodem vertice, & circa
eundem axim, & reliquam figuram &longs;olidam, ab&longs;que com
po&longs;ito ex duabus figuris circum&longs;criptis, quæ ex cylindris
componuntur, propo&longs;itum concludat.
APPENDIX.
Si &longs;int octo magnitudines quaternæ
totæ, & ablatæ proportionales, fue
rint autem, & primarum vtriu&longs;que
ordinis ablatæ ad reliquas propor
tionales; erunt vtriu&longs;que ordinis re
liquæ proportionales.
Sint octo magnitudines quaternæ
proportionales, ac primi quidem ordi
nis totæ, vt AB ad CD, ita EF ad
GH: &longs;ecundi autem ordinis ablatæ, vt
B ad D, ita F ad H: &longs;it autem vt B
ad A ita F ad E. Dico & reliquas
e&longs;&longs;e proportionales, videlicet vt A ad
C, ita E ad G. Quoniam enim com
ponendo, & conuertendo e&longs;t vt A ad
AB, ita E ad EF: &longs;ed vt AB ad
ad E ad GH: & conuertendo vt
CD ad A, ita GH ad E: & per
mutando CD ad GH, ita A ad E. Rur&longs;us quoniam e&longs;t vt A ad B ita
E ad F: & vt B ad D, ita F ad H;
erit ex æquali, vt A ad D ita E ad
H: &longs;ed vt CD ad A, ita erat GH
ad E; ex æquali igitur erit vt CD ad
D ita GH ad H: & permutando vt
CD ad GH, ita D ad H, & reli
qua C ad reliquam G: &longs;ed vt CD
ad GH ita erat A ad E; vt igitur
A ad C ita erit E ad G. Quod demon&longs;trandum erat.
Si circa datæ hyperboles communem diame
trum parabola de&longs;cripta illius ba&longs;im ita diuidat,
vt quadratum dimidiæ ba&longs;is parabole ad reli
quum quadrati dimidiæ ba&longs;is hyperboles eam
habeat proportionem, quam tran&longs;uer&longs;um latus
ad diametrum hyperboles; omnes in hyperbole
ad diametrum ordinatim applicatas ita &longs;ecabit,
vt exce&longs;&longs;us, quibus quadrata in hyperbole appli
catàrum &longs;uperant quadrata in parabola ex &longs;ectio
ne applicatarum, inter &longs;e &longs;int vt quadrata diame
tri partium inter applicatas, & verticem inter
iectarum.
E&longs;to hyperbole ABC, cuius diameter BD, tran&longs;uer-
ctis quibuslibet GH, ordinatim applicentur MG, NH:
& circa diametrum BD &longs;it de&longs;cripta parabola KBL tali
ter vt ip&longs;ius dimidiæ ba&longs;is DK quadratum ad reliquum
quadrati AD, &longs;it vt EB ad BD, & rectas MH, NG
in infinitum productas &longs;ecet parabola KBL in punctis
OP. Dico puncta OP intra hyperbolem cadere: & reli
quum quadrati MG dempto quadrato GO ad reliquum
quadrati NH dempto quadrato PH, e&longs;&longs;e vt quadratum
BG ad quadratum
BH. Quoniam enim
ponitur vt EB ad B
D, hoc e&longs;t vt rectan
gulum EBD ad qua
dratum BD, ita qua
dratum DK ad reli
quum quadrati AD,
erit componendo, &
conueniendo, vt
gulum BDE ad re
ctangulum EBD, ita
quadratum AD ad
quadratum DK: &longs;ed
vt rectangulum BGE
ad
ita e&longs;t quadratum MG ad quadratum AD; ex æquali
igitur, vt rectangulum BGE ad rectangulum EBD, ita
e&longs;t quadratum MG ad quadratum DK: &longs;ed vt rectan
gulum EBD ad rectangulum EBG, ita e&longs;t quadratum
DK ad GO quadratum; ex æquali igitur vt rectangu
lu m BGE ad rectangulum EBG, ita erit quadratum
MG ad quadratum GO: &longs;ed rectangulum BGE maius
e&longs;t totum parte rectangulo EBG; quadratum igitur MG
quadrato GO maius erit, & recta MG maior quàm
O. Similiter o&longs;tenderemus candem parabolam &longs;ecare
