ARISTOTELIS
LOCA MATHEMATICA
Ex vniuer&longs;is ip&longs;ius Operibus collecta,
& explicata.
hactenus de&longs;ideratum.
Acce&longs;&longs;ere de Natura Mathematicarum &longs;cientiarum Tractatio;
Mathematicarum in Gymna&longs;io Parmen&longs;i Profe&longs;&longs;ore.
Ad Illu&longs;tri&longs;&longs;imum, ac Nobili&longs;&longs;imum
PETRVMFRANCISCVM MALASPINAM
Aedificiorum Marchionem, apud Cæ&longs;. Maie&longs;tatem
pro Sereni&longs;s. Parmen&longs;ium Duce Legatum.
BONONIÆ M. D C. X V.
Apud Bartholomæum Cochium.
Superiorum permi&longs;&longs;u.
Sumptibus Hieronymi Tamburini.
ILLVSTRISSIMO
AC NOBILISSIMO
PETROFRANCISCO
MALASPINAE
ÆDIFICIORVM MARCHIONI.
strum de Locis Mathematicis apud Ari
stotelem, vnà cum Tractatione de natura
&longs;cientiarum Mathematicarum, necnon
Clarorum Mathematicorŭ Chronologia;
quod tibi Mecœnati meo munificenti&longs;simo iure meritò
dicare, ac &longs;ub clarißimi tui nominis patrocinio in lucem
dare con&longs;titui. primùm quidem, vt mei perpetui erga te
amoris, & ob&longs;eruantiæ hoc vnum &longs;altem specimen exta
ret: tùm vt idoneum, æquumque propo&longs;itæ rei iudicem
nanci&longs;cerer. cùm enim adiu&longs;tum arbitrŭ duo potißimùm
requirantur, rerum &longs;cilicet cognitio, atque prudentia, quem
te rei, de qua agitur peritiorem, quemuè prudentiorem
inuenire potuerim? tu enim cùm Phy&longs;iologiæ, ac Mathe
maticarum omnium Encyclopædiam mirum in modum
per&longs;ua&longs;i&longs;ti, vt Archimedis, & Apollonij admirandis, ac
&longs;ubtilißimis Demon&longs;trationibus detinearis. Quanta por
rò in rebus agendis prudentia valeas, toti penè Europæ
innotuit, cùm pro no&longs;tris Sereniß. Ducibus, non &longs;olùm ad
omnes ferè Italiæ, atque Germaniæ Principes, verùm etiam
ad Cæ&longs;aream Maie&longs;tatem rebus fœliciter ge&longs;tis Legatus
decimùm extiteris; ac demùm à Sereniß. Duce Ranutio
inter primarios de Rep. Con&longs;iliorum Authores ad&longs;citus
fueris. Cæterùm in Clarorum Mathematicorum Chro
nologia perlegenda, &longs;æpißimè tibi nobilißimi æquè, ac do
ctißimi Viri, tui omnino per&longs;imiles occurrent, quod tibi
nonni&longs;i gratißimum accidere po&longs;&longs;e arbitror. Complectere
igitur ea benignitate, atque clementia, qua &longs;oles no&longs;tra stu
dia promouere, mea hæc quantulacumque munu&longs;cula. quæ &longs;i tibi accepta e&longs;&longs;e intellexero, iam tandem ma
ximorum munerum loco habenda e&longs;&longs;e cen&longs;e
bo. incolumem tibi, ac fœlicem D. Opt.
Max. longæuitatem tueatur. Vale.
Liber de &longs;e ip&longs;o.
Nec &longs;ine me totum di&longs;cet Aristotelem.
Ego Iordanus Ca&longs;&longs;ini Præpo&longs;itus Prouincialis Prouinciæ Venetæ Societatis
Ie&longs;u, ex auctoritate Adm. Reuer. P. nc&longs;tro Præpo&longs;iti Generalis P. Claudij
Aquæuiuæ, facultatem concedo, vt hoc opus P. Io&longs;ephi Blancani eiu&longs;dem
Societatis, quod in&longs;cribitur, Ari&longs;t. Loca Mathematica ex vniuer&longs;is ip&longs;ius
operibus collecta, & explicata, à deputatis Patribus recognitum, & ap
probatum typis mandari po&longs;&longs;it. Parmæ die 15. Ianuarij 1615.
& Reuerendi&longs;s. Archiepi&longs;c. Bonon
Imprimatur
Fr. Hieronymus Onuphrius pro Reuerendi&longs;s. P. Inqui&longs;itore Bonon
LECTORI.
Qvod pri&longs;cis olim temporibus (humani&longs;&longs;ime Lector) &longs;um
mi duo Philo&longs;ophi, Philippus Mendeus, ac Theon Smyr
næus in Platonis Dialogis egregiè perfecerunt, vt videli
cet quæ pa&longs;&longs;im &longs;ummus hic Philo&longs;ophus de Mathemati
cis &longs;cripta reliquit, eadem ip&longs;a ab illis &longs;electa, & in vnum
qua&longs;i corpus redacta lucubrationibus illu&longs;trarent: idem ego quoque in
Ari&longs;totelis operibus efficere &longs;um conatus, vt quæ de Mathematicis re
bus in vniuer&longs;is eiu&longs;dem monumentis &longs;par&longs;a leguntur, eadem in vnum
à me collecta, & explicata ijs Philo&longs;ophiæ &longs;tudio&longs;is maxime &longs;eruirent,
qui pri&longs;ca illa con&longs;uetudine relicta, Mathematicarum omnium ignari
non &longs;ine graui &longs;tudiorum &longs;uorum detrimento Philo&longs;ophiæ curriculum
aggrediuntur. Vt autem huius operis nece&longs;&longs;itas,
nius cogno&longs;cantur operæpretium erit initio illius cau&longs;as exponere; quæ
me poti&longs;&longs;imum ad illud con&longs;cribendum compulerunt, quarum
Prima &longs;it, quod hæc Ari&longs;t.
loca Mathematica, quæ quidem ferè 408.
numerantur, pe&longs;&longs;imè latinis literis con&longs;ignata &longs;unt v&longs;que adeò, vt Ari
&longs;totelem ip&longs;um, vel inuitum (quod po&longs;tea multis in locis planum fiet) in
ab&longs;urdi&longs;&longs;ima errata &longs;æpi&longs;&longs;imè compellant.
Secunda, quòd plurima huiu&longs;modi loca à nemine, quod &longs;ciam, adhuc
declarata in tenebris magno no&longs;trorum malo delite&longs;cunt: cuiu&longs;modi
&longs;unt ad &longs;exaginta problemata, libellus de lineis in &longs;ecabilibus, libellus
de mundo, &longs;i tamen Ari&longs;totelis e&longs;t, & Mechanicæ quæ&longs;tiones, quamuis
enim Picolomineus in eas paraphra&longs;im ediderit, loca tamen earum dif
ficiliora non &longs;atis illu&longs;trauit. Vt autem dixi 408. in vniuer&longs;um loca mi
nimùm numerantur, quibus illud Platonis in&longs;criptum e&longs;t
ud
&longs;equuntur ducem Ari&longs;t.
eum &longs;æpe de&longs;erere non &longs;ine turpi dedecoris no
ta coguntur; quo fit vt exempla illa Mathematica luc
do allatura, t
obducant.
Tertia, quia Græcieorumdem locorum commentatores breuiter, &
ob&longs;curè admodum ea, quæ ad Mathematicum &longs;pectant, attingunt, hoc
enim ab ip&longs;is
Philo &longs;ophorum, Mathematicis imbutum; at verò no&longs;tra ætate magna
cum Philo&longs;ophiæ iactura, quamplurimi earumdem di&longs;ciplinarum de&longs;ti
tuti præ&longs;idijs, ne Græcorum quidem Interpretum explanationes, ne
dum Ari&longs;t.
ob&longs;curè dicta intelligunt.
Quarta.
Adde, quod etiam &longs;i quis leuiter &longs;it erudito illo Mathemati
corum puluere con&longs;per&longs;us, adeò tamen peruer&longs;a e&longs;t eorumdem Græco
rum in Latinum tran&longs;latio,
confu&longs;io, & deprauatio, vt nec abeo, qui &longs;it Mathematicarum &longs;cientia
excultus, &longs;ine magno labore percipi po&longs;&longs;int. Quin etiam figuræ illæ, quæ
omnino nece&longs;&longs;ariæ &longs;unt ob Scriptorum, & Typographorum in&longs;citiam,
aut inertiam pluribus in locis de&longs;iderantur. Latini verò multo minus,
quàm Græci Mathematicæ periti, qua ratione eadem loca pertractaue
rint, facilius e&longs;t conijcere, quàm vt dici oporteat.
Quinta.
Ex his omnibus in aliud incommodum, vel maximum Phi
lo&longs;ophi quidam incidebant; aut enim horum locorum expo&longs;itionem ta
citi declinabant: aut eam minime nece&longs;&longs;ariam ad Ari&longs;t.
percipiendam
&longs;ententiam a&longs;&longs;erebant; quo quid ab&longs;urdius, quid &longs;tudio&longs;orum progre&longs;
&longs;ibus pernicio&longs;ius excogitari pote&longs;t? Eorum verò nonnulli eorumdem
locorum expo&longs;itionem audacter nimis aggrediebantur,
les illæ, ac ridiculæ expo&longs;itiones pa&longs;&longs;im auditæ, cuiu&longs;modi e&longs;t illa, quan
do Ari&longs;toteles ait, quod illi frequenti&longs;&longs;imum e&longs;t, omnis triangulus ha
bet tres; nihil aliud &longs;ignificari volunt, quàm omnem triangulum habe
re tres angulos. quod &longs;i dicat, omnis triangulus habet tres æquales duo
busrectis: hic hærent, hinc anguntur:
&longs;e minimè expedire valeant, aurea verba illa, quibus ingentes &longs;apientiæ
the&longs;auri continentur, alto &longs;ilentij velo contegere Mathematicarum eos
cogit in&longs;citia: vnde illud, quod Græcæ linguæ imperitis mutata oratio
ne acclamandum illis foret, Mathematicum e&longs;t, non legitur. Nec mi
nus elegans illa altera expo&longs;itio; Diametrum e&longs;le incommen&longs;urabilem
co&longs;tæ; quod &longs;æpe apud Ari&longs;t.
legentibus occurrit, nihil aliud &longs;ibi velle,
quam Diametrum e&longs;&longs;e longiorem co&longs;ta, quam quidem a&longs;ymetriæ huius
ignorantiam Plato de legibus dial. 7. non hominum, &longs;ed &longs;uum,
rumqueQuid illa?
cum Ari&longs;t.
ait duo cubi, cu
bus, ip&longs;um loqui putant de duplatione Geometrici cubi, nondum in
uenta; non intelligentes, eum ibi de numeris cubis &longs;ermonem habere. Auerroes ip&longs;e tantus vir 5. Phy&longs;. commen. 15. quàm &longs;e Mathematicis,
reliqui&longs;que Philo&longs;ophis irridendum præbet dum à permutata propor
tione putat &longs;erectè in hunc modum pluribus apud ip&longs;um verbis explica
tum, argumentari,
It a voluntas antiqua ad effectum antiquum. Ergo permutatim, vt &longs;e habebit voluntas noua ad effectum
antiquum,
nego, ait; qui&longs;piam con&longs;equen
inferre, &longs;ic, ergo ita &longs;e habebit voluntas noua ad antiquam, quemad
modum effectus nouus ad antiquum. quæ vitio&longs;a argumentatio quan
tumuis læuis &longs;it, & manife&longs;ta, quo&longs;dam tamen magni nominis philo&longs;o
phantes adeò tor&longs;it, vt adhuc torqueat.
Quanta autem mi&longs;eratione digni, qui publicè aliquando apud &longs;uos
auditores totam Per&longs;pectiuam, qua nihil iucundius e&longs;t, de medio tolle
re conati &longs;unt, propterea quod illæ vi&longs;uales lineæ, illi anguli, illæ pyra
mides, aut coni, quibus vi&longs;io perficitur nullibi extarent, &longs;ed e&longs;&longs;ent vana
quædam opticorum figmenta. Quì verò fieri potuit, vt non aduerterint
i&longs;ti &longs;e Ari&longs;toteli &longs;uo manife&longs;te repugnare, qui &longs;æpius de lineis vi&longs;ualibus
per&longs;pectiuum pertractare a&longs;&longs;erit,
opticam a&longs;&longs;ignat,
&longs;æpius facit.
Alij ex altera parte contra A&longs;tronomos in &longs;urgunt, eccentricos,
epiciclos omnes de cœlo detrahere cupientes. Verum id i&longs;ti nulla ex
pre&longs;&longs;a nedum probabili ratione faciunt, falsò exi&longs;timantes A&longs;tronomos
admirandam illam Cœlorum fabricam a&longs;&longs;erere, non autem &longs;upponere:
&longs;ed a&longs;tronomi illam &longs;upponunt,
non a&longs;&longs;erunt. Quod &longs;i aliqua probabili ratione id facerent, vti nonnulli
ex recentioribus, quorum Ticho Coripheus e&longs;t, laudandi potius, quam
vituperandi e&longs;&longs;ent. Impugnant
a&longs;&longs;ertione;
tor, aut Zodiacus, ne dum quid Epiciclus, aut Eccentricus. Nec defuit
qui viginti duo argumenta excogitarit,
bus contra A&longs;tronomos probare conatus e&longs;t, nullo modo Solem, aut
Lunam moueri po&longs;&longs;e motibus contrarijs, ide&longs;t, ab oriente in
motu diurno, & proprio ab occidente in orientem. Sed exi&longs;timandum
e&longs;t
recedere, nunquam animaduerti&longs;&longs;e; ab ea enim hanc motuum concor
diam didici&longs;&longs;et.
Quid tandem
nium tamen
ma illud Mathematicum, omne totum e&longs;t maius &longs;ua parte, in &longs;en&longs;u in
quo à Mathematicis effertur negare non erubuit, eò, quod in infinito,
vt aiebat non concederetur ab omnibus. &longs;cilicet non intellig
thematicum tantummodo tractare de Quantitate finita, ac terminata,
in qua axioma prædictum ab omnibus conceditur. Neque vero hic
nonnullorum infen&longs;us in Mathematicas animus quieuit, verum etiam
cò progre&longs;&longs;us e&longs;t, vt eas omnes omnino conuellere, atque ex albo &longs;cien
gnante, expungere conati &longs;int;
quàm quod eas non &longs;atis calerent; non &longs;ecus
quæ cum cauda mutilata e&longs;&longs;et, caudarum mutilationem reliquis vulpi
bus vafrè per&longs;uadere conabatur. Verum enim verò optimè &longs;cio, ea,
qu&ecedil; hactenus dicta &longs;unt non in omnes no&longs;tri temporis Philo&longs;ophos qua
drare, cum non pauci hodie quoque &longs;int, qui more antiquorum Mathe
maticis &longs;uffulti, optimè &longs;uis &longs;tudijs con&longs;ulentes, reliquam Philo&longs;ophiam
non &longs;ine magno compendio aggrediuntur. Quo fit, vt cæteros ageo
metretos ita antecellant, vt eorum Magi&longs;tri appellari po&longs;&longs;int, &
tales fuerunt in primis Vicomercatus, Cardanus, Zabarella, Toletus,
Bonamicus, & alij, quibus paulo antiquiores fuerunt D. Tho. & Scotus,
Hiomnes Mathematicarum auxilio, quantum inter reliquos philo&longs;o
phantes excelluerint, nemo e&longs;t qui non nouerit. Illud hoc loco minimè
tacendum, Iacobum Zabarellam in &longs;uis logicæ commentarijs te&longs;tari &longs;e
bis &longs;umma diligentia totum Euclidem perlegi&longs;&longs;e, vt perfectè Ari&longs;t.
de
demon&longs;tratione &longs;ententiam a&longs;&longs;equi po&longs;&longs;et.
Hi ridiculas illas, ac pueriles expo&longs;itiones &longs;uperius allatas minimè
effutierunt, neque reliquis &longs;upra recen&longs;itis incommodis obnoxij fue
runt, quibus magno etiam cum dedecore alij Mathematicarum ope ca
rentes afficiuntur.
In horum igitur gratiam operam diligenter dedi, vt quantum in me
e&longs;&longs;et damna à me &longs;upra enarrata aliqua ex parte re&longs;arcirem. Quapr
diligenter prius expendi. Deinde claritate, quàm potui max ma eadem
loca interpretatus &longs;um, & in horum, de quibus dixi gratiam, quædam
fanè tenuia pro&longs;equutus &longs;um, quæ alioquin libenter omi&longs;i&longs;&longs;em. Tum fi
guras omnes, aut correxi, aut re&longs;titui, aut nouas appo&longs;ui. Hocigitur
mo&longs;tro qualicunque labore poterit qui&longs;que omnia illa facile intelligere,
thematicarum experte requiram, vt principia &longs;altem illa, &longs;cilicet defini
tiones, po&longs;tulata, & axiomata, quæ primò Euclideis libro præponuntur,
diligenter prius perlegat cum illa &longs;ua per&longs;picuitate ommbus &longs;int obuia;
cætera ego explicanda recipio. Obiter etiam auctaria nonnulla partim
mathematica, partim naturalia in&longs;erui, quæ ob nouitatem, ac pulchri
cudinem grata Lectori, atque iucunda fore exi&longs;timaui.
Sciat præterea Lector no&longs;trum in&longs;titutum e&longs;&longs;e loca hæc mathemati
ca, quatenus mathematica &longs;unt declarare, &longs;iue ea &longs;upplere, quæ ex ma
thematicis petenda e&longs;&longs;ent: reliqua autem me tantum attingere, quan
tum harum rerum cum illis connexio po&longs;tulat.
His omnibus placuit appendices opportune nonnullas addere, qua
rum prima de natura mathematicarum &longs;cientiarum: altera, qua omnes
demon&longs;trationes primi libri Euclidis breuiter ad Logicam normam ex
penduntur, vt pateat, quonam demon&longs;trationis genere
debeat, & ex illis de cæteris iudicium fiat. Tandem in gratiam etiam
Mathematicorum tertiam appendicem appendi, qua omnia loca Ari&longs;t.
Geometrica ad Euclidis ordinem referuntur; vnde & ip&longs;i ad &longs;uas pr&ecedil;le
ctiones exornandas aliquid &longs;ubinde depromere queant.
Fruere igitur amice Lector hoc no&longs;tro qualiquali labore, quo ad ple
nam totius Ari&longs;t.
intelligentiam, cui adhuc mathematicarum ignoratio
ob&longs;titit peruenire tandem po&longs;&longs;is:
dam Philo&longs;ophus, cum totum hunc librum perlegi&longs;&longs;et, effatus e&longs;t, vide
licet, opus hoc
diem de&longs;ideratum
Illud demum tanquam parergon addam, quod ego his elucubran
dis experientia didici, ad veram &longs;cilicet, ac perfectam to
tius Ari&longs;totelis intelligentiam linguæ in primis
græcæ, necnon mathematicarum om
nium di&longs;ciplinarum haud medio
crem cognitionem ne
ce&longs;&longs;ariam e&longs;&longs;e. Vale.
Pr&ecedil;cipua qu&ecedil;dam, aut noua, autre&longs;taurata,
quæ obiter pertractantur.
ip&longs;ius ex vulgata editione Lugdunen&longs;i.
In Prædicamentis.
de Relatione, vbi de Quadratura circuli.
In Primo Priorum Re&longs;olutoriorum.
æquales duobus rectis: Aequalitas Geometrica, quæ.
In &longs;ecundo Priorum Re&longs;ol.
de Paralellis, & de triangulo.
In primo Po&longs;teriorum.
Item De Mathematicarum Principijs.
Item de recto, & circulari.
Item de numero pari; impari
primo, & compo&longs;ito; æquilatero, & altera parte longiore.
De I&longs;o&longs;cele.
De Alterna Proportione,
Item quod omnis triangulus habet tres, & c.
Item, quod non duo cubi cubus.
Item de
Mathematicis &longs;ubalternatis.
Item per&longs;ectam illam e&longs;&longs;e Demon
&longs;trationem, qua Geometræ o&longs;tendunt, quod Omnis triangulus habet tres, & c. Per&longs;pectinam, & Mechanicam &longs;ubalternari Geometriæ, Mu&longs;icam, Arithmeticæ.
In Geometria quid irrationale,
refrangi, concurrere. Quid Astronomia con&longs;ideret.
Item quid multiplicata propor
tio. Quid Cæneus dixerit.
Cur Affectiones Mathematicorŭ maximè
Quid &longs;tereometria.
& De &longs;ubalternatione, &c.
& Ma
thematicorum e&longs;t &longs;cire Propter quid: &longs;en&longs;itiuorum verò &longs;cire Quod.
Item, quod omnis figura habet &longs;uos angu
los externos æquales quatuor tantum rectis.
De Eclyp&longs;i.
De principijs &longs;cientiarum.
In 2. Po&longs;teriorum.
Terram e&longs;&longs;e in medio mundi ab A&longs;tronomis per
fectè demon&longs;tratur. Item Quid con&longs;onantia.
Item de Definitionibus Mathematicarum.
lem. Zabarella correctus.
In primo lib. Topicorum.
Vox acuta velox, cur.
&c.
Colores in
Mu&longs;ica, qui. tria genera veteris Mu&longs;icæ.
In 4. libro.
In 6. libro.
In 8. libro.
In Elenchorum lib.
1.
Quadraturarur&longs;us Hippocratis, & Bry&longs;enis.
Mathe
maticæ non contentio&longs;æ. Quadratio Antiphontis.
Ex 1. Phy&longs;ic.
Ex 2. Phy&longs;ic.
tionem.
& omnis triangulus habet tres an
gulos, &c.
Ex 3. Phy&longs;ic.
Ex 4. Phy&longs;ic.
& incommen&longs;.
Ex 5. Phy&longs;ic.
Ex 8. Phy&longs;ic.
Ex 1. de Cœlo.
quid.
Item de commen&longs;urabili.
Ex 2. de Cœlo.
Recentiorum ob&longs;eruationes.
Qua ratione grauia ad mundi centrum
aptarentur.
alio item modo.
Ex 3. de Cœlo.
Hinc de admirabili
Apum mgenio.
Num plures Pyramides locum replere valeant, vbi Ari&longs;totiles, & omnes
expo&longs;itores erra&longs;&longs;e o&longs;tenduntur.
Ex 4. de Cœlo.
Ex 2. de Generatione, & Corruptione.
Ex 1. Meteororum.
De magnitudine A&longs;trorum.
Item de Cometa: e&longs;&longs;e in Cœlo.
aperitur.
Mare extraneum, quod.
Errata quæ
dam veterum Geographorum, & Ari&longs;t.
corriguntur. Altitudo montis Cauca&longs;i.
Noua ob&longs;eruatio de rotundi
tate Terræ,
Ex 2. Meteororum.
De Zonis temperatis.
Corona Ariadnæ.
Zonam torrid
falsò putabant inho&longs;pitalem. cur habitabilis.
Ex 3. Meteor.
Noua de eadem
tractatio.
Rationes Ari&longs;totelis refelluntur.
Ex 1. De Anima.
& omnis triangulus babet tres, &c.
Ex 2. De Anima.
metricæ.
Ex 3. De Anima.
Ex lib. De Sen&longs;u.
Diapa&longs;on.
Diapen&longs;e.
Ex lib. De Memoria, & Rem.
Ex lib. De Somnijs.
Ex 1. Methaphy&longs;.
Item, Automata,
quæ &longs;olstitia. Diameter incommen&longs;.
Ex 2. Methaphy&longs;.
Ex 3. Methaphy&longs;.
Mathematicas puras carere cau&longs;is efficiente, & finali.
Ariftippus, vt Mathe
maticas &longs;ugillaret. Tetragoni&longs;mus est inuentio mediæ.
Ex 4. Methaphy&longs;.
Ex 5. Methaphy&longs;.
Quæ &longs;int proportiones Mu&longs;icales.
Quid potentia vnius lineæ.
Ex 6. Methaphy&longs;.
commen&longs;urab.
Omnis triangulus habet tres, &c.
Cur Angulus in &longs;emicir
culo rectus.
Ex 10. Methaphy&longs;.
Die&longs;is.
Diuer&longs;um in Math. quid.
Ex 11. Methaphy&longs;.
Ex 12. Methaphy&longs;.
pluralitatem Cœlorum docere.
Ex 13. Methaphy&longs;.
In Mechanicas Quæ&longs;tiones.
cur maior, exactior.
inibi Ari&longs;t.
lap&longs;us.
Piccolomineus reiectus.
figura antiquæ &longs;cytalis.
De Succula.
Securis veteris figura, & con&longs;tructio; vnà cum affectione
eius mirabili.
Veteris stateræ figura restaurata.
In libello De Mundo ad Alex.
noua de maris æ&longs;tu &longs;ententia.
In libro De Admirandis audit.
100. De
In libello De lineis in&longs;ecabilibus.
De commen&longs;urabili, & incommen&longs;urabili.
De figuris incommen&longs;.
Quæ linea rationalis, quæ irrationalis.
Binomio, Apotome.
De communi men&longs;ura.
Lineæ rectæ motus in &longs;emicirculum.
Circulorum æqualium ab inuicem motus.
Multum Mathematicis demon&longs;trationibus tribuitur ab Ari&longs;totele.
Si extarent indiuidua, omnes lineæ e&longs;&longs;ent commen&longs;.
Idem probat aliteŕ.
Idem ex triangulo.
Idem ex quadrato.
Ex lib.
9. Hi&longs;toriæ Animalium.
De Ince&longs;&longs;u Animal.
& ea quid &longs;it.
De Motu Animal.
De Generatione Animal.
Ibidem Diametrum e&longs;&longs;e incommen
&longs;urabilem co&longs;tæ, habet cau&longs;am, & demon&longs;trationem.
In Ethicis ad Nicom.
trica: Quid de&longs;ignatio.
Quid Proportionalitas.
Eam in 4. terminis con
&longs;i&longs;tere. Item quid Permutata proportio.
Item quid Geometrica proportio.
Propor
tio continuata, & di&longs;iuncta quid.
Ex 1. Magnorum Moralium.
Ex 1. lib. Moralium Eudemiorum.
Ex 1. lib. Mor. Eudemiorum.
e&longs;&longs;e.
Circuli quadratio.
Ex 7. lib. Mor. Eudemiorum.
Ex 3. lib. Politicorum.
Ex 4. lib. Polit.
Ex 5. lib. Polit.
Ex 8. Polit.
Rithmus quid.
Rithmus quid &longs;it dicetur in Problematibus.
Ex Problematibus.
3. De ortu &longs;yderum innerrantium: Succulæ, Hypades, Atlantides,
Virgiliæ, Pleiades. num.
17. De occa&longs;u affixarum &longs;tellarum.
1. Diametri ethymon.
2. Iterum Diametri ethymologia.
3. Denarius numerus cur perfectus.
eius dignitates.
Petri Apponen&longs;is deceptio.
diantur.
vbi de illuminatione Lunæ,
quæ experientia docetur.
modus commodè
videndi eclyp&longs;im Solis.
1. Cur ba&longs;es bullarum in aquis &longs;unt albæ?
reflexio radiorŭ pulchrè comparatur corporŭ re&longs;ultationi.
Ex &longs;ectione 19. De Mu&longs;ica.
2. Lineæ duplæ quadratum quadruplum.
Hoc loco &longs;equentium probl.
cau&longs;a,
præmittitur totius Mu&longs;icæ ortus breuis tractatio.
Punicum quid.
Genera, Diatonicum, Chromaticum, Encharmonium.
Tetrachorda quæ.
Magadis quid.
Magadare.
Rithmus quid.
ris lyræ.
erat in v&longs;u.
re&longs;onant.
Ex &longs;ectione 23.
Ex &longs;ectione 30.
Ex &longs;ectione 31.
Cur aliquando rei vi&longs;æ gemina
tio accidat.
Auctarium De Oculi Pupilla.
De pupillæ voce.
Additamentum de natura Mathematicarum di&longs;ciplinarum.
De &longs;ubiecto Mathem. &longs;eu de materia intelligibili: vbi o&longs;tenduntur definitio
nes Mathematicæ e&longs;&longs;e perfecti&longs;&longs;imæ.
2.
3.
4.
5.
6.
7.
ad Euclidem, &longs;ecundum propo&longs;itionum ordinem refe
runtur; vt Mathematicarum Profe&longs;&longs;ores habeant,
vnde &longs;uas prælectiones aliquando valeant locupletare.
Ad verbum ip&longs;um
Methaph.
Ad principia primi elementorum, vide infra tex. 5. pri. Po&longs;ter.
Ad definitionem 10. pri. pro angulo recto, vide 30. quæ&longs;t.
Mecha
nic. & cap.
7. lib.
1. Eth.
Ad axioma 10. quamuis Ari&longs;toteles nihil hac de re dicat; &longs;cias tamen velim
hoc vno axiomate qu&ecedil;&longs;tionem
facile di&longs;&longs;olui. ea e&longs;t, vtrum marmor, aut adamas, aliudue quidpiam infle
xibile &longs;ucce&longs;&longs;iuè findi, & aperiri po&longs;&longs;it. qui enim aiunt, &longs;ic refelluntur, quia
nimirum &longs;equeretur, duas rectas lineas habere &longs;egmentum commune: in
telligantur enim duæ lineæ, vna in vna &longs;uperficie, altera vero in altera, quæ
antequam incipiat apertio, congruant; Quoniam igitur tota illa apertio
non fit in in&longs;tanti, &longs;ed &longs;ucce&longs;&longs;iuè, facta iam aliqua apertionis parte con&longs;i
derentur prædictæ lineæ, erit igitur earum pars aliqua ab inuicem &longs;epara
ta, altera verò adhuc alteri congruens, ergo &longs;equetur, duas lineas habere
&longs;egmentum commune, quod e&longs;t impo&longs;&longs;ibile, quia contra 10. axioma.
Ad Calcem axiomatum primi accommodetur tex. 1. primi Po&longs;ter.
Ad primam primi, po&longs;t ip&longs;ius explicationem, commodè declarari pote&longs;t, cur
Ari&longs;t. Demon&longs;trationes Geometricas appellet De&longs;criptiones, & De&longs;igna
tiones, vide cap.
de Priori, & cap.
24. &longs;ecti primi, libri primi Priorum, &
tex. 4. quinti Methaph. & tex. 20. &longs;exti Methaph. & cap.
3. lib.
3. Ethic. Item ad primam primi, vide tex. 7. &longs;ecundi Po&longs;ter. loco 2.
Ad 5. primi, vide cap.
24. &longs;ecti 1 lib.
1. Priorum.
Ad 21. primi, vide tex. 20. primi Po&longs;ter. loco 2.
Ad 22. primi, vide locum 10. de lineis in&longs;ecabilibus.
Ad 28. primi, vide cap.
21. & cap.
22. &longs;ecundi
Ad 32. primi, vide cap.
1. &longs;ecti 3. lib.
1. Prior. & cap.
26. &longs;ecundi
primi Po&longs;ter. loco 4. & tex. 23. primi Po&longs;ter. vbi ait hanc e&longs;&longs;e poti&longs;&longs;imam & tex. 37. primi Po&longs;ter. & tex. 39. primi Po&longs;ter.
Ibidem
loco 4. & tex. 43. primi Po&longs;ter. & tex. 2. &longs;ecundi Po&longs;ter. bis. & tex. 89. &longs;e
cundi Phy&longs;. & tex. 15. octaui Phy&longs;. & tex. 119. primi de Cœlo. & tex. 25.
&longs;ecundi de Cœlo. tex 11. primi de Anima. & cap.
1. de mem.
& remini&longs;c.
& tex. 35. quinti Methaphy&longs;. & tex. 20. &longs;exti Methaphy&longs;. & tex. 22. &longs;exti
Methaphy&longs;. & cap.
4. lib.
2. de Generat. animal. & cap.
5. lib.
6. Ethic. &
cap.
