Alterum de Compo&longs;ito.
Antonij Pi&longs;arij Superiorum permi&longs;&longs;u.
CAROLO.
genibus opu&longs;culum exhibeo, in quo naturam motuum, pleniori
methodo, quàm puto antea &longs;it actum, geometricè exequor. Neceße habui hæc præmittere, quò viam aperirem, & quo
dammodo alueum &longs;ternerem aquarum doctrinæ, quarum
argumentum vtili&longs;&longs;imum, & profundæ indaginis iam diu
meditor. Quam arduum &longs;it, & per quas &longs;alebras eun
dum, vt nouum aliquid luce dignum è latebris naturæ eruarur
vtinam Cel&longs;itudini tuæ aliquis veritatum non vulgarium
indagator fidem faceret; &longs;cio equidem, & laboris improbitas
tangeret benigni&longs;&longs;imum animum tuum, & &longs;imal naturæ inge
nium &longs;u&longs;piceres, quæ mentibus aliquorum vim inuentricem
in&longs;eruit, vt eorum iugi cogitatione humanis v&longs;ibus prouide-Et verò (&longs;i in hoc genere de me quidquam confiteri decet)
ni&longs;i aduer&longs;æ valetudinis experimento prudentior factus indo
lem meam huiu&longs;cemodi &longs;tudijs intemperanter addictam ali
quot ab hinc annis compe&longs;cuißem; nec non quotidie munus à
Cel&longs;itudine Tua &longs;ummo cum honore & beneficentia demanda
tum (adeo vt hoc etiam nomine Te&longs;eruatorem meum appella
re po&longs;&longs;im) inde me reuoca&longs;&longs;et; eorum, credo equidem, ponderi,
a&longs;&longs;iduæque contemplationi &longs;uccumbere nece&longs;&longs;e erat. Vnde au
tem, Cel&longs;i&longs;&longs;ime dux, huic &longs;cientiæ tanta vis, vt quos &longs;ibi &longs;emet
adiunxerit, nonni&longs;i altiori ratione queat a &longs;e ip&longs;a dimittere? An quod forta&longs;&longs;e vbi animus publicæ vtilitati de&longs;eruire cæpe
rit, veluti in nataræ concilium admi&longs;&longs;us, &longs;ui quodammodo
oblitus, propriam humilioremque &longs;edem reui&longs;ere dedignetur; an
quia, cùm inter cæteras &longs;cientias Geometria demon&longs;trationem,
hoc e&longs;t veritatem &longs;inceram, & quandam primi veri particu
lam profiteatur, hinc ne&longs;cio quid diuinum habent &longs;ibi
vnde nonni&longs;i Deo impellente, vbi nimirum officia, potiorque
ratio id po&longs;tulant, ab eius intuitu retrahatur. Hoc equidem
puto; atque hinc diuina Geometria iure optimo a docti&longs;&longs;imis, &
clari&longs;&longs;imis viris pa&longs;&longs;im nuncupatur. Quamobrem nemo non
eam
huic etiam qualicunque opu&longs;culo benignè annuas &longs;pero, adeo
vt iam Te in terris Dominum, Altorem, Seruatorem, Patro
numque appellare non dubitem, quam vna cum Cel&longs;i&longs;&longs;ima do
mo mihi, tot tibi nominibus deuincto, &longs;uperi vt &longs;eruent &longs;o&longs;pi
tentque, enixè oro, ac omnibus votis exopto.
Ioannes Ceua.
CVrrat mobile ab A in D &longs;ecundùm rectam
AD, & linea BHI &longs;it naturæ illius, vt dedu
ctis ad AD perpendicularibus AB, CH, DI
ex punctis quibu&longs;cunque A, C, D; veloci
tatum gradus, quos mobile &longs;ortitur in ij&longs;
dem punctis A, C, D men&longs;urentur ab ip&longs;is
rectis AB, CH, CI. Figuram planam BADIHB apellabi
mus gene&longs;im motus ab A in D.
II&longs;dem manentibus, &longs;it etiam alia linea EFG talis natu
ræ, vt protractis rectis BA in E, HC in F, & ID in G ha
beat DG ad CF eandem reciprocè rationem, quam HC
ad ID. Item &longs;it CF ad HE vt reciprocè BA ad HC, vo
cabimus figuram planam ADGIEA imaginem tempo
ris motus ab A in D iuxta gene&longs;im prædictam.
ADhuc po&longs;ita illa gene&longs;i, intelligatur linea PON eius
naturæ, vt &longs;i &longs;it KL ad LM vt tempus lationis ab A
in C ad tempus ab eodem C in D, habeat &longs;emper KP ad
LO eandem rationem, quam AB ad CH; & LO ad NM
eandem, quam HC ad ID: Figuram planam PKMNOP
A in D.
ras eße parallelogrammas.
SI &longs;int duæ gene&longs;es, aut imagines ABCD, FEG, ita vt
cum gene&longs;es &longs;int, habeat AB ad FE eandem rationem,
quam velocitas in A ad velocitatem in F, & cum imagines
velocitatum, quarum tempora AD, FG, velocitas, quam
habet mobile in&longs;tanti A ad velocitatem alterius mobilis
in&longs;tanti F, &longs;it vt AB ad FE, & demum ip&longs;is figuris vt imagi
nibus temporum con&longs;ideratis habeat velocitas in A ad
velocitatem in F rationem eandem, quam AB ad FE, vo
cabuntur tum gene&longs;es illæ, cum imagines inter &longs;e homo
geneæ.
EAm planam Figuram, in qua ductæ quoteunque
&ecedil;quidi&longs;tantes eò deinceps decre&longs;cunt, quò ad idem
extremum propiores fiunt, acuminatam nuncupabimus.
INter maximam, & minimam eiu&longs;dem imaginis veloci
tatem cadit quædam media, qua tantùm velocitate, &longs;i
conciperetur motus æquabilis, nihilominùs eodem tem
pore idem &longs;patium curreretur, ac iuxta imaginem propo&longs;i
tam: eam ergo mediam velocitatem dicimus propo&longs;itæ
imaginis æquatricem.
SPatium iuxta imaginem velocitatum quamcunque
exactum, vel iuxta æquatricem imaginis e&longs;t maius eo
&longs;patio, quod curreretur eodem tempore minima eiu&longs;dem
imaginis velocitate; &longs;ed minus eo, quod velocitate ma
xima.
TEmpus, quo curritur &longs;patium iuxta quamlibet tem
poris imaginem, maius e&longs;t eo, quo idem &longs;patium
curreretur maxima velocitate, &longs;ed contra minus eo altero,
quo ip&longs;um &longs;patium minima velocitate exigeretur, earum
videlicet, quæ &longs;unt in gene&longs;i, aut imagine velocitatum pro
po&longs;iti motus, cuius nempe illa e&longs;t imago temporis. Fit er
go, vt tempus æquale ei, quo illud ip&longs;um &longs;patium currere
tur iuxta propo&longs;itam imaginem, &longs;it inter vtrumque dicto
rum temporum maximi, & minimi.
QVæcunque excogitetur figura plana, vel e&longs;t paralle
logrammum, vel acuminata figura, aut ex his com
po&longs;itum. Has tamen figuras inter binas volu
mus parallelas, ita vt vnum latus &longs;it ip&longs;as nectens normali
ter parallelas, quanquam etiam loco parallelarum po&longs;&longs;int
e&longs;&longs;e puncta, nempè vbi de&longs;inunt in acuminatas pror&longs;us
figuras.
TEmpora, quibus duo motus complentur &longs;unt in ra
tione imaginum homogenearum ip&longs;orum
Motus &longs;int primò æquabiles, curratque mobile &longs;patium
AB tempore, cuius imago CAB, curratur item ab alio mo
bili &longs;patium DE tempore, cuius imago DEF, & &longs;int ip&longs;æ
temporum imagines inter&longs;e homogeneæ, &longs;cilicet FD ad
AC eandem habeat rationem, quam velocitas in A ad
velocitatem in D. Dico, tempus per AB ad id per DE e&longs;
&longs;e vt figura ABC, ad DEF. Cum motus æquabiles &longs;int
erunt figuræ dictarum imaginum rectangula, propterea il
lorum ratio componetur ex rationibus altitudinum AB ad
DE, & ba&longs;ium AC ad DF, ex ij&longs;dem verò rationibus &longs;pa
tiorum &longs;cilicet, & reciproca velocitatum (&longs;unt enim ima
gines inter &longs;e homogeneæ) nectitur etiam ratio temporum,
quibus
ginum ACB, DEF, ergo e&longs;t eadem ratio inter illa tempo
ra, ac inter imagines &longs;uas.
Def.
pr. S de
motu æquab. Def.
2. Sit motus vnus æquabilis, alter verò quicunque; &longs;it
tamen imago huius temporis figura acuminata vt ALGE,
& alterius temporis prædicti motus æquabilis, &longs;it HFM,
quæ rectangulum erit: Dico, imaginibus homogeneis exi
&longs;tentibus, fore inter has eandem rationem, ac homologè
inter tempora decur&longs;uum ab A in E, & ab F in M iuxt&atail;
gene&longs;es imaginum temporum propo&longs;itarum. Si enim non
e&longs;t ita, &longs;it quædam alia magnitudo Y, maior, vel minor
imagine acuminata ALGE, quæ ad imaginem FHM ha
beat eandem rationem, quam tempus per AE iuxta imagi
nem ALGE ad tempus per FM iuxta imaginem alteram
FHM; &longs;it verò magnitudinis Y differentia ab imagine ma
gnitudo Z. Secetur AE bifariam in C, pariterque &longs;eg
menta AC, CE bifariam in B, D, & &longs;ic vlteriùs progredia
tur, donec, &longs;i compleatur rectangulum po&longs;tremum, & ma
ximum DG, hoc minus exi&longs;tat quam Z. Tum ductis reli
quis æquidi&longs;tantibus CI, BK, & à punctis N,
etiam æquidi&longs;tantibus rectæ AE, efficiatur ip&longs;i ALGE cir
cum&longs;cripta figura, con&longs;tans ex rectangulis æquealtis AK
pariter æquealtis BL, CR, DI, EN. Cum circum&longs;cript&atail;
figura differat ab in&longs;cripta exce&longs;&longs;u, quo rectangulum DG
&longs;uperat BL; (nam reliqua circum&longs;cripta AK, BI, CN, re
liquis in&longs;criptis æqualia &longs;unt) &longs;e quitur, exce&longs;&longs;um illum e&longs;&longs;e
minorem magnitudine Z. Si ergo magnitudo Y ponatur
maior magnitudine ALGE pro exce&longs;&longs;u Z, maior etiam erit
circum&longs;cripta AK, BI, CN, DG. Quòd &longs;i contrà Y intelli
gatur minor ip&longs;a ALGE ex defectu Z, erit quoque eadem
Y minor, quàm in&longs;cripta figura BL, CK, DI, EN. Itaque
nunc, &longs;i fieri pote&longs;t, &longs;it Y maior magnitudine ALGE per ip
&longs;um exce&longs;&longs;um Z, & intelligantur tot motus, quot &longs;unt re
ctangula in circum&longs;cripta figura, &longs;cilicet &longs;int ip&longs;i motus ab
A in B, à B in C, à C in D, & à D in E &longs;ecundum deinceps,
temporum imagines AK, BI, CN, DG rectangula, quæ
&longs;int inter&longs;e, & propo&longs;itis imaginibus homogeneæ, qui
motus erunt proptereà æquabiles. His po&longs;itis, tempus
per FM iuxta imaginem MH ad tempus per AB iuxta ima
ginem rectangulum AK eandem habet rationem, quam re
ctangulum MH ad rectangulum AK, &longs;imiliter idem tem
pus per FM &longs;ecundùm ip&longs;am imaginem rectangulum MH
ad &longs;ingula reliqua tempora per BC, CD, DE imaginibus
deinceps rectangulis BI, CN, DG habet eandem rationem,
quam rectangulum MH ad &longs;ingula eodem ordine rectan
gula BI, CN, DG. Quo circa totidem rectangula ex MH,
quot &longs;unt illa, ex quibus con&longs;tat circum&longs;cripta figura, ha
bebunt ad ea ip&longs;a circum&longs;cripta rectangula, &longs;eu ad eandem
circum&longs;criptam figuram AK, BI, CN, DG eandem ratio
nem, quam totidem tempora eiu&longs;dem imaginis MH ad &longs;i
mul tempora, quorum imagines &longs;unt illa ip&longs;a circum&longs;cripta
rectangula AK, BI, CN, DG. Quare etiam vnicum re
ctangulum MH ad circum&longs;criptam figuram AK, BI, CN,
DG erit in eadem ratione, in quo vnicum tempus per FM
iuxta imaginem MH ad omnia &longs;imul illa tempora iuxt&atail; Et
quoniam figura imaginis e&longs;t acuminata, habetque vi def.
2. huius, applicatas, quæ &longs;unt in ratione reciproca veloci
tatum, quibus nempe mobile afficitur in punctis &longs;patij, à
quibus deducuntur ip&longs;æ applicatæ; hinc fit, vt earum ve
locitatum, quas mobile habet in decur&longs;u rectæ AB, ea, qu&ecedil;
in A maxima &longs;it, & quæ in B minima. Eodem modo iuxta
reliquas imagines BKIC, CIND, DNGE, quæ itidem acu
minatæ &longs;unt, velocitates in fine decur&longs;uum C, D, E (&longs;unt
enim omnes versùs A acuminatæ) minimæ erunt, & ma
ximæ initio dictorum &longs;patiorum. Ideo tempora, qu&ecedil; im
penduntur iuxta illas imagines, &longs;eu ip&longs;am
cuius illæ &longs;unt omnes partes, minora erunt temporibus,
quæ decurrerent, &longs;i illi decur&longs;us forent æquabiles ex mini
mis illis velocitatibus exacti, vel quod in idem recidit, &longs;i
illi decur&longs;us e&longs;&longs;ent iuxta imagines rectangulorum circum
&longs;criptorum AK, BI, CN, DG; itaque rectangulum MH ad
figuram circum&longs;criptam AK, BI, CN, DG habebit mino
rem rationem, quàm tempus per FM imagine MH ad tem
pus per AE imagine ALGE, &longs;eu quàm rectangulum MH
habet ex hypothe&longs;i ad magnitudinem Y; igitur circum&longs;cri
pta figura, quæ priùs minor o&longs;ten&longs;a fuit magnitudine Y;
nunc maior concluditur; quod cum &longs;it ab&longs;urdum, &longs;equi
tur falsò nos po&longs;ui&longs;
ALGE. At &longs;i Y minor ponatur,
fectu Z; in&longs;cripta, vt &longs;upra, figura con&longs;tante ex rectangulis
æquè altis BL, CK, DI, EN, vt &longs;cilicet differentia ab ima
gine &longs;it minor magnitudine Z, liquebit, magnitudinem Y
minorem e&longs;&longs;e in&longs;cripta figura BL, CK, DI, EN; deind&etail;
procedendo vt &longs;upra, inueniemus rectangulum MH ad in
&longs;criptam figuram BL, CK, DI, EN in eadem ratione, i&ntail;
quo tempus per FM imagine MH ad omnia &longs;imul decur
&longs;uum tempora per AB, BC, CD, DE iuxta imagines re
ctangula in&longs;cripta BL, CH, DI, EN; Hæc verò tempor&atail;
CIND, INGE (nam velocitates initio decur&longs;uum per
dictas rectas diximus e&longs;&longs;e maximas, & quibus
tur
ctangula in&longs;cripta) ergo rectangulum MH ad in&longs;cripta&mtail;
figuram BL, CK, DI, EN habebit maiorem rationem,
tempus per FM iuxta imaginem MH ad tempora &longs;imul
imaginibus ALKB, BKIC, CIND, DNGE, &longs;iue ad tempus
iuxta imaginem ALGE ex illis compo&longs;itam. Ideoque re
ctangulum MH ad ip&longs;am in&longs;criptam figuram habebit ma
iorem rationem, quàm ad magnitudinem Y, idcirco Y, quæ
minor o&longs;ten&longs;a fuit in&longs;criptà figura BL, CK, DI, EN, nunc
hac alia via maiorem inuenimus; ergo cum rur&longs;us hoc &longs;it
ab&longs;urdum, nece&longs;&longs;e e&longs;t magnitudinem Y neque minore&mtail;
e&longs;&longs;e magnitudine ALGE, propterea æquales inter &longs;e
atque adeo tempus per FM imagine MN ad tempus per
AE imagine ALGE habebit eandem rationem, quam ima
go MH ad imaginem ALGE. Quod &c.
Def.
Def.
parte.
ric. lem.
libro de dim. parabolæ.
3. Imagines propo&longs;itæ &longs;int duæ acuminatæ.
Dico ni
hilominus, tempora iuxta illas imagines per AE, HI e&longs;&longs;e vt
ip&longs;æ imagines ALGE ad HIK, quæ &longs;int inter &longs;e homoge
neæ vt &longs;emper &longs;upponetur. Nam &longs;i intelligatur alius mo
tus per MF iuxta imaginem rectangulum MFN, qui æqua
bilis erit, manife&longs;tum e&longs;t ex &longs;ecundo ca&longs;u, tempus per AE
iuxta imaginem ALGE ad tempus per FM iuxta
rectangulum MH, habere eandem rationem, quam imago
ALGE ad imaginem rectangulum MH; & &longs;imiliter tem
pus per FM imagine rectangulum MN ad tempus per HI
iuxta imaginem HKI habet eandem rationem, quam ima
go NM ad imaginem HKI, ergo ex æquali tempus per AE
ad tempus per HI &longs;ecundùm imagines propo&longs;itas erit vt
imago ip&longs;a ALGE ad imaginem HKI. Quod &c.
Def.
4. Demum imagines &longs;int quæcunque, modò &longs;int ho
mogeneæ, ADFB, GHKL: Dico rur&longs;us inter &longs;e e&longs;&longs;e vt tem-Vel enim hæ ima
gines &longs;unt &longs;implices, hoc e&longs;t tantùm parallelogramm&ecedil;, aut
tantùm acuminatæ, & tunc &longs;upra o&longs;tendimus propo&longs;itum,
quemadmodum etiam &longs;i vna acuminata, altera parallelo
gramma; vel non &longs;unt huiu&longs;modi & componentur ex illis.
Sint ergo in imagine ADFB partes ab æquidi&longs;tantibus di
&longs;tinctæ ADEN, OFB acuminatæ & NEFO paralellogram.
mum, erunt hæ procul dubio inter &longs;e, totique imagini ho
mogeneæ; &longs;int pariter in alia imagine partes GHCM,
MCKL, per æquidi&longs;tantem MC di&longs;tinctæ inter &longs;e acumi
natæ, quæ itidem inter &longs;e, & imagini, cuius &longs;unt partes, ho
mogeneæ erunt. His acceptis, quoniam tempus per AN
iuxta imaginem ADEN acuminatam ad tempus per HC
iuxta aliam imaginem item acuminatam HGMC, habet
eandem rationem, ac imago ADEN ad
&longs;imiliter tempus per HC iuxta imaginem GHCM ad tem
pus per CK iuxta imaginem acuminatam MCKL e&longs;t vt
illa ad hanc imaginem; componendo, inde per conuer&longs;io
nem rationis, & conuertendo, tempus per HC &longs;ecundùm
imaginem GHCM ad tempora &longs;imul per HC, CK,
imagines GHCM, MCKL, hoc e&longs;t ad tempus per HK iux
ta imaginem GHKL habebit
go GHCM ad imaginem GHCL; & ideo ex æquali tem
pus per AN, cuius imago ADEN, ad tempus per HK, iux
ta imaginem GHKL, erit in eadem ratione, in qua e&longs;t ima
go ADEN ad imaginem GHKL. Præterea tempus per
AN iuxta imaginem ADEN ad idem ip&longs;um tempus habet
eandem rationem, quam imago ADEN ad eandem ip&longs;am;
tempus per NO iuxta imaginem rectangulum NEPO ad
tempus prædictum per AN e&longs;t in eadem ratione
NEPO ad ADEN, & &longs;imiliter tempus per OB iuxta ima
ginem OPFB habet ad tempus per AN eandem rationem,
ac imago OPFB ad imaginem &longs;æpè dictam ADEN;
lem. 18. Toric. in lib.
de dim: parabolæ, erunt tria
OPFB, hoc e&longs;t erit tempus per AB iuxta imaginem ADFB
ad &longs;imul tria tempora per AN iuxta eandem imaginem
ADEN, vt imago ADFB ad triplum imaginis ADEN, &
cum tria æqualia tempora per AN ad vnicum ex illis &longs;it
vt triplum imaginis ADEN ad vnicam imaginem; &longs;equi
tur ex æquali tempus per AB ad tempus per AN iuxt&atail;
imaginem ADEN habere eandem rationem, quam imago
ADFB ad imaginem ADEN: & o&longs;ten&longs;um fuit tempus per
AN iuxta imaginem ADEN ad tempus per HK iuxta
imaginem GHKL habere eandem rationem, quam imago
ADEN ad imaginem GHKL, ergo rur&longs;us, & tandem ex
æquali, tempus per AB iuxta imaginem ADFB ad
per HK iuxta imaginem GHKL habebit eandem
quam imago ADFB ad imaginem GHKL. Quod &c.
parte huius.
huius.
tertia ad quartam, item alia prima ad aliam &longs;ecundam vt
alia tertia ad aliam quartam, & &longs;ic vlteriùs quoad vi&longs;u&mtail;
fuerit, &longs;int præterea omnes primæ, item omnes tertiæ inter&longs;e
æquales, con&longs;tat, inquam, primarum vnam ad omnes &longs;ecun
das habere eandem rationem, ac vna tertiarum ad omnes
quartas.
SSpatia, quæ curruntur iuxta qua&longs;cunque homogeneas
gines. Sint primùm motus æquabiles, curraturque &longs;pa
tium AB iuxta imaginem velocitatum, quæ rectangulum
erit ILMK, &longs;patium verò DE tran&longs;igatur iuxta imagine&mtail;
prædictæ homogeneam rectangulum FHNG (nam erunt
erit vt velocitas in&longs;tanti I ad velocitatem mobilis in&longs;tanti
F) Dico &longs;patium AB ad DE e&longs;&longs;e vt imago rectangulu&mtail;
ILMK ad imaginem rectangulum FHNG. Componuntur
ip&longs;a illa rectangula ex ratione altitudinum IK ad FG, & ex
ea ba&longs;ium IL ad FH; verùm ex ij&longs;dem, ea nempe
IK ad FG, atque ea velocitatum IL ad FH componitur
etiam ratio &longs;patiorum AB ad DE, ergo ip&longs;a &longs;patia erunt vt
propo&longs;it&ecedil; imagines.
Dif.
æquabili.
2. Sint nunc motus iuxta imagines, quarum altera acu
minata, altera rectangulum &longs;it. Dico rur&longs;us &longs;patium AB,
quod curritur iuxta imaginem ABCD ad &longs;patium DE,
quod curritur iuxta alteram imaginem, e&longs;&longs;e vt imago
ABCD ad imaginem PHNG. Ni&longs;i ita &longs;it, erit alia magni
tudo Y maior, vel minor imagine ABCD, quæ quidem ad
alteram imaginem HPGN habebit eandem rationem,
&longs;patium AB ad DE. Sit primùm maior exce&longs;&longs;u Z. Cir
cum&longs;cribatur; vt egimus in &longs;ecunda parte primæ huius, fi
gura imagini ABCD con&longs;tans ex rectangulis æquè altis,
excedatque imaginem ABCD exce&longs;&longs;u minori, quam Z; &longs;it
ergo circum&longs;cripta illa AE, HF, IG, KG, quam primò fa
cilè o&longs;tendemus minorem magnitudine Y; nam hæc exce&longs;
&longs;u magis di&longs;tat ab imagine, quàm circum&longs;cripta illa. Præ
terea &longs;i intelligantur tot motus æquabiles, quot &longs;unt
gula
HI, IK, KD iuxta deinceps imagines ip&longs;a rectangula AE,
HF, IG, KC inter&longs;e, & propo&longs;itis imaginibus homogeneas,
velocitates, quibus ijdem motus con&longs;iderarentur, forent
HE, IF, KG, DC, nimirum maximæ imaginum ABEH,
HEFI, IFGK, KGCD; Cumque ita &longs;it, longiora &longs;patia cur
rerentur iuxta imagines rectangula circum&longs;cripta, quam
ij&longs;dem temporibus, imaginibu&longs;que po&longs;tremis, hoc e&longs;t
tempore AD iuxta imaginem ABCD; obidque &longs;patium
AB ad DE, &longs;eu magnitudo Y ad imaginem HPGN habe-
&longs;eu quam circum&longs;cripta figura AE, HF, IG, KC ad ean
dem imaginem HPGN; quare Y, quæ priùs o&longs;ten&longs;a fuit
maior, nunc reperitur minor eadem circum&longs;cripta, quod
cum fieri nequeat, impo&longs;&longs;ibile etiam e&longs;t magnitudinem Y
maiorem e&longs;&longs;e magnitudine imaginis ABCD. Sit ergo mi
nor, &longs;i etiam fieri pote&longs;t, & defectus ip&longs;ius Y &longs;upra ABCD
&longs;it Z. In&longs;cribatur imagini figura ex rectangulis æquealtis, vt
nempe deficiat ab imagine defectu minori Z; &longs;ic enim ip&longs;a
in&longs;cripta, quæ &longs;it AB, IE, KF, DG erit magnitudine pro
pinquior imagini ABCD, quàm Y, ideoque Y minor erit
dicta in&longs;cripta figura. Deinde, quoniam, &longs;i ponantur mo
tus æquabiles, quorum imagines rect angula in&longs;cripta HB,
IE, KF, DG, quæque inter &longs;e, & propo&longs;itis imaginibus &longs;int
homogeneæ; velocitates, quibus efficerentur dicti motus,
e&longs;&longs;ent AB, IE, KF, DG, minimæ &longs;cilicet imaginum ABEH
HEFI, IFGK. KGCD, & ideo &longs;patia, quæ percurrerentur
temporibus HA, HI, IK, KD imaginibus illis, maiora e&longs;
&longs;ent, quàm quæ ij&longs;dem temporibus tran&longs;igerentur iuxt&atail;
imagines prædictas rectangula circum&longs;cripta, hinc fit vt
&longs;patium AB ad DE, &longs;eu magnitudo Y ad imagine HPGN
habeat maiorem rationem, quàm in&longs;cripta figura ad ean
dem imaginem HPGN; quare Y, quæ minor erat in&longs;cripta
figura, modò re&longs;ultat maior, non ergo Y minor e&longs;&longs;e pote&longs;t
imagine ABCD, &longs;ed neque maior vt o&longs;tendimus, ergo &longs;pa
tium AB ad DE erit, vt imago ABCD ad imaginem
PHNG. Quod &c.
pr.
ius.