quamcumque aliam in hyperbole ABC ordinatim ad dia
metrum applicatarum. Quoniam igitur &longs;unt octo magni
tudines quaternæ totæ, & ablatæ proportionales; ac pri
mi quidem ordinis, vt rectangulum BDE ad rectangu
lum BGE, ita quadratum AD ad quadratum MG: &longs;e
cundi autem ordinis, vt rectangulum EBD ad rectangu
lum EBG ita quadra
tum DK ad quadra
tum OGD: &longs;ed vt
EB ad BD, hoc e&longs;t
vt ablata primæ in pri
mis rectangulum EB
D ad reliquum BD
quadratum, ita poni
tur ablata primæ in &longs;e
cundis, quadratum D
K ad reliquum exce&longs;
&longs;um, quo quadratum
AD &longs;uperat quadra
tum DK; vt igitur e&longs;t
reliqua primæ ad reli
quam &longs;ecundæ in pri
mis, ita erit in &longs;ecundis; videlicet vt quadratum BD ad
quadratum BG, ita reliquum quadrati AD dempto qua
drato DK, ad reliquum qua rati MG dempto quadra
to GO. Similiter o&longs;tenderemus reliquum quadrati AD
dempto quadrato DK ad reliquum quadrati NH dem
pto quadrato PH, e&longs;&longs;e vt quadratum BD ad quadra
tum BH; conuertendo igitur, & ex æquali erit vt qua
dratum BG ad quadratum BH, ita reliquum quadra
ti MG dempto quadrato GO, ad reliquum ouadiatQuod demon&longs;trandum
erat.
Omne conoides hyperbolicum diuiditur in
conoides parabolicum circa eundem axim, & re
liquam figuram quandam, ad quam conoides pa
rabolicum eam habet proportionem, quam&longs;e&longs;qui
altera tran&longs;uer&longs;i lateris hyperboles, quæ conoides
de&longs;cribit, ad axem conoidis.
Sit conoides hyperbolicum ABC, cuius axis BD: hy
perboles autem, quæ conoides de&longs;cribit tran&longs;uer&longs;um latus
EB, cuius &longs;it &longs;e&longs;quialtera BEF: & ab&longs;ci&longs;&longs;a DG, ita vt
quadratum ex ip&longs;a ad reliquum quadrati AD &longs;it vt EB
ad BD, vertice B circa diametrum BD de&longs;cripta &longs;it
Dico conoides GBH comprehendi à conoide ABC &
e&longs;&longs;e ad illius reliquum, vt FB ad BD. Ab&longs;ci&longs;&longs;a enim
DK ita potentia &longs;it ad DG, vt DB ad BE longitudine,
circa axim BD de&longs;cribatur conus KBL: & &longs;ecta BD in
multas partes æquales, ducto&longs;que per ea puncta planis
quibu&longs;dam ba&longs;i parallelis, &longs;ecentur tria dicta &longs;olida, conus
&longs;cilicet & vtrumque conoides: & &longs;uper &longs;ectiones circulos
de&longs;cribantur cylindri æqualium altitudinum terni cuca
communes axes partes æquales, in quas axis BD diui&longs;us
fuit, & inter eadem plana parallela: & omnino triplex figura
ex cylindris, quos diximus &longs;it tribus dictis &longs;olidis circum&longs;cri
pta: &longs;intque circa duos axes infimos DM, MN terni cylin
dri AO, GP, KQ: & proxime ordine ip&longs;is re&longs;pondentes
cylindri TX, SV, RZ, quorum ba&longs;es circa diametros
TI, S
cum tribus &longs;olidorum &longs;ectionibus per axem, triangulo &longs;cili
cet, parabola, & hyperbole in eodem plano, atque ideo tres Quoniam
igitur e&longs;t vt EB ad BD, ita quadratum DG ad
quadrati AD, &longs;ecabit parabola GBH omnes in hyperbo
le ABC ad diametrum ordinatim applicatas, quare conoi
des ABC comprehendet conoides GBH: atque ita para
bola &longs;ecabit, vt exce&longs;&longs;us quibus quadrata in hyperbole ap