2. Magnorum Moral. & cap.
10. Mag. Moral. & cap. 16. Mag. Moral.
& cap. 7. &longs;ecundi Eudem. & cap. 12. &longs;ecundi Eudem. & problema 6. &longs;ectio
Ad &longs;cholion præcedentis 32. primi, vide tex. 39. primi Po&longs;ter. loco 3. Item
tex. 25. &longs;ecundi Po&longs;ter. loco vlt.
Ad 45. primi, vide locum 9. de lineis in&longs;ecabilibus.
Ad 46. primi, vide locum 11. de lineis in&longs;ecabilibus.
Ad 47. primi, vide locum 11. de lineis in&longs;ecab.
Item locum 14. de ij&longs;dem.
Ad 2. definitionem 2. Gnomonis, vide cap.
de Motu in Po&longs;tprædicam.
Qua
dratum augetur Gnomone circumpo&longs;ito.
Ad 14. propo&longs;.
2. opportunum e&longs;t Auditores de Quadratura circuli erudire,
vide igitur cap.
de relatione in prædicam. & cap.
31. &longs;ecundi Priorum, &
tex. 23. primi Po&longs;ter. & finem 1. cap. primi Elenchorum. lege primam Ar
chimedis de dimen&longs;ione circuli.
Ad primam 3. vide cap.
9. lib.
2. Ethycorum.
Ad 2. tertij, vide tex. 13. lib.
1. de Anima. & locum 16. de lineis in&longs;ecab.
Ad 31. tertij, vide tex. 11. &longs;ecundi Po&longs;ter. & tex. 20. &longs;exti Methaph. loco 2.
Ad commentarium P. Clauij extremum lib.
4. elementorum.
lege tex. 66.
tertij de Cœlo.
Ad 4. definitionem 5. vide cap.
3. lib.
2. Ethyc.
Ad 9. definitionem 5. vide cap.
3. lib.
5. Ethyc. loco 4. & cap. 31. primi Ma
gnorum Moralium.
Ad 10. definitionem 5. vide tex. 29. primi Po&longs;ter. loco 2.
Ad 12. definitionem 5. vide tex. 13. primi Po&longs;ter. loco 3. & tex. 25. &longs;ecundi
Po&longs;ter. & tex. 32. tertij de Anima. & cap.
3. lib.
5. Ethyc. loco 4.
Ad 16. propo&longs;.
5. vide tex. 25. &longs;ecundi Po&longs;ter. loco 2. ex hac Euclidis demon
&longs;tratione patet, vitio&longs;am e&longs;&longs;e illam Auerrois argumentationem, 8. Phy&longs;.
comm. 15. &longs;cilicet.
Vt &longs;e habet voluntas antiqua ad antiquum effectum,
Ita &longs;e habet etiam voluntas noua ad effectum nouum:
Ergo permutando, ita &longs;e habebit voluntas antiqua ad effectum nouum. Quemadmodum voluntas noua ad effectum antiquum.
Non enim in permutando confert antecedentem ad antecedentem, & con
&longs;equentem ad con&longs;equentem, vt par erat, &longs;ed confert antecedentem ad
con&longs;equentem, quod non licet.
Ad 2. propo&longs;it.
6. vide cap.
2. lib.
8. Topicorum loco 41.
Ad 13. &longs;exti, vide tex. 12. &longs;ecundi de Anima, & tex. 3. tertij Methaphy&longs;.
Ad primam definitionem 7. vide tex. 5. primi Po&longs;ter.
Ad 8. definitionem 7. vide cap.
1. lib.
1. Magnorum Moral.
Ad 4. propo&longs;.
9. vide tex. 20. primi Po&longs;ter. loco 2.
Ad 8. propo&longs;.
9. vide problem.
3. &longs;ectionis 15. loco 4.
Ad primam definitionem 10. vide cap.
23. &longs;ecti 1. primi Priorum.
& tex. 48.
primi de Cœlo.
Ad 118. decimi, vide cap.
23. &longs;ecti 1. libri 1. Priorum.
& &longs;ecto 2. cap.
23. li
bri 1. Priorum. & cap. 22. lib.
2. Priorum.
& tex. 5. primi Po&longs;ter. & tex. 44.
primi Po&longs;ter. & cap.
15. primi Po&longs;ter. & tex. 119. primi de Cœlo. & tex.
120. quarti Phy&longs;. & tex. 21. tertij de Anima. & cap.
1. primi Methaphy&longs;.
& tex. 28. quarti Met. & tex. 34. quinti Met. & tex. 8. &longs;exti Met. & cap.
4.
lib.
2. de Generat. animal. & lib.
3. cap.
3. Ethyc. & cap. 10. &longs;ecundi Eu
dem. tot Ari&longs;t.
loca ab hac vna Euclidis Demon&longs;tratione illu&longs;trantur.
Ad primam propo&longs;.
13. &longs;ecundum editionem Commandini, aut Zamberti.
vide initio Priorum, in verbum (Re&longs;olutio)
Atqne hæc &longs;unt, quæ ex Elementorum opere Ari&longs;toteles pa&longs;&longs;im v&longs;urpauit,
quæque nos infra explicabimus.
Quæ verò ad alias Mathematicas, Mu&longs;icam &longs;cilicet, Per&longs;pe
ctiuam, Mechanicam, & A&longs;tronomiam pertinens, facilè
poterunt ex primo Indice ad vnamquamque earum &longs;eor
&longs;um cum libuerit, &longs;ecerni.
Finis Tertij Indicis.
Vi&longs;um e&longs;t etiam opportunum Lectori fore, ea &longs;imul in vnum
loca colligere, in quibus Ari&longs;toteles mihi vi&longs;us e&longs;t in Ma
thematicis &longs;copum non attigi&longs;&longs;e, vt alij pr&ecedil;&longs;ertim Peripa
tetici facilius ea inuenire,
po&longs;&longs;int.
LOCA
MATHEMATICA
EX LIBRO
PRÆDICAMENTORVM
Per ordinem declarata.
Ex c. 3. De his, quæ ad aliquid.
Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita
angulus B A C, vbi ait
nihil probibet e&longs;&longs;e &longs;cibile, vt circuli quadratura, &longs;i e&longs;t &longs;cibilis,
&longs;cientia quidem eius nondum e&longs;t)
nó omnia correlatiua &longs;imul e&longs;&longs;e natura, id de &longs;cibili, & &longs;cien
tia variè probat, præ&longs;ertim verò, quia multa &longs;int &longs;cibilia,
quæ tamen nondum &longs;ciantur, vt patet, inquit, in Quadratu
ra circuli, & &longs;cientia ip&longs;ius, quia quamuis ip&longs;a circuli quadratura &longs;it &longs;cibi
lis, nondum tamen &longs;imul cum ip&longs;a, &longs;cientia illius extat. Quæ vt perfectè
intelligantur, &longs;ciendum e&longs;t, quadraturam circuli, quæ à Græcis tetrago
ni&longs;mus dicitur, nihil aliud e&longs;&longs;e, quàm propo&longs;ito cuilibet circulo exhibere
quadratum æquale. Quæ æqualitas debet intelligi de areis, &longs;eu &longs;patijs, ita
vt area circuli, &longs;eu &longs;patium illud, &longs;iue &longs;uperficies illa circularis, &longs;it æqualis
areæ, &longs;eu &longs;uperficiei quadratæ. Qua in re plurimi decipiuntur exi&longs;timantes
per quadraturam cir culi inquiri æqualitatem linearum, ita vt circumferen
tia circuli debeat e&longs;&longs;e æqualis ambitui, &longs;eu quatuor lateribus quadrati:
quod omnino fal&longs;um e&longs;t.
Quadratio porrò circuli dupliciter proponi pote&longs;t, vel tanquam Theo
rema, vel tanquam Problema
ciendum proponitur; problema verò aliquid fseri expo&longs;cit)
pore Ari&longs;t.
erat adinuentum nam theorema inuentum e&longs;t po&longs;t ip
tis circiter annis ab Archimede: problema verò nondum à quoquam per
fectè potuit reperiri. qua di&longs;tinctione &longs;aluari po&longs;&longs;unt nonnulli, vt Boetius
hocloco, qui aiunt, &longs;e vidi&longs;&longs;e Demon&longs;trationem quadraturæ huius, &longs;i nimi
rum intelligant theorema. & alij etiam verum a&longs;&longs;erunt, dum negant hacte
nus repertam e&longs;&longs;e, &longs;i nimirum de problemate loquantur, theorema Archi
tem huiu&longs;modi. Quilibet circulus æqualis e&longs;t triangulo rectangulo, cuius
quidem &longs;emidiameter vni laterum, quæ circa rectum angulum &longs;unt, ambi
tus verò ba&longs;i eius e&longs;t æqualis.
Sit, v.g. datus circulus, cuius &longs;emidiameter A B; & fit trian gulum rectangu
lum A B C, cuius angulus B, &longs;it rectus, & latus B A,
ctum B, cum ba&longs;i B C, &longs;it æquale &longs;emidiametro A B; ba&longs;is verò B C, &longs;it æqua
lis peripheriæ eiu&longs;dem circuli dati. demon&longs;trat iam ibi Archimedes acuta
æquè, ac euidenti demon&longs;tratione triangulum i&longs;tud æquale e&longs;&longs;e circulo illi. quod perinde e&longs;t, ac &longs;i o&longs;tendi&longs;&longs;et cuinam quadrato &longs;it æqualis, cum per vl
timam 2. Eucl. po&longs;&longs;imus triangulo huic quadratum æquale con&longs;truere, quod
con&longs;equenter dato circulo æquale erit. Quod &longs;i in modum Problematis ita
proponatur: Dato circulo æquale quadratum con&longs;truere, nondum inuenta
e&longs;t ratio, quæ demon&longs;tratione confirmetur, qua id geometricè penitus, hoc
e&longs;t ad æqualitatem mathematicam, &longs;eu exacti&longs;&longs;imam effici po&longs;&longs;it,
ficultas po&longs;ita e&longs;&longs;e videtur in inue&longs;tigando, quonam modo exhibeamus li
neam rectam B C, æqualem peripheriæ circuli dati. quam nullus hactenus
geometricè illi æqualem potuit exhibere,
tione comprobare; Quamuis Archimedes acumine &longs;anè mirabili in lib.
de
lineis &longs;piralibus, eam
&longs;tigauit. nam propo&longs;itione 18. illius
æqualem circumferentiæ primi circuli &longs;piralis lineæ; propo&longs; verò 19. repe
rit aliam rectam æqualem circumferentiæ &longs;ecundi circuli. tu ip&longs;um con&longs;ule,
&longs;i admirandarum rerum contemplatione delectaris. Multa hac de re Pap
pus Alexandrinus lib.
4. Math. coll. & Ioannes Buteo vnico volumine om
nes quadraturas tain pri&longs;corum, quam recentiorum Qua
re qui plura cupit, eos adeat; nos tamen infra &longs;uis locis explicabimus tres
illas celebres antiquorum Antiphontis, Bri&longs;&longs;onis, & Hippocratis quadra
turas, quamuis fal&longs;as, &longs;olet autem à non
nullis di&longs;putari, vtrum quadratura i&longs;ta problematica &longs;it po&longs;&longs;ibilis, nec ne,
cum videant eam à nemine, quamuis diu magno labore perqui&longs;itam, hacte
nus adinuentam e&longs;&longs;e. ego quidem e&longs;&longs;e po&longs;&longs;ibilem exi&longs;timo, quis enim dubi
tare pote&longs;t, po&longs;&longs;e exi&longs;tere quadratum æquale circulo propo&longs;ito? Quod &longs;i po
te&longs;t fieri, quare non etiam demon&longs;trari? pr&ecedil;fertim cum videamus ab Archi
mede iam inuentam e&longs;&longs;e, quatenus Theorema e&longs;t. & præterea con&longs;tet
pocratem quadra&longs;&longs;e lunulam, vt &longs;uo loco dicemus, & Archimedem in libel
guræ, lunula &longs;cilicet, & parabola &longs;unt curuilineæ.
Ex cap.
de Priori
elementa enim priora &longs;unt ijs, quæ de&longs;cribuntur, nam principia prior a &longs;unt theore
matibus ordine)quæ non &longs;unt in antiqua tran
&longs;latione de&longs;ump&longs;imus ex ca&longs;tigati&longs;&longs;imo græco codice editionis Francfor
dien&longs;is, propterea quod totum hunc locum declarant; &longs;unt autem i&longs;ta,
tiuas intelligendas e&longs;&longs;e hoc loco ip&longs;as Mathematicas ex eo patet, quod illis
a&longs;&longs;ignet Ari&longs;t. De&longs;criptiones; nam hoc verbo, De&longs;criptiones, &longs;eu figuratio
nes, &longs;olet ip&longs;e Mathematicas Demon&longs;trationes innuere, quod in ip&longs;is figu
rationes, & De&longs;criptiones adhibeantur, vt alijs locis patebit: idcirco ver
ba illa à nobis addita ex græco, optim è
menta intelligantur principia, qualia &longs;unt initio Euclidis, & per de&longs;criptio
nes exponant theoremata. quod autem principia illa ordine priora &longs;int de
mon&longs;trationibus, &longs;iue ip&longs;as præcedant, ex ip&longs;a primi Euclidis in&longs;pectione
patere pote&longs;t.
Ex cap.
de motu
græca inter alia &longs;ignificat in&longs;trumentum illud, quod Latini tum amu&longs;&longs;im,
tum normam appellant, Itali verò, Squadra, ad
cuius &longs;imilitudinem Geometræ denominarunt fi
guram quandam, &longs;eu portionem cuiu&longs;uis paralle
logrammi, vt videre e&longs;t in definitione &longs;ecunda
2. elem. & in præ&longs;enti figura, in qua quadratum
A B C D, circumpo&longs;ito gnomone E F G, augetur,
& fit maius quadratum H B I L.
Idem etiam verum e&longs;t in quadrato arithmeti
co, &longs;iue in numero quadrato: is enim pariter ad
dito Gnomone augetur. i.
addito numero impari.
quemadmodum infra 3. Phy&longs;. tex. 26. fusè explicabimus.
Aliquorum opinio e&longs;t, Ari&longs;totelem ho&longs;ce libros appella&longs;&longs;e re&longs;olu
torios, quod per illos doceat &longs;yllogi&longs;mum, ac demon&longs;trationem
iam factam in &longs;ua immediata principia re&longs;oluere, quam opinio
nem meum non e&longs;t, nunc refellere. per&longs;ua&longs;um tamen mihi e&longs;t, rem
multo aliter &longs;e habere, veram rationem huius tituli petendam e&longs;&longs;e ex peni
tiori Mathematicorum eruditione. Sciendum
Alex. initio &longs;eptimi Mathem. collect. antiqui&longs;&longs;imos videlicet Geometras,
Euclidem, Apollonium Pergæum, & Ari&longs;t&ecedil;um &longs;crip&longs;i&longs;&longs;e libros de re&longs;olutio
ne, in quibus ars tradebatur, qua propo&longs;ito quouis theoremate, aut proble
mate po&longs;&longs;ent facile ex eo, tanquam vero accepto inue&longs;tigare aliquam veri
tatem, per quam deinde componerent illius, quod quærebatur, Demon&longs;tra
tionem; inue&longs;tigationem illam appellabant re&longs;olutionem: compo&longs;itionem
verò nominabant di&longs;cur&longs;um Porrò Diogenes Laert. huius re&longs;olutionis in
uentorem facit Platonem: à quo eam Leodamas Tha&longs;ius didicit, cuius be
neficio, pluries deinde Geometricas demon&longs;trationes adinuenit. definitio
13. Elem. iuxta tran&longs;latio
nem Zamb
mò per re&longs;olutionem, deinde per compo&longs;itionem demon&longs;trantur, quæ tan
quam per&longs;picua exempla rei propo&longs;itæ in&longs;eruire po&longs;&longs;unt. &longs;unt præterea fre
quentes huiu&longs;modi re&longs;olutiones in operibus Archimedis, Apollonij, & Pap
pi. extat adhuc liber Datorum Euclidis, qui geometricis re&longs;olutionibus in
&longs;eruiebat. vtinam extarent etiam alij de re&longs;olutione, quorum auxilio non
tantopere recentiores Mathematici in inueniendis Demo&longs;trationibus la
borarent; hanc re&longs;olutionem, &longs;ic Pappus fu&longs;ius, quam Euclides explicat;
re&longs;olutio e&longs;t via à quæ&longs;ito tanquam conce&longs;&longs;o per ea, quæ exip&longs;o con&longs;equun
tur ad aliquod certum, & conce&longs;&longs;um: in re&longs;olutione enim id, quod quæritur
tanquam factum, & verum &longs;upponentes, quid ex hoc &longs;equatur, con&longs;idera
mus, quou&longs;que incidamus in aliquod iam cognitum, vel quod &longs;it è numero
principiorum. Quod quidem erat fignum euidens, quæ&longs;itum quoque verum
e&longs;&longs;e. eadem omnino habet Proclus in comm. ad &longs;extam primi elem.
Quod
porrò Ari&longs;t. ip&longs;e hanc re&longs;olutionem Mathematicam cognouerit e&longs;&longs;e medij
inqui&longs;itionem manife&longs;tum e&longs;t ex cap. 3. lib. 3. Ethyc. vbi &longs;ic ait
con&longs;ultat, quærere videtur, & re&longs;oluere prædicto modo, quemadmodum de&longs;igna
tiones)
&longs;upra innuimus, & infra probabimus; cum ergo con&longs;ultatio nihil aliud &longs;it,
quam medij idonei ad finem in rebus agendis inqui&longs;itio, eamque dicat e&longs;&longs;e
&longs;imilem re&longs;olutioni Geometricæ, manife&longs;tum e&longs;t, ip&longs;am
e&longs;&longs;e medij in rebus &longs;peculatiuis idonei perue&longs;tigationem. Exi&longs;timo igitur
cum docti&longs;&longs;imis Zabarella, Burana, Toleto, & alijs, Ari&longs;totilem non &longs;olum
hanc &longs;uam logicam ad mathematicarum &longs;cientiarum typum compegi&longs;&longs;e,
verum potius imitatum e&longs;&longs;e opus illud Euclidis de re&longs;olutione, atque ex eo
non &longs;olum plurima exempla Geometrica, verum etiam titulum de&longs;ump&longs;i&longs;&longs;e,
præ&longs;ertim cum argumentum e&longs;&longs;et ferè idem vtrobique, &longs;ed Ari&longs;t.
intentio
fuerit accommodare re&longs;olutionem omnibus
rum Geometriæ &longs;oli. hinc patere pote&longs;t, cur hi libri re&longs;olutorij in&longs;cribantur,
quod &longs;cilicet tradunt methodum, qua valeamus quæ&longs;itum quoduis re&longs;olue
re, ide&longs;t, ex quæ&longs;ito tanquam vero inue&longs;tare aliquam veritatem, per quam
deinde propo&longs;itæ quæ&longs;tionis rationem methodo compo&longs;itiua reddamus. Et
verò cum reliquas appellationes Problematis, Theorematis, Propo&longs;itionis,
definitionum, po&longs;tulatorum, axiomatum, & alia huiu&longs;modi ex Geometri
cis ad omnes &longs;cientias tran&longs;tulerit, quid ni etiam re&longs;olutionem? maximè
verò, quia &longs;i horum lib.
intentio e&longs;&longs;et docere iam factum &longs;yllogi&longs;mum in &longs;ua
principia re&longs;oluere, parum e&longs;&longs;et vtilis; imò nec vtilis, &longs;ed &longs;uperfluum quid. at verò vbinam docuit hanc re&longs;olutionem?
profecto nullibi.
quid opus e&longs;t
iam factum &longs;yllogi&longs;mum re&longs;oluere? at verò propo&longs;itam quæ&longs;tionem re&longs;ol
uere veterum mathematicorum more, hoc opus, hic labor e&longs;t.
Hanc porrò re&longs;olutionem attendendam e&longs;&longs;e primò penes formam, quam
docet primis duobus analyticis; &longs;ecundò penes materiam, quam tradit duoreliquas duas logicæ partes, Topicam &longs;ci
licet, & Elenchos, quæ &longs;yllogi&longs;mos probabilem, & apparentem docent, no
luit appellare re&longs;olutorios, quamuis inuentionem mediorum doceant, quia
iam mos i&longs;te inoleuerat apud Philo&longs;ophos, & Mathematicos, vt illa &longs;ola
pars, quæ ex materia nece&longs;&longs;aria doceret &longs;yllogi&longs;mum demon&longs;tratiuum con
&longs;truere, diceretur re&longs;olutio: cum Mathematici, qui primi de re&longs;olutione
&longs;crip&longs;erunt, talem materiam &longs;olum con&longs;iderent.
Ex cap.
23. &longs;ecti primi lib.
1.
imparia æqualia paribus fiant, &longs;i fuerit po&longs;ita commen&longs;ur abilis. æqualia igitur fieri
imparia paribus ratiocinantur, diametrum vtrò incommen&longs;urabilem e&longs;&longs;e ex &longs;uppo
&longs;itione
mis duabus definitionibus 10. elem. definit, quæ nam &longs;int magnitudines
commen&longs;. & quæ incommen&longs;.
&longs;ic; commen&longs;.
magnitudines dicuntur, quas
eadem men&longs;ura metitur, vt &longs;i fuerint duæ magnitu
dines, A, & B, quas eadem men&longs;ura C, ide&longs;t quan
titas C, metiatur, ide&longs;t
titati A, & per ip&longs;am aliquoties replicata ip&longs;am ad
æquatè ab&longs;umat, vt &longs;i linea C, quinquies &longs;uper li
neam A, replicata eam præcisè, & perfectè omninò
adæquaret: & eadem linea C, applicata lineæ B, & &longs;uper illam ter, v.g. re
petita ip&longs;am con&longs;umeret, diceretur
A, & B, e&longs;&longs;e comm. definit po&longs;tea
rum nullam contingit communem men&longs;uram reperiri; vt &longs;i duarum linea
rum, A, B, nunquam po&longs;&longs;et reperiri aliqua men&longs;u
ra, quæ
C, men&longs;uraret A, quater &longs;umpta, ter autem &longs;umpta
non adæquaret omnino
cederet aliquantulum,
men&longs;ura, loco ip&longs;ius C, a&longs;&longs;umpta, &longs;iue maior, &longs;iue
minor ip&longs;a C, vt
incommen&longs;. Extare porrò tales lineas, & &longs;uperficies, & corpora,
plurima, ac penè infinita ex 10. Elem. manife&longs;tum e&longs;t. inuentum autem hu
ius a&longs;ymmetriæ, quod Pythagoricis veteres attribuunt, mihi &longs;emper vi&longs;um
e&longs;t omni maius admiratione, cum nulla experientia,
&longs;ius cognitionem potuerit pri&longs;cos illos Geometras inducere. Quapropter
non immeritò diuinus ille Plato lib.
7. de legib. huius a&longs;ymmetriæ ignora
tionem, adeo dete&longs;tatus e&longs;t, vt eam non hominum, &longs;ed &longs;uum, pecorumque
ignorantiam cen&longs;uerit. inter lineas incommen&longs;.
&longs;unt diameter, & latus eiu&longs;
dem quadrati, quia nulla pote&longs;t reperiri men&longs;ura quantumuis exigua, vti
e&longs;t lineola E, in præ&longs;enti quadrato, etiam&longs;i illam in
infinitum &longs;ubdiuidas, quæ
trum &longs;cilicet A C, & latus quoduis ex quatuor, v. g.
latus B C, præcisè omnino metiatur. theorema
i&longs;tud demon&longs;tratur in vltima 10. Elem. eodem me
dio, quod ab Ari&longs;totele hic innuitur; Euclides ex
&longs;uppo&longs;itione alterius partis contradictionis ip&longs;ius
deducit ad impo&longs;&longs;ibile, &longs;iue, vt ait hic Ari&longs;t.
fal&longs;um ratiocinatur, quod &longs;ci
licet idem numerus e&longs;&longs;et par, & impar, quod Ari&longs;t. &longs;ignificat, quando ait,
imparia æqualia paribus fiunt. ex quo ab&longs;urdo deducitur fal&longs;am e&longs;&longs;e prædi
ctam &longs;uppo&longs;itionem, quæ a&longs;truebat e&longs;&longs;e comm. & proinde altera pars con
tradictionis, quæ e&longs;t, e&longs;&longs;e incomm. vera a&longs;truitur. ex quibus &longs;atis videtur ex
plicari hic locus. videas igitur, quàm leuiter nonnulli no&longs;træ tempe&longs;tatis
ageometreti i&longs;tud exponant, dicentes diametrum e&longs;&longs;e incomm. co&longs;tæ, nihil
aliud &longs;igni&longs;icare, quam diametrum e&longs;&longs;e longiorem co&longs;ta, qua expo&longs;itione
nihil ineptius. Aduerte tandem figuram vulgatæ editionis e&longs;&longs;e ineptam,
cum habeat duo quadrata alterum &longs;uper diametro alterius, quorum maius
&longs;uperuacaneum e&longs;t.
Et cap.
24. &longs;ecti primi libri primi
ptionibus, vt quod æquicruris, qui ad ba&longs;im æquales &longs;int, ad centrum ductæ A B,
A C, &longs;i igitur æqualem accipiat A G, angulum ip&longs;i A B D, non omnino ex. &longs;timans
æquales, qui &longs;emicirculorum, & rur&longs;us G, ip&longs;i D, non omnem a&longs;&longs;umens eum, qui &longs;e
cti. amplius ab æqualibus existentibus totis angulis, & ablatorum æquales e&longs;&longs;e re
tiquos E, F, quod ex principio petet, ni&longs;i acceperit ab æqualibus demptis æqualia
derelinqui.)
&longs;is re&longs;pondente, e&longs;&longs;e mendo&longs;os; propterea ex textu græco vtrunque corri
gendum putaui in hunc, quem vidi&longs;ti modum. Secundo, per de&longs;criptiones
Ari&longs;t.
intelligere
loco euidenter confirmatur, vbi manife&longs;tè loco de&longs;criptionis &longs;upponit li
nearem demon&longs;trationem. In hoc
&longs;trare, quod Euclides in 5. primi o&longs;tendit, alio tamen modo, &longs;cilicet I&longs;o&longs;ce
lium triangulorum, qui ad ba&longs;im &longs;unt anguli, inter &longs;e &longs;unt æquales. e&longs;t au
tem figura in omnibus textibus deprauata, quam &longs;ic puto
ex quodam græco codice, qui characteres hoc modo appo&longs;uerat. &longs;it I&longs;o&longs;ce
les C A B, cuius ba&longs;is C B, Dico angulos &longs;upra ba&longs;im,
in quibus literæ E F, e&longs;&longs;e inuicem æquales. facto centro
in A, de&longs;cribatur circulus A B C, tran&longs;iens per puncta
C B, iam &longs;ic. omnes anguli &longs;emicirculi &longs;unt æquales in
ter &longs;e, ergo anguli A C G, A B D, &longs;unt æquales. Præte
rea cùm anguli ciu&longs;dem &longs;ectionis &longs;int æquales ad inui
cem, erunt anguli &longs;ectionis C B D G, nimirum anguli,
in quibus &longs;unt G, & D, inter &longs;e æquales:
anguli &longs;ectionis &longs;int partes
A B D, &longs;i illi ab his auferantur, auferuntur æquales anguli ab æqualibus an
gulis, ergo anguli, qui remanent, &longs;cilicet E, & F, erunt æquales, quod erat
demon&longs;trandum. hinc Ari&longs;t.
infert manife&longs;tum e&longs;&longs;e oportere in omni &longs;yllo
gi&longs;mo, reperiri vniuer&longs;ales, & affirmatiuas propo&longs;itiones, vt Factum e&longs;t in
præcedenti aliter e&longs;&longs;et petitio principij. Quænam vero &longs;it æqualitas, quam
Geometræ con&longs;iderant, infra cap. 1. &longs;ecti 3. explicabicur.
Ex cap.
2. &longs;ecti 2. lib.
1.
veritatem de&longs;cribuntur ine&longs;&longs;e, ad dialecticos autem &longs;yllogi&longs;mos ex propo&longs;itionibus
&longs;ecundum opinionem)
verbo, De&longs;cribere, per quod &longs;uperius annotauimus apud Ari&longs;t. &longs;ignificari
Geometricas demon&longs;trationes, nam eas opponit dialecticis &longs;yllogi&longs;mis, &longs;e
quentibus verbis, cum dixit (ad diale cticos autem &longs;yllogi&longs;mos ex propo&longs;i
tionibus &longs;ecundum opinionem) hac adhibita con&longs;ideratione, quam inter
pres non videtur adhibui&longs;&longs;e, &longs;en&longs;us huius loci non erit ob&longs;curus.
Ex eodem loco paulo po&longs;t
quodque
a&longs;trologicæ &longs;cientiæ: acceptis enim apparentibus
logicæ demonstrationes)
quodlibet problema demon&longs;trandum; nunc docet, non omnia in &longs;eientijs
po&longs;&longs;e probari, aut demou&longs;trari: principia enim &longs;cientiarum
tur, &longs;ed &longs;ola experientia manife&longs;ta &longs;unt; vt patet in A&longs;tronomia, quæ ab ex
perientia &longs;ua &longs;olet &longs;tabilire principia: principijs autem
tutis ex ip&longs;is reliqua problemata duo autem &longs;unt apud a&longs;tro
nomos genera experimenti, primum dicitur Phænomena, ide&longs;t,
& &longs;unt ea, quæ vulgo omnibus patent, vt Solem oriri, & occidere; a&longs;tra fer
ri circulariter, diem augeri modo, modo minui: & his &longs;imilia. alterum ge
nus dicitur ob&longs;eruationes, quæ tantummodo a&longs;tronomiæ peritis per ob&longs;er
uationem innote&longs;cunt, vt Solem inæqualiter ferri proprio motu per Zodia
cum; aliquando maiorem, aliquando minorem videri; plures dies immo
rari citra æquatiorem in parte Zodiaci boreali, quam in altera vltra æqua
torem au&longs;trali. dies naturales e&longs;&longs;e inuicem inæquales, &c.
ex quibus deinde
ponunt eccentricos, & augem, ad &longs;aluandas tum apparentias, tum ob&longs;erua
tiones; & hac ratione a&longs;trologica &longs;cientia paulatim reperta e&longs;t, ac in dies
reperitur.
Ex cap.
3. &longs;ecti 2. lib.
1.
tria diametri, & co&longs;tæ eiu&longs;dem quadrati, de qua fusè egimus &longs;uperius in
cap.
23. &longs;ecti 1. huius libri; quæ &longs;i repetantur, optimè hunc
Ex cap.
1. &longs;ecti 3. lib.
1.