3. & 4. Si verò imagines acuminatæ &longs;int, aut demum
quæ cumque, eodem prorsùs modo, quo prima propo&longs;itio
ne, o&longs;tendemus hoc etiam propo&longs;itum, ergo patet omne
intentum.
po&longs;itiones, vtili&longs;&longs;imum e&longs;t ob&longs;eruare, quomodo liceat vti tem
poris in&longs;tantibus, non vt punctis prorsùs geometricis, &longs;ed vt
quantitatibus dicam minoribus quibu&longs;cunque datis. Hinc
oritur indiui&longs;ibilium method
faciliorem, ac &longs;i rigori geometrico penitus in&longs;i&longs;teremus, quam
quam eæ tamen difficiliores Geometras mihi magis decer&etail;
videantur.
SPatia, quæ curruntur iuxta quaslibet homogeneas ve
locitatum imagines, nectuntur ex rationibus tempo
rum, ac æquatricum.
Velocitates æquatrices duorum motuum, quorum ima
gines velocitatum &longs;int ABCD, EFHI ponantur AG, EL. Dico &longs;patia, &longs;eu ip&longs;as imagines componi ex ratione tem
porum AD ad EI; & ex ea æquatricum AE ad EL. Nam
&longs;i motus, qui e&longs;t iuxta imaginem ABCD per&longs;eueret velo
citate AG, e&longs;&longs;et quidem æquabilis, idemque &longs;patium illa
velocitate, & tempore AD percurreretur, ac &longs;ecundù&mtail;
imaginem ABCD; Itaque exi&longs;tente rectangulo DE, quod
e&longs;set imago velocitatum illius motus æquabilis, foret idem
æquale imagini ABCD (nam imagines ABCD, & DG
homogeneæ &longs;unt) eodem modo imago rectangulum VL
æquale e&longs;set imagini EFHI. Cum ergo duæ imagines re
ctangula DE, IL componantur ex rationibus temporum
AD ad EI, & ex ea æquatricum AG ad EL; ex ij&longs;de&mtail;
prorsùs rationibus etiam imagines propo&longs;itæ prædictis re
ctangulis æquales nectentur. Et ideo &longs;patia, quæ propo
&longs;itis imaginibus tran&longs;iguntur, quæque ip&longs;is proportionalia
bus æquatricum.
ex hibet, aplicetur rectangulum æquale propo&longs;itæ imagini ve
locitatum, fore vt latitudo eiu&longs;dem rect anguli, &longs;it velocitas
æquatrix propo&longs;itæ imaginis.
rint æquales, rationem &longs;patiorum e&longs;&longs;e eandem, quæ æquatri
cum, vel quæ temporum.
harum quælibet nectetur ex propo&longs;ita, & ex reliquis contra
riò &longs;umptis rationibus. Sit A ad B compo&longs;ita ex rationibus E
æd F; G ad H; & I ad K. Dico quamlibet ist arum puta G ad
K con&longs;tare ex rationibus A ad B, & ex reliquis reciprocè &longs;um
ptis F ad E, & I ad K. Vt E ad F, ita &longs;it A ad C, & vt D ad B
&longs;ic I ad K; erit C ad D, vt G ad H;
B ad D, &longs;iue K ad I. Quod &c.
TEmpora, quibus ab&longs;oluuntur duo motus componun
tur ex ratione &longs;patiorum, & ex reciproca æquatri
cum. Cum enim &longs;patia
& ex ea velocitatum æquatricum, &longs;equitur per prædictum
Lenima, quòd tempora nectantur ex rationibus &longs;patiorum,
& reciproca æquatricum.
æquales, e&longs;&longs;e tempora in reliqua ratione reciproca æquatri
cum, vel &longs;patiorum non reciproca.
ÆQuatrices velocitates componuntur ex rationibus
&longs;patiorum, & reciproca temporum.
Cum &longs;patia componantur ex rationibus temporum, &
velocitatum æquatricum, manife&longs;tum e&longs;t ex eodem Lem
mate, velocitates ip&longs;as necti ex rationibus &longs;patiorum, &
reciproca temporum.
&longs;umpta, vel vt &longs;patia, &longs;i altera ratio fuerit æqualitatis.
E F. VII.
SI in gene&longs;ibus homogeneis AEC, GFK exi&longs;tente AB
ad BC &longs;icut GI ad IK, habeat AE ad BD eandem ra-
vocentur inter &longs;e &longs;imiles, & ip&longs;æ gene&longs;es dicentur &longs;imilium
motuum; quod verò attinet ad rectas AE, BD, GF, IH apel
labimus applicatas ad homologa puncta A, B, G, I propor
tionales.
SI in imaginibus temporum homogeneis, applicatæ v
nius fuerint ad homologa puncta, proportionales ap
plicatis alterius imaginis, motus, quorum &longs;unt ip&longs;æ imagi
nes, &longs;imiles erunt.
Imagines temporum &longs;int &MLABC, &ONGIK, quæ
&longs;int homogeneæ, & cum GI ad IK &longs;it vt AB ad BC, habeat
quoque AL ad BM eandem rationem, ac GN ad IO. Di
co, motus, quorum &longs;unt illæ imagines temporum inter &longs;e &longs;i
miles e&longs;&longs;e.
Sint apud ip&longs;as imagines eorundem motuum gene&longs;es,
&longs;cilicet EAC, FGK inter&longs;e homogeneæ. Exi&longs;tente AL ad
BM, vt GN ad IO, erit conuertendo BM ad AL vt IO ad
GN; &longs;ed vt BM ad AL ita ob gene&longs;im EA ad DB, & vt IO
ad GN, &longs;ic FG ad HI. ergo EA ad DB e&longs;t vt FG ad HI, erat
autem vt AB ad BC ita etiam GI ad IK, ergo motus &longs;unt
&longs;imiles, & ip&longs;æ imagines &longs;imilium motuum.
SI in imaginibus velocitatum vnius, applicate fuerint ex
punctis homologè &longs;umptis proportionales applicatis
alterius imaginis, motus iuxta ip&longs;as imagines erunt &longs;imi
les, ideoque ip&longs;æ imagines &longs;imilium motuum.
Velocitatum imagines &longs;int ABCD, NPRT, &longs;itque AB
ad EF in eadem ratione, in qua NP ad TR; Dico exi&longs;tenti
bus etiam BF ad FC, vt PQ ad QR e&longs;&longs;e propo&longs;itas imagi
nes &longs;imilium motuum. Intelligantur eorundem motuum
tum ABFE ad EFCD. Sit &longs;imiliter Z
mentum NPQV ad VQRT, ducti&longs;que applicatis IM, QV,
manife&longs;tum e&longs;t, vt velocitas AB æqualis e&longs;t velocitati GH,
&longs;ic EF æqualem fore ip&longs;i IM; nam quia &longs;patium
iuxta imaginem ABFE ad &longs;patium tran&longs;actum imagine
EFCD e&longs;t vt illa ad hanc imaginem, nempe vt HI ad IK,
erit mobile in&longs;tanti F in puncto I, & ideo inibi erit veloci
tas eadem, quam habet mobile in&longs;tanti F, &longs;cilicet æquales
erunt EF, IM. Eodem modo erunt æquales QV,
&longs;unt etiam æquales NP, YZ, ergo &longs;icut &longs;e habet AB ad EF,
ita erit GH ad MI, & vt e&longs;t NP ad
Præterea concipiatur figura OPRSXO &longs;imilis ip&longs;i ABCD,
&longs;cilicet &longs;it CB ad PR vt AB ad OP, vel (cum &longs;int BF ad
FC ita PQ ad QR, vt EF ad homologam XQ, erit &longs;eg
mentum ABFE ad &longs;ibi &longs;imile &longs;egmentum OPQX in dupli
cata ratione laterum homologorum EF ad XQ, & item in
ad XQRS, &longs;ed cum etiam OPQX &longs;egmentum ad NPQV,
& XQRS ad &longs;egmentum VQRT &longs;int in eadem ratione
eiu&longs;dem QX ad QV, erit ex æquali &longs;egmentum ABFE ad
&longs;egmentum NPQV, vt &longs;egmentum EFCD ad VQRT, &
permutando, &longs;egmentum ABFE ad &longs;egmentum EFCD ha
bebit eandem rationem, ac &longs;egmentum NPQV ad VQRT
&longs;cilicet erit HI ad IK vt Z
ne&longs;ium applicatas vnius proportionales e&longs;&longs;e applicatis al
terius, quare &longs;imiles motus erunt, qui fiunt iuxta imagines
velocitatum propo&longs;itas.
SPatia, quæ curruntur &longs;imilibus motibus &longs;unt in ratione
compo&longs;ita temporum, & homologarum velocitatum,
interquas &longs;unt extremæ, aut primæ.
Imagines velocitatum &longs;imilium motuum &longs;int BCDE,
GMKI, & iuxta eas percurrantur &longs;patia A, F. Dico i&longs;ta com
poni ex rationibus temporum BE ad GI, & ex ea veloci
tatem extremarum ED ad IK. Fiat vt BE ad GI, ita BC
ad GH, intelligatur que GHLI figura &longs;imilis ip&longs;i BDE. Quo
niam &longs;patium A ad F, hoc e&longs;t imago BCDE ad imaginem
GMKI componitur ex ratione imaginis BCDE ad figu
ram &longs;ibi &longs;imilem GHLI, & ex ratione huius ad imaginem
GMKI: prior ratio e&longs;t duplicata homologorum lateru&mtail;
BE ad GI, &longs;eu e&longs;t compo&longs;ita ex BE ad GI, & ex huic &longs;imi
li ratione ED ad IL, & ratio altera, imaginis &longs;cilicet GHLI
ad imaginem GMKI e&longs;t, vt LI ad IK; ergo ex æquali ima
go BCDE ad imaginem GMKI, hoc e&longs;t &longs;patium A ad &longs;pa
tium F, componetur ex ratione temporum BE ad GI, & ex
rationibus ED ad LI, & IL ad IK, &longs;cilicet nectetur ex ra
tione BE ad GI, & ED ad IK, quæ po&longs;trema cum &longs;it ratio
velocitatum extremarum ED ad IK; con&longs;tat, quod propo
&longs;uimus, &longs;patia &longs;imilium motuum componi ex ratione tem
porum, & ex ratione homologarum velocitatum, hoc e&longs;t
extremarum.
vt extremæ, vel &longs;ummæ velocitates, & contra, &longs;i i&longs;tæ æquales
&longs;int, erunt &longs;patia vt tempora.
porum & ex ea velocitatum &longs;ummarum, &longs;eu earum, quæ
ad in&longs;tantia &longs;imiliter &longs;umpta in rectis BE, GI, const at ex
lem: infra cor.
bus &longs;patiorum &longs;imilium motuum, & ex recìproca dictarumEx eadem ratione patet e&longs;&longs;e velocitates &longs;um
mas, vel homologas vti diximus in ratione compo&longs;ita dicto
rum &longs;patiorum, & ip&longs;orum temporum.
reliqua tantùm computanda erit.
pror&longs;us libera, aut &longs;uper planis inclinatis ad horizonte&mtail;:
nec accidit veritates iam patefact as huc rur&longs;us lectoris tadio
afferre, &longs;ed libcat potius, rationem metiendarum imaginum,
quamuis longitudine immen&longs;arum, no&longs;tra methodo exponere.
SInt inter binas parallelas AB, GH, et IK, PQ planæ fi
guræ ABHG, IKQP, & in altera earum ducta altitudi
ne RV, &longs;int inter &longs;e ip&longs;æ figuræ talis naturæ, vt cum &longs;it
GABH ad &longs;egmentum EABF factum per æquidi&longs;tantem
ip&longs;i GH &longs;icut VR ad RT, verificetur &longs;emper (ducta æqui
di&longs;tanti NTO ip&longs;i PQ) e&longs;&longs;e GH ad EF vt reciprocè NO ad
finitam, cum AB fuerit punctum, & ideo &longs;imul con&longs;tat figu
ram IPQK immen&longs;am e&longs;&longs;e longitudine versùs K aut I, aut
vtrinque, &longs;i nempe producerentur nunquam co
QP, IK.
REctangulum &longs;ub altitudine, & ba&longs;i vnius auuer&longs;arum
ad ip&longs;am auuer&longs;am figuram, eandem habet
ac altera auuer&longs;a figura ad rectangulum ex ba&longs;i in altitudi
nem eiu&longs;dem huius figuræ.
Sint auuer&longs;æ figuræ ACB, GFDEG.
Dico rectangu
lum DF in DE ad figuram GFDEG, eandem habere ratio
nem ac figura ACBA ad rectangulum AB in BC. Sint pri
mùm ABC, FDE anguli recti, & ducta qualibet HI paral
lela BC, &longs;it BAC ad HIA vt DF ad KF, erit ob naturam
auuer&longs;arum KL ad DE vt BC ad HI; itaque &longs;i ponatur e&longs;&longs;e
quidam motus ab F in D iuxta imaginem
erit GFDEG imago temporis eiu&longs;dem motus; nam imago
BAC ad imaginem HIA e&longs;t vt &longs;patium DF ad &longs;patium FK
& velocitas BC ad Sit etiam alius motus, &longs;ed æquabilis, cuius imago velocita
tum æqualis &longs;it, & homogenea ip&longs;i BAC, rectangulum
pe
concipiaturque rectangulum FD in DN, erit hoc imago
temporis dicti motus æquabilis, homogenea, & æqualis
imagini GFDEG; nam
FD in DN rectangulum componuntur ex rationibus &longs;pa
tiorum, hoc e&longs;t imaginum velocitatum inter&longs;e æqualium,
ABM, ACB, & reciproca æquatricum pariter æqualium
BM, BM. Cum igitur rectangulum FD in DN æquale &longs;it
imagini, &longs;eu figuræ GFDEG, habebit eadem figur&atail;
GFDEG ad rectangulum FD in DE eandem rationem,
quam DN ad DE, hoc e&longs;t quam BC ad BM, &longs;eu quam re
ctangulum AB in BC ad rectangulum AB in BM, aut ad ei
æqualem figuram ABC; & conuertendo, manife&longs;tum e&longs;t
quod propo&longs;uimus, nempe rectangulum FD in DE ad fi
guram GFDEG habere eandem
primo loco.
pr.
ius.
2. Si verò propo&longs;itæ figuræ &longs;int quæcunque auuer&longs;æ
DAE, QPLMQ poterunt hæ reuocari ad qua&longs;dam alias
FKG, RSZX, quæ &longs;int inter ea&longs;dem parallelas, queis com
prehenduntur propo&longs;itæ figuræ, ad eo vt exi&longs;tentibus re
ctis angulis KFG, RXZ &longs;int ip&longs;æ binæ figuræ ab ij&longs;dem pa
rallelis interceptæ. inter &longs;e æqualiter analogæ hoc e&longs;t du
ctis æquidi&longs;tantibus, vt vi&longs;um fuerit IHBC, VTNO, &longs;int
&longs;emper interiectæ lineæ IH, BC, & VT, NO æquales: hoc
modo non tantùm liquet figuras FKG, DAE, nec no&ntail;
RSZX, PQML æquales inter &longs;e e&longs;&longs;e, verùm etiam FKG ad
IKH e&longs;&longs;e in eadem ratione, in qua QPLMQ ad QPNOQ,
quamobrem ex prima parte, rectangulum ZX in RM ad
figuram SRXZS, hoc e&longs;t rectangulum LM in altitudinem
figuræ QPLMQ ad hanc ip&longs;am figuram habebit eandem
rationem, quam figura FKG ad rectangulum KF in FG,
vel quam figura DAE ad rectangulum DE in altitudinem
eiu&longs;dem huius figuræ DAE; quo circa con&longs;tat omne pro
po&longs;itum.
&longs;it versùs G, & ob id manife&longs;tum e&longs;t, quòd quamuis aliqu&atail;
figura &longs;it &longs;inè fiue longa, non ideo &longs;emper magnitudinem ha
bet infinitam. Et &longs;imul illud con&longs;tat, vbi vna auuer&longs;arum, &longs;eu
vbi imago velocitatum, aut temporis &longs;it magnitudine termi
nata, etiam altera auuer&longs;arum, vel imaginum erit huiu&longs;
modi &c.
pr.
IN quouis parallelogrammo BD &longs;int deinceps diagona
les AGC, AHC, AIC, ALC, aliæque numerò infinitæ,
ita vt acta quælibet recta EF parallela BA &longs;ec
gonales in punctis G, L, H, I, &longs;it &longs;emper DA ad AF, vt CD,
aut EF ad FG; quadratum ex DA ad quadratum AF vt
EF ad FH; cubus ex DA ad cubum ex AF vt EF ad FI;
quadroquadratum ex DA ad quadroquadratum ex AF
vt EF ad FL; & &longs;ic continuò procedendo per infinitas ex
ordine pote&longs;tates: Stephanus de Angelis Author &longs;ubtilis,
ac celeberrimus, libro &longs;uo infin. parabolarum vocat trian
gulum rectilineum ABC parabolam primam, BAHC &longs;e
cundam; tertiam BAIC, quartam BALC, & ita in infini
tum: His definitis docet ex Cauallerio parallelogrammum
BD ad quancunque dictarum parabolarum &longs;ibi in&longs;cripta
rum e&longs;&longs;e vt numerus, vel exponens parabolæ vnitate au
ctus ad ip&longs;um exponentem, &longs;iue numerum parabol&ecedil;, qua
re ad primam habebit ip&longs;um parallelogrammum eandem
rationem, ac 2 ad 1; ad &longs;ecundam vt 3 ad 2; ad tertiam vt
4 ad 3, & ita deinceps de reliquis; itaque per conuer&longs;io
nem rationis habebit ip&longs;um parallelogrammum ad exce&longs;
&longs;um illius &longs;upra quancunque parabolarum dictarum, &longs;cili
cet ad trilineum primum AGCD eandem rationem, quam
2 ad 1, ad &longs;ecundum quam 3 ad 1, & &longs;ic deinceps quam
numerus trilinei vnitate auctus ad ip&longs;am vnitatem. Sed
e&longs;t etiam admonendum verticem dictarum parabolarum
e&longs;&longs;e punctum A, & per con&longs;equens AB diametrum, & BC
ordinatim aplicatam, &longs;eu ba&longs;im.
II&longs;dem adhuc manentibus, idem de Angelis mon&longs;trat eo
dem illo tractatu pr. 3. &longs;i quæcunque ex dictis parabo
lis &longs;ecta &longs;it qualibet recta parallela ba&longs;i BC, e&longs;&longs;e parabolam
ad re&longs;ectam portionem ver&longs;us verticem, vt pote&longs;tas ba&longs;is,
cuius exponens e&longs;t numerus parabolæ vnitate auctus ad
&longs;imilem pote&longs;tatem ex ba&longs;i re&longs;ectæ portionis; itaque i&ntail;
prima parabola e&longs;t vt quadratum ad quadratum, in &longs;ecun
da vt cubus ad cubum, & &longs;ic de cæteris. Similiter &longs;i &longs;ece
tur quodlibet ex infinitis trilineis linea GF ba&longs;i CD paral
lela, erit trilineum ad &longs;uperius &longs;ui &longs;egmentum vt pote&longs;tas
ex DA, cuius exponens e&longs;t numerus trilinei vnitate auctus
ad &longs;imilem pote&longs;tatem ex AF. quare trilineum primu&mtail;
CAD ad GAF erit vt quadratum ex DA ad quadratum
ex FA, &longs;ecundum CHAD ad &longs;egmentum HAF vt cubus
ad cubum, & ita in cæteris eodem ordine.
SIt modò ACD angulus rectus, & linea FE talis naturæ,
vt deductis ad libitum rectis AF, BE parallelis ip&longs;i
CD, pote&longs;tas ex CA ad &longs;imilem pote&longs;tatem ex CB &longs;it reci
procè vt alia quædam pote&longs;tas ex BE ad &longs;imilem huic po
te&longs;tatem ex AF; patet rectas CA, CD nondum iungi cum
EF, quamuis in immen&longs;um vnà producerentur. Ab hoc
proprietate VValli&longs;ius & Fermatius &longs;ubtili&longs;&longs;imi authores
vocauerunt curuam FE nouam hyperbolam, & eius a&longs;
&longs;ymptotos AC, CD. Omnes huiu&longs;modi hyperbolæ, quæ
infinitæ numero &longs;unt, terminantur ad vnam partem ma
gnitudine, cum hyperbola
vtranque partem magnitudine infinita. Quod ergo exi
mium e&longs;t, o&longs;tenderunt ip&longs;i authores rectangulum FA i&ntail;
fine longum, eandem habere rationem, quam differentia
exponentium pote&longs;tatum hyperbolæ ad exponentem po
te&longs;tatis minoris. Quare &longs;i in hyperbola &longs;it vt cubus CB
ad cubum CA ita quadratum AF ad quadratum BE, erit
prædictum rectangulum CA in AF dimidium Spatij &longs;inè
fine producti A & FA; at &longs;i quadratum CB ad quadratum
CA &longs;it vt recta AF ad rectam BE, rectangulum ip&longs;um CA
in AF æquale erit &longs;patio A & FA, quòd &longs;i pote&longs;tas CA vel
CB non fuerit altior pote&longs;tate ex BE, vel AF, tunc ip&longs;um
illud &longs;patium, infinitum quoque erit magnitudine, etenim
nullus exce&longs;&longs;us exponentis prædictæ pote&longs;tatis ex CA &longs;u
pra exponentem pote&longs;tatis BE, habet ad numerum expo
nentis pote&longs;tatis BE rationem infinitam.