plicatarum &longs;uperant partes quadrata in parabola applicata
rum, inter &longs;e &longs;int vt quadrata partium diametri BD inter
applicatas & verticem interiectarum, prout vt inter &longs;e
dent: vt igitur e&longs;t quadratum BD ad quadratum BM, hoc
e&longs;t vt quadratum DK ad quadratum RM, ita erit
AD quadrati dempto quadrato DG ad reliquum quadrati
TM dempto quadrato SM, & permutando. Sed quia qua
dratum DG ad reliquum quadrati AD, & ad quadratum
DK eandem habet proportionem ex vi con&longs;tructionis, reli
quum quadrati AD, dempto quadrato DG æquale e&longs;t
quadrato DK; reliquum igitur quadrati TM dempto qua
drato SM æquale erit quadrato RM: &longs;i igitur vtri&longs;que ad
dantur &longs;ingula communia, vnis quadratum DG, alteris
quadratum SM, erit & quadratum AD æquale duobus
quadratis GD, DK, & quadratum TM duobus quadra
tis SM, MR æquale. &longs;ed cum cylindri eiuidem altitudi
nis inter &longs;e &longs;int vt ba&longs;es, &longs;unt vt quadrata, quæ ab eorundem
ba&longs;ium &longs;emidiametris fiunt; cylindiusigitur AO æqualis
e&longs;t duobus cylindris GP, KQ: & cylindrus TX duobus
cylindris SEadem ratio e&longs;t de reliquis
deinceps. Tota igitur figura conoidi ABC circum&longs;cripta,
vtrique &longs;imul, conoidi GBH, & cono KBL circum&longs;cri
ptæ æqualis erit. po&longs;&longs;unt autem eæ figuræ ita e&longs;&longs;e dictis &longs;oli
dis circum&longs;criptæ per ea quæ alibi o&longs;tendimus, vt &longs;uperent
in&longs;criptas minori &longs;pacio quantacumque magnitudine pro
po&longs;ita; per tertiam igitur &longs;ecundi, conoides ABC vtrique
&longs;imul, conoidi GBH, & cono KBL æquale erit. dempto
igitur Rur&longs;us quia e&longs;t vt EB ad BD, ita
quadratum GD ad quadratum DK, hoc e&longs;t circulus cir
ca GH ad circulum circa KL, hoc e&longs;t conus GBH &longs;i
de&longs;cribatur ad conum KBL: &longs;ed vt FB ad BE ita e&longs;t co
noides GBH ad conum GBH; ex æquali igitur erit vt
FB ad BD, ita conoides GBH ad conum KBL, hoc
e&longs;t ad &longs;olidum AGBHC. Manife&longs;tum e&longs;t igitur
Ex huius Theorematis demon&longs;tratione manife
&longs;tum e&longs;t, ij&longs;dem po&longs;itis cylindros deficientes, ex
quibus con&longs;tat exce&longs;&longs;us, quo figura conoidi hyper
bolico circum&longs;cripta &longs;uperat circum&longs;criptam co
noidi parabolico, ita &longs;e habere, vt quorumlibet
trium inter &longs;e proximorum minor proportio &longs;it
minimi ad medium, quam medij ad maximum:
æquales enim &longs;unt &longs;inguli &longs;ingulis cylindris, ex
quibus con&longs;tat figura cono BKL circum&longs;cripta,
qui &longs;unt inter eadem plana parallela. Quod &longs;i
ita e&longs;t, &longs;imul illud manife&longs;tum erit, & ex hoc, &
ex ijs, quæ in &longs;ecundo libro demon&longs;trauimus; præ
dictum exce&longs;&longs;um ex tot cylindris deficientibus
eiu&longs;dem altitudinis, quos diximus componi po&longs;&longs;e,
vt ip&longs;ius centrum grauitatis in axe BD di&longs;tet à
centro grauitatis coni KBL, hoc e&longs;t à puncto in
quo axis BD &longs;ic diuiditur, vt pars, quæ ad ver
ticem &longs;it reliquæ tripla, ea di&longs;tantia, quæ minor
&longs;it quantacum que longitudine propo&longs;ita.