æquicrus, ip&longs;i
namque triangulus habet duos rectos)
v&longs;urpat Philo&longs;ophus, quam i&longs;tud ex Mathematicis de&longs;umptum de triangu
lo, &longs;cilicet, omnis triangulus habet tres angulos æquales duobus rectis an
gulis, cuius Demon&longs;tratio e&longs;t in 32. primi Elem.
quod, vt probè intelliga
tur, explicandum e&longs;t penes quid attendenda &longs;it æqualitas inter angulum, &
angulum, quod facile a&longs;&longs;equemur, &longs;i meminerimus angulum e&longs;&longs;e in clinatio
nem illam, quam duæ lineæ non in directum po&longs;itæ faciunt: &longs;iue etiam (vt
melius percipiamus) angulum e&longs;&longs;e acumen illud, &longs;iue mucronem
duæ lineæ non in directum con&longs;titutæ faciunt, vt duarum linearum A B, A C,
inclinatio in puncto A, &longs;iue acumen illud, &longs;iue mucro,
e&longs;t ratio anguli. &longs;olum igitur duo anguli erunt æqua
les,
etiam &longs;i lineæ con&longs;tituentes vnum angulum &longs;int lon
giores lineis alterum angulum con&longs;tituentibus, quia
quantitas anguli non attenditur penes longitudinem
duæ lineæ prædictæ A B, A C, productæ, &longs;iue etiam decurtatæ fuerint, dum
modo &longs;itus, &longs;iue po&longs;itio ip&longs;arum, quam ad inuicem habent, non varietur,
erit &longs;emper eadem quantitas anguli A. Aduertendum præterea rationem
anguli non po&longs;&longs;e &longs;aluari in &longs;olo puncto A, in quo lineæ concurrunt, &longs;ed ne
ce&longs;&longs;ariam e&longs;&longs;e aliquam quantitatem, quamuis exiguam, linearum A B, A C. Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite
ras, &longs;emper literam illam e&longs;&longs;e medio loco proferendam, quæ ad acumen ip
&longs;um po&longs;ita e&longs;t, vt in &longs;uperiori, litera A, debet &longs;emper media proferri, dicen
do angulum B A C, &longs;iue C A B,
vel C B A. Porrò quemadmodum vnus angulus vni angulo æqualis e&longs;t, ita
angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu
li partiales B A D, D A C, erunt æquales totali angulo
B A C, cum partes omnes &longs;imul &longs;umptæ &longs;int &longs;uo toti æqua
les. pariter tres anguli po&longs;&longs;unt æquari & vni, & duobus
alijs angulis, quando nimirum a cumina, &longs;iue mucrones il
li &longs;imul ad vnum punctum con&longs;tituti
niilli, quem con&longs;tituerent alij duo anguli, quibus illi tres
&longs;unt pares, v.g. &longs;int tres anguli trianguli A B C,
quos linea perpendicularis D E, facit cum li
nea F G; &longs;it
tunc tres anguli illius
les duobus hi&longs;ce rectis, &longs;i tres illi mucrones
trianguli fimul &longs;umpti, & vniti ad punctum
E, ad quod duo
rectorum coeunt, congruent omnino duobus
prædictis angulis rectis, &longs;iue duobus illis mu
cronibus angulorum rectorum, &longs;iue con&longs;ti
tuent lineam rectam F E G, &longs;icuti faciunt
etiam duo illi anguli recti; &longs;iue etiam dica
mus, occupabunt idem &longs;patium omninò, &
præcisè, quod occupant duo recti: v.g. &longs;i mucro B, ibi poneretur, faceret
angulum F E H, & &longs;i ibi iuxta ip&longs;um apponeretur mucro A, faceret angulum
H E I. quem &longs;i deinceps &longs;ub&longs;equetur reliquus angulus C, con&longs;titueret
quum
les duobus rectis ad E, pariter con&longs;titutis, cum illi tres fiant partes duorum
rectorú, vel quia occupant idem &longs;patium, vel eandem lineam rectam F E G,
con&longs;tituant. habet igitur omne triangulum &longs;iue &ecedil;quilaterum, &longs;iue &longs;calenum,
&longs;iue I&longs;o&longs;celes mirabilem hanc proprietatem, vt tres anguli, cuiu&longs;uis trian
guli &longs;int æquales duobus rectis angulis. Quam demon&longs;trationem primi om
nium Pythagorici perfecerunt, vt refert Proclus ad 32. primi Elem. Eucli
des deinde ibidem aliter, quam Pythagorici idem demon&longs;trauit. Quod &longs;i
quis huius rei
litas mathematica non cadat &longs;ub &longs;en&longs;um, &longs;ed &longs;ola intelligentia percipiatur,
quippe quæ in materia intelligibili, non autem &longs;en&longs;ibili ver&longs;atur, & cuius
vitare ob &longs;ui imperfectione
&longs;unt æqualia, nullus intellectus aliquam valeat reper
inquam triangulum quodpiam materiale, vt ex charta, quantum fieri po
ce&longs;t perfectum, deinde ducat lineam vnam perpendicularem &longs;uper aliam,
quæ &longs;cilicet faciat, cum illa duos angulos rectos. po&longs;tea ab&longs;cindat tres an
gulos trianguli materialis,
&longs;int vniti, & contigui ad punctum lineæ perpendicularis cum altera, vti e&longs;t
in &longs;uperiori figura punctnm E; & illicò apparebit tres illos angulos mate
riales obtegere adæquatè totum illud &longs;patium duorum rectorum, quos per
pendicularis con&longs;tituit. Hoc autem experiri poteris in diuer&longs;is admodum
triangulis Scalenis, Rectangulis, I&longs;o&longs;celibus, Aequilateris, &c. non &longs;ine de
lectatione, atque hic e&longs;t &longs;en&longs;us illorum verborum, omnis triangulus habet
tres &ecedil;quales duobus rectis. Ab&longs;tineo à demon&longs;trationibus geometricis, quo
niam ij, qui Mathematicis &longs;unt imbuti, no&longs;tra hac opera parum indigent. &longs;i quis tamen volet, con&longs;ulat 32. primi Elem.
Ex hac igitur declaratione
licet cogno&longs;cere nonnullos ageometretos locum hunc, & &longs;imiles &longs;ub&longs;equen
tes non &longs;atis intelligere, dicentes, nihil aliud verba illa Ari&longs;t.
velle &longs;ignifi
care, quàm omnem triangulum habere tres angulos, quod inquiunt, noti&longs;
&longs;imum e&longs;t. Sed &longs;i incidant in &longs;equentia; æquales duobus rectis, tunc, cum
hæc non intelligant, ab&longs;tinent etiam à priorum declaratione, quibus præ
mi&longs;&longs;is facile e&longs;t Ari&longs;t.
textum percipere. &longs;it A, duo recti, ide&longs;t, duo anguli
recti &longs;int pa&longs;&longs;io demon&longs;tranda, in quo B, triangulus, in quo C, æquicrus. ip&longs;i
itaque C, ide&longs;t triangulo æquicru&longs;i, ine&longs;t A, &longs;cilicet duo recti, hoc e&longs;t, ine&longs;t
æquicru&longs;i hæc, pa&longs;&longs;io habere tres angulos æquales duobus rectis per B, ide&longs;t
per
tit
medium ip&longs;ius A, quia prædicta pa&longs;&longs;io. A, non competit triangulo B, per
aliud, &longs;ed per &longs;e, de eo enim primo, & per &longs;e demon&longs;tratur in 32. primi Elem.
optimè Aegydius, & Niphus in hunc locum.
Ex eodem cap.
quid accidere ab&longs;urdum, nihil enim vtimur eo, quod e&longs;t hoc aliquid e&longs;&longs;e. &longs;ed &longs;icut
Geometra pedalem, & rectam hanc, & &longs;ine latitudine dicit, quæ non &longs;unt. verum
non &longs;ic vtitur, tanquam ex his ratiocinans)
pro rebus characteres, A, B, C, po&longs;&longs;et qui&longs;piam &longs;u&longs;picari aliquod propterea
ab&longs;urdum accidere: cui &longs;u&longs;picioni Ari&longs;t.
re&longs;pondet, dicens, nihil inde ab&longs;ur
di accidere po&longs;&longs;e, quoniam ip&longs;e vtitur hi&longs;ce literis,
&longs;ed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum
etiam Geometræ faciunt, qui lineam, quæ pedalis non e&longs;t, pedalem, & quæ
non e&longs;t recta, rectam; & quæ lata e&longs;t, non latam, &longs;upponunt, & tamen nihil
inde ab&longs;urdi contingit. Ex quibus intelligimus per lineas illas &longs;en&longs;ibiles, &
phy&longs;icas, quas Geometræ in &longs;uis figuris ducunt, intelligendas e&longs;&longs;e lineas ve
rè Mathematicas omni latitudine carentes; vtitur enim inquit Ari&longs;t. Geo
metra lineis phy&longs;icis, non tanquam phy&longs;icis, nec de eis tanquam de phy&longs;icis
lineis ratiocinatur, &longs;ed ijs vtitur tanquam verè mathematicis. idem dicen
dum e&longs;t de &longs;uperficiebus, necnon de corporibus, quæ ijdem Goometræ de
&longs;cribunt, vt per ea, de verè mathematicis di&longs;currant.
Ex cap.
21.
&longs;e ip&longs;os talia accip entes, quæ non est po&longs;&longs;ibile monstrare uon exiftentibus
titio principij. vbi per coalternas intelligit parallelas lineas, vox
enim græca quoad
exempli explicationem vtor figura textibus apponi &longs;olita, quæ e&longs;t præ&longs;ens.
probat Euclides in 28. primi Elem.
quod &longs;i
linea recta quædam, vti E F, cadens &longs;uper
duas rectas, vti &longs;unt A B, C D, fe cerit angu
los alternos &ecedil;quales, angulos
G H D, ij enim dicuntur alterni; &longs;iue alios
dnos, nimirum B G H, G H C, hi enim &longs;unt
neas A B, C D, e&longs;&longs;e inuicem parallelas. Iam &longs;i quis vellet probare, &longs;e duas
parallelas duxi&longs;&longs;e, hac ratione, quia &longs;cilicet fa ciunt prædictos angulos al
ternos æquales; & probaret facere angulos alternos æquales, quia &longs;unt pa
rallelæ, hic peteret principium, ide&longs;t, illud, quod principio
afferret pro ratione, & cau&longs;a, quod dicitur peti principium, quia tunc pe
timus, vt concedatur nobis, id, quod principio, & primo omnium demon
&longs;trare propo&longs;ueramus. aduerte, quod characteres, qui &longs;unt in &longs;equentibus
verbis huius loci, non appellant characteres figuræ appo&longs;itæ; in quo quidam
decepti, nullo pacto poterant locum hunc intelligere.
Ex cap.
22. lib.
2. Priorum
men&longs;. argueret Zenonis rationem, quod non e&longs;t moueri)
fusè explicauimus hanc a&longs;ymmetriam, quam &longs;i quis vellet demon&longs;trare ea
dem illa ratione, qua Zeno motum impugnabat, quia &longs;cilicet
munis, quæ debet vtramq, quantitatem men&longs;urare, debet in men&longs;urando
infinitas partes pertran&longs;ire, uimirum medietates medietatum in in&longs;initum,
e&longs;t autem impo&longs;&longs;ibile pertran&longs;ire infinitas huiu&longs;modi partes, & propterea
non poterit metiri,
tur commen&longs;urabiles, afferret hic, inquit Ari&longs;t.
non cau&longs;am pro cau&longs;a.
Ex eodem cap.
nihil forta&longs;&longs;e inconueniens, veluti coalternas coincidere; & &longs;i maior e&longs;t extrin&longs;ecus
angulus intrin&longs;eco; & &longs;i triangulus habet plures rectos duobus)
tiones &longs;ubaudi fal&longs;as. per coalternas intellige lineas æquidi&longs;tantes, &longs;eu pa
rallelas, vt in &longs;uperiori cap.
monuimus. Cæterum Euclides propo&longs;.
28. pri
mi Elem.
o&longs;tendit, quod &longs;i fuerint duæ parallelæ veluti in præcedenti figura,
A B, C D, &longs;uper quas alia recta E F, incidat, nece&longs;&longs;ario faciet angulum ex
trin&longs;ecum E G B, v. g. æqualem interno, & oppo&longs;ito, & ad ea&longs;dem partes,
angulo videlicet G H D. &longs;i ergo inquit Ari&longs;t &longs;upponamus i&longs;tud fal&longs;um, an
gulum &longs;cilicet E G B, externum e&longs;&longs;e maiorem angulo interno G H D, &longs;equi
tur etiam fal&longs;um, videlicet lineas & pro
batur con&longs;equentia hoc modo, quia &longs;i angulus E G B, maior e&longs;t angulo
E G B, B G H, maiores, quam &longs;int duo B G H, G H D, quia &longs;i inæqualibus
æqualia addantur, tota erunt inæqualia, vt prius per 4, axioma: hoc loco
communis angulus additur &longs;emel maiori angulo, & &longs;emel minori; & ideo
totum illud, in quo e&longs;t maior angulus, adhuc maius e&longs;t altero toto, in quo
minor angulus continetur. at illi duo E G B, B G H, per 13. primi, &longs;unt
æquales duobus rectis angulis, ergo duo
internis B G H, D H G, &longs;iue hi duo interni erunt minores duobus rectis. At quando hi duo interni &longs;unt minores duobus rectis, tunc lineæ A B, C D,
&longs;unt concurrentes, &longs;i protrahantur ad partes prædictorum quod
P. Clauius luculenti, & hactenus de&longs;iderata demon&longs;tratione ad 28. primi
demon&longs;trauit.
lum illum externum e&longs;&longs;e maiorem interno, & oppo&longs;ito; &longs;equitur fal&longs;um, ni
mirum lineas parallelas concurrere.
Præterea &longs;i &longs;upponamus aliam fal&longs;itatem, &longs;cilicet triangulum habere tres
angulos maiores duobus rectis, &longs;equetur eadem iterum fal&longs;itas, &longs;cilicet pa
rallelas coincidere, & probatur &longs;ic; &longs;int enim
recti anguli, & per punctum C, ducta &longs;it recta
C D, parallela lateri B A. quia ergo angulus
A, æqualis e&longs;t angulo &longs;ibi alterno A C D, per
29. primi, & quia totalis angulus B C D, æqua
lis e&longs;t duobus angulis B C A, A C D, quos tanquam &longs;uas partes adæquatas
continet, quorum alter, &longs;cilicet A C D, e&longs;t æqualis angulo A. erit idem to
talis angulus B C D, æqualis duobus angulis A, & A C B, trianguli propo&longs;i
ti. ergo totus i&longs;te angulus B C D, &longs;imul cum reliquo
flabit compo&longs;itionem ex tribus angulis trianguli dati: & con&longs;equenter ta
lis compo&longs;itio trium angulorum erit maior, quam &longs;int duo anguli recti. ex
quo &longs;equitur duas rectas B A, C D, &longs;uper quas cadit linea B C, faciens duos
angulos internos, & ad ea&longs;dem partes, &longs;cilicet A B D, maiores duobus re
ctis non e&longs;&longs;e parallelas, &longs;ed concurrentes (vt patet ex nuper citata demon
&longs;tratione P. Clauij) quod fal&longs;um e&longs;t. & &longs;equitur ex &longs;ecunda fal&longs;a &longs;uppo&longs;itio
ne. ex quibus textus Ari&longs;t.
videtur &longs;atis clarus.
Ex cap.
26.
&longs;en&longs;ibuis triangulus, &longs;u&longs;picari
triangulus habet duos rectos: quare &longs;imul no&longs;cet, & ignorabit idem. no&longs;ce enim
omnem triangulum, quod duobus rectis, non &longs;implex e&longs;t: &longs;ed hoc quidem eo, quod
vniuer&longs;alem habet &longs;cientiam: illud autem eo, quod &longs;ingularem. &longs;ic igitur, vt vni
uer&longs;ale nouit C, quod duo recti; vt autem &longs;ingulare non nouit, quare non habebit
contrarias)
thematicum e&longs;t hic, clarum redditur. reliqua verò, quæ ad Logicum &longs;pe
ctant, huius loci commentatores pro&longs;equuntur.
In cap.
31. de Abductione.
Notandum hic cum eruditi&longs;&longs;imo Burana, Abductionem hanc, de qua in hoc
cap.
agitur e&longs;&longs;e vocem mathematicam,
alia à Mathematicis mutuatum ad omnes alias &longs;cientias tran&longs;tuli&longs;&longs;e. e&longs;&longs;e
in comm. Elem.
Euclidis ad primam propo&longs;itionem primi Elementi, pag. 121. &longs;ic ait, Abductio verò e&longs;t tran&longs;itus à propo&longs;ito problemate, vel theo
remate ad aliud, quo cognito, aut comparato Propo&longs;itum quoque per&longs;pi
cuum e&longs;t. Exempli cau&longs;a, cum cubi duplicatio propo&longs;ita e&longs;&longs;et ad inue&longs;ti
gandam quæ&longs;tionem in aliud tran&longs;tulere, quod illud propo&longs;itum con&longs;equi
tur, ad duarum nempe mediarum linearum inuentionem tran&longs;lata e&longs;t quæ
&longs;tio, & &longs;ic quærebant deinceps, quonam modo datis duabus rectis lineis,
duæ mediæ proportionales reperirentur. Primum autem dicunt Hippocra
tem Chium pr&ecedil;dictorum titulorum, Abductionem feci&longs;&longs;e, qui & lunulæ qua
dratum fecit æquale, & alia multa in Geometria inuenit. hæc Proclus.
vbi
non di&longs;&longs;imulandum nos re&longs;titui&longs;&longs;e verbum, Abductionem, cuius loco inter
pres Procli vtitur inductionis voce, &longs;equuti & rationem, & græcum textum,
qui no&longs;tram hanc expo&longs;itionem euidenter po&longs;tulat,
inductionem, & abductionem, &longs;ed abductio omnino rei propo&longs;itæ quadrat.
Notandum præterea Hippoetatem Chium fui&longs;&longs;e auctorem huius Abdu
ctionis,
culi, vnde manife&longs;tè apparet, Ari&longs;totelem ex Mathematicis hunc terminum
mutuò accepi&longs;&longs;e, quandoquidem ex ij&longs;dem accepit etiam exemplum Abdu
ctionis Mathematicæ, imò etiam exemplum ip&longs;ius authoris Abductioni
Mathematicæ. &longs;yllogi&longs;mus autem Hippocratis, quo o&longs;tendebat circuli qua
draturam reducebatur ad has propo&longs;itiones, omnis rectilinea figura qua
dratur, &longs;ed circulus reducitur ad figuram rectilineam, ergo circulus qua
dratur. in probatione minoris facta e&longs;t Abductio, cum enim ip&longs;e vellet re
ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam,
quandam quadratricem, vt e&longs;t apud Pappum
uium in fine &longs;exti Elem.
& alij aliter fru&longs;tra conarentur, facta e&longs;t Abductio
circa probationem minoris, in qua adhuc Mathematici verfantur; quæ pro
batio, &longs;i tandem inueniri po&longs;&longs;et, mox &longs;equeretur principale
blema, nimirum circulus quadraretur; vide quæ &longs;crip&longs;
dicam. de hac re, quia plurimum hunc conferunt. &longs;ed iam ad textus expli
cationem veniamus.
Ex eodem cap.
quo verò F, circulus, &longs;i ip&longs;ius E F, vnum &longs;olum e&longs;&longs;et medium, hoc, quod e&longs;t, cum
lunulis æqualem fieri circulum rectilineo, e&longs;&longs;e po&longs;&longs;et propè ip&longs;um cogno&longs;cere, cum
vero B C, neque credibilius &longs;it, quam A C,
nem:
vulgatæ editionis e&longs;&longs;e mendo&longs;am, & propterea re&longs;tituendam e&longs;&longs;e, qualis pri
ma &longs;equens ex Simplicio ad tex. 11. primi Phy&longs;ic. hoc modo Hippocrates
Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua
drandus; con&longs;tituatur
cuius diameter B D, &longs;ecatur bifariam in G, à circumferentia circuli dati,
quod patet ducta &longs;emidiametro H G, perpendiculari ex B C, quæ &longs;uo extre
mo puncto G, &longs;ecat bifariam, &
facto ergo centro G, de&longs;cribatur alter circulus per puncta B C D F,
ctaturque
angulo recto C, ergo quadratum eius ex eorol
lario 47. primi, duplum erit quadrati B C, quare
etiam circulus B C D F, duplus erit circuli A B
G C, per 2. duodecimi, & &longs;emicirculus B C D,
duplus erit &longs;emicirculi B A C: & quadrans B E
C G, æqualis erit &longs;emicirculo B A C: ablato igi
tur communi &longs;egmento B E C H, remanet lunu
la B A C E, æqualis triangulo B C G, quod trian
gulum &longs;i per vltimam &longs;ecundi quadretur, erit lu
nula B A C, con&longs;equenter quadrata.
nè procedit Hippocrates. &longs;ed vt reliquum circu
li quadret, &longs;ic pergit, ponatur recta L M, dupla
ip&longs;ius B C, &longs;upra quam &longs;emicirculus de&longs;cribatur
L O M, cui in&longs;cribatur hexagoni
æquilateri dimidium L Q S M, & &longs;u
per tribus hexagoni lateribus, &longs;int
tres &longs;emicirculi, vt in figura. &
niam
cuiu&longs;que
S M, erit &longs;emicirculus L O M, &ecedil;qua
lis quatuor &longs;emicirculis prædictis
per 2. duodecimi, & per 4. &longs;ecundi
ablatis igitur tribus
munibus L N Q, Q O S, S P M, relinquetur trapezium L Q S M, æquale &longs;e
micirculo B A C, & tribus lunulis L H Q N, Q R S O, S X M P, ab&longs;cindan
tur
pra in prima figura factum e&longs;t, & quod relinquetur æquale erit &longs;emicirculo
B A C. quod deinde quadretur per vlt. &longs;ecundi, &longs;ed aduerte, quod quando
ait, ab&longs;cindantur de trapezio tria triangula æqualia lunulis, eo modo, quo
&longs;upra, committit deceptionem, quia eodem modo, quo &longs;upra minimè id fa
cere po&longs;&longs;umus, quia in &longs;uperiori figura triangula erant con&longs;tituta &longs;uper la
tus B C, quadrati B C D F, intra circulum de&longs;cripti, qui circulus facit cum
B C, maius
Q S, S M. & propterea &longs;emicirculus i&longs;te non habet eandem proportionem
ad vnamquamque lunularum &longs;uarum, quam habet &longs;emicirculus &longs;uperior
B C D, ad lunulam B A C E.
nimè latui&longs;&longs;e putandum, cuius Ari&longs;t. &longs;æpius mentionem in &longs;equentibus fa
ciet : quì enim fieri pote&longs;t, vt tam acutus inuentor, adeo manife&longs;tum erro
rem non vidi&longs;&longs;et, verum propter adinuenti excellentiam, authori &longs;uo pla
cuit paralogy&longs;mus. mirabilis tamen &longs;emper habita e&longs;t illa &longs;uperior lunulæ
quadratio. Ex quibus &longs;atis clara e&longs;&longs;e po&longs;&longs;unt ea, quæ ad
tinent, ad locum hunc de Abductione declarandum. facta e&longs;t igitur abdu
ctio ab Hippocrate in quadratione trium po&longs;teriorum lunularum, in qua
rum quadratione diu immoratus, nunquam ni&longs;i cum paralogy&longs;mo quadra
re valuit. Hæc pluribus, vt &longs;equentibus etiam textibus, in quibus huius te
tragoni&longs;mi fit mentio &longs;atisfacere po&longs;&longs;imus. Hippocrates i&longs;te Chius e&longs;t alter
Aphrod. in Primum Meteororum de Cometis.
Textu primo
&longs;tenti fit cognitione. manife&longs;tum autem hoc &longs;peculantibus in omnibus,
Mathematicæ
do Mathematicæ fiant ex præcedenti cognitione, &longs;cilicet Princi
piorum per&longs;picuè quilibet videbit, qui &longs;altem primum
dis, vel è ianuis in&longs;pexerit; pr&ecedil;cedunt enim primo principiorum tria gene
ra, quorum primum continet definitiones &longs;ubiecti Geometriæ, vt definitio
nes lineæ, &longs;uperficiei, trianguli, &c: Secundum continet Po&longs;tulata. Tertium
Axiomata, &longs;eu communes omnium conceptiones, & &longs;ententias, ex quibus
tanquam ex vberrimis, & chri&longs;taltinis fontibus Demon&longs;trationes Geome
tricæ deriuantur. Idem vìdere licet in operibus aliorum Geometrarum,
Archimedis, Apollonij, Pappi, & cæterorum. Aliæ &longs;iniliter mathematicæ,
vt Arithmetica, Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia, non ni&longs;t ex
præmi&longs;&longs;is, ac manife&longs;ti&longs;simis principijs &longs;uas demon&longs;trationes deducunt. Nulla porrò alia &longs;cientia tam di&longs;tinctè &longs;ua præmittit principia,
&longs;picua, &longs;icuti Mathematicæ, vt non immeritò Philo&longs;ophus eas, tamquam
veræ &longs;cientiæ
po&longs;uerit, ex quo veræ &longs;cientiæ de&longs;criptionem hi&longs;ce libris complecteretur.
Tex. 2.
quod autem hoc, quod e&longs;t in &longs;emicirculo triangulum e&longs;t, &longs;imul inducens cognouit)
vide primo, quæ &longs;upra libro 1. Prior. &longs;ecto 3. cap.
1. explicaui de angulis
trianguli. deinde &longs;cias, quod quando Ari&longs;t.
ait, hoc, quod e&longs;t in &longs;emicir cu
lo triangulum, &c. alludit ad demon&longs;trationem quandam, quam ip&longs;e infe
rius in exemplum adducet, & quæ e&longs;t in 3. Elem.
Euclidis 31. in qua talis fi
gura proponitur qualis e&longs;t præ&longs;ens, in qua vides triangulum A B C. in &longs;e
micirculo. tunc autem dicitur triangulum in
&longs;emicirculo, quando ba&longs;is ip&longs;ius e&longs;t diameter
&longs;emicirculi, & reliqua duo latera ita concur
runt &longs;imul in angulum B, vt ip&longs;um paricer in
circumferentia con&longs;tituant, quibus pr&ecedil;mi&longs;sis
&longs;ic textum explicaueris: quod enim omne
triangulum habet tres angulos æquales duo
bus rectis angulis præ&longs;ciuit vniuer&longs;aliter per
32. primi; quod autem hoc particulare triangulum A B C, quod e&longs;t in &longs;e
micirculo habeat eandem proprietatem, &longs;imul, ac qui&longs;piam animaduertit
illud e&longs;&longs;e triangulum cogno&longs;cit,
te illius maioris propo&longs;itionis; omne triangulum habet tres, &c.
Tex. 5.
diameter &longs;it commen&longs;urabi is)
Priorum, &longs;ecto 1. &longs;ine quibus locus hic &longs;atis intelligi nequit; ijs autem per
ceptis &longs;ic
men&longs;urabilem prædicto lateri, quod autem fal&longs;um e&longs;t, illud non e&longs;t; igitur
impo&longs;sibile e&longs;t &longs;cire diametrum e&longs;&longs;e commen&longs;urabile.
Hoc eodem cap.
plura dicuntur de Principijs Demon&longs;trationis, &longs;iue &longs;cien
tiæ, vt &longs;unt Dignitates, Po&longs;itiones, Definitiones, & &longs;imilia, quæ quo modo
&longs;e habeant, & quo modo illis Demon&longs;trationes innitantur, optimè ex con
templatione primi libri Elem.
Euclidis percipi pote&longs;t. vt propterea benè ij
&longs;entiant, inter quos præcipui &longs;unt Toletus, & Zabarella, qui a&longs;&longs;erunt, Ari&longs;t.
Mathematicas &longs;cientias tamquam typum perfecti&longs;simarum &longs;cientiarum
&longs;ibi ob oculos propo&longs;ui&longs;&longs;e; ex quo typo veræ &longs;cientiæ de&longs;criptionem his li
bris complectaretur.
Eodem tex. 5.
dum quantum)
&longs;upponitur tamen ab eis: nu&longs;quam enim Euclides in totis tribus Arithme
ticis libris, infra vnitatem de&longs;cendit, vt propterea appareat, ip&longs;am in quan
titate di&longs;creta e&longs;&longs;e minimum, & indiui&longs;ibile. Verum dubitabit forrè qui&longs;
piam hoc modo, &longs;i vnitas minimum,
qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien
tem, in quadrantem, & alijs &longs;imiliter modis, vnde numeri illi, qui fractio
nes appellantur, exurgunt? Re&longs;pondemus,
Arithmeticis, tunc ip&longs;i eam accipiunt tanquam totum quoddam
in plures partes diui&longs;ibile: &longs;iue tanquam aggregatum quoddam vnitatum,
quæ vnitates &longs;unt partes illius, vt quando dicunt, vnum horæ quadrantem,
vel duos horæ quadrantes, vel tres horæ quadrantes, accipiunt horam tan
quam aggregatum quatuor quadrantum, & propterea numeri illi 1/4. 2/4. 3/4.
& &longs;imiles fractiones, nihil aliud &longs;unt, quam numeri partium vnius horæ: ex
quo patet huiu&longs;modi fractiones omnes reduci ad numeros integros, qui
enim dicit tres quadrantes
ligitur diui&longs;um e&longs;&longs;e in 4. æquales partes, ex quibus illæ tres tantummodo
numerat.
Tex. 9.
nea, & lineæ punctum; &longs;ub&longs;tantia
te, quid e&longs;t, in&longs;unt)
tur: quorum primus e&longs;t, ea &longs;cilicet,
per &longs;e de aliquo &longs;ubiecto dici,
iu&longs;modi &longs;unt linea, & punctum, quæ per &longs;e prædicantur, illa de triangulo,
i&longs;tud de linea; in de&longs;initione enim trianguli ponitur linea recta, quia linea
recta dum terminat illam &longs;uperficiem, quæ dicitur triangulus illi trianguli
naturam impertitur, & ideo triangulus definitur &longs;ic, triangulus e&longs;t figura
tribus lineis rectis terminata. &longs;imiliter in definitione lineæ, non in&longs;initæ,
&longs;ed finitæ, & terminatæ ponitur punctum, quia duo puncta, quæ &longs;unt extre
ma illius, faciunt, vt ea &longs;it line a finita, & definitur &longs;ic, linea finita e&longs;t lon
gitudo, caius extrema &longs;unt puncta. quamuis autem hæc definitio apud Eu
clidem expre&longs;&longs;a non habeatur, tamen ex definitionibus ip&longs;ius præ&longs;ertim &longs;e
cunda, tertia, & quarta elici pote&longs;t.
Eodem tex. 9.
numero, & primum, & compo&longs;itum, & æquilaterum, & altera parte longius. &
quia locus hic benè exponitur à Toleto, & melius etiam à Conymbr. addam
tantummodo quædam, quæ ad perfectam eius intelligentiam de&longs;iderantur. Sciendum igitur primò, nu&longs;quam ab Euclide definiri rectum, circulare,
impar, par, primum, compo&longs;itum, æquilaterum, nec altera parte longius:
cularem expre&longs;sè. in definitionibus deinde &longs;eptimi definiri
& imparem, item numerum primum, & compofitum, & æquilaterum, & al
tera parte longiorem. ex quibus definitionibus po&longs;&longs;unt erui definitiones re
cti, circularis, imparis, & cæterorum, quorum hic Ari&longs;toteles meminit. Cæterum Euclides definitione 11. &longs;eptimi, &longs;ic definit numerum primum:
primus numerus e&longs;t, quem vnitas &longs;ola metitur. numerus autem, vel vnitas
metiri dicitur alium numerum, quando &longs;æpius repetita ip&longs;um omnino ad
æquat, vt ternarius metitur nouenarium, quia ter repetitus ip&longs;um ad vn
guem explet. illi igitur numeri dicuntur ab Arithmeticis primi, qui à nullo
alio, præterquam ab vnitate men&longs;urantur, quales &longs;unt, 2. 3. 5. 7. &c. Defi
nitione verò 13. definit numerum compo&longs;itum &longs;ic; compo&longs;itus numerus e&longs;t,
quem numerus qui&longs;piam metitur, vt &longs;enarius erit compo&longs;itus, quia ip&longs;um
binarius metitur, nam ter repetitus, ip&longs;i perfectè adæquatur.
Per æquilaterum, intelligit quadratum, quadratus autem numerus defi
nitione 18. &longs;eptimi &longs;ic explicatur: Quadratus numerus e&longs;t, qui &longs;ub duobus
æqualibus numeris continetur, ide&longs;t, qui fit ex ductu vnius numeri in &longs;e ip
&longs;um, vt &longs;i ducantur 3. in 3. fient 9. qui continetur &longs;ub duobus
ternarijs; omnes autem ternarij &longs;unt æquales. is autem nu
merus dicetur quadratus, quia, vt apparet in figura, nouem
ip&longs;ius vnitates po&longs;&longs;unt in plano ita ad inuicem collocari, vt
referant quadratum; & &longs;icuti quadratum geometricum ha
bet latera æqualia, ita etiam quadratum arithmeticum: &longs;i
ue numerus quadratus, habet &longs;ua latera æqualia, quot enim vnitates &longs;unt
in vno, tot etiam &longs;unt in reliquis, vt in præ&longs;enti &longs;unt tres vnitates in &longs;ingulis
lateribus. pr&ecedil;terea quemadmodum quadratum geometricum re&longs;olni pote&longs;t
in plura quadrata, ita etiam arithmeticum, vt præ&longs;ens, qui re&longs;oluitur in
quatuor quadrata arithmetica.
nantur ageometreti, in hunc modum di&longs;poni, &longs;ed &longs;olum ij, qui producuntur
ex multiplicatione numeri alicuius in &longs;e ip&longs;um.