SVpradictum propo&longs;itum habetur in commercio epi
&longs;tolico Ioannis Valli&longs;ij Epi&longs;tola quarta, quem libellum
vnà cum alijs docti&longs;&longs;imis &longs;uis operibus Vincentius Viuia
nus ingens æui no&longs;tri Geometra, antequam &longs;umma cu&mtail;
humanitate mi&longs;i&longs;&longs;et, eidem ip&longs;i quadraturam vnius ex di
ctis hyperbolis ex no&longs;tris principijs deductam, ac excogi
tatam, indicauimus. Cum verò po&longs;tea nobis eueni&longs;&longs;et
vniuer&longs;aliorem ad alias hyperbolas (&longs;emper communi ex
cepta) accomodatam reperij&longs;&longs;e, huc debemus afferre, pri
mùm vt quendam fructum &longs;cientiæ huius; deinde cum di
ctorum authorum ip&longs;am propo&longs;itionis demon&longs;trationem
non habuerimus, & demum quia ip&longs;arum hyperbolarum
men&longs;ura, ac quadratura in aquarum rationibus erunt po
ti&longs;&longs;imum ex v&longs;u. Sit igitur BC vna ex infinitis hyperbolis,
quarum a&longs;lymptoti AE, EL; Sint etiam quæcunque apli
catæ AB, DC alsymptoto EL æquidi&longs;tantes, & habeat
DE ad EA eandem rationem v. g. quam cubus ex AB ad
cubum DC. Patet &longs;i proponeretur illi auuer&longs;a figur&atail;
FGK, e&longs;&longs;etque AE ad DE vt figura GFK ad figuram IHK
e&longs;&longs;e etiam FG ad IH vt DC ad AB, e&longs;t autem cubus ex
DC ad cubum ex AB vt AE ad ED; ergo etiam figur&atail;
FGK ad IHK (&longs;unt enim FG, IH parallel&ecedil;) habebit ean
dem rationem, ac cubus ex FG ad cubum ex IH: Itaqu&etail;
GFK erit comunis parabola, hoc e&longs;t quadratica, &longs;eu
da in &longs;erie infinitarum parabolarum, & ob id eadem GFK
parabola ad rectangulum GF in FK erit vt 2 ad 3, in qua
ratione &longs;e habebit quoque rectangulum BA in AE ad &longs;pa
tium infinitè longum & BM, et erit vt 2 ad 1; &longs;cilicet vt ex
ce&longs;&longs;us exponentis maioris pote&longs;tatis, quæ cubica e&longs;t, &longs;uper
numerum exponentis, qui hoc ca&longs;u e&longs;t tantùm vnitas ra
dicis, e&longs;t ad hunc ip&longs;um exponentem, &longs;eu vnitatem lineæ
indicantem, quod concordat cum propo&longs;ita dictoru&mtail;
authorum.
guræ.
SIt etiam cubus ex DE ad cubum ex AE, &longs;icut quadra
to quadratum AB ad quadroquadratum DC, & rur
&longs;us propo&longs;ita GKF auer&longs;a huius hyperbolæ: patet &longs;i &longs;it AE
ad DE vt figura GFK ad figuram IKH, e&longs;&longs;e etiam FG ad
IH vt DC ad AB; cumque &longs;it cubus ex AE ad cubum ex
DE &longs;icut quadroquadratum ex DC ad
ex AB, erit etiam quadroquadratum ex FG ad quadro
quadratum ex IH, vt cubus ex AE ad cubum ex DE; &longs;i
igitur intelligatur quædam ratio, quæ &longs;it &longs;ubduodecupla
tam rationis quadroquadratorum quàm huic &longs;imilis cu
borum prædictorum, erit porrò FG ad IH triplicata, &
AE ad ED quadruplicata eiu&longs;dem dictæ &longs;ubduodecuplæ;
quamobrem etiam ratio figuræ GFK ad
e&longs;&longs;e debet vt AE ad ED, erit quadruplicata eiu&longs;dem &longs;ub
duodecuplæ: & ideò &longs;i ponamus IK ad KI in ratione
&longs;it &longs;emper cubus ex FK ad cubum ex KI &longs;icut GF ad IH, &
hoc modo eadem illa figura erit trilineum tertium, &longs;eu cu
bicum, ex quo ergo &longs;equitur, GFK ad HIK &longs;it in eadem ra
tione, in qua quadroquadratum ex FK ad quadroqua
dratum ex KI, hoc e&longs;t &longs;it vt AE ad ED; &longs;equiturque etiam
ob hoc figuram GFK &longs;ubquadruplam e&longs;le circum&longs;cripti
rectanguli GF in FK; e&longs;t autem vt trilineum GFK ad
gulum GF in FK circum&longs;criptum, &longs;ic rectangulum ABME
ad auuer&longs;am eidem trilineo figuram AB & EA, ergo re
ctangulum ABME &longs;ubquadruplum erit eiu&longs;dem figuræ
AB & EA longitudinis infinitæ, quare ip&longs;um rectangulum
erit &longs;ubtriplum portionis & BM & longitudinis pariter im
men&longs;æ. Cum ita &longs;it, con&longs;tat exemplo hoc quoque,
illam rationem e&longs;&longs;e exce&longs;&longs;um maioris exponentis &longs;upr&atail;
minorem exponentem ad hoc ip&longs;um, dictarum
hyperbolæ.
SVperior demon&longs;tratio effecta fui&longs;&longs;et ampli&longs;&longs;ima, &longs;i pr&ecedil;
ponere volui&longs;&longs;emus
neris parabolarum, & trilineorum, verùm cum i&longs;ta pars
&longs;it plenè tradita, vt videre e&longs;t quinto libro infinitarum pa
rabolarum eiu&longs;dem de Angelis, &longs;atius ideo duximus qua
draturam hyperbolarum à VVali&longs;io, & Fermatio acuti&longs;&longs;i
mis illis viris propo&longs;itam omnino veram admittere, vt indè
eam parabolarum & trilineorumvniuer&longs;alem, quam adhuc
ab alijs non habemus, facillimè, compendiosèque depro
meremus. Hanc igitur ita proponimus vt &longs;ubinde o&longs;ten
damus.
Si &longs;imiles pote&longs;tates applicatarum fuerint in eadem ra
tione, ac &longs;unt inter&longs;e pote&longs;tates quædam aliæ, & eiu&longs;dem
gradus diametrorum ab ip&longs;is applicatis ab&longs;ci&longs;&longs;arum v&longs;que
lum ad parabolam &longs;ibi in&longs;criptam vt aggregatum
tium
pote&longs;tatum parabolæ; & ad trilineum vt aggregatum ex
ponentium pote&longs;tatum trilinei ad exponentem inferioris
pote&longs;tatis eiu&longs;demmet trilinei. Sic enim in expo&longs;ita figu
ra prædicta, &longs;i e&longs;&longs;et quadratum ex FG ad quadratum ex
IH, &longs;icut cubus ex FK ad cubum ex IH, e&longs;&longs;et rectangulum
GF in FK ad figuram GFK (quæ tunc foret trilineum, vt
5 ad 2; nam vbi pote&longs;tas ab&longs;ci&longs;&longs;arum maior e&longs;t illa applica. tarum e&longs;t &longs;emper GF trilineum.
Simili modo, &longs;i &longs;it vt qua
dratum ex FK ad quadratum ex KI ita cubocubus ex FG
ad cubocubum ex IH; hoc e&longs;t &longs;i &longs;it cubus ex FG ad
ex IH, vt linea FK ad KI (tolluntur enim vtrinque ex &longs;imi
libus &longs;imiles rationes) erit &longs;igura GFK parabola, ad quam
&longs;ibi circum&longs;criptum rectangulum eandem habebit
quam 4 ad 3, & &longs;ic dicendum erit de omnibus alijs para
bolis atque trilineis.
VErùm vt propo&longs;itum o&longs;tendamus, e&longs;to quælibet ex
parabolis GFK, nimirum quadratocubus ex FG ad
quadratocubum ex IH habeat eandem rationem, qua&mtail;
cubus ex FK ad cubum ex IK. Demon&longs;tro, rectangulum
GF in FK habere eandem rationem ad parabolam GFK,
quam aggregatum exponentium 8 ad maiorem exponen
tem 5. Primùm, quam rationem habet rectangulum GF in
FK ad parabolam GFK, eandem habebit rectangulum HI
in IK ad parabolam HIK (hoc enim demon&longs;trabimus in
frà) permutandoque, erit rectangulum GF in FK ad re
ctangulum HI in IK, vt parabola GFK ad parabolam HIK;
componuntur verò illa rectangula ex rationibus GF ad
IH, & FK ad IK, ergo etiam parabola ad parabolam com-
exponentibus po&longs;&longs;unt con&longs;iderari quindecim rationes in
ter &longs;e &longs;imiles, ex quibus con&longs;tet tam ratio dictorum cubo
rum, quàm huic &longs;imilis altera quadratocuborum, & tunc
GF ad IH erit triplicata, et FK ad KI quintuplicata
&longs;ubquindecuplæ rationis, quæ &longs;it A ad B; ergo &longs;imul ad
ditis ij&longs;dem rationibus, quintuplicata &longs;cilicet, & triplicata
exiliet ratio octuplicata ip&longs;ius A ad B; proptereaque pa
rabola GFK ad HIK, &longs;eu &longs;i con&longs;ideremus figuram & BAEL
auuer&longs;am parabolæ GFK, ita vt AE ad ED &longs;it vt para
bola GFK ad
plicata eiu&longs;dem A ad B; & cum &longs;it ob naturam
FG ad HI vt DC ad AB; erit DC ad AB triplicata
rationis A ad B, qnare vt cubus AE ad cubum DE, it&atail;
quadratocubocubus DC ad quadratocubocubum ex
AB: rectangulum igitur ABME ad &longs;patium hyperbolicum
infin
uer&longs;um &longs;pa ium & BAE & vt 5 ad 8, in qua nempe ratio
ne debet e&longs;&longs;e parabola GF
Quod &c.
applicatarum, quoties e&longs;t numerus exponentis pote&longs;tatis ab
&longs;ci&longs;&longs;arum eiu&longs;dem parabolæ, e&longs;&longs;e ip&longs;am parabolam ad &longs;ui por
tionem in tam multiplicata ratione A ad B, ac e&longs;t numerus
aggregati exponentium ambarum pote&longs;tatum parabola. Nam
cum e&longs;&longs;et quadratocubus ex FG ad quadratocubum ex IH, &longs;i
cut cubus ex FK ad cubum ex IK, propo&longs;ita in&longs;uper e&longs;&longs;et A ad
B. &longs;ubquindecupla alterius dictarum &longs;imilium rationum ex
pote&longs;t atibus parabola, o&longs;ten&longs;um fuit rationem A ad B &longs;ubtri
plicatam ip&longs;ius GF ad IH, & &longs;ubquintuplicatam alterius FK
ad KI, & tandem o&longs;tendimus parabolam GFK ad portionem
idem omnino diceretur &longs;i figura GFK trilineum e&longs;&longs;et. Ratio
autem A ad B dicetur impo&longs;terum logarithmica pote&longs;tatum
parabolæ, &longs;eu trilinei, aut hyperbolæ.
REliquum e&longs;t vt o&longs;tendamus, parabolam GFK ad
portionem HIK e&longs;&longs;e vt rectangulum GF ad rectan
gulum HI in IK, &longs;cilicet e&longs;&longs;e in ratione compo&longs;ita ha&longs;ium,
& altitudinum parabolarum, quod nempe &longs;ic o&longs;tendetur,
Sit vt &longs;upra FGK parabola, eiu&longs;que portio IHK; exi&longs;tenti
bus verò applicatis FG, IH, fiat EG ad IE vt FK ad KI, &longs;it
que IE ba&longs;is, et K vertex parabol&ecedil; IEK &longs;imilis ip&longs;i GFK pa
tet propter &longs;imilitudinem figurarum, e&longs;&longs;e parabolam GFK
ad parabolam IEK in eadem duplicata ratione FG ad IE,
in qua nempe e&longs;t rectangulum GF in FK ad &longs;ibi &longs;imile re
ctangulum EI in IK, ob idque rectangulum GF in FK ad
rectangulum EI in IK, cum &longs;int inter&longs;e vt parabola GFK ad
parabolam EIK, hæc verò parabola ad ip&longs;am IHK habeat
eandem rationem, ac IE ad IH; &longs;eu ob eandem altitudinem
IK vt rectangulum EI in IK ad rectangulum HI in IK, erit
ex æquali parabola GFK ad parabolam HIK vt rectangu
lum GF in FK ad rectangulum HI in IK. Quod &c.
IN quacunque hyperbola (excepta &longs;emper conica) cu
ius a&longs;&longs;ymptoti EA, EM, &longs;i &longs;it pote&longs;tas applicatarum DC
AB altior pote&longs;tate ab&longs;ci&longs;&longs;arum AE, ED (&longs;ic enim finit&atail;
erit magnitudine &longs;ecundum eam a&longs;&longs;ymptoton, quæ appli
catis parallela e&longs;t) &longs;patium ip&longs;um hyperbolæ & BAE &
ad &longs;ui portionem & CDE & habebit eandem rationem, ac
rectangulum BAE ad rectangulum CDE, &longs;eu (a&longs;&longs;umpta
pote&longs;tas ex A, cuius exponens e&longs;t differentia
pote&longs;tatum hyperbolæ ad &longs;imilem pote&longs;tatem ex B.
QVam rationem habet rectangulum BAE ad &longs;patium
& BAE &, eandem habet rectangulum CDE ad
&longs;patium & CDE, & permutando erit rectangu
lum BAE ad CDE, &longs;icut &longs;patium & BAE & ad &longs;patiu&mtail;
& CDE &; &longs;i igitur in eadem propo&longs;ita hyperbola &longs;it po
te&longs;tas applicatarum DC, AB quintuplicata ip&longs;ius A ad B,
& AE ad ED &longs;eptuplicata &longs;it eiu&longs;dem; erit &longs;eptuplicat&atail;
applicatarum in eadem ratione, ac quintuplicata ab&longs;ci&longs;&longs;a
rum; &longs;cilicet quadratoquadratocubus ex DC ad &longs;imilem
pote&longs;tatem ex AB erit vt quadratocubus ex AE ad qua
dratocubum ex DE, eritque &longs;ic maior pote&longs;tas applicata
rum, atque adeo componetur rectangulum EAB ad EDC
ex &longs;eptuplicata ip&longs;ius A ad B, qualis e&longs;t AE ad ED, & &longs;ub
quintuplicata eiu&longs;dem A ad B, quæ e&longs;t AB ad DC; nimi
rùm erit rectangulum EAB ad EDC in duplicata tantum
ratione ip&longs;ius A ad B: quare &longs;patium & BAE & ad id
& CDE &, quæ &longs;unt inter &longs;e, vt ip&longs;a rectangula, erit vt po
te&longs;tas ex A, cuius exponens e&longs;t differentia exponentium &
S pote&longs;tatum hyperbolæ ad &longs;imilem pote&longs;tatem ex B. Quod &c.
SIab exponente pote&longs;tatis applicatarum hyperbol&ecedil; de
trahatur exponens minoris pote&longs;tatis ab&longs;ci&longs;&longs;arum, po
te&longs;tas reliqui exponetis erit applicatarum auuer&longs;æ figuræ,
in ab&longs;ci&longs;&longs;is verò ade&longs;t vtrobique eadem pote&longs;tas. Itaque
cum in &longs;uperiori hyperbola re&longs;idui exponentis pote&longs;tas
catarum quadratica, & ab&longs;ci&longs;&longs;arum quadratocubica.
ESto rur&longs;us hyperbola & BAE &, et &longs;icut dictum e&longs;t
AE ad ED &longs;it in &longs;eptuplicata ratione logarithmicæ
rationis A ad B, at DC ad AB in quintuplicata, videlicet
quadratocubus ex AE ad quadratocubum ex DE eandem
habeat rationem, ac quadratoquadratocubus ex DC ad
&longs;imilem pote&longs;tatem ex AB; Dico in auuer&longs;a figura pote&longs;ta
tem aplicatarum e&longs;&longs;e quadratum, cuius
ferentia exponentium pote&longs;tatum hyperbolæ; pote&longs;tatem
verò ab&longs;ci&longs;&longs;arum eandem e&longs;&longs;e, ab&longs;ci&longs;&longs;arum eiu&longs;dem hyper
bolæ. Sit vt &longs;upra FK ad KI vt hyperbola & BAE & ad
& CDE &, hoc e&longs;t, &longs;it vt pote&longs;tas ex A, cuius exponens
e&longs;t differentia exponentium pote&longs;tatum hyperbolæ ad &longs;i
milem pote&longs;tatem ex B, & ideo FK ad KI erit duplicata ip
&longs;ius A ad B, &longs;ed DC ad AB eiu&longs;dem illius logarithmicæ
quintuplicata; e&longs;tque in hac eadem ratione etiam GF ad
IH; ergo cum duplicata huius &longs;it &longs;imilis quintuplicatæ KF
ad KI (nam vtraque ratio continet decies A ad B) pater,
quadratum ex FG ad quadratum ex IH e&longs;&longs;e eam pote&longs;ta
tem, quam propo&longs;uimus euenire in applicatis auuer&longs;æ, cum
aliàs in ab&longs;ci&longs;&longs;is &longs;it vtrobique pote&longs;tas eadem, nempe qua
dratocubi. Quod &c.
&longs;cilicet in hyperbola & BAE & (quæ tunc e&longs;t &longs;emper magnitu
dine finita iuxta a&longs;symptoton EM &) pote&longs;tatem
qua pro exponente habet &longs;ummam exponentium pote&longs;tatum
parabolæ, aut trilinei; nam cum eßet in trilineo pracedenti
ex FK ad quadratocubum ex IK, fuit equidem in hyperbol&atail;
quadratoquadratocubus ex DC
ex AB &longs;icut quadratocubus ex AE ad &longs;imilem pote&longs;tatem ex
DE, &longs;cilicet inuariata pote&longs;tate ab&longs;cr&longs;arum in ambabus au
uer&longs;is. Quare ex pote&longs;tatibus notis vnius auuer&longs;arum fa
cilè inote&longs;cent pote&longs;tates alterius, atque etiam illius magnitu
do. Nunc redeamus ad motus, nouamque adhuc methodum,
quam hoc loco re&longs;eruauimus, afferamus.
SIt quædam Gene&longs;is ACBH, cuius imago temporis
& DCB &; item &longs;it FCBK gene&longs;is alterius motus ab
eodem C in B; & actà rectà OIGE ip&longs;i AFCD parallel&atail;,
ponantur CD, GE loco minimorum temporum, ita vt
pore
currat minimum &longs;patiolum indicatum per C, cui e&longs;t æqua
le &longs;patiolum aliud indicatum per G, quodque tran&longs;igitur
tempore GE velocitate GD (nam vt e&longs;&longs;ent illa &longs;patia i&ntail;
C, G æqualia, effectum fuit vt velocitas AC ad GD ean
dem reciprocè rationem haberet, ac tempus GE ad CD,
id quod patet ex natura gene&longs;is ACBH, & imaginis &
DCB &) et hic rur&longs;us notatu digni&longs;&longs;imum e&longs;t nulli errori
obnoxium e&longs;&longs;e, quòd æquabiles in illis minimis &longs;patiolis
intellexerimus motus, quamuis potius deberet videri, in
ij&longs;dem interuallis reperiri innumeras, ac inæquales veloci
tates, queis nempe efficerentur motus inæquabiles, quòd
gene&longs;es inæquabiles &longs;int. Cur i&longs;ta &longs;e ita habeant, hic non
e&longs;t nobis di&longs;putandum, ego enim puto, non ex indiui&longs;ibili
velocitates alijs &longs;uccedere, &longs;ed reuera minutulum tempo
ris con&longs;iderari debere antequam motus diuer&longs;imodè pro
cedat, nempe ac &longs;i velocitas, quæ &longs;uccedere debet priori,
non ita &longs;it in promptu, aut non ita &longs;tatim mobile afficiat ad Sed linquamus hæc alijs di&longs;
putanda: &longs;atis nobis &longs;it, methodum no&longs;tram, quoad
e&longs;t, demon&longs;trare. Ijs igitur vt &longs;upra propo&longs;itis, concipia
tur adhuc tempore CD velocitate FC
dam, item aliud tempore EG, velocitateque GI, & &longs;ic per
omnes qua&longs;cunque applicatas: quæritur, quod &longs;patiu&mtail;
vltimò exactum e&longs;&longs;et, hoc e&longs;t quam rationem id haberet ad
illud alterum &longs;patium, quod eodem tempore tran&longs;igitur
iu ta gene&longs;im HACB, cuius imago temporis CD & B. I&longs;ti duo motus in exemplo e&longs;&longs;ent, &longs;i in quodam plano mo
ueretur formica, dum ip&longs;um planum vna eius extremitate
immobili circumduceretur, Sic formica difficiliùs
retIam motus extremitatis plani circumactæ habet gene&longs;im
ACBH, cuius temporis imago & DCB &, et altera gene&longs;is
FCBK
motus formicæ pendet ex latione plani, ideò velocitates
eiu&longs;dem (nam in plano immobili ponimus æquabiliter fer
ri) durant ij&longs;dem temporibus, quibus velocitates præcipuæ
gene&longs;is ACBH. Sit denique LMSR imago velocitatum
iuxta gene&longs;im ACBH, cuius temporis imago CD & B; pa
tet &longs;i &longs;it MP ad PS &longs;icut imago temporis CDEG ad ima
ginem & BGE &, fore LM ad PQ vt AC ad OG, & con
cepta etiam figura MNOTS inter parallelas LMN, RST
ita vt &longs;it &longs;emper MN ad PO &longs;icut FC ad GI, nec non LM
ad MN vt AC ad FC. (&longs;unt enim initio motuum in C, aut
in&longs;tanti M, velocitates gene&longs;ium AC, CF, &longs;cilicet LM, MN;
& in G, hoc e&longs;t in&longs;tanti P &longs;unt velocitates OC, GI; nimi
rum QP, PO) vocetur proinde gene&longs;is FCBK &longs;puria, ac
ad&longs;tricta imaginitemporis & DCB &, cuius imago veloci
tatum MNTS pariter &longs;puria, homogenea tamen ip&longs;i legiti
SI &longs;int duo motus iuxta gene&longs;es legitimam, & &longs;puriam,
erunt mobilium exacta &longs;patia, vt imagines inter&longs;e
homogeneæ velocitatum, legitima ad &longs;puriam.
E&longs;to gene&longs;is legitima ACBH, cuius imago temporis
& DCA &, & imago velocitatum MLRS. Sit etiam gene
&longs;is altera illi homogenea, &longs;ed &longs;puria, & ad&longs;tricta imagini
temporis & DCB &, cuius imago velocitatum &longs;puria, prio
rique legitimæ homogenea NMST. Dico, &longs;patia iuxta has
imagines tran&longs;acta e&longs;&longs;e vt ip&longs;æ imagines legitima LMSR
ad &longs;puriam NMST. Cum temporis momenta M, P in
telligantur ex minimis temporibus, quæ proponi po&longs;&longs;unt,
inter&longs;e æqualibus, & quibus æquabiliter perdurant ve
locitates, quas mobile &longs;ortitur in aduentu &longs;uo in punctis
C, G, erit vt velocitas FC ad velocitatem GI &longs;ic inter&longs;e
&longs;patia, quæ i&longs;tis velocitatibus, temporibu&longs;que illis æqua
libus percurrerentur, in qua ratione e&longs;t etiam NM ad OP. Deinde momento M peragerentur &longs;patia proportionalia
velocitatibus FC, AC, &longs;eu rectis NM, ML, momento
autem P &longs;patia proportionalia velocitatibus GI, GD,
in qua ratione e&longs;t etiam OP ad PQ, & &longs;ic deinceps
procedendo per &longs;ingula temporis MR momenta, adeo
vt, cum &longs;patium velocitate FC exactum ad id veloci
tate CA, &longs;it vt NM ad ML, &longs;patium velocitate IG ad id
exactum velocitate GD &longs;it vt OP ad PQ, & &longs;int præterea
primæ inter&longs;e, hoc e&longs;t &longs;patia velocitatibus FC, GI tran
&longs;acta, proportionalia tertijs, &longs;patijs videlicet tran&longs;actis
velocitatibus ML, PQ ergo vt omnes primæ ad omnes
tertias quantitates, hoc e&longs;t omnia &longs;patia tran&longs;acta iuxta
gene&longs;im FCBK ad omnia &longs;patia iuxta gene&longs;im ACB, ita
erit &longs;umma &longs;ecundarum ad |omnes quartas, &longs;cilicet i&longs;ta
erit imago NMST ad imaginem LMSR. Quod & c.
MOtum appellamus compo&longs;itum, vbi dum fer
tur mobile, con&longs;ideratur habere plures i&ntail;
diuer&longs;as partes, vel
conatus, ex quibus oriatur tertia vis di&longs;tin
cta ab illis. Hunc librum, cum expleueri
mus, non pauca vnà cum priori, dicta erunt de motu, erit
que ea methodus, qua &longs;imul geometrica quædam, difficil
lima &longs;citu &longs;atis breuiter o&longs;tendemus. Nam vibrationes
pendulorum exigi temporibus; quæ &longs;int in &longs;ubduplicat&atail;
ratione longitudinum eorundem, planè tandem con&longs;tabit
aliàs nobis di&longs;&longs;entientibus: aperiemus etiam, qua arte in
telligi queant anguli rectilinei curuilineis æquales; nec non
exponemus parabolas quibu&longs;dam &longs;piralibus æquales, vt
e&longs;t vulgata &longs;pirali Archimedeæ, cùm videlicet ba&longs;is para
bolæ radio circuli &longs;piralem continentis, & dimidium huius
circumferentiæ circuli altitudini eiu&longs;dem parabolæ, æqua
les &longs;int.