Si conoidi parabolico figura circum&longs;cribatur,
& altera in&longs;cribatur ex cylindris æqualium alti
tudinum, binis circa communes axes &longs;egmenta
axis conoidis, & inter eadem plana parallela, mi
nimo circum&longs;criptorum ad nullum relato; omnia
re&longs;idua cylindrorum figuræ circum&longs;criptæ dem
ptis figuræ in&longs;criptæ cylindris, & inter &longs;e, & mi
nimo cylindro æqualia erunt.
Sit conoidi parabolico ABC, cuius axis BD circum
&longs;cripta figura ex quotcumque cylindris æqualium altitu
dinum, quorum tres deinceps &longs;int EL minimus &longs;upremus,
& GQ, IR, quorum ba&longs;es eodem ordine circuli, quorum
&longs;emidiametri ad parabolæ, quæ figuram de&longs;cribit diame
trum BD ordi
natim applicatæ
&longs;int EF, GH, IK:
& in duplos cre
&longs;centibus cylin
dris circa
axium duplos a
xes BH, IK, HD,
&
quotcumque plu
res e&longs;sent; &longs;it co
noidi ABC in
&longs;cripta figura ex cylindris æqualium altitudinum inter &longs;e, &
circum&longs;criptis. Bini itaque circa communes axes inter ea
dem plana parallela interijcientur, minimo EL ad nullum
&longs;criptorum &longs;it NM ba&longs;im ip&longs;i communem habens circu
lum circa EFM: & con&longs;equenti circum&longs;criptorum GQ
&longs;it. in&longs;criptorum æqualis PO ba&longs;im habens ip&longs;i commu
nem circulum circa GHO: &longs;int autem circulorum qui
&longs;unt ba&longs;es cylindrorum diametri in parabola per axim:
quæ quoniam &longs;unt communes &longs;ectiones cum parabola per
axim planorum ba&longs;i conoidis, & inter &longs;e parallelorum,
erunt etiam ip&longs;æ inter &longs;e, & parabolæ ba&longs;i AC parallelæ,
earumque dimidiæ vt EF, GH ad diametrum BD or
dinatim applicatæ. Quoniam igitur in parabola ABC
e&longs;t vt HB ad BF ita quadratum GH ad quadratum
EF, duplum erit
quadratum GH
quadrati EF: qua
re & circulus cir
ca GO circuli
circa EM at que
adeo cylindrus
GQ cylindri E
L duplus, pro
pter <17>qualitatem
altitudinum: &longs;ed
& cylindrus NL
duplus e&longs;t cylindri EL per con&longs;tructionem; cylindrus igi
tur GQ æqualis e&longs;t cylindro NL: & ablato communi
NM cylindro, reliquus GQ deficiens cylindro NM
cylindro EL æqualis. Rur&longs;us quia e&longs;t vt KB ad BH,
ita quadratum IK ad quadratum GH, hoc e&longs;t ita IR
cylindrus ad cylindrum GQ: &longs;ed vt HB ad BF ita
erat cylindrus GQ ad cylindrum EL; tres igitur cy
lindri IR, GQ, EL, tribus lineis BK, BH, BF, eodem
ordine proportionales erunt: &longs;ed tres eædem lineæ &longs;e&longs;e
æqualiter excedunt; tres igitur dicti cylindri &longs;e&longs;e æqua-
dro PO æquale erit reliquo cylindri GQ dempto cylin
dro NM, & reliquum cylindri GQ dempto cylindro
NM æquale cylindro EL. Similiter ad reliquos cylindros
quotcumque plures e&longs;&longs;ent de&longs;cendentes o&longs;tenderemus, om
nes exce&longs;&longs;us, quibus cylindri circum&longs;cripti in&longs;criptos
&longs;uperant &longs;ibi quique re&longs;pondentes inter &longs;e & cylindro
EL æquales e&longs;&longs;e. Manife&longs;tum e&longs;t igitur propo&longs;itum.