Per altera parte longius, intelligit numerum, qui producitur à duobus
numeris inæqualibus inuicem multiplicatis, qualis e&longs;t
duodenarius, qui ex ductu trium in quatuor produci
tur, & refert figuram altera parte longiorem, &longs;iue, vt
ait Boetius longilateram, cuius vnum latus e&longs;t maius
altero, vt in appo&longs;ita figura videre licet. atque hæc
&longs;unt, quæ ex Mathematicis petenda erant, ad huius
loci intelligentiam.
Tex. 11.
triangulum duo recti: etenim per &longs;e triangulum duobus rectis æquale. Vniuer&longs;ale
autem e&longs;t tunc, quando in quolibet, & primo mon&longs;tratur, vt duos rectos habere,
&longs;ed non de qualibet figura,
enim figura a quidem est, non habet autem duobus rectis æquales. Aequicrus verò
babet quidem
prius. quod igitur quoduis primum mon&longs;tratur duos rectos habens, aut
aliud, huic primo ine&longs;t vniuer&longs;ale, & demonstratio de hoc vniuer&longs;aliter e&longs;t, de alijs
verò quodammodo, non per &longs;e,
quorum intelligentia nece&longs;&longs;aria &longs;unt ea, quæ primo Priorum &longs;ecto 3. cap.
1.
&longs;crip&longs;imus. deinde memineris figuram vniuer&longs;aliorem e&longs;&longs;e triangulo, & tri
angulum vniuer&longs;alius æquicrure. quando ait (vt duos rectos habere) vult
dicere, habere duos angulos rectos non actu, &longs;ed potentia; quæ affectio e&longs;t
trianguli, quia, vt &longs;uperius diximus, habet tres angulos æquales duobus
rectis angulis: quæ proprietas vniuer&longs;aliter, & primò competit triangulo. non autem figuræ, quia figura e&longs;t vniuer&longs;alior.
re&longs;trictius triangulo. omittimus reliqua &longs;ingillatim exponere, tum quia &longs;a
tis clara &longs;unt, tum quia ab interpretibus benè explicantur.
Tex. 13.
buius e&longs;&longs;e demonstratio, eo quod in omnibus e&longs;t rectis; non e&longs;t autem: &longs;i quidem
non quoniam &longs;ic æquales, fit hoc, &longs;ed &longs;ecundum quod
ponit tres errores, qui circa demon&longs;trationem de vniuer&longs;ali contingunt,
quos omnes Geometricis exemplis illu&longs;trat; affert autem primo pro tertio
errore duo exempla, quorum primum in præmi&longs;&longs;is verbis continetur,
ex 28. primi Elem.
de&longs;umitur, quam propterea primo loco exponendam
cen&longs;ui. Quando igitur duæ rectæ con&longs;titu
tæ fuerint, vt A B, C D, in quas alia recta,
vt G F, incidens, faciat duos angulos in
ternos, re&longs;pectu rectarum A B, C D, & ad
ea&longs;dem partes rectæ E F, vt &longs;unt ex parte
&longs;ini&longs;tra anguli A G H, C H G; exparte ve
rò dextra B G H, D H G; &longs;i
fecerit duos illos angulos ex parte &longs;ini&longs;tra &longs;imul &longs;umptos, æquales duobus
rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro
bat Euclides rectas A B, C D, non concurrere, &longs;iue parallelas e&longs;&longs;e. Verum,
quia linea E F, pote&longs;t facere aliquando prædictos angulos non
les duobus rectis, verum etiam rectos, quo etiam modo
lineæ e&longs;&longs;e parallelæ, vt in &longs;equenti figura, cum &longs;int anguli A G I, C I G, re
cti, probabitur de rectis A B, C D, æquidi&longs;tan
tia. Ex his facile textum in hunc modum expo
nemus; &longs;i quis igitur mon&longs;trauerit, quod rectæ
A B, C D, nunquam coincidunt, etiam&longs;i in in&longs;i
nitum producantur, &longs;eu quod &longs;unt æquidi&longs;tantes,
quando anguli prædicti interni &longs;unt duo recti,
videbitur
omnibus habentibus prædictos angulos rectos. non autem de omni, &longs;ecun
dum quod ip&longs;um, &longs;i quidem non competit affectio hæc, e&longs;&longs;e parallelas, li
neis habentibus illos angulos rectos actu; &longs;ed primò, & vniuer&longs;aliter, & &longs;e
cundum quod ip&longs;um competit lineis habentibus illos angulos æquales duo
bus rectis,
&longs;iue vnus acutus, alter obtu&longs;us, &longs;ed tamen ambo &longs;imul æquentur duobus re
ctis, quales &longs;unt lineæ primæ figuræ. In tertio igitur errore, vniuer&longs;ale exi
&longs;tit quidem, & habet nomen, &longs;ed tamen prætermittetur, &longs;eu &longs;trictius &longs;ume
tur, quam oportet. alij latini, quos quidem viderim, præter Zabarellana
perperam omnino ob mathematicarum ignorantiam, exemplum i&longs;tud in
terpretantur.
Ibidem
&longs;celes videretur Por
rò cum tres &longs;int &longs;pecies triangulorum, æquilaterum, I&longs;o&longs;celes, Scalenum, &longs;i
accideret, vt ex illis tribus vna tantum &longs;pecies, v. g. I&longs;o&longs;celes in mundo re
periretur;
putans &longs;e
aifectio illa competeret I&longs;o&longs;celi, non vt huic &longs;peciei I&longs;o&longs;celis, &longs;ed quatenus
e&longs;t triangulum, cui primo, & per &longs;e, & &longs;ecundum quod ip&longs;um conuenit. hoc
loco di&longs;ce&longs;&longs;imus à Zabarella, qui putat i&longs;tud e&longs;&longs;e exemplum primi erroris,
cum verba textus adeo clara &longs;int, vt expo&longs;itionem illius nullo modo admit
tant. &longs;unt autem hæc textus verba
&longs;celes, &longs;ecundum quod I&longs;o&longs;celes videretur
apparet Ari&longs;t.
accipere pro &longs;ubiecto vniuer&longs;ali non indiuiduum vnum, vt in
primo errore contingit, &longs;ed &longs;peciem loco generis, &longs;cilicet I&longs;o&longs;celes, quod
e&longs;t &longs;pecies trianguli pro genere ip&longs;o, nimirum pro Triangulo. ait enim, &longs;i
non e&longs;&longs;et aliud, quam I&longs;o&longs;celes,
rè &longs;peciem, non indiuiduum, &longs;igni&longs;icat, ex his duobus exemplis manife&longs;tus
e&longs;t tertius error, qui erat, quando erat
expo&longs;uerat
quæ &longs;unt in parte inerit quidem demon&longs;tratio, & erit de omni, &longs;ed tamen non erit
buius primi vniuer&longs;aliter demon&longs;tratio. dico auttm huius primi, &longs;ecundum quod
buius demonstrationem, quando &longs;it primi vniuer&longs;aliter)
&longs;ubiectum exi&longs;tit quidem, &longs;ed tamen non de ip&longs;o &longs;it demon&longs;tratio, &longs;ed de ali
qua parte ip&longs;ius, v. g. de &longs;pecie aliqua demon&longs;tratur aliquid, quod deberet
o&longs;tendi primò de ip&longs;o vniuer&longs;ali, cum illi primò competat.
Ibidem
dum quod lineæ, & &longs;ecundum quod &longs;olida, & &longs;ecundum quod tempora: quemad
modum & mon&longs;trabatur aliquando &longs;eor&longs;um, contingens
demon&longs;tratione mon&longs;irari; &longs;ed quia non &longs;unt nominatum quidam omnia hæc vnum,
numeri, longitudines, tempora &longs;olida, & &longs;pecie differunt à &longs;einuicem &longs;cor&longs;um
cipiebanturnunc autem vniuer &longs;aliter mon&longs;tratur,
aut &longs;ecundum quod numeri, inerat; &longs;ed &longs;ecundum quod boc, quod vniuer &longs;ale &longs;up
ponunt e&longs;&longs;e)
le exi&longs;tit quidem, &longs;ed tamen e&longs;t innominatum, pro cuius explicatione &longs;cienAlternam igitur proportionem definit Eu
clides definitione 12. quinti, &longs;ic, e&longs;t &longs;umptio antecedentis ad
& con&longs;equentis ad con&longs;equentem. Explico, exponantur qua
tuor quantitates proportionales, v.g. vt 6. ad 3. ita &longs;int 4. ad
2. &longs;i igitur argumentemur &longs;ic, vt 6. ad 3. ita 4. ad 2. ergo al
ternatim erit, vt 6. ad 4. ita 3. ad 2. &longs;iue dixerimus, vt pri
mum ad &longs;ecundum, ita tertium ad quartum, igitur alterna
tim erit, vt primum ad tertium, ita &longs;ecundum ad quartum: valebit con&longs;e
quentia; quæ quidem probatur deinde propo&longs;itione 16. quinti de magnitu
dinibus, hoc e&longs;t in vniuer&longs;um de lineis, &longs;uperficiebus, & &longs;olidis. quando igi
tur Ari&longs;t.
ait, mon&longs;tramus proportionale, ide&longs;t, qua&longs;uis quatuor quantita
tes proportionales, habere hanc proprietatem, vt &longs;int etiam alternatim
proportionales, & non mon&longs;tramus vnica demon&longs;tratione de omni quouis
proportionali, &longs;ed &longs;eparatim de magnitudinibus in 16. quinti, de numeris
in 13. &longs;eptimi, & &longs;eor&longs;um de temporibus in a&longs;tronomia, vel phy&longs;ica; hoc
modo non o&longs;tendimus vniuer&longs;aliter de primo &longs;ubiecto, quia talis affectio
conuenit &longs;ingulis, non vt numeri, aut ma gnitudines, aut tempora &longs;unt, &longs;ed
&longs;ecundum quandam naturam illis omnibus communem, cui primò illa pa&longs;
&longs;io debetur; quæ quidem natura communis nomine caret, & propterea e&longs;t
cau&longs;a erroris.
uenio hanc demon&longs;trationem vniuer&longs;alem de illo communi omnibus præ
dictis, quare dicendum cum Zabarella, illud, nunc, e&longs;&longs;e intelligendum &longs;ic,
nunc autem, ide&longs;t, in præ&longs;entia autem deberet vniuer&longs;aliter demon&longs;trari,
quod tamen cum non &longs;iat, contingit nos decipi putantes vniuer&longs;aliter de
mon&longs;tra&longs;&longs;e. vel dicendum i&longs;tud verificari tantum de lineis, &longs;uperficiebus, &
&longs;olidis, de quibus &longs;imul in vnica natura communi, quæ e&longs;t magnitudo, de
mon&longs;tratur in 16. quinti vniuer&longs;aliter.
plum &longs;ecundi erroris, qui verbis illis
rebus &longs;pecie differentibus)
Ibidem
demon&longs;tratio
ne aut vna, aut altera, quod duos rectos habet vnumquodque,
& &longs;calenum, & æquicrus: nondum nouit triangulum, quod duobus rectis,
phi&longs;tico modo,
triangulum alterum. non enim &longs;ecundum quod triangulum,
ni&longs;i &longs;ecundum numerum, &longs;ecundum &longs;peciem autem non omne; & &longs;i nullum e&longs;t, quod
non nouit)
lis
vt benè intelligamus, opus e&longs;t ea, legere, quæ libro primo Priorum &longs;ecto 3.
cap.
1. &longs;crip&longs;imus de propriet
gulos æquales duobus rectis angulis, quibus præmi&longs;&longs;is, &longs;ic deinde locum
hunc interpretaberis; Propter hoc, quod præcedenti textu dictum e&longs;t; no
tandum in primo errore vniuer&longs;ale, tanquam &longs;i non e&longs;&longs;et vniuer&longs;ale o&longs;ten
ditur de &longs;ingulari, &longs;i quis igitur mon&longs;trauerit &longs;ingillatim de
gulo in &longs;ingulari, &longs;cilicet de vno æquilatero, tantum, & de vno Scaleno, &
de vno I&longs;o&longs;cele, &longs;eparatim, vtens auteadem demon&longs;trationc dum de
tero, altera pro I&longs;o&longs;cele, tertia pro Scaleno, o&longs;tendens, quod
illorum habet tres angules æquales duobus rectis angulis; i&longs;te nondum no
uit triangulum omne habere talem affectionem, ni&longs;i modo &longs;ophi&longs;tico, quia
non cogno&longs;cit hanc affectionem illis
munem trianguli, cui primo, & per &longs;e competit; & neque vniuer&longs;aliter co
gno&longs;cit triangulum omne e&longs;&longs;e tale, etiam &longs;i nullum aliud reperiatur trian
gulum, præter illud æquilaterum, vel illud I&longs;o&longs;celes, vel illud Scalenum, de
quibus &longs;eparatim
quatenus e&longs;t vnum numero. non nouit autem &longs;ecundum &longs;peciem, ideft fecun
dum naturam, & formam communem illis tribus indiuiduis, quæ e&longs;t natu
ra trianguli. hoc autem e&longs;&longs;e exemplum primi erroris manife&longs;tè conuincitnr,
tum ex verbis illis, quando nihil &longs;it &longs;uperius, præter &longs;ingulare, tum ex hu
ius textus verbis illis
rum)propterea nos de &longs;in gulari
triangulo omi&longs;&longs;a Zabarellæ &longs;ententia explicauimus tandem in confirma
tionem no&longs;træ expo&longs;itionis in hæc tria errata illud non omittendum, &longs;atiu
e&longs;&longs;e dicere, Ari&longs;t.
attuli&longs;&longs;e pro tribus erratis tria exempla ordine retrogra
do, quàm, quod facit Zabarella, primum e&longs;&longs;e pro tertio, &longs;ecundum pro pri
mo, tertium verò pro &longs;ecundo; eo enim modo, Ari&longs;t.
confu&longs;ionem nulla ra
tione, imò contra omnem rationem imponimus.
Textu 14. continet quidem quædam mathematica, &longs;ed ferè eadem cum
&longs;uperioribus, quæ quia tum ex prædictis facile intelligi po&longs;&longs;unt, tum quia
benè ab expo&longs;itoribus explicantur, ne actum agamus, prætermittimus.
Tex. 20.
fcretæ, ita vt cadant &longs;ub numernm, vt &longs;i linea quæpiam diuidatur in partes
decem, vel duodecim, tunc euadit quantitas di&longs;creta, &longs;iue numerus. & tunc
linca numerus e&longs;t. idem de &longs;uperficie, ac &longs;olido intelligendum.
Ibidem
e&longs;e &longs;cientia, &longs;ed neque quod duo cubi cubus)
quæ ad nos pertinent, vult Ari&longs;t.
docere
dere numerorum affectiones (per enbos enim intelligit numeros quo&longs;dam
&longs;ic dictos, vt paulo po&longs;t o&longs;tendam) vt &longs;i quis vellet geometricè o&longs;tendere id,
quod o&longs;tenditur in 4. noni Elem.
&longs;cilicet, &longs;i cubus numerus cubum numerum
multiplicauerit, productus numerus erit pariter cubus. nonnulli latinorum
perperam textum hunc expo&longs;uerunt putantes reperiri &longs;olummodo cubos
geometricos, at Euclides definit. 19. &longs;eptimi, &longs;ic arithmeticum cubum de
finit, cubus numerus e&longs;t, qui &longs;ub tribus numeris æqualibus continetur, qua
lis e&longs;t. 8. qui e&longs;t ad in&longs;tar cubi geometrici, & continetur&longs;ub tribus binarijs
multiplicatis inuicem, quæ multiplicatio &longs;ic in&longs;tituitur, exponuntur tres bi
narij, 2, 2, 2, primus ducitur in &longs;ecundum, & producitur. 4. qui e&longs;t numerus quadratus huius figuræ,
tertius binarius ducitur in prædictum quadratum 4. & pro
ducitur 8. qui dicitur cubus, quia &longs;i intelligantur duo qua
zerna
runt cubicam figuram, cuius tam longitudo, quam latitudo,
rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis &longs;it 9.
qui e&longs;t quadratus. quo deinde ducto in tertium ter
narium, producitur 27. qui e&longs;t cubus, & refert &longs;igu
ram cubicam hanc. Iam verò &longs;i cubus 8. multipli
cet cubum 27. procreabitur 216. qui pariter cubus
e&longs;t.
ide&longs;t, &longs;i duo numeri cubi multiplicentur mutuò, cu
bus alter producetur; ex quibus videas, quam in
eptè illi
&longs;totilem velle dicere non pertinere ad Geometram
probare duos cubos geometricos &longs;ibi additos face
re alium cubum, quod erat problema Delphicum de
duplatione cubi, nondum inuentum; bis enim i&longs;ti peccant, primo in Logi
cam, quia &longs;ic non tran&longs;iret Geometra de genere in genus, ip&longs;ius enim e&longs;t
agere de duplatione cubi; &longs;ecundò in Mathematicas, cum nondum noue
rint arithmeticos cubos; & præterca ignorent duos cubos &longs;ibi additos, non
facere alium cubum. Quod præterea hoc loco intelligendi &longs;int cubi arith
metici certò certius con&longs;tat, ex &longs;equenti 24. textu, vbi &longs;ic dicitur
Arithmetica quidem, quid impar, aut par, aut quadrangulum, aut cubus.)
Ibidem
vt &longs;it alterum &longs;ub altero, vt per&longs;pectiua ad Geometriam, & harmonica ad Arith
meticam)
nam falcem immittere) &longs;cientias &longs;ubalternatas, quæ propriè in Mathemati
cis reperiuntur, Per&longs;pectiua enim propriè &longs;ubalternatur Geometriæ, quia
vtitur Demon&longs;trationibus linearibus, quas applicat lineis vi&longs;ualibus, & Mu
&longs;ica &longs;ubalternatur Arithmeticæ, quia ab ip&longs;a mutuatur
merorum, quas applicat numeris &longs;onoris. v.g. Per&longs;pectiua dicit, ea, quæ vi
dentur eminus videri minora, quam quæ videntur cominus, quia illa viden
tur &longs;ub angulo minori, hæc verò &longs;ub angulo maiori, quod verò remotiora
videantur &longs;ub angulo minori, quam propinquiora cæteris paribus probat
per 21. primi Elem.
&longs;it enim ma
gnitudo vi&longs;a A B, remotior ab o
culo in C, po&longs;ito, & vi&longs;a propin
quior ab oculo in D. ductis lineis
vi&longs;ualibus C A, C B: D A, D B; ab
oculis C, & D, ad extremitates
&longs;pectatæ magnitudinis, erit remo
tioris vi&longs;ionis angulus C, minor
angulo D, propinquioris, vt ex præallegata Demon&longs;tratione pater. Hine
per&longs;picuè vides, qua ratione Per&longs;pectiua Geometriæ &longs;ubalternetur, &longs;iue
quid &longs;it ip&longs;a &longs;ubalternatio, vbi medium e&longs;t Geometricum, conclu&longs;io autem
optica. Exemplum &longs;ubalternationis Muficæ &longs;it,
vulgò octauam appellant in data chorda collocare, hoc e&longs;t, vocem grauio
rem facere duplam vocis acutioris &longs;umatur chorda A B, & diuidatur bifa
riam, &longs;ine in æqualia in C; tota igitur chorda A B, ad dimidium A C, haber
proportionem, quam 2. ad 1.
&longs;iue duplam, ergo etiam &longs;o
nus totius chordæ A B, ad
num
bebit eandem rationem, &longs;ed &longs;onus chor
dæ A B, ad &longs;onum chordæ A C, con&longs;onat diapa&longs;on, &longs;eu octauam, ergo in
data chorda collocata e&longs;t con&longs;onantia diapa&longs;on, quod oportebat. vides me
dium e&longs;&longs;e arithmeticam, conclu&longs;ionem verò harmonicam. Aliud exemplum
Tonus, quod e&longs;t
lia &longs;emitonia diuidi nequit, ratio e&longs;t Arithmetica, quia proportio &longs;uper
particularis in duo æqualia arithmeticè &longs;ecari nequit; at Tonus con&longs;i&longs;tit in
ratione &longs;uperparticulari, nempè in &longs;e&longs;quioctaua, ergo Tonus bifariam diui
di nequit. de&longs;umptum e&longs;t ex Boetio.
Tex. 23.
dum enim commune mon&longs;trant tales rationes)
mon&longs;trationem con&longs;tare debere ex proprijs, non autem ex communibus;
primum affert exemplum demon&longs;trationis cuiu&longs;dam Bry&longs;onis, quæ ex com
munibus procedat, vt autem benè intelligamus, quale&longs;nam &longs;int huin&longs;modi
demon&longs;trationes, quæ per communia o&longs;tendunt, legenda prius ea &longs;unt, quæ
&longs;crip&longs;imus de quadratura circuli in pr&ecedil;dicamento relationis. Bry&longs;o itaque,
vt tradit Alexander, in hunc modum conabatur quadrare &longs;it qua
drandus circulus A B C D, cui circum&longs;eribatur quadratum E F G H. per
7 quarti, & alterum quadratum I L M N, eidem in&longs;cribatur per 6. quarti,
quid autem &longs;it circum&longs;cribere, & in&longs;cribere figuram circulo, ex definitione
3. & 4. eiu&longs;dem libri petatur, quamuis
ex in&longs;pectione figuræ
cipi po&longs;&longs;it; deinde aliud
dium inter prædicta duo con&longs;tituatur, Iam &longs;ic o&longs;tendebat i&longs;tud
medium quadratum e&longs;&longs;e æquale circu
lo propo&longs;ito.
iora eodem, & minora eodem, &longs;unt in
uicem æqualia, &longs;ed circulus, & quadra
tum medium, &longs;unt ambo maiora qua
drato in&longs;cripto, & ambo minora qua
drato circum&longs;cripto, ergo circulus, &
quadratum medium, &longs;unt æqualia. vte
batur, inquit Ari&longs;t pr&ecedil;dicto principio,
etiam numeris, lineis, temporibus, &
qualitatibus communi,
bus erat demon&longs;tratio. præterea aduertendum e&longs;t, illud e&longs;&longs;e fal&longs;um, nam &longs;ex,
& quinque, ambo &longs;unt maiores, quam quatuor, & minores, quam &longs;eptem,
& tamen non &longs;unt æquales.
In codem textu
&longs;ecundum illud cogno&longs;camus, &longs;ecundum quod ine&longs;t ex principijs illius, &longs;ecundam
quod illud; vt duobus rectis æquales, habere, cui ine&longs;t per &longs;e, quod dictum e&longs;t ex
ex communibus, vt præcedens Bry&longs;onis, &longs;ed ex proprijs principijs o&longs;tendit
affectionem de &longs;ubiecto proprio. E&longs;t autem illud exemplum toties decan
tatum de triangulo habente tres angulos æquales duobus rectis angulis; id
circo operæpretium e&longs;&longs;e puto explicare demon&longs;trationem, 32. primi Eucli
dis, quæ i&longs;tud ex proprijs principijs demon&longs;trat, & quam hoc loco Ari&longs;to
teles innuit, hoc enim modo ip&longs;ius Ari&longs;t.
mentem probè penetrare poteri
mus. &longs;it ergo
Dico ag
gregatum
e&longs;&longs;e æquale aggregato ex duobus angu
lis rectis (vt autem melius intelligas, quæ
&longs;equuntur, lege prius ea, quæ dicta &longs;unt
in lib.
1. Priorum &longs;ecto 3. cap.
1.) produ
catur latus B C,
externus A C D; Iam &longs;ic, quoniam
batum
facit linea A C, cum linea B D, &longs;cilicet angulos A C B, A C D, e&longs;&longs;e pares
duobus rectis: & quia pariter in prima parte huins propo&longs;. 32. probatum
e&longs;t ab Euclide duos angulos A B, e&longs;&longs;e æquales externo angulo A C D: &longs;i ter
tius angulus reliquus A C B, &longs;umatur bis, &longs;emel cum duobus angulis A, B,
& &longs;emel cum externo A C D,
anguli A, B, A C B, &longs;imul &longs;umpti, erunt æquales duobus A C D, A C B, &longs;imul
&longs;umptis; &longs;ed his duobus &longs;unt æquales duo recti, ergo cum quæ &longs;unt æqualia
vni tertio, &longs;int etiam æqualia inuicem, erit aggregatum trium angulorum
A, B, A C B, æquale aggregato duorum rectorum; quod erat demon&longs;tran
dum. Medium
pendatur, e&longs;t, quod partes aggregati
les partibus aggregati
le e&longs;t. quod medium e&longs;t in genere cau&longs;æ materialis.
quod verò partes illius
&longs;int æquales partibus huius, probatur, per dignitatem
vni tertio, &longs;unt etiam inter &longs;e. partes porrò aggregati trium angulorum
erant hæ, anguli A, B, vna; altera verò angulus A C B; partes verò aggre
gati duorum rectorum erant A C B, A C D, quibus partibus, illæ &longs;unt æqua
les, & ideo totum toti æquale. quod medium e&longs;t omnino intrin&longs;ecum, & ex
proprijs ip&longs;ius trianguli, &longs;iue ex proprijs angulorum ip&longs;ius, cum &longs;int ip&longs;ius
partes. quod pariter medium ex parte pa&longs;&longs;ionis, quæ demon&longs;tratur, e&longs;t ex
proprijs, cum &longs;int partes illius materiales. per materiam autem oportet
hoc loco intelligere materiam intelligibilem, ide&longs;t quantitatem à qualita
tibas ab&longs;tractam, & terminatam, de qua pluribus agemus infra in tractatu
de natura mathematicarum. Hinc videas eos magnopere decipi, qui pu
tant, hanc demon&longs;trationem e&longs;&longs;e per extrin&longs;eca, eò quod ad demon&longs;tran
dum producatur linea B C, in D, putantes lineam illam productam C D,
e&longs;&longs;e demon&longs;trationis medium; lineæ
tionibus geometricis con&longs;truuntur, nunquam &longs;unt media propria demon
&longs;trationum, &longs;ed tantummodo a&longs;&longs;umuntur ad probandum medium iam ex
cogitatum e&longs;&longs;e veram cau&longs;am conclu&longs;ionis. Hinc etiam manife&longs;tè colligas
ageometreti negare &longs;olent, &longs;ed audacter aiunt exempla Ari&longs;t.
non e&longs;&longs;e vera:
illud v&longs;urpari &longs;olet, & debet de exemplis moralibus. at vero requiri confor
mitatem exemplorum cum regulis traditis, nemo &longs;anæ mentis dubitabit. Vernm i&longs;ti confundunt conformitatem cum veritate.
Veritas exemplo tunc
ine&longs;t, quando illud, quod in exemplo narratur, verè extitit, vt &longs;i quis in
exemplum pudicitiæ afferret hi&longs;toriam Io&longs;ephi, quæ veritas in exemplis moralibus non &longs;emper e&longs;t nece&longs;&longs;aria, talia exempla
&longs;unt &longs;æpè parabolæ, & fabulæ, quæ nunquam extiterunt, v. g. narratur ab
Ari&longs;t.
de quodam filio, qui patrem crudeliter traxerat, qui po&longs;tea grandior
factus, cum filium procrea&longs;&longs;et, ab eodem pariter raptatus e&longs;t ip&longs;e, v&longs;que ad
eundem locum, quo ip&longs;e patrem &longs;uum impiè raptauerat. non e&longs;t nece&longs;&longs;e, ta
lem extiti&longs;&longs;e filium, Verumtamen &longs;emper conformitas exem
pli cum regulis, & præceptis, quæ traduntur nece&longs;&longs;aria e&longs;t, alioquin exem
pla de&longs;truerent id, quod præceptio con&longs;truit,
nino ab&longs;urdum foret. non &longs;ecus, ac &longs;i quis vellet alium docere characteres
latinos, re
quiritur igitur &longs;emper in omni exemplo conformitas cum eo, quod doce
tur; in moralibus tamen non &longs;emper requiritur veritas, vti diximus; Alij
verò dicunt non requiri in exemplis determinatam veritatem, &longs;ed &longs;atis e&longs;&longs;e,
&longs;i exemplum verum &longs;it &longs;ecundum opinionem aliquorum:
non improbamus. Exempla igitur ab Ari&longs;t.
pa&longs;&longs;im ex mathem aticis allata,
congrua,
mentientem facimus. Po&longs;tremò illud etiam e&longs;t aduertendum, fortè Ari&longs;t.
in
præ&longs;enti textu &longs;pecta&longs;&longs;e
tius ad Pithagoricam. Pithagorei enim eam aliter, quamuis per idem me
dium, &longs;cilicet à cau&longs;a materiali, demon&longs;trabant; con&longs;truebant enim aliter, quod dictum velim propter nonnullos, qui ab
huiu&longs;modi diui&longs;ionibus abhorrent,
ni per eas plurimum derogetur. Pithagoreorum demon&longs;trationem vide
apud Clauium in &longs;cholio 32. primi Euclidis, quam ex Eudemo etiam Pro
clus in comm. eiu&longs;dem recitat.
Ibidem
Ibidem
e&longs;t geometricæ demon&longs;trationes in Per&longs;pectiuas, aut Mcchamcas, & arithmeticæ in
harmonicas)
tulimus; nunc Mechanicæ &longs;ubalternationis, quam hic Ari&longs;t.
in&longs;inuat, exem
plum &longs;it illud, quod Archimedes prop.
14. primi Aequep. demon&longs;trat, ni
mirum centrum grauitatis omnis trianguli e&longs;&longs;e punctum illud, in quo rectæ
lineæ ab angulis trianguli ad dimidia latera oppo&longs;ita ductæ concurrunt. &longs;it
triangulum A B C, à cuius angulis A, & B, ducantur duæ rectæ A D, B E, ita
vt bifariam &longs;ecent latera A C, B C, in punctis D, & E, & concurrant in F. Dico F, e&longs;&longs;e centrum grauitatis propo&longs;iti trianguli.
Quoniam enim in 13.
Aequep. probauit centrum grauitatis e&longs;&longs;e in ea linea, quæ ducta ab angulo
quouis &longs;ecat oppo&longs;itum latus bifariam, crit in linea A D,
&longs;ed eadem ratione erit etiam in linea B E, er
go non ni&longs;i in puncto F, quod
que, quod erat demon&longs;trandum. ex quibus ap
paret, qua ratione mechanica conclu&longs;io Geo
metriæ &longs;ubiaceat, dum lineari di&longs;cur&longs;u ip&longs;a
demon&longs;tratio perficitur. Scias præterea cen
trum grauitatis e&longs;&longs;e tale punctum, ex quo &longs;i &longs;u
&longs;pendatur corpus triangulare vniformis cra&longs;
&longs;itici, manet &longs;emper horizonti parallelum, &longs;i
tamen antequam &longs;u&longs;penderetur, iacebat plano horizontis, æquidi&longs;tans;
Tex. 24.
lum, aut cubus)
&longs;pecies numerorum, &longs;icuti &longs;upra tex. 9. & 20. explicauimus, quò nunc te vi
ci&longs;&longs;im, vt præ&longs;entem locum intelligas, remittimus.
Ibidem
verbum, irrationale, non videtur Ari&longs;t.
intellexi&longs;&longs;e proprietatem illam duo
rum linearum incommen&longs;urabilium longitudine, & potentia, quia v&longs;us fui&longs;
&longs;et verbo,
ne, &longs;ed v&longs;us e&longs;t verbo,
Per verbum
in directum rendere, &longs;ed in aliquo puncto frangi, &longs;eu declinari à rectitudine,
ita vt con&longs;tituat angulum.
Per verbum
quod punctum coire, &longs;i protrahantur.