SI in eadem recta linea currantur &longs;patia temporibus
æqualibus, & &longs;int motus &longs;implices, ac ad ea&longs;dem par
tes tendentes, eadem illa &longs;patia &longs;imul motu compo&longs;ito, ab
eodemque mobili duabus illisgene&longs;ibus affecto, vnicoque
ex dictis temporibus æqualibus, excurrentur.
Curratur LI iuxta imaginem velocitatum HAEF, et IO
iuxta aliam dictæ homogeneam BAED. Dico LO &longs;um
mam dictorum &longs;patiorum LI, IO exactum iri vnico tem
pore AE, &longs;i nempe mobile feratur
ginem.
Per quodlibet punctum, &longs;eu temporis momentum M
agatur recta GMC parallela HB, vel FD. Habebit mobi
le momento A,
&longs;imul velocitates AH, AB, ide&longs;t vnicam HB. Similiter mo
mento M habebit GC, & momento E ip&longs;am FD. Itaque
erit HBDF imago velocitatum compo&longs;iti motus, qui fiet
tempore AE iuxta imaginem, quæ aggregatum e&longs;t
HAEF, ABDE. E&longs;t verò LI ad IO vt imago HAEF ad
imaginem ABDE; ergo conuertendo, componendoqu&etail;
erit vt LI ad LO, &longs;ic imago HAEF ad imaginem HBDF;
propterea quemadmodum &longs;patium LI currebatur iuxt&atail;
imaginem HAEF, &longs;ic LO percurretur imagine HBDF &longs;olo,
eodemque tempore AE. Quod &c.
minimè ad terram venturum aggregato virium, quarum vna
e&longs;t ab impellente impreßa, altera verò à grauitate Nam ex impartita vt celerior fit ca&longs;us, quam vt graue in de
cur&longs;u &longs;uo po&longs;&longs;it ex acceler atione naturali eum gradum acqui
rere, quem certè &longs;ponte &longs;ua tantùm de&longs;cendens in fine eiu&longs;dem
altitudinis adeptum e&longs;&longs;et. Hoc ita verum e&longs;t, vt aliquando
minimum inter&longs;it, inter impetum ab ambabus cau&longs;is proue
nientem, & eum, qui a &longs;ola oritur grauitate, quamobrem pa
rum is proficeret, qui conaretur maiorem impetum componere
in ca&longs;u grauis, illi nempe adiecta vi, mobile idem in decur&longs;u
impellente, vltra nat
dubiè po&longs;&longs;et, &longs;i ca&longs;us obliquus eßet.
ter de&longs;cendens eò concitatiùs ferri, quoad potentia re&longs;i&longs;tentis
aeris (validior namque i&longs;ta fit, vbi mobilis ca&longs;us e&longs;t celerior)
vi granitatis mobili inhærenti exaquatur, tunc enim cau&longs;&atail;
vlterioris accelerationis adempta e&longs;t, con&longs;umiturque in lucta
tione aeris contranitentis: quare tunc grane progrederetur
aquabili motu, id quòd citiùs euenire deberet &longs;i grane intr&atail;
aquam de&longs;cendat.
SI in eadem recta duos motus &longs;ibi contrarios, fimplices,
ac eodem tempore peractos intelligamus, mobile di
ferentiam illorum &longs;patiorum, &longs;i vtroque motu e&longs;&longs;et affe
ctum, percurreret.
Curratur à puncto L &longs;patium LO imagine velocitatum
ABFG, & codem tempore curratur etiam recta OM ex
puncto altero O, &longs;cilicet contrario motu, & iuxta
AHIG prædict&ecedil; homogeneam. Dico mobile,
vtri&longs;que motu, & tempore ip&longs;o AG cur&longs;urum differentiam
LM dictorum &longs;patiorum LO, OM.
Primùm intra parallelas AB, GF non &longs;e &longs;ecent lineæ
BF, HI, & ducatur quælibet DC æquidi&longs;tans AB, vel GF,
quæ fecet HI in E. Manife&longs;tum e&longs;t, mobile, compo&longs;ito
motu feratur habere duplicem velocitatem, vnam AB al
teram illi oppo&longs;itam AH, ob idque moueri ver&longs;us O &longs;ol&atail;
velocitate HB differentia dictarum inter&longs;e pugnantium
velocitatum: pariter momento D feretur mobile veloci
tate EC differentia duarum DE, DC, & in&longs;tanti G habebit
differentialem IF; ex quo &longs;equitur figuram BHEIFCB, dif
ferentiam imaginum ABFG, HAGI, aptatam tempori AC
imaginem e&longs;&longs;e velocitatum compo&longs;iti motus. Hoc po
&longs;ito habebit LM ad LO eandem rationem, ac BHIF ad
ABFG; Propterea LM, quæ e&longs;t differentia &longs;patiorum LO,
motu, & tempore AG.
huius.
2. Se nunc &longs;ecent lineæ BF, HI in C.
Ducatur CD pa
rallela alteri æquidi&longs;tantium AB, GF. Con&longs;tat ex prima
parte, quòd mobile compo&longs;ito motu, & iuxta imaginem
HBC feretur ver&longs;us O tempore AD; &longs;it ergo &longs;patium, quod
curreretur illa imagine, PR, & ob id LO ad PR eande&mtail;
habebit rationem quam imago ABFG ad imagine&mtail;
HBC.
Similiter dum mobile mouetur tempore DG iuxta ima
gines DCIG, DCFG, feretur verè &longs;ecundùm imagine&mtail;
FCI ver&longs;us L, quamobrem &longs;i &longs;patium, quod exigeretur
hac imagine &longs;it RQ, habebit i&longs;tud ad LO eandem rationem,
quam imago CFI ad imaginem ABFG, & ideo ex æquali
QR ad PR &longs;e habebit vt imago CFI ad imaginem HBC; &longs;i
igitur ponatur ABFG maior imagine AHIG, demptà co
muniter AHCFG relinquetur HBC maior imagine CEI, &
ideo etiam PR maior QR: curritur verò PR versùs R tem
pore AD, & RQ versùs P tempore DG, ergo toto tempo
re AG curretur PQ differentia &longs;patiorum PR,
verò HBC ad CFI, &longs;it vt PR ad RQ, erit diuidendo vt ex
ce&longs;&longs;us imaginis HBC &longs;upra imaginem FCI ad imagine&mtail;
i&longs;tam, ita PQ ad QR, & o&longs;ten&longs;um e&longs;t QR ad LO, &longs;icut ima
go FCI ad imaginem ABFG, ergo ex æquali exce&longs;&longs;us ima
ginis HBC &longs;upra imaginem AHIG habebit eandem ratio
nem ad imaginem AHIG, ac PQ ad LO, at e&longs;t in illa
ratione etiam LM ad LO (e&longs;t enim LO ad MO vt imago
ABFG ad imaginem AHIG) ergo PQ erit æqualis LM,
atque adeo mobile dum currit vtroque motu, hoc e&longs;t iux
ta &longs;imul duas imagines propo&longs;itas contrariorum motuum,
peraget &longs;patium LM versùs O &longs;ecundùm imaginem, quæ
differentia e&longs;t propo&longs;itarum ABFG, AHIG, tempore AG. Quod &c.
parte.
gines &longs;implicium motuum fuerint aquales.
REperire eam velocitatem, eamque directionem, quæ
orirentur, &longs;i mobile pluribus eodem momento velo
citatibus, &longs;eu conatibus affectum e&longs;&longs;et. Opportet autem
non &longs;olum has velocitates, verùm etiam earum directio
nes manife&longs;tas e&longs;&longs;e.
Habeat mobile A, eodem momento conatum AB, quo
tendat in R; AC; quo in C; & AD, quo in D. Quæritur ve
locitas, & directio, quas mobile habiturum e&longs;&longs;et in multi
plici illa affectione (Nam actu vnam velocitatem, vnam
que tantùm directionem &longs;ortiri debet) Ex duabus qui
bu&longs;que AD, AC intelligatur perfici parallelogrammum
ACED, & ducta diametro AE fiat itidem aliud parallelo
grammum ABFE, cuius agatur diameter AF. Dico AF
e&longs;&longs;e quæ&longs;itam velocitatem, ac directionem, quibus mobile
ex illis pluribus conatibus motum &longs;uum in&longs;titueret.
Si mobili A currendum e&longs;&longs;et æquabili motu &longs;patium
AE, pertran&longs;iret eodem tempore tam rectam AD, quàm
ip&longs;am AC; nam cum fertur ab A in E verè de&longs;cendit ab A
in C, & ab A in D motu pariter æquabili; ergo AD ad
AC, erit vt velocitas, qua curritur per AD ad velocitatem,
qua curritur per AC. Itaque &longs;i mobile dum e&longs;t in A in
telligatur affectum velocitatibus AD, AC habentibus di
rectiones ip&longs;as rectas AD, AC, perinde e&longs;&longs;et, ac &longs;i &longs;ola fo
ret mobili velocitas vnâ cum directione AE. Eadem ra
tione AF velocitas habens directionem AF, æquipollebit
duabus velocitatibus AB, AE iuxta directiones rectas ea&longs;-Mo
bile igitur ex affectione trium illorum conatuum, vt &longs;up
po&longs;itum fuit, nitetur &longs;ecundùm AF velocitate ip&longs;a AF
Quod &c.
pr. de mo
ACcelerationem alicuius motus, tunc intelligimus,
velocitates, quæ &longs;ubinde mobili adueniunt, non de
lentur, &longs;ed pror&longs;us integræ, atque indelebiles mobili in ip&longs;o
motu per&longs;euerant. Ex quo &longs;equitur motum &longs;implicem di
ci, cum præteritæ velocitates protinus euane&longs;cunt, illæ
que tantum con&longs;iderantur, quæ mobili &longs;ubinde oriun
tur.
IMaginem accelerationis cuiu&longs;cunque &longs;implicis motus
exhibere.
Imago velocitatum &longs;implicis motus e&longs;to rectangulum
AFDC: &longs;ic motus e&longs;t æquabilis, vt acceleretur debent in
&longs;tanti C vigere omnes velocitates in imagine AFDC
prehen&longs;æ
CD, erit mobile momento B affectum omnibus antece
dentibus velocitatibus, comprehen&longs;is nempe ab imaginis
portione AFEB; quare &longs;i ponamus HLG imaginem e&longs;&longs;&etail;
accelerationis, itaut nempe tempus GL æquale &longs;it tempo
ri AC; item KL æquale tempori AB, erit vt figura CAFD
ad figuram BAFE, &longs;ic velocitas, qua mobile fertur
to
ponitur imago &longs;implicis motus rectangulum AFDC, erit
rectangulum CF ad BF, hoc e&longs;t recta CA ad AB immò
LG ad LK, vt GH ad KI; quamobrem GLH imago velo
citatum huiu&longs;modi motus, erit triangulum. Quod &longs;i ima-
accelerationis foret trilineum &longs;ecundum, & ita pro
portionaliter de infinitis numero accelerationibus.
def.
mi.
rum naturaliter de&longs;cendentium triangulum &longs;it. Nam quo
libet momento &longs;ui ca&longs;us habet graue idem in&longs;e principiu&mtail;
motus, &longs;eu grauitas, ex qua concipitur imago &longs;implicis motus
&longs;i nempe priores gradus velocitatis &longs;ubinde deperirent, at
quia in eius de&longs;cen&longs;u pror&longs;us per&longs;euerant (id enim &longs;upponi
tur ab&longs;trabendo ab aere) inde motus concitatur, & fit vti di
ximus imago accelerationis triangulum.
QVælibet linea, vt fluxus puncti concipi po
te&longs;t.
VT propo&longs;ita linea ex fluxu puncti exarètur, duò tan
tùm nece&longs;&longs;aria &longs;unt, &longs;cilicet motus, & puncti di
rectio.
REcta, quæ priùs de&longs;eripta e&longs;t, pote&longs;t alijs à primis
velocitatibus, rur&longs;us exarar
Nam punctum pote&longs;t fluere &longs;ecundum quamcunque
rectam, quocunque motu, ergo illam pote&longs;t etiam quibu&longs;
cunque velocitatibus affectum rur&longs;us exarare.
VT eadem recta ex fluxu puncti renouetur, opportet in
quocunque illius puncto &longs;eruari pri&longs;tinas directio
nes,
Cum, vti diximus, ad de&longs;criptionem lineæ duo tantùm
exigantur, nempe motus, & puncti directio; motus verò po
te&longs;t e&longs;&longs;e quilibet, &longs;equitur ergo directionem, alteram de
duobus, &longs;eruari debere.
pr.
LIneam dicimus curuam, in qua &longs;umptis duobus ad
libitum punctis, recta, quæ ip&longs;a puncta coniunge
ret, nullam cum propo&longs;ita linea partem &longs;it habitura com
munem.
DIrectiones puncti de&longs;cribentis lineam, iuxta rectas
lineas concipi debent.
Dum punctum fluere intelligimus, ine&longs;t in eo &longs;ingulis
momentis certus, ac præfixus gradus velocitatis, quo tan
tùm attento, rectà,
tenderet; at huiu&longs;modi iter, aliud non e&longs;t, quàm directio
puncti, qua eius temporis momento profici&longs;citur; ergo iux
ta rectas lineas, directiones omnes con&longs;iderari opportet.
TAngens, & directio motus in quouis curuæ puncto
e&longs;t vna,
Nam in de&longs;criptione
quentes, aliò tendentes ni&longs;us, & ob id di&longs;trahitur punctum
ip&longs;um à priori tendentia, idem accidit ex alia parte &longs;i re
flaxiffet idem punctum, nempe hinc inde vnicam rectam
eandemque, continuantibus oppo&longs;itis ad idem punctum
directionibus, ergo directio, & tangens vna, & eadem e&longs;t
recta.
cunque illius puncto vnica tantùm ex vtraque parte egre
ditur tangens.
QVòd &longs;i ex aliquo puncto duæ tangentes hinc inde
egredientes angulum efficiant; tunc propo&longs;itam li
neam inflexam dicemus, & punctum, in quo &longs;unt
contactus, inflexionis appellabitur.
manat artificium
componendi duas curuas, vel curuam & rectam, adeout vni
cam lineam efforment, nullumque angulum; nempe cum &longs;ic
inuicem tuxgamus, vt taugentes ad punctum connexus, vnam
tantùm rectam efficiant.
cem compo&longs;it
tium ob&longs;eruauerimus, &longs;unt enim inter&longs;e æquales, licèt diuer
&longs;a &longs;peciei,
TAngens, &longs;eu directio motus in quocunque curuæ
puncto e&longs;t illa recta, quæ vtrinque &longs;tatim cadens
extra curuæ conuexum ad eandem, quàm fieri pote&longs;t ex
vtraque parte accedit.
Nam alia quæque recta tran&longs;iens per punctum conta
ctus ad &longs;ectionem magis accedere nequit, quin ip&longs;am illinc
&longs;ecet, ob id extra conuexum eius non cadet, ab altera ve
rò parte magis à propo&longs;ita curua &longs;eparabitur, quamobrem
nulla alia recta, quàm tangens poterit &longs;imul extra curuam
e&longs;&longs;e, & quàm fieri pote&longs;t ad ip&longs;am accedere.
LIneæ AC, AD occurrant &longs;ibi in A, quod punctum in
telligatur transferri ab A in C vnà cum linea AD
&longs;emper &longs;ibi parallela, quo tempore punctum A currat ip
&longs;am latam lineam ex A in D. Manife&longs;tum e&longs;t idip&longs;um
punctum A de&longs;cripturum e&longs;&longs;e motu compo&longs;ito lineam
quandam AB diagonalem &longs;uperficiei parallelogrammæ
ABCD. Vocamus ergo diagonalem illam &longs;emitam com
po&longs;iti motus, & AC, AD latera illius.
AD, licèt curuæ &longs;int, nam verè transfertur illo tempore, tam
ad lineam CB quam ad DB.
ius diameter AB; quamobrem ex datis punctis C, A, D repe
riretur &longs;tatim punctum B, &longs;cilicet extremum &longs;emitæ compo
&longs;iti motus, cuius latera ip&longs;æ curuæ, aut rectæ AC, AD
EX datis
&longs;emitæ terminum exhibere.
Si latera compo&longs;iti motus e&longs;&longs;ent duo tantùm AB, AC.
Facto parallelogrammo vt dictum e&longs;t, inueniretur pun
ctum E extremum motus: &
tus, pote&longs;t idem E &longs;upponi tanquam extremum alterius la
teris, adeoque, &longs;i motus con&longs;tet ex tribus lateribus AC,
AB, AD, perinde &longs;it ac &longs;i foret duorum laterum AE, AD;
nam AC, AD valent &longs;imul ac &longs;olum AE; cum ita &longs;it, facto
etiam parallelogrammo EADF ex datis punctis E, A, D,
habebitur F extremum &longs;emitæ, cuius &longs;unt tria latera CA,
AD, AB —
nempe a&longs;&longs;umptis partibus AG, AH, AI in dictis lateribus,
quæ quidem &longs;ciantur percurri temporibus æqualibus, &longs;i per
ip&longs;as &longs;ingulas mobile punctum ferretur eo modo, quo in com
po&longs;ito motu nititur per ea&longs;dem directiones; reperietur in
quam punctum K in &longs;emita AE, atque L in &longs;emita AF: qua
re hoc modo &longs;umptis alijs, atque alijs partibus in ip&longs;is lateri
bus, reperientur alia, atque alia puncta ad ip&longs;am &longs;emita&mtail;
pertinentia, quorum tandem beneficio, facile erit qua&longs;itam
fermè &longs;emitam exarare.
EX datis imaginibus velocitatum, iuxta quas &longs;implici
motu currantur latera compo&longs;iti motus; datis item
tangentibus ad quæcunque puncta ip&longs;orum laterum, repe
rire &longs;emitam compo&longs;iti motus, nec non directiones,
tate&longs;que
Opportet tamen latera ip&longs;a,
in imperatas &longs;ecari po&longs;&longs;e rationes, quamquam nos non la
teat, in lateribus curuis hoc effici non po&longs;&longs;e, præterqua&mtail;
aliquatenus in periphærijs circulorum.
Sint AB, AF latera compo&longs;iti motus, quæ quidem &longs;eor
&longs;im currantur eodem tempore QM, &longs;cilicet AB iuxta ima
ginem MNPQ, et AF iuxta imaginem alteram ei homoge
neam TMQR. Ponatur AB circuli arcus, quem tangat re
cta BC æqualis QB, at AF lineam', quæ parabola &longs;it, con
tingat recta FG æqualis
H extremum &longs;emitæ compo&longs;iti motus; &longs;unt enim data pun
cta A, F, B. Cum igitur mobile venerit in H. Dico, eo
temporis momento velocitatem, ac directionem HL, quæ
recta diameter e&longs;t parallelogrammi, cuius duo latera &longs;unt
dictæ lineæ HI, HK; Iam vti diximus punctum H e&longs;t ex
tremum compo&longs;iti motus, quare eo momento, quo pun
ctum mobile e&longs;t in H, habet inibi ea&longs;dem illas velocitates,
quas haberet in B, et F, dum &longs;eor&longs;im illa latera excurri&longs;&longs;et;
&longs;cilicet con&longs;ideratur ip&longs;um mobile habens &longs;imul velocita
tem HI æqualem, ac æquedirectam, &longs;eu æquidi&longs;tantem
ip&longs;i CB, cui e&longs;t æqualis alia QP; & velocitatem HK æqua
lem, &longs;imiliterque directam, ip&longs;i GF æquali
&longs;it erit HL velocitas, & directio quæ&longs;ita momento
dem modo, &longs;i &longs;it, vel fiat vt imago PNMQ ad ONMV
(ducta &longs;cilicet applicata SVO) ita BA ad AX, et ONMV
ad imaginem VMTS, vt XA ad AI, percurrentur AX, AI
ctio, tangens ip&longs;a ZX æqualis VO, & in I velocitas, & di
rectio, tangens 2 I æqualis VS; Itaque datis punctis X, I, A
dabitur etiam Y extremum &longs;emitæ compo&longs;iti motus, cuius
latera AX, AI, & ideo mobile dum e&longs;t in Y momento V
affectum erit duplici velocitate, hoc e&longs;t Y 4 æquali ve
locitati ZX, &longs;eu VO, ac æquidi&longs;tante eidem ZX, et veloci
tate altera Y 3 æquali, & æquèdirecta ip&longs;i 2 I: quare ex
datis punctis 4, Y, 3 inuenietur punctum S quartus angu
lus parallelogrammi habentis diametrum YI, quæ quidem
erit directio, & velocitas mobilis currentis compo&longs;ito mo
tu in&longs;tanti V. Cumque alia quotcunque puncta eadem
methodo reperire queamus, per quæ duci po&longs;&longs;it linea ferè
quæ&longs;itam &longs;emitam repræ&longs;entans,
co, quod propo&longs;uimus.
huius.
CVm imagines velocitatum, iuxta quas curruntur du&ecedil;
rectæ, quæ &longs;int latera compo&longs;iti motus, &longs;unt paral. logrammum, & triangulum; tunc &longs;emita compo&longs;iti motus
erit communis parabola.
Tempore HM curratur latus AC iuxta imaginem velo
citatum HILM rectangulum, & latus AB iuxta imaginem
triangulum HMN; erit CA ad AB, vt imago
mumFiat
po&longs;iti motus, quæ &longs;i ponatur AFC; Dico e&longs;&longs;e parabolam. Sumatur in ip&longs;a linea quoduis punctum F, ab ip&longs;o dedu-
AE, AG latera compo&longs;iti motus, cuius &longs;emita AF: Con
cipiatur modò P momentum, quo mobile ade&longs;t in F, &
ducta OPK parallela alteri HI, vel NL, erit imago MHIL ad
ad GF. Pariter erit imago NHM ad
quadratum ex MH ad
illud ex GF, vt BA ad AG; quamobrem punctum F cadet
in curuam parabolicam communem, cuius diameter AB,
& ba&longs;is, &longs;eu ordinatim applicata BD, &longs;cilicet AFD erit ip&longs;a
curua parabolica. Quod &c.
huius.
berum ab omni obice, ni&longs;i turbaretur eius motus à propri&atail;
grauitate pergerct moueri æquabiliter iuxta directionem, ve
locitatemque ei traditam; habet verò coniunctam grauita
tem, qua, ni&longs;i ab impre&longs;&longs;o impetu flecteretur motus, de&longs;cen
deret iuxta perpendiculum motu naturaliter concitato, cuius
imago velocitatum, triangulum e&longs;t; Hinc propterea gran&etail;
vltra perpendiculum proiectum de&longs;cribit in cur&longs;u &longs;uo, motu
&longs;cilicet compo&longs;ite, parabolam vulgatam. Verùm enim verò
de&longs;criptionem i&longs;t am nece&longs;&longs;e aliquo pacto e&longs;t ex duabus cau&longs;is
vitiari, hoc est ab aeris re&longs;i&longs;tentia, & perpendiculis non in
ter&longs;e parallelis, quippe in idem,
centrum, conuergentibus.
SI ab a&longs;&longs;umpto hyperbolæ puncto, recta axi primo pa
rallela deducatur, quæ ad &longs;ecundam diametrum per
tingat; Quadrilineum comprehen&longs;um ab ip&longs;a curua hy
perbolica. & dictis tribus rectis, erit imago velocitatis il-
ad axem eius habet eandem rationem, quam duplus axis
propo&longs;itæ hyperbolæ ad ductam illam
eiu&longs;dem hyperbolæ a&longs;&longs;ymptotos interiectam.