Dato conoide hyperbolico, & ip&longs;ius conoi
de parabolico circa eundem axim, quod ad
reliquum hyperbolici conoidis eam proportio
nem habeat, quam &longs;e&longs;quialtera tran&longs;uer&longs;i late
ris hyperboles, quæ conoides de&longs;cribit, ad axim
conoidis; fieri pote&longs;t vt conoidi parabolico fi
guræ quædam in&longs;cribatur, & altera circum&longs;cri
bantur vt &longs;upra factum e&longs;t, & hyperbolico alio cir
cum&longs;cribatur omnes ex cylindris æqualium al
titudinum multitudine æqualibus exi&longs;tentibus
ijs, ex quibus con&longs;tant figuræ conoidibus cir
cum&longs;criptæ, ita vt exce&longs;&longs;us, quo figura conoidi
parabolico circum&longs;cripta in&longs;criptam &longs;uperat,
quem breuitatis cau&longs;a voco exce&longs;&longs;um primum,
ad exce&longs;&longs;um, quo figura conoidi hyperbolico cir
cum&longs;cripta &longs;uper
quem voco exce&longs;&longs;um &longs;ecundum, minorem habeat
proportionem quacum que propo&longs;ita.
Sit conoides hyperbolicum ABC, & pars eius para
bolicum EBF circa eundem axim BD: & conoides
EBF ad reliquum conoidis ABC eam habeat proportio
nem, quam &longs;e&longs;quialtera tran&longs;uer&longs;i lateris hyperboles per
axim ABC ad axim BD. Dico fieri po&longs;&longs;e quod proponitur.
Habeat enim DL ad LB quamcumque proportionem: &
conoides ABC reliquo &longs;olido AEBFC dempto conoi
de EBF. &longs;it conus circa axim BD æqualis GBH: &
de&longs;cribatur conus GLH: & &longs;ecta BD bifariam in pun
cto K, & rur&longs;us BK, KD in multitudine, & longitudi
ne æquales in&longs;cribatur conoidi EBF, & altera cirum&longs;c
batur, vt in antecedenti factum e&longs;t, figura ex cylindris æ
qualium altitudinum, ita vt exce&longs;&longs;us, quo circum&longs;cripta
&longs;uperat in&longs;criptam fit minor cono GLH; & cylindris cre
&longs;centibus in latitudinem ab&longs;oluatur figura conoidi ABC
circum&longs;cripta ex cylindris altitudine, & multitudine æqua
libus ijs, qui &longs;unt circa conoides EBF. Quoniam igitur
primus exce&longs;&longs;us e&longs;t minor cono GLH, multo minor crit
pars eius communis &longs;olido AEBFG, quàm conus GLH:
&longs;ed &longs;olidum AEBFC æquale e&longs;t cono GBH; reliquum
igitur &longs;olidi AEBFC dicto communi ablato, maius erit
coni GBH reliquo BGLH; minor igitur proportio e&longs;t
&longs;olidi AEBFC, quàm coni GLH ad reliquum coni
GBH: &longs;ed &longs;ecundus exce&longs;&longs;us maior e&longs;t prædicto reliquo
&longs;olidi AEBFC, ctenim illud comprehendit; multo igitur
minor proportio erit primi exce&longs;&longs;us ad &longs;ecundum, quàm
coni GLH ad reliquum BGLH, hoc e&longs;t minor propor
tio quàm DL ad LB: ponitur autem proportio DL ad
LB quali&longs;cumque. Fieri igitur pote&longs;t, quod proponitur.
Omnis re&longs;idui conoidis hyperbolici dempto
conoide parabolico, vt &longs;upra diximus, centrum
grauitatis e&longs;t punctum illud, in quo axis &longs;ic diui
ditur, vt pars propinquior vertici &longs;it tripla re
liquæ.