Ibidem
diciariam, quamuis à recentioribus hoc nomine vocetur, &longs;ed quam hodie
dicunt A&longs;tronomiam,
tum, & locum totius Mundi, ac partium ip&longs;ius integrantium, vt &longs;unt Cœli,
& Elementa.
Tex. 25.
centes, quod non oportet fal&longs;o vti: Geometram verò mentiri dicentem pedalem, non
pedalem, aut rectam de&longs;criptam, non rectam
cludit eò, quod bæc e&longs;t linea, &longs;ed quæ per hæc e&longs;tenduntur)
materiam intelligibilem, quæ e&longs;t &longs;ubiectum Geometriæ: eam &longs;cilicet, quæ
&longs;ub figuris Geometricis &longs;en&longs;ibilibus, &
metra vtitur linea quadam &longs;en&longs;ibili pro recta, quæ verè nec e&longs;t linea mathe
matica, nec recta; &longs;upponit aliquando talem lineam e&longs;&longs;e pedalem, quæ ve
rè non e&longs;t pedalis: Verumtamen non mentitur, quia re&longs;picit ad veram li
neam mathematicam, quæ &longs;ub illa intelligitur, & quæ recta concipitur; &
quidem hæc omnia verè concipiuntur, quoniam ita e&longs;&longs;e re vera po&longs;&longs;unt.
Tex. 28.
rallelas lineas, alias, & nunc
Tex. 29.
diŭ e&longs;t &longs;emper, quod duplex, de hoc enim omni, & hoc rur&longs;us de alio dicitur omni)
aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in
plmæ; verbum autem Porrò non
e&longs;t in mathematicis, &longs;icut in alijs paralogi&longs;mus, quia in omni demon&longs;tra
tione maius extremum dicitur de omni medio, & rur&longs;us medium dicitur de
omni minori extremo, ac &longs;i diceret mathematicæ demon&longs;trationes &longs;unt in
primo modo, qui barbarè à latinis recentioribus Barbara appellatur. Hæc
e&longs;t autem pulcherrima mathematicarum commendatio, quippe præclarum
e&longs;t à laudato laudari. In mathematicis, inquit, non accidit &longs;imiliter para
logi&longs;mus, ide&longs;t, tam frequenter, quemadmodum in &longs;yllogi&longs;mis dialecticis,
quia modus argumentandi mathematicarum e&longs;t perfecti&longs;&longs;imus, quippe in
primo modo primæ figuræ.
Eodem tex.
que con&longs;equentia accipiunt, quemadmodum & Cæneus facit, quod ignis in multi
plici proportione: etenim ignis celeriter gignitur, vt ait: & hæc est proportio. &longs;ic
autem non e&longs;t &longs;yllog &longs;mus, ni&longs;i celerrimam proportio &longs;equatur multiplex: & ignem
celerrima in motu proportio)
&longs;e habent,
tiplici proportione, quemadmodum fecimus, quam in multiplicata, quem
admodum in vulgata editione. porrò quid inter multiplicem, & multipli
catam rationem inter&longs;it, optimè declarat no&longs;ter Clauius ad 4. definit. lib.
5.
Elem.
ex quo etiam loco pauca decerpam, quæ huic loco declarando con
ducunt. Proportio igitur multiplex e&longs;t habitudo inter duas quantitates in
æquales, quarum maior continet minorem, bis, vel ter, vel quater, &c. vn
de proportio multiplex habet &longs;ub &longs;e genera infinita, quando enim maior
continet minorem bis, dicitur Dupla: quando ter, Tripla: quando quater,
Quadrupla: & &longs;ic in infinitum: v. g. 2. ad 1. e&longs;t proportio dupla; 3. ad 1. tri
pla; 4. ad 1. quadrupla, &c. omnes tamen continentur &longs;ub genere multipli
cis rationis. porrò &longs;i qu&ecedil;piam proportio ex genere multiplici progrediatur
per plures terminos, v. g. proportio quadrupla progrediatur hoc modo,
1. 4. 16. 64. 256. &c. fit, vt &longs;ub&longs;equentes termini mirum in modum augean
tur. hic vides primum ip&longs;am quadruplam rationem in di&longs;po&longs;itis terminis
progredi, quia quilibet &longs;equens terminus ad præcedentem e&longs;t quadruplus. cernis etiam in paucis terminis, quinque &longs;cilicet magnum factum e&longs;&longs;e incre
mentum, cum Cæneus igitur dicens ignem augeri
&longs;ecundum multiplicem rationem, vnam ex prædictis intelligebat aliquam,
quia quælibet illarum magnopere cre&longs;cit, &longs;i propagetur, vt ad 10. quinti
definit. traditur: & vt paulo ante exemplo licuit per&longs;picere.
argumentaba
tur igitur Cæneus in hunc modum; quod in multiplici ratione augetur, ce
lerrimè augetur: ignis celerrimè augetur, ergo
augetur, quæ argumentatio vitio&longs;a e&longs;t, ex duabus quippe affirmatiuis in &longs;e
cunda figura procedens, vt colligitur ex verbis illis tex.
quentia accipiunt
Ibidem (
lum accidens accipiunt (m quo quidem ijs præ&longs;tăt, quæ di&longs;putationibus traduntur)
&longs;ed definitiones
præ&longs;tantia elucet, quia &longs;cilicet mathematicæ pro medijs vtuntur definitio
v. g. in prima Euclidis demon&longs;tratione per definitionem &longs;ubiecti probantur
tres lineæ e&longs;&longs;e æquales, quia nimirum &longs;int &longs;emidiametri circulorum æqua
lium, quæ e&longs;t ip&longs;arum definitio. & in 4. primi probantur ba&longs;is, & anguli
vnius trianguli æquales e&longs;&longs;e ba&longs;i, & angulis alterius trianguli per formalem
definitionem pa&longs;&longs;ionis, videlicet æqualitatis, quæ traditur in octauo axio
mate &longs;ic, quæ &longs;ibi mutuo congruunt, ea inter &longs;e &longs;unt æqualia. probat igitur
Euclides in quarta ba&longs;im, & angulos vnius trianguli e&longs;&longs;e æqualia ba&longs;i, & an
gulis alterius trianguli, quia o&longs;tendit, quod, &longs;i ba&longs;is illa huic ba&longs;i, & illi an
guli hi&longs;ce angulis &longs;uperponantur, congruunt; ex qua congruentia mutua,
quæ e&longs;t æqualitatis definitio, infert æqualitatem ip&longs;arum ba&longs;ium, necnon
angulorum. eadem deinde æqualitatis definitione totam demon&longs;trationem
concludit, &longs;cilicet totum triangulum toti triangulo æquale e&longs;&longs;e, quia vnum
alteri congruat. A&longs;tronomi
rerpo&longs;itionem terræ inter Lunam, & Solem, quæ interpo&longs;itio e&longs;t definitio
cau&longs;alis ip&longs;ius eclyp&longs;is, &longs;cilicet pa&longs;&longs;ionis. huiu&longs;modi
Geometras, Arithmeticos, A&longs;tronomos,
&longs;trationes: ita vt meritò dixerit Ari&longs;t. Mathematicas alias omnes natura
les &longs;cientias, quæ di&longs;putabilibus rationibus traduntur ex hac parte antecel
lere. a&longs;&longs;umunt igitur terminos conuertibiles, quia adhibent &longs;æpè definitio
nes ad demon&longs;trandum. Reliqua logici expo&longs;itores declarant.
Tex. 30. (
menta: &longs;i enim quod ita augetur, e&longs;t &longs;phæricum; augetur autem Luna; planŭ quod
&longs;phærica
ex a&longs;tronomia; A&longs;tronomi enim demon&longs;trant Lunam e&longs;&longs;e &longs;phæricam ab ef
fectu ip&longs;ius &longs;phæricitatis, qui e&longs;t illuminatio &longs;phærica: &longs;ic enim ratiocinan
tur: ea, quæ &longs;phæricè illuminantur &longs;unt &longs;phærica, Luna &longs;phæricè illumina
tur, ergo &longs;phærica e&longs;t: quæ argumentatio fu&longs;ius explicanda e&longs;t; quod ait,
quod ita augetur, ide&longs;t, &longs;phæricè, e&longs;t &longs;phæricum, ide&longs;t, quia lumen nouæ Lu
næ augetur &longs;phæricè, hoc e&longs;t, ad eum modum, quo quæuis &longs;phæra obiecta
corpori lumino&longs;o &longs;olet illuminari. illuminatio porrò Lunæ in &longs;e &longs;emper e&longs;t
eadem, quia &longs;emper dimidium Lunæ quod Solem a&longs;picit, illuminatur; dici
tur tamen augeri re&longs;pectu oculi no&longs;tri, quia &longs;cilicet initio facto à nouilunio
pars illuminata incipit quotidie magis vergere ad oculum no&longs;trum, ita vt
in dies maiorem, ac maiorem illuminationem videamus, donec opponatur
Soli, in qua oppo&longs;itione totum ferè Lunæ Vt autem
huius illuminationis non iniucundam f cias experientiam; cape &longs;phæram
quampiam &longs;olidam manu, cum qua recede ad medium cubiculi, & pone lu
men &longs;eor&longs;um ad partem aliquam: deinde brachio exten&longs;o oppone &longs;phæram
lumini, quo &longs;itu nihil de illuminatione videbis, quamuis dimidium ferè il
lius illuminetur. po&longs;tea conuerte te ip&longs;um ibidem paulatim, ita vt aliquid
illuminationis oculo tuo appareat; & videbis partem illam illuminationis,
falcatæ, &longs;eu nouæ Lunæ &longs;imilem. Deinde adhuc magis te conuerte, & cer
nes illuminationem dimidiatæ Lunæ &longs;imilem: verte adhuc te ip&longs;um donec
&longs;it &longs;phæra ita lumini oppo&longs;ita, vt inter ip&longs;am, & lumen oculus tuus &longs;it me
dius; apparebit tunc tota illuminatio, quæ erit in&longs;tar plenilunij. perge ad
non aliter ac in Luna &longs;ene&longs;cente.
&longs;phærica illuminationis augmenta. cum ergo videamus Lunam eo modo lu
mine augeri, quo &longs;phæra, hinc ip&longs;am
Po&longs;t nonnulla (
tricam, & Harmonica ad Arithmeticam, vt Apparentia ad A&longs;trologicam
tex. 20. exempla &longs;ubalternationum Per&longs;pectiuæ, & Mechanicæ cum Geo
metria &longs;unt allata. hic primo notandum Stereometriam non ef&longs;e &longs;cientiam
di&longs;tinctam à Geometria, ni&longs;i &longs;icuti partem à toto: nam cum Geometria
con&longs;ideret quantitatem, &longs;ecundum tres dimen&longs;iones, longitudinem, latitu
dinem, & profunditatem, oritur triplex illius diui&longs;ro, de lineis, de &longs;uperfi
ciebus, de &longs;olidis. pars igitur, quæ de &longs;olidis tractat,
11. 12. 13. 14. & 15. Euclidis, partim aliorum Geometrarum libris, vt li
bro Archim. de Sphæra, & Cyl. & &longs;imilibus, dicitur Stereometria à græco Porrò cur malit Ari&longs;t. Mechanicam &longs;ubalternari Ste
reometriæ, quam toti Geometriæ, qua tamen, vt videre e&longs;t apud Archime
dem, innititur, fortè ea ratio e&longs;t, quia Mechanica præcipuè con&longs;iderat ma
chinas, quæ corpora &longs;unt, & propterea præcipuè, & primò debet Stereome
triæ, quæ corpora pariter contemplatur, &longs;ubalternari. Quod ait Apparen
tia ad A&longs;irol. inteiligit per Apparentia vulgarem quandam Nautarum, &
Agricolarum a&longs;tronomiam, quæ quodammodo &longs;ubaiternatur, & pendet ex
&longs;cientia A&longs;trologiæ; indiget enim cognitione ortus, & motus a&longs;trorum,
præ&longs;ertim Lunæ, Hyadum, Pleiadum, & Canis. Reliqua
pitis optimè à Zabarella explicantur,
maticis agant, quatenus ad Logicum &longs;pectant.
Po&longs;t nonnulla (
rò Propter quid Mathematicorum; hi
&c.
appellat, quia de rebus &longs;en&longs;ibilibus &longs;unt, vt in Per&longs;pectiua de obiectis vifibi
libus, & in Mu&longs;ica de &longs;onis cogno&longs;citur Quod, ide&longs;t effectus: cuius effectus
cau&longs;a, &longs;eu Propter quid &longs;citur auxilio Mathematicarum, ide&longs;t, traditur à
&longs;cientijs &longs;ubalternantibus. v. g. alicuius effectus in Per&longs;pectiua cau&longs;a inqui
ritur, & inuenitur ope Geometriæ, cuiilla &longs;ubiacet. Hic obiter notandum,
Ari&longs;t.
fateri manife&longs;tè Mathematicas &longs;ubalternatas, &longs;eu medias o&longs;tendere
per cau&longs;as, quas &longs;ubalternantium ope perue&longs;tigant.
Et po&longs;tea (
hanc, vt quoæ e&longs;t de Iride ip&longs;um enim quod Naturalis e&longs;t &longs;cire, ip&longs;um vcrò Prop
ter quid Per&longs;pectiui
&longs;pectiuam, ita Per&longs;pectiua ad Geomettiam. qua verò ratione cau&longs;a Iridis
pertineat ad opticam,
in Meteoris, cum ip&longs;ius demon&longs;trationem afferemus.
Tex. 37. (
&longs;ecandum commune aliquod ine&longs;t
&longs;atis explicatum e&longs;t lib.
r. Priorum &longs;ecto 3. cap.
r.
nunc igitur paraphra
&longs;olum huius loci dabo. Triangnlo I&longs;o&longs;celi, & Scaleno connenit pa&longs;&longs;io i
habere tres angulos æquales duobus rectis angulis &longs;ecundum aliquod
tenus
petit habere tres angulos æquales duobus rectis.
Tex. 38. (
vbique, &longs;ed in pondere quidem mina, in cătu verò die&longs;is
pars Toni. Tonus autem e&longs;t interuallum duarum vocum, quale e&longs;t inter pri
mam vocem, Vt, & &longs;ecundam Rè, vt modo loquuntur. i&longs;tud interuallum
diuidunt Mu&longs;ici primum in &longs;emitonia, non tamen æqualia, &longs;ed vnum maius
altero. minus iterum in duas partes æquales &longs;ubdiuidunt, quarum
veteres harmonici die&longs;im dixerunt. & h&ecedil;c die&longs;is e&longs;t minima vox ab eis con
&longs;iderata; & quæ prima cadit &longs;ub &longs;en&longs;um; & propterea veluti &longs;implex prin
cipium, & clementum, ex quo alia maiora interualla conftent; & in quod
re&longs;oluuntur. igitur interual
lum i&longs;tud minimum dictum e&longs;t die&longs;is, quod &longs;it quædam diui&longs;io, &longs;eu &longs;egmen
turn Toni (
ctant, a&longs;&longs;erunt, Minam fui&longs;&longs;e maiorem libra per &longs;emunciam, æquipondera
bat enim centum drachmis: quæ refragantur huic loco. &longs;ed fortè
Ari&longs;t.
con&longs;idera&longs;&longs;e, Minam re&longs;pectu Talenti, re&longs;pectu enim illius dici pote&longs;t
principium, cum &longs;ex millia minarum in Attico talento continerentur.
Tex. 39.
quantum æquicrus, &longs;ed in quantum triangulus, no
&longs;cens, &c.)
bere tres angulos æquales duobus rectis conuenit
æquicruri, non quatenus æquicrus e&longs;t, &longs;ed quate
nus triangulus e&longs;t, &c. quid &longs;it habere tres æqua
les duobus rectis, &c. fusè explicatum e&longs;t in lib.
1.
Priorum &longs;ecto 3. cap.
1. quò te nunc mitto.
Po&longs;t pauca
duos, &c.)
duos angulos rectos non actu, &longs;ed per æquiualen
tiam trium angulorum trianguli. Vide quæ im
mediatè &longs;upra de hac re dixi, & quò te remi&longs;r.
Eodem tex
quatuor exteriores &longs;unt æquales, quoniam I&longs;o&longs;celes,
adhuc defseit, propier quid I&longs;o&longs;celes? quoniain trian
gulus: & hoc quoniam figura rectilinea, &c.)
plo geometrico vult o&longs;tendere demon&longs;trationem
vniuer&longs;alem e&longs;&longs;e particulari præ&longs;tantiorem: e&longs;t
autem exemplum de pulcherrima,
bili proprietate, quæ omnibus figuris rectilineis
conuenit, e&longs;t
neæ anguli externi omnes &longs;imul &longs;umpti, &longs;unt æqua
les quatuor rectis angulis, quæ affectio demon
&longs;tratur in &longs;cholio 32. primi Elem.
dicuntur autern
anguli externi, qui productis lateribus fiunt, vt in
triangulo pra&longs;enti anguli externi &longs;unt, B D C,
habet latera; cum exproductis lateribus oriantur. Vt autem propo&longs;itio ve
rificetur, &longs;ingula latera ordinatim &longs;unt producenda, hoc e&longs;t, ver&longs;us eandem
partem, vt in figuris appo&longs;itis vides. Quæuis igitur figura rectilinea, &longs;iue
trilatera &longs;it, &longs;iue quadrilatera, vel etiam millelatera, & proinde mille quo
que angulos externos habeat, hanc tamen mirabilem proprietatem (quod
vix credi pote&longs;t) po&longs;&longs;idet, vt omnes illi anguli externi &longs;imul &longs;int æquales
quatuor rectis angulis. vnde tres externi anguli trianguli, & quatuor exter
ni quadranguli, & quinque externi &longs;unt æquales quatuor tan
tum rectis, nec aliter res &longs;e habet in figura millelatera. Ex quo fit, vt an
guli externi cuiu &longs;uis figuræ &longs;int æquales angulis omnibus externis alterius
cuiu&longs;libet figuræ. Ari&longs;t.
igitur inquit, quando cogno&longs;cimus, quod quatuor
angulis rectis &longs;unt æquales exteriores omnes anguli alicuius figuræ, quo
niam figura illa e&longs;t triangulum &longs;calenum, adhuc talis cognitio e&longs;t defecti
ua, quia non illi competit illa pa&longs;&longs;io, quia &longs;it triangulum &longs;calenum, neque
competit &longs;caleno, quia &longs;it triangulum; &longs;ed his omnibus competit, quia &longs;unt
figuræ rectilineæ, cui hæc proprietas ine&longs;t primo, & vniuer&longs;aliter: qui igi
tur &longs;cit, &longs;calenum habere prædictam affectionem, ex eo, quod &longs;it figura re
ctilinea, perfectius &longs;cit, quia nihil amplius quæri pote&longs;t, quia illa figura re
ctilinea illud vniuer&longs;ale e&longs;t, cui primo competit; reliquis autem per illam. qui igitur vniuer&longs;ale &longs;cit, perfectius &longs;cit; quod volebat Ari&longs;t.
demon&longs;trare.
Eodem tex.
æquales)vide &longs;upra lib.
1. Priorum &longs;ecto 3. cap.
1.
Tex. 43.
æquales habet angulos)
Po&longs;t pauca
non
terra inter Lunam, & Solem po&longs;ita, impedit, ne lumen Solis feratur in Lu
nam, &longs;ed efficit, vt vmbra ip&longs;ius terræ eam contegat.
Et paulo po&longs;t
trum perforatum videremus, & lumen
permeans, planum vtique e&longs;&longs;et propter
quid comburit)
bu&longs;tione, cuæ fit per refractionem
media &longs;phæra vitrea. de qua Vitel
48. decimi libri; non au
tem de ea, quæ fit per reflexionem
ex &longs;peculo concauo quando combu
&longs;tio fit per refractionem, cau&longs;atur à
radijs Solis vitrum permeantibus,
in quo ita franguntur, vt egredien
tes è vitro &longs;imul vniantur, ex qua
vnione ita calor intenditur, vt ibi
comburat. vt in appo&longs;ita figura cer
nere facile e&longs;t; in qua radij à Sole
manentes, &longs;phæram vitream perua
&longs;int, &longs;i quid combu&longs;tibile occurrat, comburere. Si igitur, inquit Ari&longs;t.
vide
remus illos radios &longs;ic permeare, & refrangi, planum
pter quid incendant.
Ad finem tex. 43.
quæ quidem igitur, ex quibus, communia &longs;unt: quæ autem circa quod propria, vt
numerus, magnitudo)
tudine) &longs;ed ex græco tex. corrigendi &longs;unt, vti fecimus. Cæterum per prin
cipia, ex quibus intelligit Dignitates, quia ex illis di&longs;currimus. per princi
pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli
dem, explicatur &longs;ubiectum, circa quod &longs;cientia ver&longs;atur; vt in definitioni
bus primi Elem.
docemur, quid &longs;it linea, quid triangulum, quid circulus,
quid magnitudines reliquæ, quæ &longs;unt materia, circa quam Geometria &longs;pe
culatur. In &longs;eptimo verò traduntur definitiones numerorum, quid &longs;it nu
merus, quid impar, quid compo&longs;itus, quadratus, cubus, & reliquæ nume
rorum &longs;pecies, quæ &longs;unt materia &longs;eptimi, octaui, & noni, in quibus de Arith
metica tractatur.
Tex. 44.
vide, quæ de
lib.
1. Priorum &longs;ecto 1. cap.
23. ait igitur Ari&longs;t.
ab&longs;urdum e&longs;&longs;e opinari dia
metrum e&longs;&longs;e commen&longs;urabilem co&longs;tæ, &longs;eu lateri eiu&longs;dem quadrati, reli
qua &longs;unt Logica.
Tex. 1.
ram, aut Solem, aut triangulum; aliqu
inæqualitatem. &longs;i in medio, aut non)
quatenus ad Mathematicum attinet, optimè declarat. In quæ
&longs;tionibus, & demon&longs;trationibus duo &longs;unt, &longs;ubiectum, &
cau&longs;æ exi&longs;tunt, & quæruntur: v. g. Luna, terra, Sol, & triangulum &longs;unt &longs;u
biectum in demon&longs;tratione, quorum prædicata &longs;unt, Lunæ quidem, & So
lis, eclyp&longs;is. terræ autem e&longs;&longs;e in medio mundi, quod ab A&longs;tronomis ratione
ab eclyp&longs;ibus de&longs;umpta, euidentius, quam ab alio quoquam demon&longs;tratur,
vt patet ex tractatu de &longs;phœra. in quo Zabarella non probatur, qui &longs;olum
ait, terram e&longs;&longs;e in medio mundi, à Phy&longs;icis demon&longs;trari.
&longs;eu angulorum ip&longs;ius
primi Elem.
demon&longs;trat Euclides, omne triangulum habcre tres angulos
æquales duobus rectis.
Ibidem
ratio numerorum in acuto, & graui, &c)
git breuiter Ari&longs;t.
cau&longs;am formalem con&longs;onantiæ, & con&longs;equenter defini
tionem ip&longs;ius. definiunt igitur Mu&longs;ici con&longs;onantiam hoc modo; Con&longs;onan
tia e&longs;t compo&longs;itio &longs;oni grauis, & acuti, quæ &longs;uauiter auribus accidit; & quo
rum &longs;onorum proportio ad inuicem &longs;it &longs;icuti proportio numerorum, qui
quaternario includuntur: vt e&longs;t proportio 2. ad 1. vel 3. ad 1. vel 4. ad 1.
vel 3. ad 2. vel 4. ad 3.
tantum &longs;onus efficiatur; &longs;onus ille erit concordans, & auribus gratus.
hæc e&longs;t &longs;ententia pri&longs;corum præ&longs;ertim Pythagoreorum, qui propterea di
cebantnon licere Mu&longs;ico vltra quaternarium pertran&longs;ire, eò quod &longs;olæ pro
portiones, vt diximus, numerorum quaternario contentorum, concordem,
ac con&longs;onantem concentum efficere poterant: quod vt adhuc melius per
cipiamus, accipe exemplum. Sint duæ chordæ
A, & B, æqualis cra&longs;&longs;itici, & æquè ten&longs;æ. qua
rum A, dupla &longs;it ip&longs;ius B, quia igitur corpora
&longs;onantia &longs;unt in dupla proportione, erunt pa
riter corum &longs;oni in ratione dupla (vt patet ex
principijs harmonicæ) hoc e&longs;t,
vt 2. ad 1. quia &longs;cilicet &longs;onus maioris chordæ A, erit duplus ad &longs;onum mi
noris chordæ B. hoc e&longs;t, erit, vt 2. ad 1. & propterea, &longs;i &longs;imul ambæ chordæ
pul&longs;entur, &longs;onus, quem ex duobus mixtum edent, con&longs;onans,
mus auribus no&longs;tris perueniet. huiu&longs;modi porrò con&longs;onantia, quæ e&longs;t in
proportione dupla,
pafon.
blematum 19. quæ tota e&longs;t de Mu&longs;ica, dicenda &longs;int.
Tex. 2.
tata lib.
1. Priorum &longs;ecto 3. cap.
1.
Eodem tex.
ip&longs;um quid e&longs;t, vt Mathematicæ, quid vnitas, quid par, & impar)
finitiones 7. Elem.
vbi agitur de numeris. Quæ verò hoc loco de principijs
dicuntur, luculenti&longs;&longs;imè patent con&longs;ideranti definitiones, & axiomata, quæ
Mathematicis demon&longs;trationibus in omnibus ferè libris præmittuntur; ex
quibus &longs;tatim demon&longs;trationes deriuantur.
Et paulo po&longs;t
gura planum)
lus, cuius definitio e&longs;t inter definitiones primi Elem.
15. & e&longs;t huiu&longs;modi:
circulus e&longs;t figura plana, &longs;ub vnica linea comprehen&longs;a, quæ periphæria ap
pellatur, ad quam ab vno puncto eorum, quæ intra figuram &longs;unt po&longs;ita, ca
dentes omnes rectæ lineæ inter &longs;e &longs;unt æquales: in qua quidem definitione
non prædicatur planum de figura, nec figura de plano:
&longs;au plana &longs;uperficies e&longs;t figura &longs;ecundum &longs;e, ni&longs;i terminetur;
plana &longs;uperficies, cum plurimæ &longs;int figuræ curuæ, & præterea &longs;olidæ quam
plurimæ.
Ibidem
bus rectis, &longs;i id de omni triangulo mon&longs;tratum &longs;it)
3. cap.
1. petatur huius loci declaratio.
Tex. 7.
e&longs;t in 20. dednitione primi Elem.
Ibidem
tione
illud e&longs;&longs;e triangulum æquilaterum. Certum tamen e&longs;t, Geometram luppo
nere triangulum in communi, cum inter definitiones ip&longs;ius contineatur,
& e&longs;&longs;e aliquod particulare triangulum, vt fit in prædicta demon&longs;tratione,
Euclidis.
Tex. 11.
affert exemplum demon&longs;trationis per cau&longs;am materialem,
Mathematicis petitum, e&longs;t enim apud Euclidem 31. demon&longs;tratio 3. Elem.
vbi ip&longs;e o&longs;tendit angulum in &longs;emicirculo e&longs;&longs;e rectum. Vbi aduertendum e&longs;t
propo&longs;itionem hanc 31. ab Euclide demon&longs;trari duobus modis; ex quibus
&longs;ecundum innuit hoc loco Ari&longs;t.
cui a&longs;cripta e&longs;t figura &longs;imilis huic no&longs;træ;
in editione Clauiana. quod fortè non benè aduertens Iacobus Zabarella,
alioquin in his &longs;atis oculatus incidit in errorem, dicens, &longs;e nullo pacto vi
dere medium Euclidianæ demon&longs;trationis e&longs;&longs;e cau&longs;am materialem; quod
tamen nos mox aperiemus. per angulum in &longs;emicirculo intelligas eum, qui
fit à lineis ductis ab extremitatibus diametri, & &longs;imul in quoduis punctum
circumferentiæ coeuntibus, vt in figura
præ&longs;enti vides lineas A C, B C, ad C, pun
ctum conuenire,
A C B, qui dicitur angulus in &longs;emicircu
lo, quia de&longs;criptus e&longs;t in &longs;emicirculo A
C B.
proprietas, cum
periphæria &longs;umptum fuerit, &longs;emper ta
men angulus A C B, fiat rectus. quod Euclides eodem pror&longs;us medio, quod
Ari&longs;t.
hic innuit, hoc modo demon&longs;trat. ducta enim recta D C, à centro D,
ad punctum C, exurgunt duo l&longs;o&longs;celia triangula A D C, C D B, ergo per
5. primi, anguli D C A, D A C, &longs;unt æquales: pariter anguli D C B, D B C,
æquales &longs;unt. & quia per 32. primi, anguli D A C, D C A, &longs;imul &longs;unt æqua
les angulo externo C D B, & inter &longs;e æquales, erit angulus A C D, dimidium
anguli C D B. eadem ratione probatur angulus D C B, e&longs;&longs;e dimidium an
guli C D A. ergo totus angulus A C B, dimidium erit duorum angulorum
A D C, C D B, qui per 13. primi, &longs;unt vel recti, vel duobus rectis Sequitur igitur, angulum A C B, in &longs;emicirculo e&longs;&longs;e dimidium duorum re
ctorum; & quia omnes recti &longs;unt æquales, &longs;equitur dimidium duorum re
ctorum, nihil aliud e&longs;&longs;e, quam vnum rectum angulum, ergo angulus in &longs;e
micirculo, cum &longs;it &longs;emi&longs;&longs;is duorum
erat probandum. ex quibus vides medium illud, quod Ari&longs;t.
a&longs;&longs;ump&longs;it, e&longs;&longs;e
omnino idem cum eo, quo Euclides vtitur, &longs;cilicet, e&longs;&longs;e dimidium duorum
rectorum, & propterea e&longs;&longs;e rectum: quod etiam medium in toto demon
&longs;trationis decur&longs;u e&longs;t vltimum, & principale, quod proximè conclu&longs;ionem
attingit, & propterea dici meretur e&longs;&longs;e medium huius demon&longs;trationis. Cæterum, quod medium i&longs;tud &longs;it in genere cau&longs;æ materialis, patet ex co,
quod e&longs;t, e&longs;&longs;e dimidium; nam e&longs;&longs;e dimidium, vel e&longs;&longs;e tertiam partem, & &longs;i
milia, nihil aliud e&longs;t, quam e&longs;&longs;e partem; e&longs;&longs;e autem partem e&longs;t e&longs;&longs;e materiam
totius, etiam ex &longs;ententia ip&longs;ius Ari&longs;t.
ex hac præterea materia conflatur
definitio minoris extremi, vel &longs;ubiecti; d
lo e&longs;t dimidium duorum rectorum. &longs;yllogi&longs;mus enim reducitur tandem ad
culo e&longs;t dimidium duorum vides in minori propo&longs;itione contineri definitionem &longs;ubiecti materialem?
adeò vt hæc &longs;it demon&longs;tratio omnibus numeris ab&longs;oluta per cau&longs;am mate
rialem, vt benè &longs;entit Ari&longs;t. Reliqua ad logicum pertinent, etiam&longs;i per cha
racteres more mathematicorum exponantur.
Tex. 24.
aut propter quid apparet?
aut propter quid
Iris? omnia enim hær idem problemata &longs;unt genere, omnia enim &longs;unt refractio, &longs;ed
&longs;pecie altera)&longs;cilicet echo; propter quid apparet?