Hyperbolæ IRS &longs;it centrum H, &longs;emiaxis HI, a&longs;&longs;ymptoti
HT, NH, et SN parallela HI; tùm ducta HM &longs;ecunda dia
metro hyperbolæ, intelligatur de&longs;criptio parabolæ AFD;
itaut duplus axis hyperbolæ, hoc e&longs;t quadruplum ip&longs;ius
HI ad NT eandem habeat rationem, quam DB ba&longs;is pa
rabolæ ad BA axim eiu&longs;dem. Dico quadrilineum HISM
e&longs;&longs;e imaginem velocitatum, iuxta quam motu compo&longs;ito
de&longs;cribitur parabola AFD; & cum &longs;it homogenea imagi
nibus HILM, HTM, e&longs;&longs;e quoque rectangulum HDLM ad
imaginem ip&longs;am HISM vt recta CA ad curuam AFD. Fiat rectangulum ACDB, et HM &longs;it tempus, quo curritur
vtrunque latus AB, AC, nempe axis AB motu grauium
iuxta imaginem triangulum HTM, alterum verò latus AC
æquabili motu iuxta imaginem rectangulum HILM, quod
quidem erit HILM; etenim AB ad &longs;patium AC e&longs;t vt ima
go triangulum HMT ad imaginem rectangulum HILM,
&longs;cilicet e&longs;t vt MT ad duplam HI, vel vt NT ad quadru
plam HI, quemadmodum po&longs;uimus. Iam mon&longs;trauimus
lineam, quæ curritur iuxta illas imagines motu compo&longs;ito
parabolam e&longs;&longs;e, cuius diameter AB, & ba&longs;is BD; & pro
pterea erit ip&longs;a AFD (nam vnica tantum parabola ex
datis AB, BD po&longs;itione, ac magnitudine, axi &longs;cilicet, ac
ba&longs;i dari pote&longs;t) Ducatur nunc à quolibet puncto F dictæ
parabolæ rectæ FE, FG parallelogrammum con&longs;tituentes
AEFG; & P &longs;it momentum, quo mobile punctum inueni
tur in F. Habebit inibi ip&longs;o temporis momento P veloci
tatem PQ iuxta directionem GF, &longs;unt verò i&longs;tæ directiones
&longs;ibi ip&longs;is perpendiculares; ergo recta, quæ diameter e&longs;&longs;et
rectanguli AEFG, & ob id potentiâ æqualis duabus PK,
PQ erit gradus velocitatis, quem mobile habet momen-
PR &ecedil;quatur rectangulo ORQ vnà cum quadrato ex PQ, &
e&longs;t ob hyperbolam rectangulum ORQ æquale quadrato
ex HI, vel PK; ergo PR quadratum æquale erit duobus &longs;i
mul quadratis PQ, PK; itaque PR erit gradus velocitatis
prædicti mobilis in F momento P, compo&longs;itoque motu
currentis iuxta curuam parabolicam. Pariter momento
M, cum mobile e&longs;&longs;et in D velocitas compo&longs;iti motus foret
MS pote&longs;tate æqualis duabus MT, ML, ac demum in A
initio motus velocitas e&longs;t HI: quare HISM erit imago ve
locitatis motus compo&longs;iti dum mobile punctum de&longs;crip&longs;e
rit curuam parabolicam AFD, e&longs;tque illa imago imagini
bus diui&longs;orum, &longs;eu &longs;implicium, motuum homogenea; ergo
con&longs;tat ba&longs;im etiam BD ad parabolam AFD eandem ha
bere rationem, quam rectangulum HILM ad quadrili
neum HISM. Quod &c.
& pr.
mi huius.
pr.
huius.
nic.
huius.
pendicularia, tunc gradum velocitatis eìu&longs;dem motus compo
&longs;iti æqualem e&longs;&longs;e potentiâ duobus &longs;imul gradibus, quos habet
mobile eodem momento, ac &longs;i &longs;eorfim intelligatur in ip&longs;is ferri
lateribus.
inter&longs;e homogenea, erit vt quadrilineum HISM primi ad
ua illa parabolica ad hanc &longs;ecundi ca&longs;us parabolam.
huius.
HIRP ad ip&longs;um PRSM, ita AF ad FD.
PRopo&longs;itis Spirali Archimedea primæ circulationis
ABD, et AGF
æqualis radio DA, et GA &longs;it dimidium circumferenti&ecedil; cir
culi AEG; erit parabola AGF axem habens GA æqualis
propo&longs;itæ &longs;pirali.
Sit PNK communis hyperbola, cuius coniugati &longs;emia
xes &longs;int IK, IH, & a&longs;&longs;ymptotos IO. E&longs;to etiam axis hy
perbolæ huius, dupla &longs;cilicet IK, ad HO illi &ecedil;quidi&longs;tantem
vt FG ad AG. Iam con&longs;tat quadrilineum IHPK fore ima
ginem velocitatum, iuxta quam curreretur parabola AGF
tempore IH: &longs;i modo o&longs;tendimus hoc ip&longs;um
e&longs;&longs;e pariter homogeneam imaginem alterius compo&longs;iti
motus, quo videlicet de&longs;cribitur &longs;piralis propo&longs;ita ABD,
palam erit, ip&longs;am parabolam eidem illi &longs;pirali æqualem fu
turam. Ducatur recta KL, quæ æquidi&longs;tet IH; item ex
quouis puncto Q
parallela IK: erit parallelogrammum rectangulum HIKL
imago velocitatum, iuxta quam curritur FG, et HIO trian
gulum imago, qua curritur AG motu grauium de&longs;cenden
tium: Verùm quia eodem tempore IH, &longs;i mobile currat
æquabili motu DA æqualem FG, e&longs;t eius imago idem re
ctangulum IHKL, curriturque illo eodem tempore IH (&longs;pi
rali exigente) omnis circuli circunferentia AGEA æqua
bili etiam motu ab extremitate A radij AD circumducti in
de&longs;criptione &longs;piralis; ob idque factum e&longs;t, vt IK ad HO e&longs;
&longs;et vt DA ad circunferentiam ip&longs;am AGEA; nam hoc mo-
dem motus per AGEA. Ducatur nunc ex quocun
que momento Qlinea QRMN ip&longs;i IK æquidi&longs;tans, & au
&longs;picato motu ex centro D momento I, vt nempe oriatur
&longs;piralis, intelligatur momento Qventum e&longs;&longs;e in B, quamo
brem ductâ DBE, erit rectangulum, &longs;eu imago QIKR ad
imaginem rectangulum HIKL, ita DB ad DE, in qua ra
tione, cum propter &longs;piralem, &longs;it etiam circunferentia AGE
ad circunferentiam AGEA, erit rectangulum IQ in HO
imago velocitatis per AGE, e&longs;tque velocitas iuxta tangen
tem in E ad velocitatem iuxta tangentem circulum BC in
B vt ED ad DB, &longs;eu vt HO ad QM; ergo cum iuxta
tem
gentem circulum BC in B, ip&longs;a QM velocitas; propterea
que imago triangulum HIO, quæ in parabolæ de&longs;criptio
ne erat per AG, nunc erit per omnes tangentes circulos &longs;u
binde crc&longs;centes ex D in E: &longs;cilicet momento I, erit mobi
li puncto &longs;ecundùm DA, velocitas IK; momento Q du&mtail;
ade&longs;t in B, erit &longs;ecundùm BE velocitas QR, & iuxta
tem
locitates QR, QM cum &longs;int normaliter directæ, erit eidem
mobili in B iuxta &longs;piralem velocitas QN potentia ip&longs;is am
babus æqualis. Similiterque momento H cum mobil&etail;
fuerit in A, erit velocitas iuxta &longs;piralem, ip&longs;a HP æqualis
potentiâ duabus velocitatibus HL iuxta radium, et HO
iuxta tangentem; & &longs;ic omnino liquet, ip&longs;um quadrilineum
HIKP e&longs;&longs;e imaginem velocitatum tam in de&longs;criptione pa
rabolæ AGF, quàm &longs;piralis Archimedeæ DBA, & cum &longs;it
in ij&longs;dem de&longs;criptionibus homogenea &longs;ibi ip&longs;i, con&longs;tat ip
&longs;as curuis æquales e&longs;&longs;e. Nam vt imago illa ad &longs;e ip&longs;am ita
parabola ad &longs;piralem prædictam. Quod &c.
prima.
Cor. pr.
re rationem, quam quadrilineum QIKN ad quadrilineum
HIKP; pariterque rectam DA ad eandem &longs;piralem DCB ha
bere ip&longs;am rationem, ac rectangulum HIKL ad dictum qua
drilineum HIKP. Eodem ferè modo exhiberi pißet ratio &longs;pi
ralis ad &longs;piralem, licèt plurium inter&longs;e circulationum, eritque
pror&longs;us ea, quam habet vnum ad alterum eiu&longs;dem illius na
turæ, quadrilineorum.
SPiralis orta ex motu naturaliter accelerato per
circuli comprehendentis &longs;piralem ip&longs;am, & ex motu
æquabili circa
ei curuæ parabolicæ natæ ex motu compo&longs;ito, cuius vnum
latus curritur iuxta imaginem trianguli, nempe motu gra
uium, alterum verò latus iuxta imaginem trilinei &longs;ecundi,
habebitque parabola ip&longs;a axim æqualem radio, & ba&longs;i&mtail;
tertiæ parti circunferentiæ eiu&longs;dem circuli &longs;piralem com
prehendentis.
E&longs;to &longs;piralis ACB, quæ &longs;ignatur ex motu
biliter lati circa circumferentiam ADA, dum nempe
tempore IF, punctum B currit à quiete lineam BA motu
grauium de&longs;cendentium; &longs;it verò imago velocitatum dicti
motus æquabilis per ADA rectangulum HGFI, & alte
rius motus imago, (quæ triangulum erit) e&longs;to FEIM. Pa
tet, quia ip&longs;æ imagines ponuntur homogeneæ, e&longs;&longs;e rectan
gulum HGFI ad triangulum IFM vt ADA circumferentia
ad radium BA, & propterea IM ad IH erit vt BA ad dimi
dium circunferentiæ AEDA. Sumatur quodlibet
tum
po&longs;itæ BCA: agatur per ip&longs;um punctum radius BCD, & &longs;ic
illo momento extremitas A currendo circa periphæriam
reperietur in D, eritque circonferentia AED ad ip&longs;am
AEDA, vt imago rectangulum OGFK ad
hoc e&longs;t erit vt KF ad FI; at BC ad BD erit vt imago trian
gulum KFL ad triangulum FIM, nempe vt quadratum KF
ad quadratum FI, e&longs;t autem vt BD ad BC ita velocitas
iuxta tangentem in D ad velocitatem iuxta tangentem in
C circulum, cuius radius BC; &longs;cilicet ita velocitas IH ad
velocitatem KN, quadrati nempe IF ad quadratum KF, &
ob id velocitates, quæ &longs;unt iuxta tangentes circulos &longs;ubin
de
&longs;ecundo, cuius &longs;cilicet indoles e&longs;t vt ab&longs;ci&longs;&longs;arum quadrata
&longs;int vt applicatæ. His compo&longs;itis, intellecti&longs;que erit in B,
momento F, nulla velocitas, in C momento K duæ velo
citate
tera &longs;it KN iuxta tangentem circulum, cuius radius CB, ne
ctitur vna ex duabus illis, quibus
lis, & qua idem mobile mouetur iuxta &longs;piralem illo mo
mento K. Similiter cum mobile e&longs;t in D, &longs;cilicet momento
I, habebit velocitatem potentia æqualem HI, qua dirigitur
iuxta tangentem, & velocitati IM, qua &longs;ecundùm radium,
Itaque imago velocitatum mobilis de&longs;cribentis &longs;piralem
propo&longs;itis motibus tempore IF, ea erit, cuius applicatæ
&longs;unt vbique æquales potentia ijs applicatis, quæ ab
momento intelligi queunt in imaginibus &longs;implicibus, nem
pe partialium motuum, HNFI, IFM. Cum præterea OT
ponatur tertia pars e&longs;&longs;e circumferentiæ AEDA, & e&longs;t
trilineum HFI vtpote &longs;ecundum tertia pars
mi
BA, vel ei æqualis QO ad OT; curritur verò vt &longs;upponi
tur OQ tempore IF iuxta imaginem triangulum IFM, ergo
eodem tempore iuxta trilineum HNF curretur alterum la-Si itaque parabola ip&longs;a
putetur e&longs;&longs;e ORI, in qua punctum R e&longs;to vbi mobile ade&longs;t
momento K, deducantur verò ab eodem illo puncto RS
parallela axi QO, et RP æquidi&longs;tans QI, vel OT, profectò
in O, momento F, &longs;icuti in &longs;pirali, nulla erit mobili veloci
tas, &longs;ed cum e&longs;t in R momento K habebit geminam veloci
tatem, KL &longs;ecundùm SR, et KN iuxta PR perpendicularem
ip&longs;i SR, quæ duæ velocitates itidem component vnicam
potentia &longs;imul illis æqualem, & cum idem dicatur de qui
bu&longs;cunque alijs punctis parabolæ, momentis temporis FI
re&longs;pondentibus, manife&longs;tum e&longs;t &longs;pirali BCA, & parabolæ
ORI vnicam, eandemque e&longs;&longs;e imaginem velocitatum, pro
pterquam quòd ip&longs;æ curuæ, quòd &longs;int vt imagines, erunt
inter&longs;e æquales.
pr.
prop.
huius.
huius.
les &longs;uis parabolis æquales excogitari, nec ideo res minùs de
mon&longs;trabitur, &longs;i loco rectarum, &longs;eu laterum OT, OP compo&longs;iti
motus, &longs;ub&longs;tituantur circuli, aut circulorum arcus, qui ad re
ctos angulos &longs;e &longs;ecent, &longs;cilicet
xionis, &longs;eu occur&longs;us ip&longs;arum curuarum &longs;ibi ip&longs;is perpendicu
res fuerint. Quòd &longs;i ip&longs;a curua latera ad rectos angulos non
&longs;e &longs;ecent curuæ nihilominus ab ip&longs;o compo&longs;ito motu na&longs;cen
tes poterunt exhiberi curuas parabolicas exequantes, quarum
itidem latera &longs;int rectæ eundem angulum, quem pradictæ
gentesSed de his &longs;atis, nunc dicamus ea
tempora, quibus duorum pendulorum &longs;imiles vibrationes ab
&longs;oluuntur, hoc e&longs;t Galilei &longs;ententiam demon&longs;trabimus, quam
quondam haud ruditer decepti fal&longs;am credidimus.
retur Galilei &longs;ententiam, cuius digni&longs;&longs;imè &longs;e fui&longs;&longs;e di&longs;
lum profitetur, tradidit mihi per admodum Reuerendum, at-
mon&longs;trationem &longs;uam verè pulcherrimam, ac di&longs;erti&longs;&longs;imè
exaratam, qua vna potui&longs;&longs;em de Galilei aßerto &longs;atisfactus
e&longs;&longs;e; eam demon&longs;trationem, ij&longs;dem pror&longs;us verbis, ac figuris,
quibus ad me peruenit hic duxi reponendam, ne gloria&mtail;,
quam Vir tantus meretur, ip&longs;i videremur no&longs;tra, quam inde
&longs;ubdemus, demon&longs;tratione, &longs;ubripere.
TEmpora naturalium de cur&longs;uum &longs;phærarum grauium
per &longs;imiles, &longs;imiliterque ad horizontem inclinatos
arcus curuarum linearum in planis, aut verticalibus, aut
ad horizontem æqualiter inclinatis de&longs;criptarum, & quæ
totæ &longs;int ad ea&longs;dem partes cauæ, inter&longs;e &longs;unt in &longs;ubdupli
cata ratione chordarum eorundem arcuum homologè
&longs;umptarum.
3. 4.
Ex puncto A ad curuam lineam BCD extra ip&longs;am i&ntail;
plano po&longs;itam, & in totum ad ea&longs;dem partes cauam, quæ
cunque ea &longs;it (vel nimirum pars aliqua circumferentiæ
circuli, vel alicuius ex infinitis ellip&longs;ibus, aut parabolis, aut
hyperbolis, aut &longs;piralibus, aut cycloidibus, vel concoidis,
vel ci&longs;oidis, &longs;eu alterius cuiu&longs;cumque ex notis, vel ignotis
curuis educantur omnes rectæ AB, AC, AD &c. quæ à
punctis E, F, C, vel intra, vel extra eas &longs;umptis proportio
nalibus &longs;ecentur, ita vt &longs;it AB ad AE, &longs;icut AC ad AF, &
&longs;icut AD ad AG &c. & hoc &longs;emper.
Sic enim dubio pro
cul apparet, prout facillimum e&longs;t o&longs;tendere, lineam EFG
tran&longs;euntem per &longs;ingula puncta E, F, G &longs;ic inuenta, cur
uam
eique &longs;imilem, &longs;imiliterque cum ip&longs;a po&longs;itam, atque in to
tum cauam ad ea&longs;dem partes, ad quas ponitur caua ip&longs;&atail;
BCD. Concipiatur modò planum, in quo manent huiu&longs;
modi &longs;imilium curuarum &longs;imiles arcus BCD, EFG, vel e&longs;&longs;e
ad horizontem erectum, nempè verticale, vel ad ip&longs;u&mtail;
BCD, EFG inflexas e&longs;&longs;e &longs;uperficies eidem plano erectas,
ita tamen, vt &longs;uper has po&longs;itis grauibus &longs;phæris in A, E per
ip&longs;as &longs;ic inflexas &longs;uperficies eædem &longs;phæræ naturaliter
decurrere queant; id quod &longs;anè accidet, cum arcus BCD
totus fuerit infra horizontalem IL ex arcus &longs;ubli
miori puncto B ductam, fuerintque ab hac continuati re
ce&longs;&longs;us, ac totus ad vnam partem perpendiculi BH: nam &longs;ic
talis quoque erit alter arcus EFG illi BCD &longs;imilis, &longs;imili
terque po&longs;itus. His omnibus &longs;ic manentibus: Dico tem
pus decur&longs;us &longs;phæræ grauis E per &longs;imilem, &longs;imiliterque po
&longs;itum arcum EFG, e&longs;&longs;e in &longs;ubduplicata ratione chordarum
BO, EG arcus ip&longs;os &longs;ubtendentium. Secto enim b
angulo BAD per rectam AC arcum BD &longs;ecantem in C,
atque arcum EFG in F, iungantur chordæ BC, CD, et EF,
FG, quæ ex huiu&longs;modi curuarum natura cadent totæ intra
ip&longs;os arcus, &longs;ed in prima, & &longs;ecunda figura ad partes poli
A, in tertia verò, & quarta ad oppo&longs;itas.
Et quoniam, ex talium curuarum gene&longs;i, e&longs;t vt BA ad
AE, ita DA, ad AG, erit BD ip&longs;i EG parallela, hoc e&longs;t
vtraque ad horizontem æqualiter inclinata, atque in ra
tione BA ad AE. Similiter cum &longs;it, vt BA ad AE, ita CA
ad AF, etiam BC, EF inter&longs;e æquidi&longs;tabunt, &longs;eu ad hori
zontem æqualiter inclinabuntur, eruntque in ratione ea
dem, ac BA ad AE. Idemque o&longs;tenditur de chordis CD,
FG, quare ex magni Galilei &longs;ententia de motu naturaliter
accelerato indubitanter &longs;equitur tempus decur&longs;us &longs;phæræ
grauis ex B in D per binas chordas BC, CD ad tempus
decur&longs;us per vnicam BD, e&longs;&longs;e vt tempus decur&longs;us grauis
&longs;phæræ ex E in G per binas EF, FG ad tempus decur&longs;us
per vnicam EG: eadem itidem ratione demon&longs;tratur (an
gulis pariter BAC, CAD bifariam &longs;ectis per rectas, quæ
&longs;imiles arcus BC, EF, ac CD, FG duas in partes diuidant)
ex quatuor vtrinque arcuum horum cordis, illas inter&longs;e
e&longs;&longs;e æqualiter inclinatas, ac alteram alteri in ratione ea
dem, in qua &longs;unt rectæ AB, AE &c: ac propterea ex ea
dem Galilei &longs;cientia con&longs;tabit vtique, tempus decur&longs;us ex
B in C &longs;phæræ grauis B per quatuor chordas quatuor par
tes arcus BCD &longs;ubtendentes ad tempus decur&longs;us per vni
cam BD, e&longs;&longs;e vt tempus decur&longs;us &longs;phæræ grauis E ex E in
G per quatuor illis homologas chordas quatuor partes
arcus EFG pariter &longs;ubtendentes ad tempus decur&longs;us per
vnicam chordam EG: & hoc &longs;emper ita euenire demon
&longs;trabitur quantacunque, & maxima fuerit in perpetua an
gulorum bi&longs;ectione æquèmultiplicitas in vtroque arcu
talium chordarum homologè &longs;umptarum, ac inter&longs;e pro
portionalium, æqualiterque ad horizontem inclinatarum:
Propterquam quòd &longs;emper decur&longs;us ex B in D per aggre
gatum chordarum omnium in arcu BCD ad tempus de
cur&longs;us per &longs;olam chordam BD e&longs;&longs;e vt tempus decur&longs;us ex
E in G per aggregatum totidem chordarum in arcu EFG
ad tempus decur&longs;us per vnicam chordam EG; adeo vt de
nique iure optimo educi po&longs;&longs;e videatur, tempus decur&longs;us
grauis ex B in D per aggregatum infinitarum chordarum
totum arcum BCD con&longs;tituentium, &longs;eu tempus per ip&longs;um
arcum BCD ad tempus decur&longs;us per &longs;olam cordam BD
e&longs;&longs;e vt tempus decur&longs;us grauis ex E in G per aggregatum
totidem infinitarum chordarum dictis homologè propor
tionalium, æqualiterque &longs;ingulæ &longs;ingulis ad horizonte&mtail;
inclinatarum, ac totum arcum EFG conformantium, &longs;iue
vt tempus per ip&longs;um arcum EFG per &longs;olam chordam EG. Quocirca permutando, tempus, decur&longs;us &longs;phæræ grauis B
per arcum BCD ad tempus decur&longs;us &longs;phæræ grauis E per
arcum &longs;imilem, &longs;imiliterque po&longs;itum EG erit vt tempus
decur&longs;us per chordam BD ad tempus decur&longs;us per chor
dam EG; &longs;ed ex eadem Galilaica &longs;cientia de motu, tempus
decur&longs;us per chordam BD ad tempus decur&longs;us per æqua-
chordarum BD, EG; ergo tempus quoque decur&longs;us ex B
per arcum BCD ad tempus decur&longs;us ex E per arcum EFG
e&longs;t in eadem &longs;ubduplicata ratione chord&ecedil; BD ad chordam
EG, quod o&longs;tendendum propo&longs;uimus.
&longs;tum fit celeberrimum illud magni Galilei pronuntiatum,
quòd videlicet, ratio temporum &longs;imilium vibrationum pen
dulorum &longs;it &longs;ubduplicata rationis longitudinum filorum ho
mologè &longs;umptorum, non tantum verum e&longs;&longs;e de vibrationibus
pendulorum per arcus &longs;imiles, &longs;imiliterque po&longs;itos, &longs;umptos
ex circulorum quadrantibus ad perpendiculum v&longs;que termi
nantes, &longs;ed etiam de vibrationibus per arcus quo&longs;cumque &longs;i
miles quadrantum à perpendiculo &longs;eiunctos: dummodo ip&longs;i
&longs;imiles arcus &longs;int quoque &longs;imiliter po&longs;iti: quales nimirùm ap
parent in figuris prima, ac &longs;ecunda arcus BCD, EFG, dum
grauia B, E ex filis, aut ha&longs;tulis AB, AE circa punctum A
conuertibilibus appen&longs;a concipiantur.