Sit conoides hyperbolicum ABC, cuius axis BD, &
ablatum conoides parabolicum EBF circa eundem axim
BD, ita &longs;it ad reliquum &longs;olidum AEBFC, vt &longs;e&longs;quialte
ra tran&longs;uer&longs;i lateris hyperboles, quæ conoides de&longs;cribit ad
axem BD: & ponatur BG ip&longs;ius GD tripla. Dico re
Secta
enim BD bifariam in puncto H, & po&longs;ita GK ip&longs;ius GH
minori quantacumque longitudine propo&longs;ita, &longs;umptoque
in GK quolibet puncto L, intelligantur id enim (fieri po&longs;
&longs;e manife&longs;tum e&longs;t ex &longs;upra demon&longs;tratis) tres figuræ vna in
&longs;cripta conoidi EBF, & duæ circum&longs;criptæ altera alteri
conoidum, vt &longs;upra factum e&longs;t, compo&longs;itæ ex cylindris
æqualium altitudinum ita multiplicatis, vt vtrumque illud
accidat; & vt &longs;ecundi exce&longs;&longs;us centrum grauitatis quod &longs;it
M (omnium autem trium dictorum exce&longs;&longs;uum in axe
BD erunt centra grauitatis) &longs;it puncto G propinquiu
quàm punctum L: & vt primus exce&longs;&longs;us ad &longs;ecundum mi
norDein
de vt HK ad KL, ita &longs;it HN ad NM, & vt primus
exce&longs;&longs;us ad &longs;ecundum, ita MO ad OH. Quoniam igitur
cylindri omnes deficientes, & &longs;ummus integer, ex quibus
p
in axe BD centra grauitatis æqualibus interuallis à bipar
titi axis BD &longs;ectione H & inter &longs;e di&longs;tantia; totius pri
mi exce&longs;&longs;us centrum grauitatis erit H: &longs;ecundi autem ex
ce&longs;&longs;us centrum grauitatis ponitur M; cum igitur &longs;it vt pri
mus exce&longs;&longs;us ad &longs;ecundum, ita ex contraria parte MO
ti centrum grauitatis O. Quoniam igitur minor propor
tio e&longs;t primi exce&longs;&longs;us ad &longs;edundum, hoc e&longs;t MO ad OH,
quàm LK ad KH; erit conuertendo maior proportio HO
ad OM, quàm HK ad KL: &longs;ed vt HK ad KL, ita
ponitur HN ad NM; maior igitur proportio e&longs;t HO ad
OM, quàm HN ad NM; eiu&longs;dem igitur lineæ HM
minor erit MO, quàm MN, & punctum O propinquius
puncto G quam punctum N. Rur&longs;us quia vt HK ad
KL, ita e&longs;t HN ad NM; erit componen do & per con
uer&longs;ionem rationis, vt LH ad HK ita MH ad HN: &
permutando, vt HM ad HL, ita HN ad HK: &longs;ed HM
e&longs;t maior quàm HL; ergo & HN erit maior quam H
& punctum N propinquius puncto G quàm punctum K:
&longs;ed punctum O propinquius erat puncto G quàm punctum
N; multo igitur erit punctum O propinquius puncto G
quàm punctum K. ponitur autem di&longs;tantia GK minor
quantacumque longitudine propo&longs;ita: & e&longs;t O centrum
grauitatis tertij exce&longs;&longs;us reliquo &longs;olido AEBFC circum
&longs;cripti; ex ijs igitur, quæ in primo libro demon&longs;trauimus,
&longs;olidi AEBFC centrum grauitatis erit G. Quod demon
&longs;trandum erat.
Omnis conoidis hyperbolici centrum grauita
tis e&longs;t punctum illud, in quo duodecima pars axis
quarta ab ea, quæ ba&longs;im attingit &longs;ic diuiditur, vt
pars propinquior ba&longs;i &longs;it ad reliquam, vt &longs;e&longs;quial
tera tran&longs;uer&longs;i lateris hyperboles, quæ conoides
de&longs;cribit; ad axem conoidis.
Sit conoides hyperbolicum ABC, cuius axis BD:
BE, huius autem &longs;e&longs;quialtera BEF: & &longs;umpta axis BD
tertia parte DG, & quarta DH, qua ratione erit GH
axis BD pars duodecima, ordine quarta ab ea, cuius termi
nus D; e&longs;to vt FB ad BD, ita HK ad KG. Dico conoi
dis ABC centrum grauitatis e&longs;&longs;e K. Diuidatur enim co
noides ABC in parabolicum conoides LBM, & reliquum
&longs;olidum ALBMC, ita vt conoides LBM ad &longs;elidum
ALBMC &longs;it vt FB ad BD, hoc e&longs;t vt HK GK. Quo
niam igitur G e&longs;t centrum grauitatis conoidis LBM, & H
&longs;olidi ALBMC; tot us conoidis ABC centrum graui
tatis crit K. Quod demon&longs;trandum crat.
TERTII LIBRI FINIS.