&longs;cilicet imago in &longs;peculo.
dicit cau&longs;am echo, imaginis in &longs;peculo, & iridis
in nubibus e&longs;&longs;e eandem; nimirum refractionem; quamuis tres illæ refractio
nes, &longs;eu; vt melius loquamur, reflexiones differant &longs;pecie ab inuicem, illa
enim e&longs;t repercu&longs;&longs;io vocis; hæc reflexio &longs;peciei vi&longs;ibilis ex corpore ter&longs;o;
i&longs;ta qua
ratione autem i&longs;ta omnia fiant, longum e&longs;&longs;et exponere, & ab intelligentia
huius loci fortè alienum. Illud tamen non prætereundum, quod &longs;i propriè
cum Per&longs;pectiuis loqui velimus, dicendum e&longs;&longs;e, omnia illa e&longs;&longs;e reflexionem,
non refractionem. nam reflexio e&longs;t, quando linea vi&longs;ualis, per quam fertur
&longs;péecies in aliquod corpus ter&longs;um, impingit, ex quo deinde ad oculos refle
ctitur. refractio tunc e&longs;t, quando &longs;pecies obiectivi&longs;ibilis tran&longs;it per media
diuer&longs;æ cra&longs;&longs;itiei., vt quando &longs;pecies lapilli per aquam primùm, deinde per
æerem means ad oculum peruenit; tunc enim linea, per quam &longs;pecies pro
greditur, frangitur in confinio aquæ, & aeris, ita vt &longs;pecies non per vnicam
lineam rectam, &longs;ed per fractam, &longs;eu refractam in confinio illo, oculis tan
dem accidat.
In fine textus
eclyp&longs;is appellatur, &longs;ed ilium, quo paulatim lumen Lunæ minus oculis no
&longs;tris apparet: decre&longs;cente enim Luna &longs;olent humida augeri.
Tex. 25.
& c.
titates, quæ &longs;unt proportionales, &longs;int etiam alternatim, &longs;eu permutatim
proportionales explicatum e&longs;t ad tex. 13. primi Po&longs;ter. quæ etiam nece&longs;&longs;a
ria &longs;unt ad hunc locum benè intelligendum. Illud autem commune propter
quod ea, quæ &longs;unt proportionalia, &longs;int etiam permutatim proportionalia,
e&longs;t quoddam innominatum, de quo ibi dictum e&longs;t, quod cum conueniat li
neis, & numeris, & tamen &longs;eparatim de vtri&longs;que illa pa&longs;&longs;io demon&longs;tretur,
quærit cuinam primò, & per &longs;e conueniat hæc pa&longs;&longs;io, e&longs;&longs;e permutatim pro
portionale; &longs;cilicet quidnam &longs;it illud innominatum; in quo deinde commu
nicent lineæ, & numeri, vt inde habeant e&longs;&longs;e etiam permutatim propor
tionalia.
Ibidem (
indicare, in quonam con&longs;i&longs;tat &longs;imilitudo inter duas-figuras rectilineas geo
metricas, quam &longs;imilitudinem Euclides definit. 1. &longs;exti, &longs;ic explicat: &longs;imi
les figuræ rectilíneæ &longs;unt; quæ & angulos &longs;ingulos, &longs;ingulis angulis æquales
habent, vt &longs;i duo triangula appo&longs;ita habeant angulos æquales,
angulum B, angulo E. angulum C, angulo F. & præterea latera, quæ &longs;une
circa angulos æquales, v. g. cirea an
gulos A, & D, habeant proportiona
lia, hoc e&longs;t, vt latus A B, ad latus A C;
ita &longs;it latus D E, ad latus D F; & &longs;ic de
lateribus alijs circa reliquos angulos
æquales; erunt tunc prædicta duo tri
angula fimilia.
Ibidem (
guræ rectilineæ æquales e&longs;&longs;e quatuor rectis angulis: vide quæ &longs;crip&longs;imus de
hac re ad tex. 39. &longs;ecundi Po&longs;ter. quæ huic pariter loco &longs;atisfaciunt.
Cap. 13. (
quæ de hac re &longs;crip&longs;i lib.
1. Priorum &longs;ecto 1. cap.
23.
Eodem cap. (
in omnibus dicitur: nam vox acuta quidem velox (&longs;icut dicunt, qui &longs;e
cundum numeros harmonici &longs;unt) angulus autem acutus, qui minor e&longs;trecto; gla
dius verò, qui e&longs;t anguli acuti
tum, quod e&longs;t in voce acuta; aliud, quod e&longs;t in angulo acuto: aliud denique,
quod e&longs;t in gladio acuto horum enim trium acumen diuer&longs;o modo &longs;e habet. nam acumen vocis, & &longs;oni ex celeritate motus, qua aer percu&longs;&longs;us impelli
tur; grauitatem autem ex tarditate oriri tradiderunt antiqui Mu&longs;ici om
nes: quamuis non ex &longs;ola celeritate, & tarditate, &longs;ed ex alijs etiam cau&longs;is
oriri po&longs;&longs;e voluerint. Primus
phirium in harmonicis Ptolæmei, & Zarlinum pag. 58. complem.
mu&longs;ica
lium, ait, &longs;i virga celerius feriat aerem, gigni motum celeriorem in aere,
ga tardius aerem feriat, gigni motum in aere tardiorem, ex quo etiam &longs;o
num grauem, vt experientia docet. Ptolæmeus deinde lib.
1. cap.
3. Harm.
cum ex alijs, tum ex celeritate oriri &longs;onum acutum, grauem verò ex tardi
tate a&longs;&longs;erit; vt &longs;i chorda eadem parum inten&longs;a pul&longs;etur, tardius aerem ver
berat, & ideo grauiorem &longs;onum efficit: &longs;i autem magis intendatur, validius
aerem pul&longs;abit, & proinde citiorem motum illi imprimet, & propterea
acutiorem &longs;onum reddet. hæc ille.
videmus etiam, quod cannæ organo
rum maiores cum plus aeris moucant, & idcirco tardius, &longs;onum grauiorem
emittunt, quàm cannæ graciliores, quæ quia parum aeris cient, & ideo ce
lerius, &longs;onum acutum edunt. ab hac &longs;ententia po&longs;teriores Mu&longs;ici non recel
&longs;erunt, vt videre e&longs;t apud Zarlinum.
In quo po&longs;tea con&longs;i&longs;tat ratio acnti anguli, explicat
ip&longs;ius, quæ e&longs;t inter de&longs;initiones primi Elem.
huiu&longs;modi, Angulus acutus
e&longs;t, qui minor recto e&longs;t. Demum explicat, cur nam gladius dicatur acutus,
quia nimirum habet angulum acutum &longs;uperficialem, ide&longs;t, quem duæ &longs;uper
ficies &longs;imul in acie gladij concurrentes efficiunt.
Eodem cap. (
rentiæ &longs;unt; vt coloris, qui e&longs;t in corporibus, & in melodijs
tilenas omnes ad tria genera reuocarunt, videlicet Enharmonicum, Chro
maticum, & Diatonicum; quæ diftinguebantur inuicem ex varia diui&longs;ione
interuallorum, ex quibus ip&longs;orum Monochordia conftabant: &longs;iue ex varijs
vocum interuallis, v. g. quia in vno continebantur plures toni, vt in Diato
nico; in alio plures die&longs;es, vt in Enharmonico; in tertio verò plura &longs;emito
nia, vt in Chromatico: quæ vox deducitur à chroma, græco, quod latinis
e&longs;t color; quare Chromaticum latinè redditur coloratum. Hic e&longs;t igitur
color ille, quem hic Ari&longs;t.
innuit. quod genus for&longs;itan à calore denomina
batur, quòd ip&longs;ius notæ mu&longs;icales e&longs;&longs;ent coloratæ, vt hoc modo ab alijs ge
neribus digno&longs;eeretur
cundæ partis, etiam no&longs;tra tempe&longs;tate aliquo modo per&longs;euerare, cum vi
deamus in organis, & alijs huiu&longs;modi in&longs;trumentis, quæ pinnas, vulgò ta
&longs;tos, habent; illas inquam pinnas, quæ chromaticis interuallis deputatæ
&longs;unt, colore nigro tinctas e&longs;&longs;e.
Cap. 1. loco 10. (
Philo&longs;ophorum putarunt omnia ex indiui&longs;ibilibus componi, vt Demo
critus, & Leucippus, & propterea dixerunt, etiam lineas con&longs;tare ex lineis
quibu&longs;dam ade ò paruis, quæ omnino e&longs;ient in&longs;ecabiles, &longs;eu indiui&longs;ibiles: de
quibus plura in libello de line is in&longs;ecabilibus.
Cap. 2. loco 32. (
&longs;uppenimus lectorem inteil exi&longs;&longs;e definitiones &longs;altem primi Elem.
in
ter quas d
Cap. 2. loco 41. (
ob de&longs;initionis defectum non facile de&longs;cribi; vt & quoniam, quæ ad latus &longs;e
cat planum linea, &longs;imiliter diuidit & lineam, & locum: definitione autem dicta,
fi
lugitur à nonnullis (
ceptos ab æquiuoco
nem &longs;ignificat: hic autem &longs;ignificare proportionem res &longs;ubrecta &longs;atis mani
fe&longs;tat. Notandum po&longs;tea cum Alexandro (quod & &longs;uperius alias commo
nui in cap.
de Priori, & alibi) per verbum (De&longs;cribi) fignificari hoc loco
geometricè demon&longs;trare, quoniam Geometræ
nibus, &longs;Vult autem Ari&longs;t.
exemplo mathematico
o&longs;tendere, difficile e&longs;&longs;e di&longs;putare, aut Porrò exemplum mathematicum hic
allatum &longs;ic videtur explicandum: Conetur aliquis demon&longs;trare hanc pro
po&longs;itionem; &longs;i linea ducta fuerit æquidi&longs;tans lateri vnius plani trianguli, &longs;e
cabit & latera, & locum, ideft &longs;uperficiem illam triangularem &longs;imiliter, ide &longs;t
in eadem proportione, vt in triangulo A B C,
linea D E, parallela ba&longs;i B C, &longs;ecat latera A B,
& A C, in punctis D, & E, in eadem ratione,
in qua etiam fecat totum triangulum, ita vt
eadem &longs;it proportio lineæ A D, ad D B, & lineæ
A E, ad E C, quæ e&longs;t partium totalis trianguli
A B C, &longs;eilicet quæ e&longs;t partis A D E, ad partem
E D C, fiue ad partem D E B. quod con&longs;tat ex
&longs;ecunda 6. Elem. Inquit ergo Ari&longs;t.
Si quis
vellet hoc demon&longs;trare nondum præmi&longs;&longs;a defi
nitione eorum, quæ habent eandem rationem, &longs;iue nondum definitione al
lata quantitatum proportionalium, hic difficile id valeret o&longs;tendere: at ve
rò allata prins definitione quantitatum proportionalium facile demon&longs;tra
bit. Subdit verò Ari&longs;t.
dictam definitionem, dicens, tunc quantitates e&longs;&longs;e
proportionales, quando habent eandem ablationem, ide&longs;t, eandem diui&longs;io
nem, ide&longs;t, eadem diui&longs;io ne tantum proportionaliter de vna, quantum de
altera magnitudine re&longs;ecatur: Quemadmodum etiam Euclides loco cita
to probat, latera illius trianguli, & &longs;uperficiem e&longs;&longs;e &longs;imiliter diui&longs;a, ex quo
&longs;equitur e&longs;&longs;e proportionalia. Porrò Euclides definit.
&longs;eptima 5. paulo ali
ter definit quantitates proportionales e&longs;&longs;e illas, quæ eandem habent ratio
nem, v. g. &longs;i &longs;it, vt prima ad &longs;ecundam, ita tertia ad quartam. ex quibus
quoad Mathematicas &longs;pectat, huic loco &longs;atisfactum &longs;it.
Cap. 4. loco 86.
tiones, tenere, nam quemadmodum in Geometria ante opus e&longs;t circa elementa exer
citatum e&longs;&longs;e, & in numeris circa capitales promptè &longs;e habere, & multum refert ad
boc, & alium numerum cogno&longs;cere multiplicatum)
demon&longs;trationes faciliores, & &longs;impliciores, quales propriè &longs;unt omnes, quæ
&longs;ex prioribus libris Euclidianis continentur: ex illis enim tanquam ex ele
mentis ab&longs;tru&longs;iores, & difficiliores demon&longs;trationes deducebant.
e&longs;t ratio, cur Euclides &longs;uos librait igitur curan
dum e&longs;&longs;e horum elementorum cognitionem in promptu habere, quia fre
quens de ip&longs;is incidit di&longs;putatio. Per capitales numeros intelligo &longs;implices
ab v& quando ait, alium numerum cogno
&longs;cere multiplicatum, &longs;ignificat vtile valdè e&longs;&longs;e ad quotidianum v&longs;um
cogno&longs;cere, quemnam numerum producant numeri capitales,
&longs;i ad inuicem multiplicentur, quamuis huiu&longs;modi co
gnitio facilis, ac leuis &longs;it: qua de cau&longs;a vide
mus v&longs;que in hanc diem pueros diu in
Abaco memoriter perdi&longs;cen
do detineri.
Cap. 10.
&longs;ub arte &longs;unt, captio&longs;æ &longs;unt ratiocinationes)
phia circa verum, vt Hippocratis quadratura, quæ per lunulas, &longs;ed, vt
Bry&longs;&longs;o quadrauit circulum; & tamet&longs;i quadretur circulus, quia tamen
non &longs;ecundum rem, ideo &longs;ophi&longs;ticus)
exhibere æquale tentauerit, explicatum e&longs;t abundè in 2. Priorum cap.
31.
& quo itidem modo Bry&longs;&longs;o lib.
1. Po&longs;ter. tex. 23.
tandum per p&longs;eudographiam intelligere, vt apertè etiam inferius explicat,
Geometricam demon&longs;trationem fallacem, eò quod demon&longs;trationes geo
metricæ fiant adhibitis de&longs;criptionibus, &longs;eu figurationibus: p&longs;endographia
autem latinè idem e&longs;t, ac fal&longs;a de&longs;criptio; quemadmodum è contrariò, &longs;i
cuti &longs;upra in Topicis, & alibi ob&longs;eruaui, per de&longs;cribere intelligit geometri
cè demon&longs;trare, & per de&longs;criptiones intelligit demon&longs;trationes geometri
cas. Qua ratione item Hippocrates ex ijs, quæ &longs;ub arte Geometriæ &longs;unt,
procederet ibi dictum e&longs;t, propter quod non e&longs;t contentio&longs;a, quamuis fallax
ip&longs;ius demon&longs;tratio: appellat enim Ari&longs;t illas demon&longs;trationes contentio
&longs;as, quæ non procedunt ex proprijs illius &longs;cientiæ, in qua fiunt, &longs;ed ex com
munibus alijs &longs;cientijs: captio&longs;as verò, & &longs;ophi&longs;ticas, quæ ex proprijs &longs;cien
tiæ, in qua fiunt, decipiunt. At verò demon&longs;tratio, &longs;eu p&longs;eudographia Bry&longs;
&longs;onis erat contentio&longs;a, quia ex communibus, & extra Geometriam petitis
argumentabatur: quemadmodum ibi explicatum e&longs;t.
Eodem cap.
tetragoni&longs;mum, de quo in 2. Priorum, quæ non contentio&longs;a dicitur, quia ex
proprijs Geometriæ deducebatur.
Ibidem
ad Geometriam &longs;olum; eo quod ex proprijs &longs;it principijs)
dem)vide 2. Prior cap.
31. & quæ pau
lo ante in præcedentibus locis diximus.
Ibidem
per communia deducebatur. lege &longs;uperius dicta in præcedentibus locis hu
ius capituli.
Ad &longs;inem cap.
uit)
tem in orbe quadrando, ac Hippocratem,
Ari&longs;t.
his verbis videtur &longs;ignificare, ide&longs;t,
ip&longs;um, quamuis ex proprijs Geometriæ,
fal&longs;is tamen ratiocinatum e&longs;&longs;e. Cæterum
Antiphontem in hunc modum orbem ad
quadrum redigere tenta&longs;&longs;e, tradit Simpli
cius. circulo quadrando in&longs;cribebat pri
mò quadratum A B C D. deinde in &longs;ingu
lis quatuor &longs;egmentis in&longs;cribebat totidem
trigona æquilatera, vt patet in ad&longs;cripta po&longs;tea &longs;uper &longs;ingula latera horum triangulorum in reliquis &longs;egmen
tis in&longs;cribebat adhuc triangula &longs;imilia triangulo A I E. alia in&longs;uper trigona
&longs;uper latera i&longs;torum con&longs;tituebat, donec ambitus figuræ illius multilateræ
in circulo delinearæ, circumferentiæ circuli aptaretur. quod fieri po&longs;&longs;e ille
falsò contra Geometriæ principia a&longs;&longs;umebat; e&longs;t enim principium Geome
tricum continuum e&longs;&longs;e diui&longs;ibile in infinitum,
po&longs;&longs;e; cui principio aduer&longs;atur, dum putat &longs;e con&longs;umpturum rotum circu
lum, diuidendo illud in triangula &longs;emper minora; vel quia putat, lineam
curuam con&longs;tare ex minimis lineis rectis. Similiter igitur
res errauit, qúi pariter in Geometria fallebatur: Antiphon quidem contra
principia illius: Hippocrates verò a&longs;&longs;umens fal&longs;i quidpiam in Geometria. At Bry&longs;&longs;o, eo quod per communia alijs &longs;cientijs deduceret ratiocinatio
nem propterea p&longs;eudographia Antiphontis non litigio&longs;a quidem, &longs;ed
tamen fallax extitit, non enim per communia alijs &longs;cientijs
procedat; vnde nec transferri poterat ip&longs;ius fal&longs;a de
&longs;criptio, &longs;eu demon&longs;tratio extra Geometriæ li
mites, quod cau&longs;a e&longs;t contentionis.
EX PRIMO LIBRO
PHYSICORVM.
Tex. 11.
principijs aliquis demon&longs;trans
vt tetragoni&longs;mum, eum quidem, qui per &longs;ectiones Geometrici est di&longs;
&longs;oluere: illum autem, qui Antiphontis non Geometrici e&longs;t
ni&longs;mum, &longs;eu circuli quadraturam per &longs;ectiones, e&longs;&longs;e illam Hip
pocratis Chij exi&longs;timant græci expo&longs;itores, qui per lunulas, quas Ari&longs;t. &longs;e
ctiones appellat, orbem quadrare tentabat. Eius den on&longs;trationem expli
caui ad cap.
31. de Abductione in 2. Priorum, quam inibi videas. hoc &longs;olum
hic notandum pertinere ad Geometram, ip&longs;am refellere, quia ex fal&longs;a qua
dam præmi&longs;&longs;a ex Geometria de&longs;umpta, ratiocinabatur, idcirco debet (in
quit Ari&longs;t.) Geometra illius deceptionem inuenire. Tetragoni&longs;mum autem
Antiphontis non e&longs;t Geometræ
metriæ, &longs;upponebat enim circuli circumferentiam ex indiu
lineis rectis componi: cuius fal&longs;am demon&longs;trationem exp
ad cap.
10. primi Elench. po&longs;&longs;umus addere tertiam rat
Hippocrates non procedebat per communia alijs &longs;ci
tex. 23. primi Po&longs;ter. cap.
8. vbi ip&longs;ius p&longs;endographi
modum igitur Geometra di&longs;&longs;oluit fal&longs;as tantum
uatis Geometricis principijs procedunt; non au
principia conuellunt: ita Phy&longs;ico non incumbit
li&longs;&longs;um naturæ principia de&longs;truentes di&longs;ceptare, aut falla
coarguere. Hoc volebat Ari&longs;toteles inferre.
Tex. 20. (
e&longs;t phy&longs;ici: Per&longs;pectiua autem ma
quatenus phy&longs;ica e&longs;t
maticum, libuit tamen illum in ordin
ræpretium etenim e&longs;t ea, quæ in ip&longs;o continentur à nonnullis recentioribus
rectè intelligi, vt ab his moniti, ab inani quadam optices impugnatione ab
&longs;tineant, ac tandem ex Ari&longs;t.
lineas illas vi&longs;uales quas ip&longs;i de medio tollunt,
per&longs;picuè videant. cætera, quæ in præcedentibus locis Ari&longs;t.
de Natura Ma
thematicarum habet, &longs;unt præter no&longs;trum in&longs;titutum.
Tex. 28. (
quod quid erat e&longs;&longs;e, & huius genera, vt ip&longs;ius diapa&longs;on duo ad vnum, & omnino
numerus, & partes, quæ in ratione &longs;unt
textu
&longs;ter. &longs;uper verba illa (
&longs;onantiæ
quæ &longs;ub his num. 2.1. continetur: quibus per&longs;pectis facilis erit phy&longs;ico totius
loci intelligentia.
Tex. 68. (
bihbus, vt in Mathematicis, ad definitionem enim recti, aut commen&longs;urabilis, aut
alius cuiu&longs;piam reducitur vltimum
mon&longs;trare
ad definitionem reducant. quorum exempla in logicis ex Mathematicis at
tuli: &longs;ed etiam &longs;equentis loci exemplum de triangulo idem apertè manife
&longs;tat; in quo probat duos angulos A C B, A C D, e&longs;&longs;e rectos, ex definitione
ip&longs;orum, &longs;iue ex definitione lineæ perpendicularis A C, quod idem e&longs;t.
Tex 89. (
natur am fiunt qua&longs;i eodem modo; quoniam enim hocrectum e&longs;t, nece&longs;&longs;e e&longs;t, trian
gulum trcs angulos habere æquales duobus rectis; &longs;ed non, &longs;i hoc, illud; &longs;ed &longs;i hoc
non e&longs;t,
difficultatis in exemplo hoc mathematico explicando, ita vt recentiores
quidam textum
mum ex græcis codicibus interpretationem hanc veram attuli. deinde, quia
etiam græci in exemplo mathematico enodando, vel malè, vt Simplicius;
vel ob&longs;curè nimis, vt reliqui; Latini verò vel nihil, vel peius multò loquun
tur, ideò &longs;ic ego exponendum cen&longs;ui. cum velit Ari&longs;t.
o&longs;tendere nece&longs;&longs;ita
tem, quæ in &longs;cientijs inter præmi&longs;&longs;as, &longs;cu medium, & conclu&longs;ionem reperi
tur, affert exemplum illud mathematicum &longs;ibi familiare, demon&longs;trationem
&longs;cilicet illam, qua o&longs;tenditur, omne triangulum habere tres angulos æqua
les duobus rectis angulis, cuius fu&longs;i&longs;&longs;imam explicationem inuenies &longs;upra in
primo Priorum, &longs;ecto 3. cap.
1. quam nece&longs;&longs;e e&longs;t, con&longs;ulas. pro medio autem
huius pa&longs;&longs;ionis accipit lineam perpendicularem, quam innuit verbis illis
A C, &longs;it perpendiculare
ducatur B C, in D; tunc triangulum A B C,
habere tres angulos, A, B, & A C B, æquales
duobus rectis planum erit: nam
&longs;it perpendiculare (quod Ari&longs;t.
dicit, cum
ctum
A C D, recti, ex definitione lineæ perpendicu
laris, cum ergo duo anguli A, & B, externo,
32. primi, & reliquus angulus A C B, communis, ide&longs;t, &longs;it angulus triangu
li, & angulus vnus lineæ perpendicularis, & ideò rectus; manife&longs;tè apparet,
tres angulos A, B, A C B, e&longs;&longs;e æquales nece&longs;&longs;ariò duobus rectis, ex po&longs;itio
ne illius recti, &longs;iue lateris perpendicularis, quia ex verò, verum nece&longs;&longs;ariò
&longs;equitur; non tamen po&longs;ita hac pa&longs;&longs;ione, &longs;iue conclu&longs;ione, habere &longs;cilicet
tres angulos æquales duobus rectis, nece&longs;&longs;ariò &longs;equitur illud e&longs;&longs;e rectum,
idelt latus illud A C, e&longs;&longs;e perpendiculare ad latus B C, quia verum
&longs;equi pote&longs;t ex verò, & falsò. valebit tamen hæc con&longs;e quen
tia, &longs;i triangulum non habet hanc proprietatem, ne
que illud rectum e&longs;t, ide&longs;t,
ctum crit
non, ni&longs;i exfal&longs;o &longs;equitur.
Tex. 26.
ab impari terminatum tribuit ijs, quæ &longs;unt, infinitatem. &longs;ignum autem
huius id e&longs;&longs;e, quod contingit in numcris, circumpo&longs;it is enim Gnomoni
bus circa vnum, & &longs;eor&longs;um, aliquando quidcm &longs;emper aliam fieri &longs;pe
ciem, aliquando autem vnam)
prius, quæ in cap.
de Motu in po&longs;t prædicamentis &longs;crip&longs;i de Gnomone, ad
&longs;imilitudinem enim Gnomonis illius Geometrici, inueniuntur etiam in nu
meris Gnomones Arithmetici. Pythagorici enim (à quibus i&longs;ta mutuatus
e&longs;t Ari&longs;t.
numeros impares &longs;olos appellabant Gnomones, eò quod in for
mam normæ æquilateræ, &longs;iue Gnomonis con&longs;titui po&longs;&longs;int, vt patet in his
nimirum in ternario, quinario, &longs;eptenario, & &longs;ic de
reliquis imparibus. pares autem numeri, quia ne
queunt in figuram normæ æquilateræ di&longs;poni, cum
non habeant vnitatem pro angulo, & paria po&longs;tea la
tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, &longs;i di
&longs;ponatur &longs;ic
&longs;tat;
illi nece&longs;iaria e&longs;t. Pythagorici igitur dicebant, numerum parem ideò e&longs;&longs;e
infinitum ip&longs;um, quia videbant ip&longs;um e&longs;&longs;e cau&longs;am perpetuæ diui&longs;ionis, cum
quælibet res quanta &longs;it diui&longs;ibilis bifariam, ide&longs;t in duo &longs;ecundum numerum
parem, & &longs;ubdiui&longs;ibilis po&longs;tea bifariam, & &longs;ic in infinitum, vt de linea pro
blematicè probatur in 10. primi Elem.
quamuis theorematicè &longs;it axioma. hunc porrò numerum parem dicebant terminatum e&longs;&longs;e ab impari, quia ori
tur ex diui&longs;ione cuiu&longs;uis rei, quæ vna &longs;it, &longs;umentes vnitatem pro impari. &longs;ignum præterea huius finitatis ab impari, & infinitatis à pari numero pro
cedentis, aiunt e&longs;&longs;e Gnomones, numeros &longs;cilicet impares: Gnomones enim,
ide&longs;t impares numeri vnitati additi, producunt eandem perpetuò numero
rum formam, videlicet quadratum: at verò è contrariò numeri pares vni
tati additi, conflant perpetuò varias numerorum formas: quapropter vi
dentur numeri impares e&longs;&longs;e finitatis cau&longs;a; &longs;icut pares exaduersò infinitatis
principium. quæ vt melius intelligas, declaranda e&longs;t 26. propo&longs;.
7. Arith
metices lordani, vbi i&longs;tud idem demon&longs;trat, quæ e&longs;t hæc. &longs;it vnitas, & &longs;uo or
dine &longs;equantur impares, vt in &longs;equenti hac &longs;erie apparet 1. 3. 5. 7. 9. & c.
&longs;i igitur vnitati addatur ternarius in Gnomo
nis modum, vt vides in prima figura, produ
cetur quaternarius numerus, qui e&longs;t numerus
quadratus (quid &longs;it quadratus numerus expli
caui in Logicis tex. 9. primi Po&longs;ter.) etfi huic
quaternario addatur &longs;equens impar, qui e&longs;t
quinarius in modum Gnomonis, vt in &longs;ecund
figura, &longs;it numerus nouenarius, qui pariter e&longs;t quadratus. et&longs;i huic &longs;imiliter
addatur &longs;e quens impar, nimirum &longs;eptenarius, conflabitur &longs;edenarius, qui
numerus pariter quadratus e&longs;t, vt in tertia figura, & hoc modo, &longs;i in infiniVides igitur, qui
ratione Gnomonum, &longs;iue imparium additione fiat &longs;emper eadem &longs;pecies,
&longs;cilicet quadratus numerus, quod &longs;ignum e&longs;t, inquiunt, imparem numerum
non infinitatis, &longs;ed finitatis e&longs;&longs;e auctorem. Po&longs;t prædictam 26. propo&longs;itio
nem Iotdani, &longs;unt aliquot propo&longs;itiones, quarum &longs;umma hæc e&longs;t: &longs;i pares
numeri ab vnitate coaceruentur; coaceruati eru
merorum. quæ &longs;ic explicantur: &longs;int ab vnitate pares di&longs;po&longs;iti ordinatim
hoc modo, 1. 2. 4. 6. &c. &longs;i igitur vnitati binarius coaceruetur, fit numerus
triangularis, vt in prima figura. &longs;i huic ternario
coaceruetur &longs;equens par, fiet altera &longs;pecies, ni
mirum
ra. cui &longs;i &longs;equens addatur par, &longs;cilicet &longs;enarius,
fiet iterum noua numeri forma, v. g. dodecago
nus, vt in tertia figura. & &longs;ic &longs;emper in infinitum nouæ ac variæ numerorum
formæ ex hac additione parium prouenient, quod argumento e&longs;t numerum
parem infiniti naturam &longs;apere. Porrò reperiri numeros triangulares, pen
tagonos, & &longs;imiles, con&longs;tat ex Arithmetica Nicomachi, Boetij, & Iordani,
citati in definitionibus 7. &longs;uæ Arithmeticæ, atque ex tractatu Diophantis
Alex. de numeris rectangulis.
Tex. 31.
cunt lineas quantumuis longas, &longs;eu indefinitæ longitudinis, quas etiam in
finitas appellant: & hoc modo vtuntur infinito, vt infra tex. 71. ip&longs;e Ari&longs;t.
exponit. alio præterea modo vtuntur infinito, vt quando &longs;upponunt data
quauis quantitate po&longs;&longs;e &longs;umi maiorem, vel etiam minorem in infinitum, vt
patet ex 6. po&longs;tulato primi Elem.
editionis Clauianæ. numerum
geri po&longs;&longs;e in infinitum, e&longs;t &longs;ecundum po&longs;tulatum libri 7. Elem.
vel demum
quando probant quamlibet lineam po&longs;&longs;e diuidi bifariam, quia hinc &longs;equitur
po&longs;&longs;e &longs;ub diuidi in
Tex. 68. & 69. plura de magnitudine, & numero continent; &longs;ed quæ non
indigeant opera no&longs;tra.
Tex. 71.
&longs;ic e&longs;&longs;e infinitum, vt actu &longs;it ver&longs;us augmentum, vt intran&longs;ibile,
digent infinito,
phy&longs;ica tollens infinitum actu, non e&longs;t Mathematicis impedimento, quia ip&longs;i
non vtuntur infinito actu; quam enim ip&longs;i ducunt lineam infinitam, non e&longs;t
verè infinita, &longs;ed indefinit
po&longs;&longs;it ad demon&longs;trandum &longs;ufficere.
Tex. 120. ter in hoc textu meminit commen&longs;urabilitatis, & incommen
&longs;urabilitatis, quæ e&longs;t diametri ad co&longs;tam: cuius explicationem vide
primo Priorum, &longs;ecto primo, cap.
23.
Tex. 6.
dinem chordarum in mu&longs;icis in&longs;trumentis, vbi media chorda edit &longs;o
num, re&longs;pectu quidem vltimæ, & &longs;upremæ chordæ grauem: re&longs;pectu verò
primæ, & infimæ acutum.
Tex. 15.
lib.
1. Priorum, &longs;ecto 3. cap.
1. huius rei explicationem reperies.
Tex. 33.
minimum introducens, maxima
&longs;i quis, vt Democritus po&longs;uerit in magnitudinibus e&longs;&longs;e minima,
&longs;eu indiui&longs;ibilia, ex quibus entia mathematica componerentur,
hic euerteret maxima mathematicorum, ide&longs;t maxime ip&longs;orum demon&longs;tra
tiones, atque etiam effata euerterentur: v. g. 10. primi Elem.
quæ docet
quamlibet lineam po&longs;&longs;e diuidi bifariam nulla e&longs;&longs;et, quia linea illa, quæ con
&longs;taret ex tribus Democriti atomis, nulla ratione bifariam &longs;ecari po&longs;&longs;et. pa
riter totus ferè decimus liber Elem.
deceptiuus, & nullus e&longs;&longs;et, &longs;i enim da
rentur illæ atomi, ex quibus
men&longs;urabiles, quandoquidem omnes communi illa, ac indiuidua, commen
&longs;urarentur. po&longs;tulatum
minorem pror&longs;us irritum redderetur, quia data atomo, illa minor accipi
non po&longs;&longs;et.