&longs;imiles arcus ex circulis commune centrum A habentibus; ac
in verticali plano po&longs;itis, & in prima figura recta AB, AE
fuerint fila aut ha&longs;tulæ quædam circa clauum A conuertibi
les, in &longs;ecunda verò recta AB, AE concipiantur, vt ha&longs;tulæ
inflexibiles, volubile&longs;que circa imum punctum E, atque ex
huiu&longs;
graues &longs;phæræ B, E (cum eadem &longs;int tempora prout a&longs;&longs;umi
tur quoque ab ip&longs;o met Ceua) tempora inquam decur&longs;uum
liberorum granium B, E per arcus BCD, EFG, ac tempor&atail;
filis pendeant, vel ab hastulis &longs;ustineantur) erit quoque tem
pus de&longs;cen&longs;us, &longs;eu vibrationis penduli B per arcum BCD ad
tempus de&longs;cen&longs;us, &longs;eu vibrationis penduli E per arcum EFD
in &longs;ubduplicata ratione chordæ BD ad chordam EG; &longs;ed hæc
ratio chordarum BD, EG eadem e&longs;t, ac ratio filorum, aut ha
&longs;tularum AB, AE; Ergo tempus vibrationis penduli AB per
arcum BCD ad tempus vibrationis penduli AE per arcum il
li &longs;imilem, &longs;imiliterque po&longs;itum EFG est quoque in &longs;ubdupli
cata ratione longitudinum, vel filorum, aut ha&longs;tularum, ex
quibus eadem grauia pendula &longs;imiles vibrationes ab&longs;oluunt
BCD, EFG.
eua&longs;uram, &longs;i ommi&longs;&longs;a illa continua bi&longs;ectione angulorum &longs;i
miles, &longs;imiliterque po&longs;itos arcus ab&longs;cindentium ex &longs;imilibus
curuis ibidem de&longs;criptis; atque ommi&longs;&longs;a pariter continua co
niunctione chordarum, vt ibi factum fuit, horum vice, vt in
quinta figura, ex punctis B, D binæ tangentes curuam BCD
ducantur BH, DH, quæ omninò mutuò &longs;e &longs;ecabunt in puncto
H (ob conditiones in ip&longs;a Theorematis expo&longs;itione vltimo lo
co po&longs;itas) atque ex E, G ip&longs;is BH, DH agantur æquidistan
tes, quæ iunctæ, AH &longs;imul occurrent in I, curuamque EFG
contingent pariter ad E, G (quæ omnia &longs;i opus fuerit, facilè
demon&longs;trabuntur) ac in&longs;uper, &longs;i à puncto C, in quo iunct&atail;
AH &longs;ecat arcum BCD, agatur tangens LM primas BH, DH
&longs;ecans in LM; Per F verò, in quo AICH &longs;ecat arcum EFG
agatur NO parallela tangenti LM, quæ curuam pariter EFG
tanget ad F, ac tangentes EI, GI &longs;ecabit ad NO: & &longs;i iunctis
in&longs;uper AL, AM, eadem, quam nunc explicauimus, continue
tur con&longs;tructio per alias, atque alias tangentes, ac parallelas&longs;ic enim vnicuique harum curuarum circum&longs;cribetur
rectilineum, primò ex binis tangentibus, &longs;ecundò ex tribus,
tertiò ex quinque, quartò ex &longs;eptem, & &longs;ic vlteriùs iuxta re
liquos impares numeros &longs;ucce&longs;&longs;iuè &longs;umptos; atque omnia pa
ria talium æquidi&longs;tantium tangentium eam &longs;emper inter &longs;e
rationem &longs;eruabunt, quam habent chorda BD, EG, &longs;en quam
habent rectæ BA, EA,
binarum tangentium BH, HD, EI, IG, quàm per minores
&longs;ummas, ex quinque &longs;imul chordis vtrinque &longs;umptas, aut
quàm per alias &longs;emper minores &longs;ummas huiu&longs;modi tangen
tium iuxta quantumuis maiorem numerum imparem æquè
multipliciter &longs;umptarum, erunt perpetuò proportionalia tem
poribus decur&longs;uum per chordas BD, EG; & hoc &longs;emper; etiam
&longs;i per huiu&longs;modi decrementa aggregaterum ex tangentibus
vtrinque æquèmultipliciter &longs;umptis, deueniatur ad vltim
ac breui&longs;&longs;imas ip&longs;is arcubus circum&longs;criptiones pol
ex lateribus numero innumerabiliter aquèmultiplicibus, hoc
e&longs;t ad ip&longs;os &longs;imiles, &longs;imilitorque po&longs;itos arcus BCD, EFG,
quorum &longs;ingula homologorum laterum, &longs;eu punctorum paria,
vt B, & E; C et F; D, et G &c. haberi poßunt tanquam
paria parallelarum, ac proportionalium tangentium ip&longs;os &longs;i
miles, ac &longs;imiliter po&longs;itos arcus con&longs;tituentia. Quapropter
ratio
rationi temporum decur&longs;uum per chordas; &longs;ed horum decur
&longs;uum ratio &longs;ubdupla e&longs;t rationis inter ip&longs;as chordas. Quare,
& alia hac methodo con&longs;taret propo&longs;itum.
dum annuimus, veritatem eandem no&longs;tra quoque methodo,
confirmemus, vt ijs, quibus &longs;atis probat demon&longs;tratio allata,
&longs;it nostra, quam afferemus, in experimentum traditarum hùc
hæc per demon&longs;trationes no&longs;tras pror&longs;us, &longs;Illud etiam admoneo, eam rem non tantum me o&longs;ten&longs;urum,
bus aperiatur; verùm potius vt ampli&longs;&longs;ima Methodus, qua tum
vtemur, aliorum motuum demon&longs;trandorum in exemplum
veniat.
IN eadem recta CD coeant duæ planæ,
ac pror&longs;us æquales figuræ ADCA, BDCB, & quidem
ita, vt ab eodem puncto M &longs;i ducatur MH parallela CA,
et ML ip&longs;i CB, &longs;it &longs;emper MH æqualis ML, quemadmo
dum æquales &longs;unt inter&longs;e CA, CB. Dico (&longs;i concipiatur
&longs;olidum eius indolis, vt ductis rectis BA, LH cadant i&longs;tæ
omninò in &longs;olidi i&longs;tius &longs;uperficie; ip&longs;um verò &longs;olidum, quod
&longs;it BADC, &longs;ecetur plano quolibet æquidi&longs;tante figuræ
BCD) fore, vt &longs;ectio i&longs;ta KFEIK, &longs;it pror&longs;us &longs;imilis, æqua
li&longs;que alteri conterminæ AEI; &longs;ed opportet, vt palam e&longs;t,
coeuntes illæ figuræ non in eodem plano reperiantur.
Cum duo plana inuicem parallela KIE, BCD &longs;ecent
alia duo inter&longs;e item parallela ACB, HML, erunt commu
nes &longs;ectiones, inter&longs;e omnes æquidi&longs;tantes rectæ lineæ KI,
GF, ML, CB. Cum verò ob naturam &longs;olidi, &longs;ectiones
BAC, IHM triangula &longs;int rectilinea, erit vt BC ad CA,
ita KI ad IA. Sunt autem priores inter&longs;e æquales, ergo &
po&longs;tremæ KI, AI inter&longs;e æquabuntur. Eademque ratione
&longs;unt æquales HG, GF: & quoniam ob fimilitudinem figu
rarum angulus BCD æquatur angulo ACD, & angulus
BCD æqualis angulo KIE (nam etiam CD, IE &longs;unt rectæ
æquidi&longs;tantes, cum nempe &longs;int communes &longs;ectiones plani
DCA &longs;ecantis duo æquidi&longs;tantia KIE, BCD) ergo cu&mtail;
angulus pariter ACD æquet angulum AIE, erunt anguli
KIE, AIE, et FGE, HGF æquales. Quod &c.
II&longs;dem manentibus.
Dico triangula ACB, LHM e&longs;&longs;&etail;
&longs;imilia. Sunt enim parallelæ &c.
inter&longs;e tam rectæ CB,
ML, quàm CA, MH; ideo anguli ACB, HML inter&longs;e
æquabuntur, & &longs;unt circa eos proportionalia latera, nem. pe BC ad CA, vt LM, MH; ergo con&longs;tat propo&longs;itum.
II&longs;dem vt &longs;upra manentibus, ita tamen vt ACD &longs;it an
gulus rectus (&longs;ic enim DC perpendicularis erit duabus
AC, CB) Dico &longs;olidum huiu&longs;modi ad pri&longs;ma, cuius ba&longs;is
ABC, & altitudo CD eandem habere rationem, quam &longs;o
lidum rotundum ortum ex rotatione figuræ CAD circ&atail;
axem CD ad cylindrum genitum ex conuer&longs;ione rectan
guli AC in CD circa eundem axem.
Compleatur ip&longs;um pri&longs;ma, & &longs;it quidem AQDPBC,
quod &longs;ecetur vnà cum propo&longs;ito &longs;olido per quoduis pla
num ba&longs;i ACB æquidi&longs;tans: fiet in pri&longs;mate &longs;ectio trian
gulum OMN &longs;imile, æqualeque ip&longs;i ACB, & in altero &longs;o
lido triangulum LHM eidem ACB &longs;imile. Triangulum
ACB pri&longs;matis ad
mune, e&longs;t vt circulus radio CA de&longs;criptus ad circulum
eundem; Item triangulum NOM &longs;ectio pri&longs;matis e&longs;t ad
triangulum LHM &longs;ectionem propo&longs;iti &longs;olidi, vt circulus ex
radio MO de&longs;criptus ad circulum radio MH. Cum dein
de idem dicatur de alijs omnibus &longs;ectionibus pri&longs;matis, &
æquales &longs;unt, ad omnes &longs;imul &longs;ecundas vt omnes tertiæ,
his partibus inter&longs;e æqualibus, ad omnes quartas; &longs;cilicet
erunt omnia triangula pri&longs;matis, &longs;eu ip&longs;um pri&longs;ma ad om
nia triangula propo&longs;iti &longs;olidi, &longs;eu ad ip&longs;um &longs;olidum, vt om
nes circuli eius cylindri, qui oritur ex conuer&longs;ione figuræ
ADCA circa axem CD, hoc e&longs;t vt ip&longs;um &longs;olidum rotun
dum, &longs;eu cylindrus ad omnes &longs;imul circulos &longs;olidi rotundi
geniti ex rotatione figuræ AHDCA circa axem
&longs;eu ad ip&longs;um propo&longs;itum &longs;olidum. Quod &c.
libro de dim. parab.
Euang.
T
ET rur&longs;us ip&longs;a manente figura patet, &longs;i ducantur HR,
LS parallelæ MD, fore non &longs;olum figuram AHDPA,
&longs;imilem, ac æqualem BLDQB; verùm etiam APRHA ip&longs;i
BLSQB: Cum ita &longs;it, aio, eundem cylindrum ad &longs;oli
dum rotundum genitum, ex volutatione figuræ APD cir
ca eundem axem CD eandem rationem habere, ac pri&longs;ma
&longs;upere&longs;t ex ip&longs;o pri&longs;mate, dempto &longs;olido ACBLDHA.
Nam ex præterita propo&longs;itione nouimus, dictum pri&longs;ma
ad &longs;olidum eius partem ACBLDHA e&longs;&longs;e vt cylindrus or
tus ex conuer&longs;ione rectanguli CP circa axem CD ad par
tem eius rotundum circa axem eundem CD conuer&longs;a fi
gura ADC, ergo per conuer&longs;ionem rationis, erit id quod
propo&longs;uimus.
QVodcunque ex dictis propo&longs;itis &longs;olidis vocetur ab
ea figura, iuxta quam intelligitur ortum. Scilicet
ACBLDHA dicatur à figura AHDCA, & alte
rum, quod fuit re&longs;iduum prædictum dicatur à figura AH
DPA.
SI à quibu&longs;cunque figuris fuerint duo &longs;olida, hæc inter
&longs;e erunt vt &longs;olida alia genita ex conuer&longs;ione illarum
figurarum circa communem &longs;ectionem &longs;imilium, æqua
lium, ac inter&longs;e coeuntium figurarum.
Solidum à figura ABC &longs;it CAFDBC, & quod e&longs;t à fi
gura GLH e&longs;to HGILH. Dico illud ad hoc &longs;olidum e&longs;&longs;e
vt rotundum natum ex conuer&longs;ione figuræ ABC circ&atail;
axem CE ad rotundum ortum ex
circa axem HL. Opportet tamen angulos ACF, GHI
æquales e&longs;&longs;e. Intelligantur pri&longs;mata triangularia, quorum
ba&longs;es ACF, GHI, & altitudines CE, HL; hoc e&longs;t &longs;int ip&longs;a
&longs;olida pri&longs;matica AFCEBD, GIHLMK. Solidum à figu
ra ABC ad pri&longs;ma AFCEBD habet eandem rationem,
quam &longs;olidum rotundum ortum ex conuer&longs;ione &longs;iguræ
ABC circa axem CE ad cylindrum natum ex rotatione
ABEC circa eundem axem CE; hic verò cylindrus ad cy
lindrum alium natum ex rotatione rectanguli GMLH cir
ca axem HL e&longs;t vt pri&longs;ma, cuius ba&longs;is ACF, altitudineque
CE ad alterum pri&longs;ma ba&longs;em habens GHI &longs;imilem ip&longs;i CF
(nam circa angulos æquales H, C &longs;unt latera etiam pro
portionalia, nempe æqualia) & altitudinem HL. Solidum
præterea, hoc e&longs;t pri&longs;ma GKHM ad &longs;olidum, quod e&longs;t à
plano GLH habet eandem rationem, ac cylindrus, qui fit
ex conuer&longs;ione rectanguli HM circa axem HL ad &longs;olidum
rotundum ortum ex circumactione figuræ GLH circa ip
&longs;um axem HL, ergo ex æquali erit &longs;olidum à figura ABC
ad &longs;olidum à figura GLH, vt rotundum ex rotatione figu
ræ ABC circa axem CE ad rotundum alterum ex conuer
&longs;ione alterius figuræ GLH circa axem HL. Quod &c.
PRopo&longs;itis ij&longs;dem &longs;olidis, erunt inter &longs;e, vt momenta fi
gurarum a quibus &longs;unt, quæ tamen figuræ &longs;u&longs;pen&longs;æ
&longs;int ex longitudinibus deductis ab ip&longs;arum grauitatu&mtail;
centris v&longs;que ad coeuntium figurarum communes illas &longs;e
ctiones.
Figuræ, à quibus &longs;unt &longs;olida, ponantur ABC, GLH,
tra grauitatum illarum M, N; axes, &longs;iue communes &longs;ectio
nes coeuntium binarum inter&longs;e &longs;imilium, ac æqualium fi
gurarum à quibus dicuntur ip&longs;a &longs;olida; & demum MO, NP
perpendiculares &longs;int ab ip&longs;is centris ad illas communes &longs;e
ctiones deductæ CE, HL. Dico, &longs;olidum à plana figur&atail;
ABC ad &longs;olidum a plana GHL eandem habere rationem,
ac momentum figuræ ABC pendentis ex MO ad momen
tum alterius figuræ &longs;u&longs;pen&longs;æ ex NP, &longs;unt enim hæc &longs;oli
da inter&longs;e, vt rotunda, quorum genetrices figuræ ABC,
GLH circa axes CE, HL, huiu&longs;modi verò &longs;olida &longs;unt vt
momenta propo&longs;ita; ergo &longs;olidum à plana figura ABC ad
&longs;olidum à plana GLH, erit vt momentum figuræ ABC
&longs;u&longs;pen&longs;æ ex MO ad momentum GLH pendentis ex NP. Quod &c.
p
ABC, GLH, & ex longitudinibus, ex quibus pendent ip&longs;æ fi
gura (nam habentur vt grauia) ex ij&longs;dem etiam rationibus
componentur &longs;olida, qua &longs;unt ab ip&longs;is figuris—
cis,
IMagines velocitatum, &longs;eu &longs;patia, quæ curruntur accele
ratis motibus, &longs;unt vt &longs;olida ab imaginibus &longs;implicium
motuum, ex quibus ip&longs;i gignuntur accelerati.
Sint imagines &longs;implicium motuum ABC, GLH, & &longs;oli
da ab ip&longs;is imaginibus (angulis ACQ, GHD &longs;emper re
ctis, aut &longs;altem æqualibus) intelligantur ABCRQ, GLHD.
Dico, vt &longs;unt inter&longs;e i&longs;ta &longs;olida, &longs;ic e&longs;&longs;e homologè &longs;patium
exactum tempore AC motu accelerato ex &longs;implici motu
imaginis ABC ad &longs;patium tran&longs;actum tempore GH motu
item accelerato ex &longs;implici imagine priori homogene&atail;
GLH: &longs;ecetur &longs;olidum ABCRQ plano æquidi&longs;tanti QCR,
quod faciat in &longs;olido ip&longs;o &longs;ectionem TSVX: erit hæc figu
ra pror&longs;us &longs;imilis, ac æqualis conterminæ ABVI; quare
cum in accelerato motu velocitas, quæ habetur momen
to C ad velocitatem momento S &longs;it vt imago ABC &longs;im
plex ad &longs;egmentum eius ABVS: erit etiam QCR æqualis
ABC ad &longs;ectionem &longs;olidi TSVX, quæ æquatur ABVS, vt
illa eadem velocitas momento C mobili inhærens ad ve
locitatem momento S alterius accelerati motus. E&longs;t au
tem &longs;ectio TSVX ad libitum &longs;umpta; ergo &longs;olidum ABC
QR pote&longs;t &longs;umi merito vt imago velocitatum accelerati
motus, cuius &longs;implex imago ABC: & eodem modo &longs;oli
dum alterum vicem geret imaginis velocitatum alterius
motus ex &longs;implici imagine GLH, itaque erit ob homoge
neitatem &longs;patium tran&longs;actum motu accelerato iuxta &longs;im
plicem imaginem ABC ad &longs;patium tran&longs;actum motu ac
celerato iuxta &longs;implicem imaginem GLH,
GH, vt &longs;olidum ABCQR ad ALHD,
huius.
ius vnà cum
pr.
SInt nunc CE, HL communes &longs;ectiones imaginum &longs;im
plicium ABC, GLH, &longs;i extenderentur cum &longs;ujs æqua
libus, ac &longs;imilibus coeuntibus figuris. E&longs;to pariter M cen
trum grauitatis imaginis ABC, et N grauitatis alterius ima
ginis GLH; actis demùm MO, NP perpendicularibus ad
ip&longs;as CE, HL. Dico, &longs;patium accelerati motus ab imagine
&longs;implici ABC ad
gine &longs;implici GLH componi ex ratione imaginis ABC ad
imaginem GLH, & ex ea perpendicularis MO ad perpen
dicularem NP. Cum hæc ip&longs;a &longs;patia &longs;int o&longs;ten&longs;a, vt &longs;oli
da à figuris ABC, GLH; hæc verò &longs;unt vt momenta ip&longs;a
rum figurarum &longs;u&longs;pen&longs;arum ex MO, NP. Ergo quemad
modum momenta i&longs;ta nectuntur ex rationibus figurarum
tanquam magnitudinum ABC ad LGH, & di&longs;tantiarum
MO ad NP, ita pariter ex his nectentur propo&longs;ita &longs;patia.
cis.
plicatis AB, HL, quæ in imaginibus &longs;umuntur perpendicula
res rectis AC, GH. nam HL est recta, in quam coeunt figura
huius.
SI imagines &longs;implicium motuum fuerint &longs;imiles, &longs;imili
terque &longs;u&longs;pen&longs;æ, imagines velocitatum accelerato
rum motuum erunt in triplicata ratione temporum &longs;impli
cium motuum, aut in triplicata homologarum, vel extre
marum velocitatum eorundem &longs;implicium motuum.
Cum centra grauitatum &longs;imilium imaginum, &longs;eu figu
tur verò imagines &longs;imiliter &longs;u&longs;pen&longs;æ, ergo &longs;equitur ip&longs;as
longitudines e&longs;&longs;e vt latera homologa dictarum imaginum,
&longs;cilicet vt tempus AC ad tempus FG, vel vt extremæ ve
locitates BC ad KE. Quamobrem imagines ip&longs;æ, cum &longs;int
in duplicata ratione laterum homologorum, &longs;i huic dupli
catæ addatur alia ratio &longs;imilis rationi longitudinum, fiet
ratio imaginum velocitatum, &longs;eu &longs;patiorum acceleratorum
motuum ex &longs;implicibus illis deriuantium triplicata tempo
rum, vel extremarum velocitatum &longs;implicium motuum.
SI verò &longs;implices motus extiterint &longs;imiles,
temporibus ab&longs;oluantur, imagines acceleratorum
motuum erunt in &longs;ola ratione amplitudinum imaginum
&longs;implicium.
Sint imagines &longs;imilium, ac &longs;implicium motuum BAC,
KFG, quarum grauitatis centra D, H, erunt ex hypothe&longs;i
tempora AC, FG æqualia; & ideo &longs;patia, &longs;cilicet imagines
velocitatum BAC, KFG habebunt eandem rationem,
quam &longs;ummæ, aut extremæ motuum &longs;implicium velocita
tes, &longs;cilicet, quam amplitudines imaginum, &longs;eu gene&longs;um:
&longs;unt verò di&longs;tantiæ DE, HI pariter æquales, quia AC, FG
æquales &longs;unt; ergo cum &longs;patia acceleratorum motuum ne
ctantur ex imaginibus &longs;implicium motuum ABC, KFG, &
ex di&longs;tantijs DE ad HI, liquet ip&longs;a &longs;patia e&longs;&longs;e in vnica, &longs;o
laque ratione amplitudinum BC, KG, aut amplitudinum
gene&longs;um.
AT&longs;i &longs;implicium, &longs;imiliumque motuum fuerint imagi
nes æquè amplæ, imagines acceleratorum motuum,
rum, vel illorum motuum.
Amplitudines imaginum &longs;implicium, velocitatumque
BAC, KFG &longs;unto BC, KG, quæ æquales &longs;int. Dico &longs;pa
tia acceleratorum motuum ab illis &longs;implicibus imaginibus
fore in duplicata ratione temporum AC ad FG (qu&ecedil; &longs;em
per in acceleratis ponuntur eadem, ac in &longs;implicibus, nec
aliter e&longs;&longs;e po&longs;&longs;unt.) Vt FG ad GK, ita &longs;it AC ad CL, &
intelligatur LAC imago alterius motus &longs;imilis motui, cuius
imago BAC, vel KFG. Facilè demon&longs;trabitur ip&longs;am fi
guram LAC &longs;imilem e&longs;&longs;e ip&longs;i KFG, & ad BAC eande&mtail;
habere rationem, quam LC ad BC. Cum ergo imago BAC
ad imaginem KFG componatur ex ratione imaginis BAC
ad LAC (quæ &longs;unt vt BC ad CL) & ex ratione imagi
nis ALC ad imaginem KFG, quæ &longs;unt in ratione compo
&longs;ita LC ad KG, et AC ad FG: priores verò duæ rationes
componunt vnicam æqualitatis, ergo relinquitur, imagi
nem BAC ad imaginem KFG e&longs;&longs;e vt AC ad FG; &longs;patium
verò accelerati motus ex &longs;implici imagine BAC ad accele
ratum ex &longs;implici KFG nectitur ex ratione imaginum &longs;im
plicium ip&longs;arum, & ex ea di&longs;tantiarum DE, HI à centris
grauitatum deductarum D, H, et &longs;unt hæ rectæ in eadem
ratione, ac altitudines AC, FG (nam in figuris, &longs;eu imagi
nibus &longs;imilium motuum BAC, LAC centra grauitatum
&longs;unt in eadem recta parallela ip&longs;i BC, & in LAC, KFG
&longs;unt in punctis &longs;imiliter po&longs;itis, adeout, &longs;icut po&longs;itum e&longs;t,
ratio ip&longs;arum di&longs;tantiarum in ip&longs;is figuris LAC, KFG, &longs;eu
BAC, KEG eadem &longs;it, ac laterum homologorum LC ad
KG, vel AC ad FG) ergo &longs;patium accelerati motus ex &longs;im
plici imagine KFG, erit vt quadratum ex AC ad quadra
tum ex FG, nempe in duplicata ratione temporum &longs;impli
cium motuum.
huius.
DEmùm &longs;i &longs;int imagines, quæcunque velocitatnm &longs;im
plicium, &longs;imiliumque motuum, imagines accelera
torum motuum, &longs;eu &longs;patia ijs motibus exacta componen
tur ex duplicata temporum ratione, & ex ea amplitudi
num, vel applicatarum homologarum earundem imagi
num.
Imagines &longs;imilium, &longs;impliciumque motuum &longs;int BAC,
KFG. Dico, imagines acceleratorum motuum ab illis &longs;im
plicibus deriuantium habere rationem compo&longs;itam ex du
plicata temporum AC ad FG, & amplitudinum imaginum
dictarum, vel gene&longs;um. Intelligatur alius &longs;imilis motus,
cuius velocitatum imago &longs;it DFG æquèampla, ac homo
genea ip&longs;i BCA; nimirum &longs;it DG æqualis BC. Quoniam
imago accelerati motus ex &longs;implici imagine BA ad imagi
nem accelerati ex &longs;implici imagine KFG componitur ex
ratione imaginis accelerati motus, cuius &longs;implex imago
BAC ad imaginem accelerati motus ex &longs;implici DFG, &
ex imagine huius accelerati motus ad accelerati imaginem
à &longs;implici KFG; e&longs;t autem prior ratio imaginum, &longs;eu &longs;pa
tiorum acceleratis motibus percur&longs;orum ip&longs;a temporum
duplicata AC ad FG, & altera dictarum imaginum, &longs;eu
&longs;patiorum item acceleratis motibus confectorum, & quo
rum &longs;implices imagines &longs;unt DFG, KFG, e&longs;t eadem, ac ra
tio amplitudinum DG, &longs;eu BC ad KG. Ergo cum i&longs;tæ
amplitudines &longs;int eædem, ac illæ gene&longs;um, con&longs;tar propo
&longs;itam rationem acceleratorum motuum ex &longs;implicibus
imaginibus BAC, KFG habere rationem compo&longs;itam ex
duplicata temporum AC ad FG, & ex ea amplitudinum
imaginum &longs;implicium BC ad KG, &longs;eu amplitudinum gene
&longs;um. Quod &c.
SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint
æquèamplæ, imagines acceleratorum motuum erunt
in duplicata ratione temporum, vel altitudinum ip&longs;arum
gene&longs;um.
Gene&longs;es &longs;imilium, ac &longs;implicium motuum &longs;unto ABC,
DEF, quarum amplitudines æquales &longs;int AC, DF. Dico,
imagines, &longs;iue &longs;patia acceleratorum motuum e&longs;&longs;e in dupli
cata ratione temporum, vel altitudinum BC ad EF. Cum
AC, DF &longs;int gradus velocitatum in extremitatibus &longs;impli
cium decur&longs;uum, etiam imagines velocitatum, iuxta ip&longs;as
gene&longs;es, quæ &longs;int inter&longs;e homogeneæ, erunt æquèamplæ,
& &longs;unt &longs;imilium motuum; ergo imagines acceleratorum
motuum, iuxta &longs;implices illas gene&longs;es, aut imagines æquè
amplas erunt in duplicata ratione temporum: &longs;unt autem
imagines velocitatum æquèamplæ, &longs;imiliumque motuum,
hoc e&longs;t &longs;patia BC ad EF vt ip&longs;a tempora; ergo &longs;patia acce
leratorum, propo&longs;itorumque motuum erunt in ratione du
plicata altitudinum BC, EF &longs;implicium gene&longs;um, ABC,
DEF. Quod &c.