Tex. 36.
infinita, in qua
quando lineam
enim tempus, in quo circulariter latum
e&longs;t Cœlum finitum e&longs;t, & ablatum igitur,
quo &longs;ecans ferebatur; erit igitur aliqued
prmcipium, quo primum linea A G E, li
neam &longs;ed impo&longs;&longs;ibile est; non
est igitur circulariter verti infinit&
textus hic parum &longs;it mathematicus,
quia tamen &longs;upponit figuram mathe
maticam, quæ in codicibus pariter, ac
commentarijs de&longs;ideratur, illam pla
cuit apponere. in qua quidem, quamuis duæ lineæ infinitæ &longs;upponantur, vna
ad alteram
illas, ad quas deberent e&longs;&longs;e infinitæ lineolæ quædam infinitatem indicantes. debemus po&longs;tea, vt mentem Ari&longs;t.
percipiamus concipere lineam A G E,
moueri circulariter facto centro in G. quæ quia infinita &longs;upponitur ad par
tem E, &longs;ecabit nece&longs;&longs;ariò alteram
finito tempore percurret, finito enim tempore tota mundi circulatio per
agitur, &longs;patio videlicet viginti quatuor horarum. ex quo Ari&longs;t.
infert mun
dum non po&longs;&longs;e e&longs;&longs;e infinitæ magnitudinis; quia &longs;i mundus e&longs;&longs;et infinitus; &. duæ lineæ infinitæ, quales &longs;unt prædictæ in ip&longs;o,
ra earum A E, intelligatur, alteram
in diurna conuer&longs;ione pertran&longs;ibit: fieri autem nequit, vt infinita magni
tudo finito tempore percurratur; quare dicendum e&longs;t, mundum e&longs;&longs;e finita
magnitudine præditum.
Tex. 48.
&longs;ur abiles)
catum e&longs;t lib.
1. Priorum, &longs;ecto 1. cap.
23.
Tex. 119.
ne quidem, dico autem, vt triangulum impo&longs;&longs;ibile e&longs;t duos rectos habere, &longs;i hæc)
ide&longs;t, &longs;i &longs;upponantur fal&longs;a quædam, quæ &longs;upponi po&longs;&longs;unt, &longs;equetur impo&longs;&longs;i
bile e&longs;&longs;e triangulum habere tres angulos æquales duobus rectis angulis, vi
de, quæ &longs;crip&longs;i lib.
1. Priorum, &longs;ecto 3. cap.
1. de hoc, quod e&longs;t, habere tres
angulos æquales duobus rectis. v. g. &longs;i in triangulo pag.
73. producto late
re A C, in D. &longs;i &longs;upponatur externus angulus B C D, non e&longs;&longs;e æqualis duobus
internis, & oppofitis A, & B, nunquam poterimus eo modo, quo Euclides,
demon&longs;trare pa&longs;&longs;ionem prædictam de triangulo A B C. huiu&longs;modi impo&longs;&longs;i
bile, cuius oppo&longs;itum non &longs;olum po&longs;&longs;ibile, &longs;ed etiam nece&longs;&longs;arium e&longs;t, vocat
Ari&longs;t.
impo&longs;&longs;ibile ex &longs;uppo&longs;itione, quia &longs;cilicet impo&longs;&longs;ibile euadit ex quo
dam fal&longs;o &longs;uo &longs;uppo&longs;ito, vt in allato exemplo, triangulum habere tres an
gulos æquales duobus rectis, quamuis nece&longs;&longs;arium &longs;it, tamen ex fal&longs;a &longs;up
po&longs;itione, impo&longs;&longs;ibile oua&longs;it.
Ibidem
&longs;ecto 3. cap.
23. hoc &longs;olum nunc addendum
neas e&longs;&longs;e compo&longs;itas ex indiui&longs;ibilibus, con&longs;ectarium erit diametrum e&longs;&longs;e
commen&longs;urabilem co&longs;tæ, quia indiui&longs;ibile illud, ex quo vtraque linea con
&longs;tat, erit
Tex. 24.
generant, his te&longs;tes fui&longs;&longs;e videntur: &longs;olam enim figurarum &longs;olidarum
&longs;phæram non diuidunt, vt non plures &longs;uperficies. quam vnam
diui&longs;io enim in plana non perinde e&longs;&longs;icitur, vt qui&longs;piam
tes diuidat totum, &longs;ed vt in &longs;pecie diuer&longs;a: patct igitur &longs;phæram e&longs;&longs;e &longs;olidarum
primam)
&longs;icierum, quibus ambiuntur, v. g. diuidunt cubum in &longs;ex &longs;uperficies, quia
cubus &longs;ex quadratis planis &longs;uperficiebus continetur: qua ratione nequcunt
ra ambitur vnica tantum &longs;uperficie &longs;phærica. quando verò ex planis corpo
ra generant, vt facit Plato in Timæo, accipíunt primò triangulum æquila
terum, & ex quatuor triangulis æquilateris &longs;imul compactis conficiunt py
ramidem; & hoc modo alia &longs;olida à pluribus &longs;uperficiebus ambita con&longs;ti
tuunt: verum hac ratione nullo modo po&longs;&longs;unt &longs;phæram componere, quia
vnica tantum,
diuidentes, & componentes corpora fidem faciunt, &longs;phæram, cum ex nullis
componatur, &longs;olidorum e&longs;&longs;e primam.
Tex. 25.
nentibus rationabili&longs;&longs;imam, circulum quidem &longs;ecundum vnum; triangulum autem
&longs;ecundum dualitatem, quoniam duo recti. &longs;i autem &longs;ecundum triangulum, vnum.
circulus non erit figura)
facere circulum propter &longs;implici&longs;simam ip&longs;ius naturam, cum vnica, ac per
fecta circulari linea comprehendatur:
duo anguli recti, ide&longs;t, quia triangulum habet tres angulos æquales duobus
rectis angulis; quod fusè explicatnm e&longs;t lib.
1. Priorum, &longs;ecto 3. cap.
1. De
mum &longs;i primum locum dederimus triangulo, nullus alius remanet pro cir
culo, quod e&longs;t inconueniens, ergo circulus prima figura erit.
Tex. 31.
tione &longs;umpta, quod apta natura e&longs;t &longs;emper con
gis autem concauum e&longs;t, quod centro propinquius est. ducantur ergo ex centro A,
duct
quæ ex centro. magis igitur concauus locus e&longs;t, quare
influet aqua, donec æqualis e&longs;t autem eis,
quæ ex centro linea A E, quare nece&longs;&longs;e e&longs;t apud eas, quæ
ex centro, e&longs;&longs;e aquam, tunc enim quie&longs;cet. linea autem,
quæ eas, quæ ex centro tangit, circularis e&longs;t, &longs;phærica
igitur aquæ &longs;uperficies e&longs;t, in qua B E C.)
textu lineari demon&longs;tratione probat aquæ manen
tis &longs;uperficiem e&longs;&longs;e &longs;phæricam: quæ demon&longs;tratio
per&longs;picua euadit, &longs;i &longs;igura, quæ in codicibus tam
græcis, quam latinis,
dum fecimus, re&longs;tituatur. &longs;it igitur in præcedenti figura A, centrum mundi,
ex quo educantur duæ rectæ lineæ æquales A B, A C, quæ deinde alia recta
B C, coniungantur. educatur
ad B C, quæ ba&longs;is e&longs;t trianguli B A C, & producatur vlterius quantumlibet
in E. intelligatur demum circumferentia tran&longs;ire per puncta B, & C, quia
illæ duæ lineæ A B, A C, &longs;unt æquales, quæ circumferentia alteram A D, quæ
fuit protracta, &longs;ecet in E. Iam &longs;ic argumentatur: aqua natura &longs;ua &longs;emper
de&longs;luit ad locum magis concauum, ide&longs;t, ad loca centro A, terræ propin
quiora, quale e&longs;&longs;et in figura locus D, re&longs;pectu locorum B, & C, quia A D,
linea minor e&longs;t ijs, quæ ex centro eductæ &longs;unt A B, A C. quapropter aqua
debet de&longs;luere ex B, ad D, vel ex C, ad idem D, donec pertingat ad E. qui
locus non e&longs;t decliuior punctis B, & C. quare cum loca B, E, C, quæ &longs;unt ex
con&longs;i&longs;tere, tunc enim debet quie&longs;cere, aliter nunquam quie&longs;ceret; &longs;ed vide
mus aquam manentem, & quietam, ergo quie&longs;cit circa puncta B, E, C, à
centro terræ æquidi&longs;tantia, per quæ tran&longs;it linea circularis coniungens illa;
et&longs;i &longs;uperficies per eiu&longs;modi loca pertran&longs;iret, e&longs;&longs;et &longs;phærica: &longs;ed &longs;uper&longs;i
cies aquæ tran&longs;it per talia loca, ergo &longs;phærica e&longs;t. Huius etiam habes acu
ti&longs;&longs;imam Archimedis demon&longs;trationem initio libelli de ijs, quæ vehuntur
in aqua, quam in &longs;uam &longs;phæram retulit Clauius.
Tex. 46.
ip&longs;is orbibus ferri; &longs;olum enim &longs;ic nullum ab&longs;urdum accidit. celeriorem enim e&longs;&longs;e
maioris circuli velocitatem, rationabile e&longs;t circa idem centrum infixis: vt enim in
alijs maius corpus velocius fertur propria latione, &longs;ic, & in circularibus: maius
enim e&longs;t eorum, quæ auferuntur
ex intellectione vltimæ periodi textus totius intelligentia pendet: &longs;it igitur
figura præ&longs;ens, in qua cum &longs;int duo circuli concen
trici, vnus altero maior,
&longs;emidiametri A D, A E, quæ
cant, apparet maius e&longs;&longs;e
iori circulo &longs;emidiametri ex
quam &longs;egmentum B C, minoris circuli, quod ei&longs;dem
&longs;emidia metris intercipitur. Verumtamen &longs;i circuli
ambo &longs;imul moueantur, maior circulus æquali tem
pore maius illud &longs;patium D E, & minor minus B C,
pertran&longs;ibit: idem igitur de cœle&longs;tibus orbibus di
cendum, qui quamuis omnes diurnum &longs;imul motum
ab&longs;oluunt, maiores tamen celerius conuertuntur: quo fit, vt &longs;tellæ maiori
bus circulis infixæ,
que oportet eas, dum mouentur cœlum di&longs;&longs;ecare, quod accideret, &longs;i pro
prio motu veluti pri&longs;ces per aquam progrederentur.
Hæc quidem Ari&longs;t.
con&longs;entanea ob&longs;eruationibus veterum A&longs;tronomo
rum; at verò illis no&longs;træ ætatis ob&longs;eruationes repugnant; præ&longs;ertim illæ,
quæ fiunt circa &longs;tellas errantes: ex quibus fatendum e&longs;&longs;e videtur, Cœlum,
qua parte Planetas continet, liquidum e&longs;&longs;e, ac per illud Planetas proprio
motu, ceu pi&longs;ces in aqua progredi. Tycho
demon&longs;trant Cometas in regione Planetarum e&longs;&longs;e,
tran&longs;uer&longs;um moueri, quo nece&longs;&longs;ario C&ecedil;lú deberent perforare; ijdem o&longs;ten
dunt nonnullos Planetas, Martem præ&longs;ertim, ac Venerem modo &longs;upra So
lem, modo infra a&longs;cendere, & de&longs;cendere. Idem patet ex ob&longs;eruatione no
ua per nouum Tele&longs;copij i
à Sole apparet: quando nimirum e&longs;t in imo epicyclo.
luti Luna plena, cum in &longs;ummo epicyclo ver&longs;atur: quæ minimè apparerent,
ni&longs;i &longs;upra, ac infra Solem circumiret. His rationibus conantur ip&longs;i proba
re Cœlum e&longs;&longs;e liquidum;
quarum &longs;olutio mihi nulla occurrit, alijs forta&longs;&longs;is occurrct.
Tex. 57.
quædam &longs;int priora, quædam posteriora, & quomodo &longs;patijs &longs;e h
A&longs;trologiam, pro A&longs;tronomia, &longs;i iuxta recentiores loqui velimus. Dicit igi
tur ordinem cœlorum, ac &longs;yderum, item &longs;itum, & proportiones magnitu
dinum corundem, cum per naturalis &longs;cientiæ princip ia &longs;ciri nequeant, ex
rationibus A&longs;tronomorum petenda e&longs;&longs;e, apud quos i&longs;ta &longs;ufficienter
&longs;trentur& meritò quidem hæc dicuntur; po&longs;teriores enim ab Ari&longs;t.
ordines,
&longs;itus, ac magnitudines tam cœlorum, quam &longs;yderum firmis rationibus,
inuentu peracutis demon&longs;trarunt. quorum princeps fuit ptolæmeus; no&longs;tra
tamen ætate Tycho Brahe, qui certis ob&longs;eruationibus, quas maximo labo
re, ac &longs;umptu exantlauit, in nonnullis à Ptolæmeo, ac reliquis di&longs;&longs;entjt: &longs;tan
dum autem e&longs;&longs;e recentioribus ob&longs;eruationibus apud A&longs;tronomiæ peritos in
confe&longs;&longs;o e&longs;t.
Tex.
enim
altera concaua, aut
vi&longs;um, ide&longs;t per opticem probari Lunam e&longs;&longs;e &longs;phæricam: &longs;ed con&longs;ule, quæ
primo Po&longs;ter. tex. 3. de hac re &longs;crip&longs;i, & plenam etiam huius loci intelligen
tiam a&longs;&longs;equeris, præ&longs;ertim &longs;i experimentum ibi traditum inieris.
Ibidem
&longs;peciem præ&longs;eferentes. Quare &longs;i vnum est tale, palam e&longs;t, quod & alia
talia)
ita præ&longs;ens ex A&longs;tronomia, ex eò enim, quod eclyp&longs;is Solis habeat figuram
lunulæ, ide&longs;t, &longs;i in&longs;tar Lunæ falcatæ, probant A&longs;tronomi Lunam e&longs;&longs;e &longs;phæri
cam. intellige tamen partem illam Solis, quæ non eclyp&longs;atur, habere figu
ram lunulæ, pars enim à Luna obumbrata non videtur, et&longs;i videretur oua
lem quandam &longs;peciem, præ&longs;eferret: pars igitur, illa e&longs;t corniculata, quia
cum Solis defectio ex interpo&longs;itione Lunæ inter nos, &
Solem contingat, & Luna &longs;it &longs;phærica, nece&longs;&longs;ariò &longs;phæ
ricè, & circulariter Solem obumbrabit; quare pars illa
non obumbrata remanet falcata, & corniculata, vt in
præ&longs;enti figura vidcre e&longs;t; vbi cernis, Lunam Solem or
biculariter offu&longs;care in linea A D C, partem Solis de
tectam
& falcatam; cum ergo in hunc modum fiat Solis deli
quium, &longs;ignum certum e&longs;t, Lunam e&longs;&longs;e &longs;phæricam.
Tex. 107.
rum con&longs;iderauerimus, & di&longs;tinxerimus, quonam modo cen&longs;eamus quantamuis ma
gnitudinem grauem ad medium ferri. manife&longs;tum enim e&longs;t, quod non quou&longs;que ex
tremum tangat ip&longs;um centrum; &longs;ed maior pars vincat, oportet,
ip&longs;um medium compræhendat;
telis e&longs;t, debere nos exi&longs;timare, quod &longs;i quæpiam grauis magnitudo de&longs;cen
dat ad centrum mundi, eam non perman&longs;uram, &longs;latim ac ip&longs;ius extremum
centrum mundi attigent; &longs;ed cò
mundi medium, &longs;iue centrum a&longs;&longs;equutum &longs;it; maior enim ip&longs;ius pars, in qua
&longs;cilicet medium e&longs;t, minorem partem propellit, donec vtrinque à centro
mundi æquè emineat; omne enim graue
prima ip&longs;ius pars ad mundi centrum pertingeret, &longs;ed reliquæ ip&longs;ius partes
adhuc grauitarent,
áis medium, mundi medio congrueret: quo facto lapis quie&longs;ceret. quæ num
vera &longs;int, vt intelligamus, oportet prius præmittere, iuxta Mathematicos
duplex e&longs;&longs;e medium, &longs;iue centrum cuiu&longs;uis magnitudinis: aliud enim e&longs;t
centrum molis, aliud e&longs;t centrum grauitatis. centrum molis e&longs;t illud pun
ctum, à quo extrema æquidi&longs;tant: centrum grauitatis e&longs;t punctum illud, à
quo extrema æque ponderant, &longs;iue à quo graue &longs;u&longs;pen&longs;um æquè ponderat,
&longs;iue in æquilibrio manet. Porrò in corporibus regularibus, &longs;i vnifo mia &longs;int
idem, & vnum &longs;unt centrum molis, ac centrum grauitatis: vt in &longs;phæra
plumbea, idem crit
vt in &longs;phæra partim plumbea, partim lignea, diuer&longs;um erit centrum molis,
à centro grauitatis; illud enim erit in medio &longs;phæræ; centrum verò graui
tatis in parte plumbea exi&longs;tet. In corporibus deinde irregularibus, etiam&longs;i
&longs;int vniformis ponderis, aliud tamen e&longs;&longs;e pote&longs;t centrum molis à
uitatis, vt in corpore oblongo, cuius alterum extremum &longs;it reliquis parti
bus multò maius, vti e&longs;t claua: vbi centrum molis erit in medio longitudi
nis clauæ; centrum verò grauitatis, erit propinquius capiti clauæ. quando
igitur Ari&longs;t.
ait, graue de&longs;cen&longs;urum, donec ip&longs;ius medium, &longs;iue centrum,
mundi centrum attingat; benè dicit, &longs;i de medio grauitatis intelligat; ma
lè autem &longs;i de medio molis. quia grauia omnia ratione centri grauitatis
ponderant,
trum mundi &longs;emper grauitabunt, & mouebuntur. Verum enim verò ex an
tiquorum monumentis manife&longs;tum e&longs;t, Archimedem, qui multò po&longs;t Ari
&longs;totelem floruit, primum omnium de centro grauitatis e&longs;&longs;e philo&longs;ophatum,
qua ratione dicendum e&longs;&longs;et, Ari&longs;totelem de centro, molis loquutum e&longs;&longs;e,
& perinde non
Tex. 109.
enim Lunæ eclyp&longs;es tales
&longs;em fiunt, figurationibus, omnes accipit diui&longs;iones: etenim recta fit, & vtrinque
curua, & concaua)
Lunæ eclyp&longs;ibus de&longs;umpta: nam ni&longs;i terra e&longs;&longs;et rotunda, nunquam Luna in
eclyp&longs;i haberet tales deci&longs;iones, ide&longs;t non haberet falcatas, aut lunulatas
partes illas, quæ in eclyp&longs;i ob&longs;curantur, & quafi à Luna re&longs;ecantur. quam
uis enim &longs;ingulis men&longs;ibus Luna terminetur modo linea concaua, vt quan
do noua e&longs;t; modo recta, vt quando diuidua e&longs;t: modo vtrinque curua, vt
cum à diuiduæad plenilunium tendit. quod fu&longs;ius primo Po&longs;ter. tex. 30. ex
po&longs;ui. in eclyp&longs;ibus tamen &longs;emper curuam habet lineam illam, quæ partem
ec'yp&longs;atam de&longs;init; vt paulo po&longs;t explicabo. Vide precedentem textum 59.
& ca, quæ ibi annotaui,
tiam a&longs;&longs;equeris. vide etiam, quæ mox &longs;ubdam circa huius loci reliquum.
Ibidem
re qaon'am eclyp&longs;im palitur propter terræ obiectionem, terræ
rica exi&longs;tens, figuræ cau&longs;a erit)
ex eo, quod Luna &longs;phæricè eclyp&longs;etur, quod innuitur illis verbis, &longs;emper
eclyp&longs;atam à non eclyp&longs;ata, &longs;emper apparet circularis; cum autem hæc li
nea &longs;it terminus vmbræ terræ, quæ lumen obumbrat, &longs;ignum
vmbram ip&longs;am e&longs;&longs;e rotundam; nam cum Luna deficiat propter terræ obie
ctionem inter ip&longs;am, & Solem, ita, vt vmbra terræ protendatur
nam,
ad quamlibet
per&longs;picuum e&longs;t terram proijcere quoquouer&longs;us vmbram rotundam, quæ vt
in &longs;phæra o&longs;tenditur, e&longs;t rotunda ad modum coni; cum ergo vmbra terræ
ex quauis parte proijciatur, &longs;it rotunda, certò certius colligitur, hanc eandem rationem, &longs;i libue
rit, fu&longs;ius pertractatam videre poteris apud P. Clauium in &longs;phæra.
Tex.
tunda, &longs;ed & quod magnitudine non magna &longs;it; paruo enim facto ncbis tran&longs;itu ad
meridiem, & Vr&longs;am, manifa&longs;tè fit alter horizon circulus, ita vt a&longs;tra, quæ &longs;uper
caput, magnam habcant mutationem, & non eadem appareant, & ad Vr&longs;am, & ad
meridiem tran&longs;euntibus, quædam enim in Acgypto quidem stellæ
ca Cyprum, in ijs autem, quæ ad Vr&longs;am vergunt regionibus, non & a&longs;tro
rum ea, quæ &longs;emper in ijs, quæ ad Vr&longs;am vergunt, apparent, in illis locis occidunt. Quare non &longs;olum ex his manife&longs;tum e&longs;t rotundam e&longs;&longs;e figuram terræ, &longs;ed & &longs;phæræ
non magnæ: non enim tam celeriter in&longs;igne quippiam faceret, tran&longs;latis nobis adeò
parum)
rea eum breuiter &longs;ic paraphra&longs;ticè exponam. Terram e&longs;&longs;e rotundam,
re&longs;pectu cœle&longs;tium corporum non magnam, &longs;ignum e&longs;t, quod facto à nobis
paruo itinere &longs;iue ad meridionalem plagam, &longs;iue ad
Vr&longs;am dicit) magnopere mutatur horizon: quod apparet primo ex varia
tionc a&longs;trorum, nam quæ in primo loco &longs;upra no&longs;trum verticem
in &longs;ecundo loco non amplius, &longs;ed alia,
exfacto quamuis paruo itinere tran&longs;eunt. &longs;it in
præ&longs;enti figura terra, vbi A, in qua facta parua
mutatione ex loco F, in locum G, fieret magna
mutatio
tum ab inuicem di&longs;tant. &longs;i autem terra e&longs;&longs;et
maior, v. g. circulus medius, tunc facta maio
ri mutatione ex D, in E, fieret eadem a&longs;trorum
variatio ex B, in C; &longs;ed cum nos experiamur
&longs;ieri magnam a&longs;trorum mutationem, ex parua
locorum intercapedine, &longs;ignum e&longs;t magnope
re mutari horizontem, ac proinde terram e&longs;&longs;e
rotundam, ac re&longs;pectu cœle&longs;tium corporum
paruam. aliud præterea &longs;ignum hums horizontis permutationis e&longs;t, quod
&longs;tellæ, quæ in priori loco &longs;upra horizontem apparebant, mutato paululum
loco ad alterutram plagam, &longs;tatim ab&longs;conduntur; aliæ verò nouæ
vt in Acgypto, & Cypro, &longs;tella, quæ dicitur Canobus &longs;upra horizontem
a&longs;cendit; quæ &longs;i paululum Vr&longs;am, &longs;eu &longs;eptentrionem ambulaueris, &longs;tatim
latitabit. Demum ciu&longs;dem citæ mutationis &longs;initoris indicium etiam &longs;it,
quam occidunt, quamuis horizontem leuiter per&longs;iringant, quæ tamen Cy
prijs, ex quibus & rotunditas, &
paruitas terræ colligi pote&longs;t. has ea&longs;dem rationes fu&longs;ius explicatas repe
ries apud P. Clauium in &longs;phæra.
Tex. 111.
cum coniungi ei, qui circa Indiam, & boc modo mare vnum e&longs;&longs;e, nen admcdum
incredibilia exi&longs;timare videntur &c.)
&longs;tima&longs;&longs;e apertè
nauigationes; quibus nouus orbis repertus e&longs;t, qui inter columnas Hercu
lis,
Tex. 112.
cinari tentant, ad 400. dicunt ftadiorum millia, &c.)
bus inue&longs;tigauerint A&longs;tronomi quantitatem terræ, optimè, ac dilucidè ex
ponitur à P. Clauio in &longs;phæra: quem &longs;i libet, con&longs;ule, ne inani labore opu
&longs;culum i&longs;tud exere&longs;cat.
Tex. 40.
quidem ex rectilmeis: fphæra verò ex octo partibus componitur)
xander exiftimat, Ari&longs;totelem dicere &longs;phæram con&longs;tare ex octo
partibus illis, quæ de&longs;ignantur per tres circulos, quorum duo &longs;e
cant &longs;e mucuò ad angulos rectos, vt in &longs;phæra mundi faciunt duo coluri;
tertius verò medios illos diuidit æquidi&longs;tanter à &longs;ectionibus
quemadmodum æquator in &longs;phæra mundi &longs;ecat duos coluros. ex quibus &longs;e
ctionibus tota &longs;phæra in octo partes diuiditur, quibus &longs;phæram componi
vult Ari&longs;toteles. aduerte tamen hanc &longs;phæræ compo&longs;itionem nullo modo
habere partes actu, cum &longs;phæra &longs;it vnica &longs;implici &longs;uperficie terminata; &longs;ed
quæ tantum &longs;int à prædictis imaginatis circulis de&longs;ignatæ: at verò aliæ fi
guræ, quæ pluribus planis terminantur, vt cubus, octaedrum, & &longs;imilia, quæ
Ari&longs;t.
vocat rectiliheas, quia terminantur &longs;uperficiebus rectilineis actu di
&longs;tinctis ab inuicem ex natura &longs;ua, non per no&longs;tram de&longs;ignationem, ideò re
ctè dicuntur componi ex pyramidibus, v. g. dicimus cubum componi ex &longs;ex
pyramidibus, quia cum habeat &longs;ex ba&longs;es, cogitamus &longs;upra
larum &longs;ingulas pyramides erigi, quarum omnium vertices ad idem punctum
medium intra cubum imaginatum coeant. & &longs;ic de reliquis &longs;olidis.
quæ qua
ratione re&longs;oluantur in plures pyramides, con&longs;tat ex 10. 11. 12. & 13. Ele
mentorum Euclidis, at verò in &longs;phæra nullum reale compo&longs;itionis, aut di
ui&longs;ionis fundamentum reperitur.
Tex.
certi&longs;&longs;i nis &longs;cientijs; nam Mathematicæ ip&longs;um quideæ intelligibile, accipiunt diui
&longs;ibile)
quam di&longs;eretam, quam &longs;tatuunt Philo&longs;ophi e&longs;&longs;e &longs;ubiectam materiam ma
thematicarum. quam ideo appellant intelligibilem, quia cum &longs;it ab&longs;tracta
per intellectum à &longs;en&longs;ibilibus affectionibus, re&longs;tat vt &longs;it tantummodo intelHanc eandem &longs;upponunt e&longs;&longs;e diui&longs;ibilem in infinitum,
vt &longs;upra 3. Phy&longs;. textu 31. dictum e&longs;t.
Tex. 66.
nabile e&longs;t. primò quidem, quia accidit non repleri totum; nam in planis tres figuræ
videntur implere locum, Triangulus, Quadratum, & Sexangulus)
corpora intelligit quatuor elementa. Vult enim probare quatuor elemen
ta non habere figuras illas mathematicas, quas illis Plato tribuebat, vt au
tem Ari&longs;t.
rationem probè percipiamus, &longs;ciendum, quod implere totum,
&longs;iue locum, illæ figuræ dicuntur, quæ &longs;imul &longs;uis angulis in plano quopiam ad
vnum,
lud con&longs;i&longs;tit, tales &longs;unt,
quibus fieri po&longs;&longs;unt pauimenta, oportet enim, vt &longs;imul vnitæ nihil vacui in
pauimento relinquant. huiu&longs;modi &longs;unt triangula æquilatera (de his enim
intelligendus e&longs;t textus) quadrata, & hexagona, &longs;iue &longs;exilatera regularia;
nam &longs;ex triangula æquilatera &longs;imul iuncta in plano paui
re po&longs;&longs;unt, vt patet in figura præ&longs;enti; ratio huius e&longs;t,
quia omnes anguli circa idem punctum (y. g. A, in hac
figura) in plano, quotquot fuerint con&longs;tituti, &longs;unt æqua
les quatuor rectis, ex coroll. &longs;ecundo 15. primi Elemen
ti: cum igitur &longs;ex anguli, trianguli æquilateri
quatuor rectis angulis, con&longs;tituti omnes circa punctum
A, totum locum circa illud implere po&longs;&longs;unt. Quadratum etiam replere lo
cum manife&longs;tum e&longs;t, cum enim ip&longs;ius anguli &longs;intrecti, &longs;i
quatuor quadrata ad idem punctum A, copulentur, vt in
figura apparet, replebunt eadem de cau&longs;a vacuum.
Hexagonum quoque regulare, ide&longs;t æquilaterum, &
æquiangulum idem præ&longs;tare pote&longs;t; cum enim tres angu
li ip&longs;ius æquiualeant quatuor rectis, &longs;i tria hexagona ad
idem punctum A, vt in &longs;igura adaptentur, nece&longs;&longs;ariò ni
hil vacui inter ip&longs;a relinquetur, vt in figura hac o&longs;tenditur. præter has tres
figuras, nulla alia reperitur, quæ i&longs;tud efficere pol
&longs;it. cuius demon&longs;trationem perfectam videre pote
ris in fine commentarij P. Clauij &longs;uper 4. Elem.
nos
ea tantum attingimus, quæ percipi po&longs;&longs;int ab homi
ne vix mathematicis tincto: &longs;ed tamen, quæ &longs;en&longs;um
Ari&longs;totelis patefaciunt. Aliæ porrò figuræ replen
tes locum planum, quibus aliquando Architectores
vtuntur, vel &longs;unt irregulares, vel ad prædictas redu
ci po&longs;&longs;unt. cum igitur tres tantum ex figuris planis
totum repleant, hæ &longs;olæ poterunt elementis attri
bui, ac propterea non &longs;ufficient, ni&longs;i pro tribus elementis. quare quartum
Cæterum occa&longs;ione harum figurarum illud hoc loco apponere vi
&longs;um e&longs;t, quod Pappus
mathematicarum &longs;cribit, De admirabili Apum indu&longs;tria, atque
prudentia in con&longs;truendo &longs;uas cellulas figura hexagona regulari. cum enim vellent omne vacuum excludere, & præterea capaci&longs;&longs;imam
nium
que præ&longs;tat, nam & inane omne excludit, & illarum trium capaci&longs;&longs;ima e&longs;t,
cum magis ad circularem figuram accedat: vt patet ex tractatu de figuris
I&longs;operimetris, qui e&longs;t apud Clauium in &longs;phæra, necnonin Geometria pra
ctica. hoc ideò libentius recen&longs;ui, quia animaduerti naturales hi&longs;toriogra
phos omnes latere, vel ip&longs;um Aldobrandum no&longs;trum, qui quamuis indu
&longs;trio&longs;æ Apis in&longs;tar omnia delibauerit, i&longs;tud tamen de Apibus artificium tan
ta &longs;apientia plenum, ne&longs;cio quo modo prætermi&longs;it.