SI gene&longs;es &longs;imilium, &longs;impliciumque motuum fuerint
æquèaltæ, imagines, &longs;iue &longs;patia, acceleratorum mo
tuum erunt vt tempora, vel reciprocè vt amplitudines ge
ne&longs;um ip&longs;orum &longs;implicium motuum.
Gene&longs;es &longs;imilium, &longs;impliciumque motuum, ac inter&longs;e
homogeneæ &longs;int BAC, DEF, quæ habeant altitudines
AC, EF æquales. Dico, imagines acceleratorum motuum
e&longs;&longs;e inter &longs;e, vt tempora dictorum &longs;implicium motuum, vel
reciprocè vt amplitudines ip&longs;arum gene&longs;um. Concipian-
iuxta gene&longs;im BAC, et MKL iuxta
quia, vtpotè homogene&ecedil;, &longs;unt inter &longs;e vt &longs;patia &ecedil;qualia AC
ad EF,
GI ad ML, & ex ea, quam habet HI ad KL, &longs;equitur e&longs;&longs;e
GI ad ML, vt KL ad IH, & demum quia acceleratorum
motuum &longs;patia à &longs;implicibus imaginibus GHI, MKL ne
ctuntur ex duplicata temporum HI ad KL, & ex ea an pli
tudinum GI ad ML, &longs;iue ex ea, quam habet KL ad HI, re
linquitur, &longs;patia acceleratis illis motibus confecta e&longs;
&longs;ola,
li ratione, reciproca amplitudinum imaginum ML ad GI,
vel gene&longs;um DF ad BC. Quod &c,
QVæcunque fuerint gene&longs;es &longs;imilium, &longs;impliciumque
motuum, dum inter&longs;e homogeneæ, &longs;patia accelera
tis motibus ex illis &longs;implicibus exacta nectentur
ex duplicata ratione altitudinum, & reciproca amplitudi
num earundem &longs;implicium gene&longs;um,
Sint quæcunque &longs;imilium motuum gene&longs;es BAC, KFG.
Dico, &longs;patia acceleratorum motuum, ab ijs &longs;implicibus de
riuantium, componi ex duplicata ratione altitudinum AC
ad FG, & ex ratione extremarum velocitatum, &longs;eu ampli
tudinum reciprocè &longs;umptarum ip&longs;arum gene&longs;um: e&longs;to alia
gene&longs;is DFG illis homogenea, & motu pariter &longs;imilis cum
ij&longs;dem gene&longs;ibus. Eadem &longs;it amplitudine æqualis BAC,
& altitudo eius &longs;it FG, &longs;patia acceleratorum motuum ex
&longs;implicibus gene&longs;ibus æquales amplitudines haben
& &longs;imilium motuum BAC, DFG &longs;unt in duplicata 'ratione
rectarum, &longs;eu altitudinum AC ad FG, & &longs;patia accelera
dem altitudine DFG, KFG, &longs;unt in reciproca ratione am
plitudinum, &longs;eu primarum vel ocitatum KG ad DG, vel
BC; ex æquali igitur &longs;patia acceleratorum motuum ex
propo&longs;itis &longs;implicibus gene&longs;ibus BAC, KFG nectentur ex
ratione duplicata altitudinum AC ad FG, & reciproca
amplitudinum KG ad BC earundem gene&longs;um BAC,
KFG. Quod &c.
&longs;eruatur ratio altitudinum gene&longs;um &longs;implicium, ex quo ori
tur in hac methodo quædam percipiendi dificultas; ideo &longs;e
quenti problemate, alij&longs;que iam notis veritatibus, rem planè
illu&longs;trabimus, ac &longs;imul doctrina v&longs;um trademus.
EX datis &longs;patijs accelerato motu confectis, cogniti&longs;
que primis, aut po&longs;tremis &longs;imilium, &longs;impliciumque
motuum velocitatibus, reperire tempora ip&longs;orum de
cur&longs;uum.
Spatia motibus acceleratis exacta &longs;unt C, D, & velo
tates, &longs;eu amplitudines gene&longs;um ponantur e&longs;&longs;e A, B, &longs;cili
cet A principio motus per C, & B initio motus per D, quæ
ritur ratio temporum, quibus exiguntur propo&longs;ita &longs;patia. Vt A ad B, ita fiat C ad E, & inter E, et D &longs;umatur F me
dia proportionalis. Dico ip&longs;a tempora e&longs;&longs;e vt E ad F.
Componuntur &longs;patia acceleratis motibus exacta ex ratio
ne quadratorum temporum, & ex ea amplitudinum, &longs;eu
homologarum velocitatum in &longs;implicibus motibus, &longs;imili
bu&longs;que &longs;umptarum; & ideo temporum quadrata necten
tur ex ratione &longs;patiorum C ad D, & ex reciproca ampli-
D, ip&longs;a verò tempora vt E ad F. Quod &c.
pr.
mi huius.
EXdatis &longs;patijs accelerato motu tran&longs;actis, datis item
primis velocitatibus &longs;imilium, &longs;impliciumque mo
tuum, inuenire altitudines &longs;implicium gene&longs;um, ex quibus
propo&longs;ita &longs;patia effecta &longs;unt.
30.
Spatia &longs;int E, D reliquis, vt &longs;upra, manentibus: quoniam
&longs;patia accelerato motu tran&longs;acta componuntur ex ratio
nibus amplitudinum gene&longs;um &longs;implicium, &longs;imiliumqu&etail;
motuum reciprocè &longs;umptarum B ad A, &longs;iue E ad C, & ex
ea quadratorum altitudinum ip&longs;arum gene&longs;um; erit ratio
dictarum altitudinum duplicata C ad D; quare F, &longs;i &longs;it me
dia proportionalis, non inter E, & D (vt antea po&longs;uimus)
&longs;ed inter C ad D; erit &longs;anè C ad F ratio altitudinum gene
&longs;um &longs;implicium, &longs;imiliumque motuum, quam quereba
mus.
SI idem graue naturaliter cadens percurrerit à quiete
duo &longs;patia; tempora erunt in ratione &longs;ubduplicat&atail;
corundem &longs;patiorum.
Ex Cor.
pr: 4. huius con&longs;tat rectangula e&longs;&longs;e gene&longs;es &longs;im
plicium motuum grauium naturaliter de&longs;cendentium, &
ex def. 7. primi liquet ea&longs;dem gene&longs;es e&longs;&longs;e motuum &longs;imi
lium. Cumque eiu&longs;dem mobilis naturaliter cadentis ve
locitas à quiete &longs;it vna, eademque; &longs;implices motus erunt
ij, vt gene&longs;um &longs;imilium, &longs;impliciumque motuum amplitu
dines æquales &longs;int, proptereaque, vt in figura præcedentis
propo&longs;itionis æquales erunt C, E, atque adeo &longs;patiu&mtail;
C, &longs;iue E ad D erit in duplicata ratione temporum E ad F.
TEmpora &longs;imilium vibrationum &longs;unt in &longs;ubduplicata
ratione arcuum exactorum, &longs;eu longitudinum pen
dulorum, quorum &longs;unt vibrationes. Sint grauia pendula
LA, LF, quæ ab eadem recta LF di&longs;cedentia currant &longs;u&longs;
pen&longs;a ex L duos &longs;imiles arcus circulares FI, AC. Dico
tempora horum de&longs;cen&longs;uum e&longs;&longs;e in ratione &longs;ubduplicat&atail;
arcuum FI, AC, &longs;eu longitudinum filorum, aut ha&longs;tularum
FA, LA. Ducamus quamcumque rectam LBG, erit AB
ad BC, vt FG ad GI, & cum præterea velocitates pendu
lorum a quiete in A, F &longs;int æquales, pariterque velocita
tes æquales a quiete in B, G; erit velocitas in A ad veloci
tatem in B, vt velocitas in F ad velocitatem in G, quare
con&longs;ideratis arcubus ABC, FGI, vt altitudines rect&ecedil;, (quæ
item forent in B, G proportionaliter &longs;ect&ecedil;) gene&longs;um &longs;imi
lium &longs;impliciumque motuum, quarum amplitudines æqua
les &longs;unt, erunt &longs;patia in acceleratis decur&longs;ubus per FI, AC
in ratione duplicata temporum, &longs;cilicet ip&longs;i arcus, aut lon
gitudines LF, LA erunt in ratione duplicata temporu&mtail;. Quod &c.
Idem demon&longs;tratum e&longs;&longs;et beneficio imaginum, quæ vt
pote eorundem illorum motuum &longs;implicium, forent etiam
&longs;imilium, & &longs;unt amplitudines æquales, etenim eæde&mtail;
&longs;unt, ac gene&longs;um ergo rur&longs;us &longs;patia, hoc e&longs;t arcus ABC,
FGI, nempe longitudines filorum IF, AC erunt in ratione
duplicata temporum. Quod &c.
tulimus, nec dubium, quin illa extendi queat ad qua&longs;cum
que lineas decur&longs;uum, dummodo &longs;imiles, ac &longs;imiliter po&longs;itas in
ij &longs;dem, vel æqualibus ab horizonte planis elenatis, quemad
modum Dominus Viuianus pulcherrimè propo&longs;uit.
TEmpora lationum à quiete per plana eandem eleua
tionem habentia &longs;unt homologè vt longitudines
planorum.
Sint plana AB, AC eandem eleuationem AD habentia.
Dico tempus lationis per AC ad id per AB e&longs;&longs;e vt AC ad
AB. (hæc Torricellij propo&longs;itio,
eandem veritatem ex no&longs;tris principijs demon&longs;trare
e&longs;t, non vt de re illa dubitemus, immò contrà, quòd de e&atail;
plenè &longs;atisfacti &longs;imus, ex eo rur&longs;us demon&longs;trandam &longs;u&longs;ce
pimus, vt exinde methodus no&longs;tra, quàm vera &longs;it, eluce&longs;
cat) Momentum de&longs;cen&longs;us inplano AC ad id de&longs;cen&longs;us &longs;u
per plano AB e&longs;t vt AB ad AC; &longs;unt autem
grauium, etiam &longs;uper planis inclinatis motus, quos &longs;impli
ces appellamus, inter &longs;e &longs;imiles, nempe quorum gene&longs;es
&longs;unt rectangula; ergo habebimus &longs;implices gene&longs;es, vnam,
cuius altitudo AC amplitudoque AB; alteram, cuius am
plitudo AC, altitudo autem AB; itaque propo&longs;itis &longs;patijs
AC, AB, primi&longs;que velocitatibus AB, AC, &longs;i fiat AB ad AC
vt CA ad EA, erit EA ad AB duplicata
ratio temporum per AC, AB erit CA ad AB. Quod &c.
motu
II&longs;dem pror&longs;us manentibus demon&longs;trarunt Gallileus, ac
Torricellius, gradus velocitatum acqui&longs;itos in B, et C
eiu&longs;dem mobilis de&longs;cendentis à quiete in A pares e&longs;&longs;e;
idip&longs;um nos o&longs;tendemus.
Cum tempora &longs;int vt AC ad AB, & velocitates à quie
te in ratione reciproca temporum, &longs;cilicet vt AB ad AC,
&longs;int deinde velocitates eæ vt amplitudines imaginum &longs;im
plicium, &longs;imiliumque illorum motuum (nam amplitudines
imaginum velocitatum &longs;unt pror&longs;us eædem, ac illæ gene
&longs;um) erunt ip&longs;æ imagines &longs;implicium motuum æquales;
nam tempora, quæ &longs;ummuntur vt altitudines imaginum
reciprocantur, vt dictum e&longs;t, amplitudinibus, &longs;eu primis à
quiete velocit atibus, at in motibus acceleratis ip&longs;æ inte
græ imagines &longs;implicium motuum &longs;unt loco graduum ve
locitatum in extremo &longs;patiorum acqui&longs;itorum; ergo in B, et
C gradus velocitatum æquales erunt.
33.
SI æqualia pondera, &longs;u&longs;pen&longs;a &longs;int ex filis, quorum par
tes inter&longs;e æquales, præ tractione æqualiter elongen
ter tempora in reditu ip&longs;orum filorum, cum ab ip&longs;is graui
bus &longs;tatim liberantur, æqualia erunt. Hoc primùm
&longs;trabimus alia via, tum methodo no&longs;tra, vt de ea aliud
exemplum tradamus. Sint funiculi AB, DC, & ex ijs
pendeant æqualia grauia B, C, adeo vt &longs;umptis hinc indè
partibus æqualibus eorundem &longs;uniculorum, con&longs;tet ip&longs;as
æqualiter ab ip&longs;is grauibus trahi, atque produci. Dico, &longs;i
elongationes &longs;int HB, GC, & omnibus &longs;ic &longs;tantibus pon-
dem extremitates re&longs;tituantur in H, et G æqualibus tem
poribus. Sit AE æqualis DC, erit porrò elongatio facta
per idem graue B, quæ &longs;it EF, æqualis GC; propterea li
beratis funiculis ad B, et C, eodem tempore re&longs;tituetur C
in G, ac E in F, quo tempore etiam B in H re&longs;titutum fue
rit; nam vno puncto in primum &longs;uum locum redito, etiam
alia &longs;ingula in &longs;uum locum perueni&longs;&longs;e, opportebit.
HAc occa&longs;ione de funiculis erit non iniucunda di&longs;er
tatio, remque &longs;ic adhuc intactam promouebimus,
&longs;imulque demon&longs;trabimus.
Idip&longs;um propo&longs;itum no&longs;tris principijs &longs;ic demon&longs;tra
mus.
Sint
libet partibus inter&longs;e æqualibus,
plentibus, hæ &longs;ingulæ æqualiter à pondere B trahentur,
eritque BH &longs;umma omnium dictarum partium elongatio
num, & eodem pacto EF erit &longs;umma elongationum
omnium in AE contentarum, ab eodemque pondere effe
ctarum; propterea vt AB ad BH, ita erit AE ad EF; quamo
brem velocitas etiam puncti B &longs;ublato pondere B erit ad
velocit atem puncti E ob eandem detractionem, vt BH ad
EF, vel BA ad EA (nam quot &longs;unt partes concept&ecedil; i&ntail;
vtraque fili longitudine, totidem &longs;unt etiam impetus inter
&longs;e æquales) idem o&longs;tenderemus &longs;i loco ponderis B, minus
quodcumque &longs;u&longs;penderemus, vt &longs;cilicet puncta B, et E ad
quemuis locum &longs;uperius remanerent, librarenturque cum
re&longs;i&longs;tentijs
& puncti E in F &longs;ubducto pondere B erunt motus &longs;imilium
&longs;impliciumque; &longs;ed motus ex C in G exempto pondere C
e&longs;t pror&longs;us idem, ac motus E in F, ergo motus &longs;imiles, ac
rati, gene&longs;es habebunt, quarum primæ velocitates, &longs;eu am
plitudines proportionales &longs;unt altitudinibus earundem,
&longs;patijs nimirum CG, BH accelerato motu exigendis; qua
mobrem componentur ex ratione ip&longs;arum velocitatum,
&longs;eu amplitudinum CG ad BH, & ex ea quadratorum tem
porum, quæ proinde æqualitatis erit; itaque etiam huius
&longs;ubduplicata; hoc e&longs;t tempora in tran&longs;itibus accelarato
motu exactis, erunt paria.
cedentiæ &longs;u&longs;pen&longs;a &longs;int æqualia pondera, tunc primas velocita
tes, &longs;ubductis ponderibus, fore in eadem ratione
vel longitudinum filorum.
SI extremitatibus funiculorum ex vna parte
ac eandem cra&longs;&longs;itiem habentium, nec non eiu&longs;dem
cædentiæ exi&longs;tentium, fuerint &longs;u&longs;pen&longs;a æqualia pondera,
quæ inde ij&longs;dem longitudinibus &longs;eruatis, quomodo opor
tet tollantur, erunt &longs;patia recur&longs;uum, temporibus &longs;impli
cium motuum exacta in ratione longitudinum pendulo
rum.
Sit funiculus AC æquè cra&longs;&longs;us ac BD, & &longs;u&longs;pen&longs;is
hinc inde ponderibus æqualibus, elongatio primi funiculi
&longs;it CE, & alterius &longs;it DF. Dico &longs;patia temporibus &longs;impli
cium imaginum, ab extremitatibus &longs;olutis exacta, fore i&ntail;
ratione longitudinum ip&longs;orum funiculorum.
Iam con&longs;tat CE ad DF e&longs;&longs;e, vt AC ad BD, in qua ratione
&longs;unt etiam velocitates à quiete, dum pondera &longs;ubduceren
tur ex E, et F, vel ex alijs punctis quibu&longs;cunque &longs;i æqualia
&longs;ic enim concipiuntur gene&longs;es &longs;imilium, &longs;impliciumque
motuum, quarum altitudines æquantur elongationibus
funiculorum; propterea &longs;patia recur&longs;uum temporibus &longs;im
plicium motuum exacta, nectentur ex rationibus duplicata
CE ad DF, hoc e&longs;t AC ad BD, & ex reciproca filorum,
&longs;cilicet BD ad AC, quæ ratio, vti diximus, e&longs;t reciproc&atail;
primarum velocitatum, &longs;eu amplitudinum gene&longs;um &longs;impli
cium, ergo ip&longs;a &longs;patia in reditu filorum ab extremitatibus
&longs;olutis exacta, erunt vt AC ad BF, &longs;eu vt CE ad DF. Quod &c.
TEmpora &longs;implicium, &longs;imiliumque dictorum motuum
&longs;unt æqualia.
Nam cor.
2. pr. 8. huius primi demon&longs;tratum e&longs;t, tem
pora &longs;implicium, &longs;imiliumque motuum componi ex ratio
ne &longs;patiorum, &longs;eu altitudinum gene&longs;um, & reciproca pri
marum, aut extremarum velocitatum, &longs;eu amplitudinum
gene&longs;um: &longs;unt autem altitudines gene&longs;um tractiones, &longs;eu
elongationes funiculorum, quæ &longs;unt vt longitudines funi
culorum, ergo tempora æqualia erunt.
niculorum, e&longs;&longs;e compo&longs;ita ex ratione elongationum funiculo
rum, & ex reciproca primarum velocitatum.
ius, in eo tantùm variatur, quod ibi ponuntur data &longs;pati&atail;
elongitiones funiculorum, hic verò tempora &longs;impliciu&mtail;
motuum, & quia elungationes o&longs;ten&longs;æ &longs;unt proportionales &longs;pa
tijs nunc exactis, manife&longs;tum e&longs;t, no&longs;tri iuris e&longs;&longs;e modò &longs;patia
acceleratis motibus exact a ex temporibus &longs;implicium
datis concludere, modò contrà, ex &longs;patijs altitudinibus gene
&longs;um proportionalibus, qua item data &longs;unt, tempora inuenire,
qua proinde methodus mihi videtur ampli&longs;&longs;ima.
SI eiu&longs;dem cra&longs;&longs;itiei funiculis pondera dependeant, qu&ecedil;
&longs;int in ratione reciproca longitudinum ip&longs;orum funi
culorum, &longs;patia temporibus gene&longs;um &longs;implicium motuum
exacta erunt in ratione duplicata elongationum.
cra&longs;sities
tur
idque gene&longs;es
les (nam cum pondera erant æqualia, primæ velocitates
proportionabantur longitudinibus
pondera reciprocantur longitudinibus ij&longs;dem, &longs;eu viribus
funiculorum, fit vt primæ velocitates æquales reddantur)
cum ergo ita &longs;it, &longs;patia recur&longs;uum temporibus imaginu&mtail;
&longs;implicium & accelerato motu confecta erunt in ratione
duplicata elongationum.
ta temporum, erunt ip&longs;a tempera in ratione &longs;ubduplicat&atail;
elongationum.
SI funiculis æqualem cra&longs;&longs;itiem habentibus fuerint &longs;u&longs;
pen&longs;a inæqualia pondera, &longs;patia, quæ acceleratis mo
tibus, ac temporibus gene&longs;um &longs;implicium recurruntur ne
ctentur ex ratione duplicata elongationum, & ex duabus
reciprocè &longs;umptis rationibus, nempe longitudinum prima
rum funiculorum, antequam pondera &longs;u&longs;penderentur; &
ip&longs;orum ponderum.
In antecedenti figura illud primum &longs;atis patet, quòd &longs;i
loco ponderis F &longs;u&longs;pen&longs;um fui&longs;&longs;et pondus aliud grauius,
aut leuius, prior velocitas in a&longs;cen&longs;u fili, &longs;eu funiculi, aut
chordæ aucta, vel imminuta fui&longs;&longs;et pro magnitudine pon
deris &longs;ub&longs;tituti; quamobrem priores velocitates ex inæqua
litate ponderum eidem chordæ &longs;u&longs;pen&longs;orum dependentes
forent, vt ip&longs;a pondera; verùm cum &longs;uppo&longs;itis funiculis
æqualia pondera &longs;u&longs;pen&longs;a veniunt, primæ velocitates &longs;unt
vt longitudines funiculorum, ergo velocitates primæ, cum
inæqualia &longs;unt pondera, quæ &longs;ubtrahuntur, nectentur ex
ratione longitudinum funiculorum, & ex ea ponderum
inæqualium: quæcumque igitur &longs;it tractio DF, gene&longs;es ha
bebimus &longs;imilium &longs;impliciumque motuum, vnam, cuius al
titudo CE, & alteram habentem altitudinem DF, & &longs;unt
earundem gene&longs;um amplitudines, &longs;eu primæ velocitates
in ratione compo&longs;ita funiculorum AC ad BD, & ponderis
pendentis ex E ad pondus &longs;u&longs;pen&longs;um in F; ergo &longs;patia ac
celeratis motibus tran&longs;acta temporibus gene&longs;um
dinum gene&longs;um, & ex duabus rationibus reciprocè &longs;um
ptis funiculorum AC ad BD, & ponderum E ad F. Quod &c.
pr.
II&longs;dem po&longs;itis, &longs;i &longs;patia recur&longs;uum erunt ip&longs;æ elongatio
nes, tempora, quibus ab extremitatibus &longs;olutis recur
runtur, erunt in ratione &longs;ubduplicata eorundem. Nam cum
gene&longs;es &longs;imilium, &longs;impliciumque motuum &longs;int æquè am
plæ, erunt, tempora in ratione &longs;ubduplicata imaginum,
&longs;eu &longs;patiorum acceleratorum motuum, &longs;unt verò &longs;patia
ip&longs;æ elongationes; ergo &c.
CHordæ non eiu&longs;dem cra&longs;&longs;itiei, eiu&longs;dem tamen mate
riæ, ac longitudinis, tunc æquè trahentur
&longs;aNam
cra&longs;&longs;ior chorda pote&longs;t concipi compo&longs;ita ex funiculis eiu&longs;
dem cra&longs;&longs;itiei alterius chordæ, &longs;i illa huius fuerit multiplex,
& &longs;i partes exilior funiculus fuerit alterius cra&longs;&longs;ioris, erit
cra&longs;&longs;ities alicuius alterius funiculi, quæ pluries accept&atail;
con&longs;tituere poterit vtranque cra&longs;&longs;itiem funiculorum pro
po&longs;itorum (hìc enim non accidit enumerare cra&longs;&longs;ities in
ter&longs;e irrationales, quippe quia, quod de iam dictis o&longs;ten
derimus, de his quoque facilè e&longs;t iudicare, &longs;ecùs e&longs;&longs;emus
longi, quam par e&longs;t, poti&longs;&longs;imùm cum hæc præter
adijciantur, & quidem vt con&longs;tet, quomodo methodus i&longs;ta
no&longs;tra facilis &longs;it, ac vtili&longs;&longs;ima) quapropter &longs;i cuique acce
ptarum æqualium chordarum, pondera æqualia &longs;u&longs;pen&longs;a
&longs;int, porrò hæc omnes æquè trahentur ab ip&longs;is æqualibus
ponderibus, & &longs;ic etiam compo&longs;ita, nempe choidæ pro-
&longs;icut propo&longs;uimus; ergo patet propo&longs;itum.
SI fuerint eiu&longs;dem materiæ funiculi, & &longs;int illis &longs;u&longs;pen&longs;a
pondera cra&longs;&longs;itiebus proportionalia, ratio &longs;patiorum
in reditibus accelerato motu exactorum,
plicium gene&longs;um, erit eadem ac funiculorum.
42.
Nam, vt in præcedenti figura, erit tractio CE ad DF ita
AC ad BD, vel AE ad BF, &longs;unt autem primæ velocitates,
&longs;eu amplitudines gene&longs;um &longs;implicium, &longs;imiliumque
in ratione funiculorum, ergo decur&longs;uum &longs;patia motibus
acceleratis exacta nectentur ex ratione duplicata altitu
dinum gene&longs;um &longs;implicium, nempe duplicata
& reciproca amplitudinum, &longs;untque ip&longs;æ amplitudines
homologè vt longitudines funiculorum, ergo relinquitur
vt ip&longs;a &longs;patia &longs;int in vnica ratione longitudinum funicu
lorum.
pr.
Quòd &longs;i &longs;patia recur&longs;uum ponantur ip&longs;æ tractiones, vel
longitudines funiculorum, o&longs;tendetur tempora e&longs;&longs;e æqua
lia, quemadmodum æqualia &longs;unt tempora &longs;uperius pro
po&longs;ita &longs;implicium gene&longs;um.
SI eiu&longs;dem materiei quibu&longs;cunque funiculis aligentur
quæcunque pondera, ijs &longs;ublatis a&longs;cen&longs;uum &longs;patia ab
extremitatibus &longs;olutis exacta temporibus gene&longs;um &longs;impli
cium, ijs nempe quæ impenderentur in motibus iuxta &longs;im
plices gene&longs;es, erunt in ratione compo&longs;ita quadratorum
elongationum chordarum, ex ea cra&longs;&longs;itierum, & ex duabus
reciprocè &longs;umptis rationibus, nempe longitudinum fu
niculorum antequam traherentur; & &longs;u&longs;pen&longs;orum ponde
rum.
Funiculi AB, GH trahantur à ponderibus quibu&longs;cunque
C, I in C, et I. Dico &longs;i exempta &longs;int pondera, fore, vt &longs;patia
quæ acceleratis motibus exiguntur ab extremitatibus &longs;o
lutis C, I &longs;int in ratione compo&longs;ita ex duplicata IH ad BC,
cra&longs;&longs;itudinis ad cra&longs;&longs;itudinem funiculorum AB, GH; dein
de ex funiculi longitudine HG ad longitudinem AB, pon
deri&longs;que I ad pondus C. Intelligatur funiculus, &longs;eu chor
da, æque cra&longs;&longs;a, ac &longs;imiliter cedens, quàm GH (id quod
&longs;emper intelligimus quoties funiculi, inter&longs;e comparantur)
&longs;ed æquè longa, ac AB, &longs;itque illi pondus F adiectum, ad
quod C eandem habeat rationem, ac cra&longs;&longs;ities AB ad cra&longs;
&longs;itiem DE, con&longs;tat elongationem EF æqualem fieri ip&longs;i
CB, & cum primæ velocitates, &longs;eu amplitudines æquè al
tarum gene&longs;um &longs;imilium, &longs;impliciumque motuum &longs;int
æquales, &longs;patia decur&longs;uum acceleratis motibus exacta
pror&longs;us æqualia; &longs;unt verò funiculi DE, GH eiu&longs;dem cra&longs;
&longs;itiei, ei&longs;que &longs;unt &longs;u&longs;pen&longs;a duo'pondera inæqualia F, I; ergo
decur&longs;uum &longs;patia ab extremitatibus &longs;olutis exacta
tur
ex ratione, quam habent longitudines funiculorum HG ad
DE, &longs;eu AB, & ex ea ponderum I ad F; verùm pondera I
ad F nectuntur ex rationibus ponderum I ad C et C ad F,
quæ po&longs;trema e&longs;t ratio cra&longs;&longs;itiei funiculi AB ad craffitiem
funiculi DE, &longs;eu GH; ergo vt propo&longs;uimus &longs;patia accele
ratis motibus exacta, nectentur ex rationibus
CB ad HI; cra&longs;&longs;itudinum funiculorum AB, GH;
I ad C, & longitudinum HG ad AB. Quod &c.
TEmpora gene&longs;um &longs;implicium, dum chordis &longs;u&longs;pen
duntur quæcunque grauia, nectuntur, ex ratione
elongationum funiculorum, & ex contrariè &longs;umptis ratio
nibus, cra&longs;&longs;itudinum, longitudinumque funiculorum, nec
Nam Cor: 2. pr. 8. primi demon&longs;tratum e&longs;t, tempor&atail;
&longs;implicium &longs;imiliumque motuum componi ex ratione &longs;pa
tiorum, &longs;eu altitudinum gene&longs;um, & reciproca primarum
velocitatum, &longs;eu amplitudinum gene&longs;um, &longs;unt autem alti
tudines gene&longs;um tractiones, &longs;eu elongationes funiculorum;
velocitatesverò primæ nectuntur ex rationibus cra&longs;&longs;itudi
num, & ex ea longitudinum funiculorum antequam tra
herentur (hoc enim &longs;ubinde o&longs;tendemus) ergo tempora
propo&longs;ita &longs;implicium gene&longs;um, dum chordis
cunque inæqualia pondera, componentur ex rationibus
elongationum funiculorum, & ex contrariè &longs;umptis cra&longs;&longs;i
tudinum, longitudinumque funiculorum, & ponderum.
VErùm primæ velocitates in ij&longs;dem chordis componi
ex ratione cra&longs;&longs;itudinum, longitudinum
& &longs;u&longs;pen&longs;orum ponderum, &longs;ic o&longs;tendemus,
Quoniam in eadem po&longs;trema figura velocitas, qua&mtail;
haberet funiculus AB ex liberatione ponderis e&longs;t æqualis
velocitati, quam haberet alius funiculus, vbi hic etiam li
beraretur à pondere, &longs;cilicet cum pondera cra&longs;&longs;itiebus fu
niculorum proportionalia &longs;unt, & ip&longs;i funiculi æquè longi;
velocitas funiculi DE à pondere F ad velocitatem
funiculi, &longs;i loco ponderis F &longs;ub&longs;titutum e&longs;&longs;et aliud æquale
ip&longs;i I, e&longs;&longs;et vt pondus F ad &longs;ub&longs;titutum, &longs;eu ad I, e&longs;t autem
velocitas eiu&longs;dem funiculi DE, dum fui&longs;&longs;et pondus ei &longs;u&longs;
pen&longs;um æquale I ad velocitatem funiculi GH a pondere I
vt longitudo DE ad GH; ergo patet propo&longs;itum.
grauatis, non ab&longs;imili modo præst abimus in chordis ad
que
ne, vt &longs;i in ijs pondere &longs;ublato, motus extremitatis &longs;olutæ at
tendebatur, hìc media parte attractâ chordâ, & &longs;ubinde &longs;ui
iuris relictâ, vibrationem eius ob&longs;eruamus, & equidem illa
omnia in hunc finem o&longs;tendimus, quippe ab hac re, plurima
vtili&longs;&longs;imæque veritates manere po&longs;&longs;unt. Nam de arcubus po&longs;&longs;es
pulcherrima in&longs;titui ratio, & qui vellet armonicorum &longs;ono
rum, vel vocum per chordarum vibrationes editarum, tempo
ra, cum &longs;oni ad aures perueniunt, inue&longs;tigare, reor non aliam
viam, quàm hanc ingredi nos debere, atque indè con&longs;onantia
rum forta&longs;&longs;e naturam pertipere po&longs;&longs;e, vt primus
lileus quamquam vibrationes ten&longs;arum chordarum
ab ijs pendulorum.
IN que&longs;ta figura &longs;i e&longs;prime vna nuoua
tione
in grande, per tirar granate, ò &longs;a&longs;&longs;i può e&longs;
&longs;ere di gran con&longs;eguenze nella militare, co
me dimo&longs;trera&longs;&longs;i.
Dalle &longs;ue parti &longs;i verrà in cognitione del
modo di fabbricarla, e &longs;ono le &longs;eguenti.
AM, MN &longs;ono amendue le braccia.
Il punto M è il cen
tro della machina. Per la cauità M deue pa&longs;lar la pall&atail;
&longs;cagliata dalla corda; e per di &longs;otto M &longs;i ferma & inca&longs;lra
nel manico, al modo delle bale&longs;tre communi. Ai due capi,
ò &longs;iano e&longs;tremità A, N &longs;i annette la fune. I punti A, E, F,
G, I, K &longs;ono in vna linea retta. Gl' interualli AE, EF, FG,
GI, IK, &longs;ono, benche non di nece&longs;&longs;ità, eguali. Le altezze,
ò comme&longs;&longs;ure KL, IH, GD, FC, EB perpendiculari, nell'
incuruar&longs;i dell' arco, &longs;i aprono intorno a' centri K, I, G, F,
E. Donde ne &longs;iegue, che prendendo la corda dal &longs;uo mez
zo, e tirandola ver&longs;o O; amendue le braccia &longs;i aprono nel
le predette comme&longs;&longs;ure, come compare nell' vno d'e&longs;&longs;i &longs;e
gnato a punti con le lettere corri&longs;pondenti. Cia&longs;cuna
delle predette comme&longs;&longs;ure viene &longs;trettamente rin&longs;errat&atail;
da vna molla, come &longs;i vede in L, H, D, C, B; e que&longs;te mol
le, quanto più &longs;i auuicinano al centro M, deuono e&longs;&longs;ere più
grandi e più ma&longs;&longs;iccie, in modo che, per cagione della
dezza
dell'altre, e per cagione della gro&longs;&longs;ezza, habbiano nel &longs;er
rar&longs;i maggior forza, ò &longs;ia momento, per la ragione, che &longs;ot
to &longs;i dirà.
Ciò pre&longs;uppo&longs;to, è facil co&longs;a dimo&longs;trare i vantaggi di
Primieramente nel triangolo ALK, e&longs;&longs;endo le altezz&etail;
EB, FC, GD, IH, KL perpendicolari, e perciò paralelle;
ne &longs;iegue che le proportioni di AE ad EB, di AF ad FC, di
AG a GD, di AI ad IH, di AK a KL &longs;ieno tutte eguali; e
douendo e&longs;&longs;ere parimente eguali le re&longs;i&longs;tenze delle molle
in B, C, D, H, L, che &longs;i &longs;uppongono di egual neruo nell'
aprir&longs;i; ne &longs;iegue (&longs;econdo i principij della Meccanica) che
attraendo&longs;i con la fune l'e&longs;tremità A, nel mede&longs;imo tempo
e con la mede&longs;ima facilità vincera&longs;&longs;i l'equilibrio di tutte le
molle; la re&longs;i&longs;tenza delle quali &longs;i con&longs;idera in ragione di
pe&longs;o, &longs;i come le linee AE, EB; AF, FC; AG, GD; &c. &longs;i con
&longs;iderano come vetti, ò lieue, che hanno i loro ippomoclij, ò
&longs;iano centri in E, F, G, I, K, e la potenza in A, la quale è
comune a tutte.
In &longs;econdo luogo, hauendo il braccio AE al braccio EB
(il &longs;imile dica&longs;i degli altri) hauendo, dico, gran proportio
ne, re&longs;terà molto ageuolato il moto.
Terzo e&longs;&longs;endo molte le molle, e a prendo&longs;i tutte, ne deue
&longs;eguire vn notabile incuruamento d'amendue le braccia;
onde la&longs;ciando l'arco in libertà, e chiudendo&longs;i tutte le &longs;u
det te molle nel mede&longs;imo tempo, cioè qua&longs;i in vn'attimo;
dourà la corda, che era tirata ver&longs;o O, pa&longs;&longs;are qua&longs;i in
te
locità, per la grandezza dello &longs;patio; e a que&longs;ta corri&longs;pon
dendo la forza, ne &longs;eguirà vn colpo molto con&longs;iderabile, e
vantaggio&longs;o, come cia&longs;cuno può arguire.
Re&longs;tano hora a &longs;eior&longs;i alcune difficoltà.
La prima è,
che, quantunque &longs;ia vero, che quella forza ba&longs;tante in A
per vincer l'equilibrio della molla B, quella mede&longs;ima al
tre&longs;i &longs;ia &longs;ufficiente a vincer l' equilibrio di tutte l'altre, per
e&longs;&longs;ere eguali le proportioni delle vetti; ciò non o&longs;tante,
&longs;iderando&longs;i
KLA &longs;egnato a punti, le proportioni rie&longs;cono alterate; do-
più le lunghezze di prima, ma bensi le applicate di detto
arco, cioe af, ag, ai, ak; delle quali aK, e l' altre a lei più
vicine &longs;i abbreuiano molto più quando l' arco è incurua
to, che quando non è: Onde per tal ragione dourebbero
le parti più vicine al centro M aprir&longs;i meno dell'altre più
vicine alle e&longs;tremità. A ciò &longs;i ri&longs;ponde, che per e&longs;&longs;er la
corda a o più obliqua alla lunghezza a e di quel che &longs;ia all'
altre più vicine al centro M, quindi ne &longs;iegue, che per quel'
altra cagione s' aprono più ageuolmente le parti vicine al
centro; onde, temperata vna ragione con l' altra (quando
l' arco non &longs;ia e&longs;tremamente incuruato) &longs;i con&longs;egui&longs;ce vno
&longs;tato d'apertura opportuna.
La &longs;econda difficoltà è che cia&longs;cuna molla nel &longs;uo re
&longs;tringer&longs;i, par che cagioni qualche effetto contrario all'in
tento. Imperoche, per e&longs;empio, nella molla B il mezzo
anello, che ri&longs;guarda l'e&longs;tremità A, nello &longs;tringer&longs;i fà ben&longs;i
il &longs;uo douere, perche il &longs;uo moto è ver&longs;o il centro M; ma l'
altra metà, che ri&longs;guarda il &longs;udetto centro M, nello &longs;trin
ger&longs;i, hauendo il &longs;uo moto ver&longs;o A, &longs;i oppone al chiudi
mento della molla &longs;eguente C; e il &longs;imile dica&longs;i dell' altre. A ciò &longs;i è po&longs;to rimedio col far più grandi, e più mafficcie
le molle più vicine al centro M, accre&longs;cendole, e ingro&longs;&longs;an
dole di mano in mano opportunamente. Quindi ne &longs;egue
che per la maggior grandezza
aprir&longs;i con facilità; ma all' incontro nel &longs;errar&longs;i, per e&longs;&longs;ere
più ma&longs;&longs;iccie, e di maggior corpo, vengono ad hauere
maggior momento delle men corpulenti, &longs;uperando co&ntail;
ciò non &longs;olo il detto moto oppo&longs;to, ma etiandio impri
mendo maggior moto al ferro dell'arco, con cui &longs;i acco
muna il moto.
Auuerta&longs;i, che quanto &longs;aranno di maggior numero le
corpo, tanto riu&longs;cirà il colpo a di&longs;mi&longs;ura maggiore, per l'
mento delle molle; e ciò con adoperare la mede&longs;ima
forza.
Auuerta&longs;i parimente, che il braccio AE, è il &longs;uo corri&longs;
pondente deuono e&longs;&longs;ere alquanto più corti, cioè A vna
delle e&longs;tremità dell'arco deue e&longs;&longs;ere più ver&longs;o il centro di
quel che &longs;ia il concor&longs;o delle linee LB, KE, come pure dall'
altra parte; perche &longs;i vede che aprendo&longs;i meno le parti vi
cine ad A, l'altre molle fanno miglior effetto.
Finalmente la &longs;perienza ha mo&longs;trato, che e&longs;&longs;endo&longs;i la
uorata vna tal machina con pochi&longs;&longs;imi nodi, ageuoli&longs;&longs;ima
ad aprir&longs;i, e &longs;enza hauer ingrandite e ingro&longs;late le molle,
che più &longs;i vanno auuicinando al centro M, come &longs;i è det
to; con tutto ciò l' ordigno è riu&longs;cito di forza molto &longs;upe
riore a vna bale&longs;tra grande, e difficilif&longs;ima a inarcar&longs;i. On
de non dubito, che, facendo&longs;i con tutte ie regole accenna
te, non debba riu&longs;cire vna machina di effetto marauiglio&longs;o
aggiungendo che per tirar granate dourebbero i bracci
e&longs;&longs;er di legno, armati di ferro &longs;ol doue &longs;i richiede.
Explicatio.
IN hac figura exprimitur nouum genus Bali
&longs;tæ, quæ machina præ&longs;ertim in mole maio
ri, non parum vtilitatis afferre pote&longs;t rei mi
litari ad eiaculanda mi&longs;&longs;ilia, vt demon&longs;tra
bitur. Ex eius verò partibus, quas &longs;ubinde
recen&longs;eo, etiam modus &longs;tructuræ apparebit.
AM, MX &longs;unt brachia.
Punctum M centrum machi
næ. Per
Infra M in
&longs;eritur manubrium, vt in bali&longs;tis vulgaribus. Extremis
capitibus A, N adnectitur funis. Puncta A, E, F, G, I, K
&longs;unt in linea recta. Interualla AE, EF, FG, GI, IK &longs;unt (li
cèt non nece&longs;&longs;ariò) æqualia. Altitudinis, &longs;eu commi&longs;&longs;uræ
KL, IH, GD, FC, EB &longs;unt perpendiculares rectæ occultæ
KA. Singulæ autem, dum curuatur arcus, aperiuntur cer
ca centra K, I, G, F, E. Hinc &longs;equitur vt funis ex medio
dum attrahitur in O, aperiantur prædictæ commi&longs;&longs;uræ, &longs;eu
nodi, & curuentur vtraque brachia, vt in eorum altero ap
paret punctis notato. Quilibet ex his nodis arcti&longs;limè &longs;trin
gitur &longs;upernè, a &longs;uo elaterio, vt videre e&longs;t in L, H, D, C, B. Elateria autem quò propinquiora centro M tanto maiora,
& cra&longs;&longs;iora debent e&longs;&longs;e remotioribus: Hinc fit vt, propter
molem opportunè auctam, æquè facilè aperiantur, ac cæ
tera; & vice ver&longs;a, propter cra&longs;&longs;itiem maiorem, &longs;ibi relicta
validiùs re&longs;tringantur. Cuius rei paulo infra rationem
dabimus.
His po&longs;itis facile e&longs;t o&longs;tendere, quantum præ&longs;tet hu
iu&longs;cemodi machina vulgaribus & communibus bali&longs;tis.
Primùm, in Triangulo ALK cùm altitudines EB, FC,
GD, IH, KL &longs;int perpendiculares, ideoque parallelæ, hinc
IH, AK ad KL &longs;int æquales. Sunt pariter æquales re&longs;i
&longs;tentiæ elateriorum in B, C, D, H, L (po&longs;uimus enim ela
teria ita opportunè aucta vt æquè facile &longs;ingula aperian
tur) ergo (ex primis principijs mecanicorum) dum attra
huntur fune extrema capita A, N, eodem tempore, eadem
que facili ate vincetur æquilibrium omnium elateriorum,
quorum re&longs;i&longs;tentia in &longs;ingulis con&longs;ideratur in ratione pon
deris, quemadmodum lineæ AE, EB; AF, FC; AG, GD
&c. con&longs;iderantur vt vectes, quorum hippomoclia &longs;eu
tra
communis omnibus.
Scundò, cùm AE ad EB (idem die de cæteris) habeant
magnam proportionem, facilè aperientur nodi, & curuabi
tur arcus; quantumuis augeatur numerus nodorum.
Tertiò Cum &longs;int plures nodi, atque omnes aperiantur,
nece&longs;&longs;e e&longs;t vt brachia arcus valdè incuruentur;
&longs;i idem arcus &longs;ibi relinquatur, prædicti nodi omnes, vi ela
teriorum, ictu oculi claudentur; e odemque puncto tempo
ris corda ex O percurret totum &longs;patium v&longs;que ad M: Quòd
cùm fieri nequeat ni&longs;i &longs;umma velocitate, propter magni
tudinem prædicti &longs;patij, & velocitati re&longs;pondent vis, atque
impetus, nece&longs;&longs;e e&longs;t vt hinc &longs;equatur ictus valde notabilis,
vt facilè e&longs;t vnicuique conijcere.
Super &longs;unt nunc difficultates nonnullæ &longs;oluendæ.
Prima
e&longs;t, quòd licèt vis &longs;ufficiens in A ad vincendum
elaterij B, illa eadem quoque &longs;ufficiat ad vincendum æqui
librium cæterorum, propter æquales proportiones
his tamen non ob&longs;tantibus, &longs;i con&longs;ideretur brachium iam
incuruatum, vt apparet in KLA punctis notato, proportio
nes illæ cernuntur notabiliter variatæ. Neque enim pro
longitudinibus vectium &longs;umi po&longs;&longs;unt longitudines priores,
&longs;ed loco ip&longs;arum accipiendæ &longs;unt applicatæ arcus, videli
cet af, ag, ai, ak quarum ak, eidemque propinquiores, Re&longs;pondeo, quòd corda ao cùm &longs;it obliquior re&longs;pectu
longitudinis ae, quàm re&longs;pectu cæterarum centro propin
quiorum, hinc fit vt, quantùm e&longs;t ex hac ratione, faciliùs
aperiantur partes propinquiores centro; quamobrem, vtra
que ratione inuicem temperata, dummodo arcus non &longs;it
&longs;ummè incuruatus omnes partes aperientur, quantum &longs;a
tis e&longs;t ad intentum.
Altera difficultas e&longs;t, quod elaterium quodlibet dum
re&longs;tringitur videtur ob&longs;tare motui elaterij &longs;equentis. Nam,
exempli gratia, in elaterio B &longs;emiannulus re&longs;piciens extre
mum A, dum &longs;tringitur, optim è præ&longs;tat &longs;uum effectum,
eius motus &longs;it versùs centrum M At è contrario reliqua
pars, &longs;eu &longs;emiannulus re&longs;piciens prædictum centrum M,
habeat &longs;uum motum ver&longs;us A videtur ob&longs;tare, quo minus
liberè claudatur &longs;equens elaterium C. Aque idem de cæte
ris dicendum. Huic incommodo con&longs;ultum e&longs;t augendo
magnitudinem, & cra&longs;&longs;itiem elateriorum, quò magis acce
dunt ad centrum M. Hinc enim &longs;equitur vt propter ma
gnitudinem facilè con&longs;entiant arcui, dum incuruatur; at
dum idem arcus liberè &longs;ibi relinquitur, cum &longs;int corpulen
tiora & cra&longs;&longs;iora habent maius momentum, quàm cætera
graciliora, ideoque non modo vincunt motum illum op
po&longs;itum, &longs;ed etiam imprimunt maiorem motum ferro ar
cus, cui ille motus communicatur.
Aduerte quod commi&longs;&longs;uræ &longs;eu nodi, quò plures fuerint,
elateria autem maioris ponderis, arcus denique corporis
gracilioris equæ expeditioris, tanto ictus longat præ&longs;tan
tior &longs;equetur, tum propter notabilem curuaturam brachio
rum, tum propter momentum maius elateriorum, &
po&longs;ita eadem potentia, aut etiam minori, pro vt longitudi
nes vectium &longs;tatuuntur.
Aduerte etiam, longitudinem brachij AE, eiu&longs;demqu&etail;
corre&longs;pondentis debere cæteris paribus nonnihil imminui,
tro M, quàm &longs;it concur&longs;us linearum LB, KE. Idem dicen
dum de altero extremo N. Nam cùm minus aperiantur
partes propinquiores puncto A, cætera elateria, vt com
pertum e&longs;t, meliorem effectum præ&longs;tant.
Fauet denique experientia.
Nam huiu&longs;cemodi machi
na pauci&longs;&longs;imis nodis con&longs;tructa, facillimæ curuaturæ, cum
elaterijs eiu&longs;dem pror&longs;us molis & cra&longs;&longs;itudinis; nihilomi
nus longè &longs;uperauit vim bali&longs;tæ communis maximæ, & dif
ficillimæ flexionis. Quamobrem non dubito quin, &longs;i præ
cepta &longs;uperiùs data exactè &longs;eruentur, elaborari po&longs;&longs;it ma
china miræ vtilitatis. Adde po&longs;tremo ad iacienda
mi&longs;&longs;ilia, vt e&longs;t genus quoddam bolidum, vulgo
portuniora e&longs;&longs;e brachia lignea, tantummodo, vbi nece&longs;&longs;i
tas po&longs;tulat, armata ferro.
ctor Pœnitent. in Metropol. Bonon.
pro Illu&longs;tri&longs;s. & Reverendi&longs;s. Domino
D. Iacobo Boncompagno Archiepi&longs;
copo, & Principe.
reui&longs;or, & imprimi po&longs;&longs;e cen&longs;uit.
Sancti Offi c ij Bononiæ.
BVLA I.
TABVLA I.
TABVLA II.
TABVLA III.
TABVLA VI.
TABVLA V.
TABVLA IV.
TABVLA VII.
TABVLA VIII.
TABVLA VIIII.