Ibidem
&longs;olidum. nullum reperi, qui in hoc loco explicando non errauerit; nam Græ
ci, qui alioqui &longs;olent mathematica probè intelligere, hic omnes lap&longs;i &longs;unt, communis ferè error omnium fuit, pyramides plures &longs;imul compactas po&longs;
&longs;e replere &longs;olidum locum. quod vt melius intelligamus, &longs;ciendum e&longs;t, reple
re locum
punctum coaptata, ita con&longs;tipentur, vt totum &longs;patium, quod e&longs;t circa pun
ctum illud omninò occupent, hoc e&longs;t, nihil vacui inter ip&longs;a relinquatur: &longs;i
cut enim prædictæ tres &longs;iguræ planæ, de quibus paulò ante, replent locum
planum, ide&longs;t &longs;uper&longs;iciem; ita cubi replent &longs;olidum, ide&longs;t &longs;oliditatem &longs;imul
vniti con&longs;tituunt, ita vt &longs;i octo cubi &longs;imul ad idem punctum
&longs;tituant corpus &longs;olidum ex octo illius con&longs;latum,
cubos relinquatur. & &longs;icuti planæ illæ figuræ erant conficiendis pauimentis
aptæ, ita &longs;olidæ hæ muris, qui corpora &longs;unt &longs;olida,
regularem, quæ dicitur etiam Tetraedrum,
pora regularia rectilinea, quæ alias Platonica corpora dicuntur.
defraitiones &longs;unt in 11. Elem. Tetraedrum autem &longs;ic definitur, e&longs;t figura &longs;o
lida &longs;ub quatuor triangulis æquilateris,
de hac inquam e&longs;t &longs;ermo. quia &longs;i liceret intelligere de irregularibus figuris,
infinitæ reperir entar figuræ tam planæ, quam &longs;olidæ, quæ vtrumque locum
complerent. Aduertendum tandem Ari&longs;t.
videri loqui de repletione loci
&longs;olidi, quia tran&longs;it à planïs figuris ad &longs;olidas. & quia &longs;i hæ duæ pyramis, &
cubus replent locum &longs;olummedo &longs;ecundum &longs;uas &longs;uperficies, quæ &longs;unt trian
gulum, & quadra
&longs;et idem po&longs;t modum repetere. ad hæc &longs;i in medium &longs;olida hæc duo profert,
liter, ide&longs;t de ip&longs;is, vt &longs;oh da &longs;unt. Quare Ari&longs;t.
videretur &longs;ibi non con&longs;tare,
vel perperam exi&longs;tima&longs;&longs;e plura Tetraedra complere &longs;oliditatem. deceptus
bus, verùm illæ non &longs;unt regulares, ide&longs;t
dam. Verum quidem e&longs;t octo cubos &longs;imul adactos &longs;oliditatem conficere,
quia ad id nece&longs;&longs;arij &longs;unt octo anguli &longs;olidi, quos octo cubi præbere po&longs;&longs;unt,
cum anguli ip&longs;orum &longs;int recti, & &longs;olidi. Verum enim verò plures pyramides
regulares, &longs;iue plura Tetraedra non po&longs;&longs;e replere vacuum,
&longs;tituere, ex eo patet, quia &longs;i id præ&longs;tarent, conflarent nece&longs;&longs;ariò, vel vnum
ex
piam; non aliud, nam, vt patet ex &longs;cholio 13. Elem.
non dantur, ni&longs;i illa. quinque;
poneretur ex illis pyramidibus, e&longs;&longs;et dupla lateris eiu&longs;dem, vt patet, quia
pyramides illæ omnes concurrerent ad centrum &longs;phæræ illas omnes com
plectentis, quare latus vnius pyramidis à &longs;uperficie &longs;phæræ incipiens de&longs;i
neret in centrum, ergo latus i&longs;tud e&longs;&longs;et &longs;emidiameter, quapropter tota dia
meter illius &longs;ph&ecedil;ræ, & con&longs;equenter huius corporis in illa in&longs;cripti, e&longs;&longs;et du
pla lateris eiu&longs;dem figuræ &longs;olidæ in&longs;criptæ, &longs;ed nullo talis proportio diame
tri alicuius ex illis
quæ &longs;it nimirum dupla, vt patet ex vltimis demon&longs;trationibus 13. Elem.
ini
tio facto à 13. demon&longs;tratione, in quibus nulla reperitur proportio dupla
inter diametrum, & latus eiu&longs;dem alicuius ex illis &longs;olidis; ex quibus mani
fe&longs;tum e&longs;t, plures regulares pyramides quouis pacto &longs;imul vnitas nullo mo
do replere locum &longs;olidum. cum igitur animaduerterem, &longs;en&longs;um Ari&longs;t.
nullo
modo po&longs;&longs;e verificari de repletione &longs;olidi per plura Tetraedra, & omnes
tamen commentatores auctoritate Ari&longs;t.
decepti pro ip&longs;o &longs;tarent, dubius,
Clauium præceptorem meum per literas con&longs;ului, qui in hunc modum hu
mani&longs;&longs;imè re&longs;pondit; cubus implet locum quater &longs;umptus, ad idem enim
punctum quatuor cubi coaptantur: &longs;ic etiam pyramis &longs;exies &longs;umpta, &longs;eu &longs;ex
pyramides ad idem punctum iunctæratione &longs;ub&longs;tantium
laterorum. Verum hac ratione non videntur implere locum lolidum, fa
teor; &longs;ed tamen Ari&longs;t.
in co tex. non loquitur de repletione loci &longs;olidi. hæc
ip&longs;e. &longs;i igitur libeat Ari&longs;totelem, quod fortè Clauius intendebat defendere,
dicendum e&longs;t cum eo Ari&longs;t non loqui de repletione loci &longs;olidi:
de cubo, & Tetraedro, quatenus &longs;unt corpora, &longs;ed quatenus habent &longs;uper
ficies, cubus quidem &longs;ex quadratas, Tetraedrum autem quatuor æquilate
ras &longs;uperficies, quæ duæ figuræ, vt &longs;upra in hoc textu vidimus, replent lo
cum: ex
aduersò ne videamur magis Ari&longs;t.
quam veritatem &longs;equi, videtur dicen
dum, Ari&longs;totilem formaliter locutum e&longs;&longs;e, & vt patet ex rationibus &longs;upra
allatis de repletione &longs;olidi e&longs;&longs;e intelligendum, vt etiam intellexerunt omnes
huius loci expo&longs;itores; Verumtamen ip&longs;um erra&longs;&longs;e, dum plures pyramides
replere &longs;olidum exi&longs;timauit. Vtrumuis dixerimus, non tamen Ari&longs;t.
ab om
ni crrore vindicabimus. Hoc tamen certum e&longs;t, ex prædictis, Græcos om
nes pariter, ac Latinos, illos &longs;equentes, lapos e&longs;&longs;e, a&longs;&longs;erentes duodecim py
ramides complere &longs;olidum locum,
ramides Dodecaedron con&longs;tituentes non &longs;unt regulares, ide&longs;t, non &longs;unt TeIndul
geas Lector, &longs;i hoc loco nece&longs;&longs;e fuit in Geometriæ penetralia ingredi: ope
ræpretium enim e&longs;t aliquando ip&longs;is Mathematicis &longs;atisfacere. tu verò, &longs;i
adeo es mathematicis imbutus, con&longs;ule po&longs;tremas demon&longs;tra. 13. Elem.
&
præcipuè &longs;cholium vltimum, vbi plura de his corporibus &longs;citu digni&longs;&longs;ima,
partes, per&longs;picuè &longs;atis expo&longs;uimus.
Multo po&longs;t tempore, quàm hæc &longs;crip&longs;eram incidi fortè in cap.
38. &longs;pecu
lationem 10. Benedicti de placitis Ari&longs;t.
erroris notari, dum a&longs;&longs;eruit duodecim pyramides replere
ide&longs;t, vt exponit ip&longs;e, &longs;ex pyramides &longs;uper hexagonam aliquam figuram
&longs;uperficialem, & &longs;ex &longs;ub eadem, id præ&longs;tarent, cum potius maius vacuum
remaneat ad quamlibet partium &longs;upra, & infra, quam plenum. hæc ip&longs;e.
&longs;ed
expo&longs;itio i&longs;ta puerili, ne dum Ari&longs;t.
ingenio pror&longs;us indigna e&longs;t: vt propte
rea exi&longs;timem ca&longs;u potius eum Ari&longs;t.
rectè reprehendi&longs;&longs;e, quam ex certa
&longs;cientia, cum illius erratum maiori errato conetur corrigere. Incidi po
&longs;tremò in Indicem librorum, quem Maurolyius &longs;uæ Co&longs;mographiæ præpo
nit, vbi &longs;ic ait: Demon&longs;tramus autem in libello de figuris planis,
locum replentibus, cubos per &longs;e, pyramides verò cum octacdris compactas
dumtaxat implere locum, qua in re Auerroem erra&longs;&longs;e pueriliter manife&longs;tum
erit. Vides igitur tanti viri auctoritate confirmari no&longs;tram &longs;ententiam, py
ramides videlicet per &longs;e, non replere vacuum. cum igitur con&longs;tet vnam tan
tum ex figuris &longs;olidis, &longs;iue etiam dicas, vt perperam Ari&longs;t. & alij plures exi
&longs;timarunt, replere totum &longs;olidum; nulla ratione poterunt
quatuor diuer&longs;is figuris indui, &longs;ed vnum tantummodo, quare reliqua
figura remanere nece&longs;&longs;e e&longs;&longs;et: quod e&longs;t omnino inconueniens.
Tex. 71
admodum faciles, ibi reperies de&longs;initiones quinque corporum regularium,
quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha
bes in &longs;phæra Clau. Simpl. etiam hoc loco &longs;atisfacit.
Tex. 33.
terra autem deor&longs;um, & omninò quod grauitatem babet, quare nece&longs;&longs;e
est ferri ad medium. boc autem vtrum accidit ad ip&longs;um tcrræ medium,
an ad vniuer&longs;i, quoniam idem ip&longs;orum &longs;it, alius &longs;ermo e&longs;t)
probare Ari&longs;toteles dari
mundi, ad quod grauia de&longs;cendant, & concurrent:
& à quo leuia a&longs;c
ratione aliqua ex parte mathematica; quæ e&longs;t huiu&longs;
modi. videmus ignem, & cætera l&ecedil;uia a&longs;cendere à
terra &longs;ur&longs;um ad angulos æquales; &longs;imiliter videmus
terram, & c&ecedil;tera grauia de&longs;cendere ad terram dcor
&longs;um ad angulos æquales, quod &longs;ignum e&longs;t omnia i&longs;ta
idem mundi medium re&longs;picere: v.g. &longs;it terra in &longs;igu
ra præ&longs;enti circulus E C D, cuius medium, &longs;ine c
perficie terræ æquales, nimirum angulos B C D, B C E. &longs;imiliter terra per
candem lineam linea autem, quæ
facit tales angulos tendit ad centrum &longs;phæræ A, vt patet ad &longs;en&longs;um in figu
ra, & probari pote&longs;t geometricè ex primis tertij Elem.
ex quibus patet tam
læuia, quam grauia, quæ per talem lineam ferantur, re&longs;picere centrum A,
&longs;phæræ. Vtrum autem i&longs;tud centrum &longs;it idem cum
inquit, e&longs;t &longs;ermo, hoc e&longs;t, ad a&longs;tronomum pertinet. vide igitur hac de re
pulchram de&longs;&longs;ertationem apud Clauium in &longs;phæra: qui probat euidenter
e&longs;&longs;e vnum, & idem.
de Cœlo.
cuius loco ìn-
Tex. 56.
&longs;ed quæ &longs;ecundum obliquum circulum, in hac enim & continuum vnum
e&longs;t & moueri duobus motibus)
tum primi mobilis, qul &longs;it &longs;uper polis mundi, quo Stellæ omnes
ab oriente in occidentem rectà feruntur. per obliquum verò circulum in
telligit Zodiacum, qui obliquus e&longs;t, quia poli eius &longs;unt alij à polis mundi, &
quia non tendit rectà ab ortu ad occa&longs;um, &longs;ed in &longs;phæra mundi tran&longs;uer
&longs;us e&longs;t, & deflectit à &longs;eptentrione in meridiem, quamuis non rectà, vt in
&longs;phæra explicari &longs;olet. motus ergo Planetarum, qui fit &longs;ecundum hunc cir
culum, & ip&longs;e obliquus, & tran&longs;uer&longs;us codem modo erit; ferrentur que per
eum à Borea ad Au&longs;trum, & è conuer&longs;o; ex quo acce&longs;&longs;u, & rece&longs;&longs;u efficiunt
æ&longs;tatem, & hyemem, item generationes, & corruptiones. Sol porrò, & pla
netæ, qui motibus proprijs hunc circulum peragunt, dicuntur moueri duo
bus motibus, & quidem contrarijs: quoniam dum Sol. v. g. per Zodiacum
graditur motu proprio, interim etiam à primo mobili fertur ab ortu in oc
ca&longs;um: ex quibus duobus motibus fit vnus tantum Solis motus &longs;piralis, qui
mixtus e&longs;t, ide&longs;t, qui fit à duobus motoribus; vnde re vera Sol non mouetur
duobus motibus contrarijs re ip&longs;a di&longs;tinctis; hoc enim impo&longs;&longs;ibile e&longs;t: &longs;ed
motu mixto ex duobus, qui &longs;piralis e&longs;t, circa mundum de&longs;cribens &longs;piras ab
v
Sol ip&longs;e, mouens &longs;e ip&longs;um per Zodiacum: & primum mobile mouens in&longs;u
per ip&longs;um Solem, & Zodiacum ab ortu in occa&longs;um circa mundum.
Svmma 1. cap.
3.
nes, non immamfestum, iam enim vi&longs;um est per a&longs;trologica theoremata,
quod multò etiam quibu&longs;dam a&longs;tris est minor)
lum ab&longs;olutè con&longs;iderata, ab A&longs;tronomis explorata habetur, vt vi
dere e&longs;t in &longs;phæra Clauij; &longs;ed etiam re&longs;pectiuè con&longs;iderata, ide&longs;t re&longs;pectu
aliorum elementorum, & ip&longs;orum etiam a&longs;trorum; cuius demon&longs;trationes
&longs;unt partim in libello Ari&longs;tarchi Samij, de magnitudine, & di&longs;tantia Solis,
& Lunæ, partim apud Ptolæmeum in magna Syntaxi, &longs;iue Almage&longs;to: par
tim apud Albategnium de &longs;cientia &longs;tellarum: partim demum apud Ticho
nem Brahe. Porrò facile e&longs;t demon&longs;trare Solem e&longs;&longs;e terra multò maiorem,
terram verò maiorem Luna,
in figura &longs;equenti; vbi vmbra terræ e&longs;t D B E, in quam Luna nigricans im
mergitur, ac lumine deficit, reliqua cognitu &longs;unt facilia: quia igitur A&longs;tro
nomi ob&longs;eruarunt vmbram terræ paulò &longs;upra Lunam pertingere, cum &longs;upe
riora a&longs;tra non adeat, hinc collegerunt eam nece&longs;&longs;ariò e&longs;&longs;e acuminatam, &longs;eu
conicam, vt figura refert. Cum ergo terra vmbram proijciat turbinatam,
nece&longs;&longs;ariò corpus Solis, quod ip&longs;am illuminat, eadem maior erit: quoti
diana enim experientia docemur, corpore illuminante exi&longs;tente maiore
quà &longs;it illuminatum, vmbram proijci fa&longs;tigiatam: cum deinde Solem val
de a terra di&longs;tare certum &longs;it, optimè infertur, eum re&longs;pectu terræ e&longs;&longs;e maxi
mum: quanto enim duæ lineæ, &longs;iue radij B A, B C. à terra ad partes Solis
magis elongantur, tan
to maius corpus
minansha
ctenus de magnitudine
terræ ad Solem. Cum
verò Luna eclyp&longs;atio
nis tempore, aliquan
do non &longs;olum tota in
vmbræ vertice lateat,
verùm etiam
moram trahat, euidens
e&longs;t, eam e&longs;&longs;e multò mi
norem illa vmbræ par
te, in quam immergi
tur; quæ pars cum &longs;it
conicæ vmbræ media,
crit multò gracilior
quàm &longs;it ip&longs;a terra. Ex quo manife&longs;tè apparet, Lunam, quæ illa vmbra minor e&longs;t, e&longs;&longs;e à fortio
ri multò minorem ip&longs;a terre&longs;tri mole. Atque hæc de comparatione terræ
ad Lunam. harum rerum demon&longs;trationes exactio
huius loci.
Eodem cap.
e&longs;t pu
detur
pa&longs;&longs;im apud Ari&longs;t.
occurrunt,
ratione probari po&longs;&longs;e exi&longs;timant, Solem, v. g. terra e&longs;&longs;e centies &longs;exagies &longs;e
xies maiorem; &longs;ed c
&longs;trationes autem a&longs;tronomicas dicunt &longs;e exi&longs;timare eas e&longs;&longs;e fallaces; at que
impo&longs;libile e&longs;&longs;e nos res adeo à nobis di&longs;taptes &longs;ufficienter perue&longs;tigare:
quanto &longs;apientius, ac prudentius eorum Magi&longs;ter Ari&longs;t.
alibi &longs;æpius, &longs;ed hoc
præcipuè loco; quippe qui Mathematicis &longs;ufficienter excultus erat; quibus
i&longs;ti de&longs;tituti, nullo vnquam modo ve&longs;tigia præceptoris a&longs;&longs;equi poterunt.
Summa 1. cap.
4.
verò Lunæ deor&longs;um quidem, tarda autem: quæautem Solis ambo hæc babet &longs;uffi
cienter)
ra; Lunæ verò latio terræ quidem proxima, tarda tamen: at verò Solis la
tio medio modo &longs;e habet inter vtrumque, ide&longs;t, quia
&longs;tat, exi&longs;timo Ari&longs;t.
loqui de motu diur
no, quia &longs;ecundum hunc a&longs;tra inerrantia &longs;unt Sole citatiora, Sol verò ip&longs;a
Luna citior. Verumenimuerò illud non prætereundum, quod plurium inua
luerit opinio exi&longs;timantium Ari&longs;t.
his verbis, Solem &longs;upra Lunam proximè
colloca&longs;&longs;e; quod tamen ex ip&longs;is nullo pacto deduci pote&longs;t; &longs;ed &longs;olummodo
ip&longs;um &longs;upra Lunam colloca&longs;&longs;e. quod &longs;i ita &longs;en&longs;i&longs;&longs;et venia dignus haberetur,
cum tunc temporis nondum fortè adinuentæ e&longs;&longs;ent demon&longs;trationes illæ
a&longs;tronomicæ, quibus ordo Planetarum certi&longs;&longs;imè con&longs;tat,
ter Planetas collocatur. At verò nulla ratione ferendi &longs;unt
hac tempe&longs;tate non &longs;olum Ari&longs;t.
ita &longs;en&longs;i&longs;&longs;e, &longs;ed etiam contra firmi&longs;&longs;imas
aftronomorum demon&longs;trationes, quibus adeò Ari&longs;t.
deferebat, vnica, vt pu
tant ip&longs;ius auctoritate fulti, Solem &longs;ecundum à Luna locum occupare om
ni ope defendunt.
Summa 2. cap.
3.
curius non ni&longs;i rarò con&longs;pici po&longs;&longs;it, cau&longs;a e&longs;t, quia parum à Sole elongatur,
&longs;iue ip&longs;um antecedat, &longs;iue &longs;ub&longs;equatur. ex quo fit, vt diu ferè &longs;imul cum So
le cit cumferatur, & propterea &longs;iue oriatur, &longs;iue occidat, parum &longs;upra ho
rizontem eleuatus apparere pote&longs;t, quod Ari&longs;t.
ait modicum vnde fit tum propter nimiam Solis vicinitatem, cuius lumine tegitur; tum
propter vapores, qui horizonti vt plurimum incumbunt, vt rarò, & po&longs;t ma
gna temporis interualla con&longs;piciatur. non me fugit hæc omnia ab a&longs;trono
mis per epiciclum excu&longs;ari; &longs;ed ego mediocritati eorum, in quorum gra
tiam hæc &longs;cribo, con&longs;ultum volo.
Eodem cap.
humiditatis, &longs;ed quia parua e&longs;t &longs;ictio circuli, quæ &longs;uper terram, quæ autem deor
&longs;um multiplex, non po&longs;&longs;e vi&longs;um hominum fractum ferri ad Solem,
au&longs;trino appropinquanti; quapropter in
lis quidem locis neque fieri cometem ip&longs;um. quando verò ad Boream &longs;ubdefecerit,
accipere comam, quia magna e&longs;t circun&longs;erentia, quæ e&longs;t &longs;upra horizontem; quæ au-
Solem)
non appareret, cau&longs;am referebat Hippocrates paruitatem circuli, qucm
motu diurno cometa de&longs;cribebat, ob quam adeò parum &longs;upra horizontem
attolleretur, vt
dum ip&longs;um erat nece&longs;&longs;arium ad cometarum apparitionem. I oquitur igitur
Hippocrates de circulis, quos diurna conuer&longs;ione cometes circumducir, qui
omninò &longs;imiles &longs;unt ijs, quos etiam Sol,
gnant. qui quidem omnes in no&longs;tr a &longs;phæra obliqua ita &longs;e habent, vt ij, qui
&longs;unt vltra æquatorem ad Capricorm tropicum, minus &longs;upra horizontem
extent, quàm infra de primantur, & tanto minus, quanto magis ab æquato
re in auftrum recedunt: contra verò f
cri conuer&longs;ionem co&longs;&longs;ocantur, quanto enim magis ab æquatore in boream
remouentur, tantò eorum &longs;ectio, quæ e&longs;t &longs;upra horizontem, maior e&longs;t ea,
quæ infra horizontem latet. quæ quidem omnia clara &longs;unt adhibita &longs;phæra
materiali, quam &longs;i ad tuam poli eleuationem accommodaueris, illicò vi
debis tropici, Cancri &longs;ectionem, quæ e&longs;t &longs;upra horizontem multo maiorem
ea, quæ e&longs;t infra. oppo&longs;itum verò in altero Capricorni tropico, cuius mini
mam portionem &longs;upra, maximam verò infra horizontem exi&longs;tere videbis. Idem proportionaliter imaginari debes de circulis, quos cometa tam vltra
Capricornum, quàm citra Cancrum delineat; nam eorum, qui &longs;unt vltra
Capricornum ad au&longs;trum minores adhuc &longs;ectiones &longs;upra horizontem exi
&longs;terent, quàm opus &longs;it ad cometen &longs;pectandum.
tentia Hippocr. cur in illa au&longs;trali plaga è con
trario autem, quia ad boream &longs;ectiones illæ maximæ &longs;unt,
ctionem vi&longs;us no&longs;tri v&longs;que ad Solem, idcircò in hac mundi parte cometas
con&longs;picere &longs;olemus. Reliqua Vicomercatus,
cant, quos tu con&longs;ule, ne actum agatur.
In præ&longs;enti cap. Ari&longs;t. &longs;uam de Cometis &longs;ententiam exponit: Come
tam nimirum infra Lunam in elementari mundo procreari, & ignitum
quoddam Meteoron, ex lenta, pingui,
prem
vel elementi ignis (quod illic e&longs;&longs;e putat) vicinitate, vel etiam vi a&longs;trorum
incendi, Hanć porrò opinionem & &longs;i probabilibus tantum ra
tionibus confirmatam vulgò tamen
&longs;am e&longs;&longs;e a&longs;tronomi exi&longs;timent, non erit abs re rationes eas ex &longs;ecundo pro
gymn. Tichonis volumine, de&longs;umptas hic breuiter referre, quibus a&longs;trono
mus ille eos &longs;upra Lunam in ætherea regione collocauit: quas quidem ra
tiones ille ex diuturnis ob&longs;eruationibus per exqui&longs;ita organa factis adinue
nit: ea&longs;que Mathematicis linearum, ac numerorum demon&longs;trationibus
explicauit.
Prim
&longs;ed vt ab auctoritate, in quam obiter incidimus
non e&longs;t exi&longs;timandum nonnuilos &longs;olum ex recentioribus id con&longs;tanter a&longs;&longs;eCardan. libro de &longs;ubtili
tate conatus e&longs;t,
&longs;e ip &longs;o &longs;en&longs;i&longs;&longs;e ait Albumazar. quibus etiam ex antiquis Seneca annumeran
dus e&longs;t. pr&ecedil;dicti autem recentiores omnes varijs demon&longs;trationibus ex ac
curata ob&longs;eruatione erutis illud certò certius con&longs;irmare contendunt:
non in vno dumtaxat, &longs;ed in
Tychonem partim in progymn. partim in epi&longs;t. fu&longs;ius explicatas reperies.
2. Quarum poti&longs;&longs;ima illa e&longs;t, quæ ex parallaxi, &longs;eu a&longs;pectus diuer&longs;itate
de&longs;umitur, certi&longs;&longs;imum enim e&longs;t lumen illud e&longs;&longs;e altero &longs;ublimius, quod mi
norem exhibet parallaxim: expertos autem &longs;e e&longs;&longs;e hi omnes, affirmant ho
&longs;ce quinque cometas multò minorem pati parallaxim, quam Lunam; imò
quempiam minorem, quàm Sol ip&longs;e patiatur, quo po&longs;ito manife&longs;tè conuin
ceretur eos omnes &longs;upra Lunam in ætherea regione efful&longs;i&longs;&longs;e.
3. Ratio, qua etiam ante nouas ob&longs;eruationes vti &longs;olebant, de&longs;umitur
ex motu cometæ diurno, quo &longs;cilicet oritur, & occidit, quemadmodum cæ
tera &longs;ydera, hoc e&longs;t &longs;patio 24. horarum diurnam conuer&longs;ionem circa totam
terram ab&longs;oluit. &longs;i igitur comete e&longs;&longs;et in &longs;ublimiori aeris regione, vbi cæte
ra ignita meteora collocantur,
queretur nece&longs;&longs;ariò eum tanta velocitate videri à nobis circumferri, vt po
tius fulgor quidam, &longs;eu radius pertran&longs;iens ab oriente in occidentem appa
reret, quam &longs;tella qu&ecedil;dam:
miam di&longs;tantiam videntur tardè moueri, quamuis veloci&longs;&longs;imè moueantur.
Quod melius ex &longs;equenti figura
terra, cuius &longs;emidiameter A B. cir
culus verò exterior e&longs;t cometæ gy
rus, quem ip&longs;e &longs;patio 24. horarum
percurrit, qui &longs;ecundum veram pro
portionem deberet adhuc ip&longs;i terræ
propinquior, ac proinde minor e&longs;&longs;e,
iuxta aeris &longs;upremam partem. hori
zon e&longs;t recta D C, tangens terram in
B, vbi e&longs;t oculus no&longs;ter, qui nihil in
fra ip&longs;am D C, videre pote&longs;t; quare
&longs;i cometa 24. horarum totum gyrum
D C E, percurrit, non videbitur, ni&longs;i
quando percurret portionem D C,
&longs;upra horizontem; quæ quidem por
tio,
gura iuxta veram proportionem con&longs;trueretur. experientia tamen con&longs;tat,
cometas videri &longs;upra horizontem tot horis, quot &longs;tellæ fixæ, &longs;ub quibus mo
uentur: non ergo e&longs;t in &longs;upremo aere. Quod &longs;i &longs;iat figura, in qua exterior
cometæ ambitus adeò magnus &longs;it, vt ip&longs;ius portio D C, &longs;upra horizontem
exi&longs;tens, re&longs;pondeat tempori, quo cometa &longs;upra no&longs;trum pariter horizon
tem &longs;pectatur, ea figura terræ &longs;emidiametrum A B. toties multiplicabit, vt
ip&longs;i Lunæ circuitui proximè accedat.
Præterea aiunt, quis &longs;anæ mentis dixerit, Meteoron vlium ex materia
vaga, ac fluxa con&longs;tans, po&longs;&longs;e tanta pernicitate moueri, vt diurnam con
uer&longs;ionem ab&longs;oluat? vnde illi motus i&longs;te?
præ&longs;ertim cum videamus cætera
ignita m
4. Comprobationem nobis &longs;upp
toto durationis tempore proprio cur&longs;u de&longs;ignarunt: prædicti
que cometæ motu &longs;ibi proprio, quo ab occidente non omninò orientem
ver&longs;us, &longs;ed ad aquilonem deflectentes ab initio &longs;uæ apparitionis,
timum fiuem exqui&longs;iti&longs;&longs;imè portionem circuli maximi in c&ecedil;lo de&longs;ignarunt;
non aiiter quàm Sol proprio motu per eclypticam in cœlo mundi &longs;phæram
in duo æqualia diuidentem de&longs;cribit. necnon aliter ac Luna &longs;uum iter per
circulum maximum cœlum bifariam diuidentem perficit. quapropter co
metas ho&longs;ce
tendunt. qui enim, aiunt, fieri potui&longs;&longs;et, &longs;i in mundo elementari flagra&longs;&longs;ent,
vt tam regulari,
delinea&longs;&longs;ent, quam quidem inter elementa vagum,
teriæ in&longs;tabilitate exercere debui&longs;&longs;ent?
5. Adde, quod in maximo hoc circulo de&longs;cribendo, etiam &longs;i inæquali ve
locitate vi&longs;i &longs;int moueri, inæqualitatem tamen illam regularem
per &longs;eruauerunt, in principio quidem velociores, deinde &longs;ucce&longs;&longs;iuè, & pro
portionaliter velocitatem illam &longs;imili analogia &longs;emper &longs;eruata
nullo igitur pacto inordinatam inæqualitatem, qua à tardiore motu &longs;ubito
in celeriorem, & rur&longs;us &longs;tatim ab hoc in
omnia Meteora, quæ in mundi parte elementari ex flammanti materia ge
nerantur, talem di&longs;parem,
6. Argumento præterea e&longs;t cometas ho&longs;ce minimè elementares fui&longs;&longs;e,
quod hic eorum proprius motus, quo maximo illo tramite ferebantur, nua
quam tantus fuit, vt proprium Lunæ motum, vel tardi&longs;&longs;imum adæquauerit,
quæ quidem cum lenti&longs;&longs;ima e&longs;t plus denis gradibus vna die promouetur;
cum tamen cometæ initio cum veloci&longs;&longs;imi &longs;unt non multum vltra quinos
gradus diurno motu progre&longs;&longs;i &longs;int, vt ob id longè &longs;upra Lunam cur&longs;um &longs;uum
ab&longs;olui&longs;&longs;e manife&longs;tè comprobari po&longs;&longs;it: quo enim &longs;ydera magis à terra at
tolluntur,
tionibus proferuntur: ita vt &longs;teilæ i&longs;tæ cœlo ad&longs;cititiæ &longs;upra Lunam admo
dum euehendæ videantur. Quod &longs;i in &longs;uprema aeris regione con&longs;lagrarent,
qua nam ratione vnà cum toto cœlo diurnam conuer&longs;ionem ab&longs;olui&longs;&longs;ent:
qua cœle&longs;tes orbes, verum minori admodum imò tardi&longs;&longs;imè à diurno mo
tu, &longs;i tamen eo rapitur circumduci.
7. Tandem argumentum ex ip&longs;orum duratione de&longs;umatur.
cætera nam
que meteora &longs;tatim
nida extinguuntur: At verò cometæ ad men&longs;em aliquando integrum per
&longs;euerant. quì igitur fieri potuerit, vt in hac corruptibili
teria adeò &longs;luxa, & vaga, quam illis Ari&longs;teteles &longs;upponit, tandiu perdura
re potui&longs;&longs;ent.
natus e&longs;t, obtinui&longs;&longs;e, munifeftum e&longs;&longs;e volunt; ac proinde eorum locum, &
cur&longs;um in cœle&longs;ti mundi parte extiti&longs;&longs;e, &longs;e comproba&longs;&longs;e exi&longs;timant: qua de
re prudentis Lectoris e&longs;to iudicium:
tantas componere lites.
Verumenimuerò Peripatetica omnis &longs;chola reclamat; Cœlum e&longs;t inge
nerabile, & incorruptibile, mhil igitur noui cœlo pote&longs;t accidere. &longs;ed age
re&longs;pondent, nonne omnium a&longs;tronomorum con&longs;en&longs;u &longs;tellæ tres nouæ no&longs;tro
hoc &longs;æculo in cœlo toti mundo con&longs;picuæ illuxerunt?
ra re&longs;edi&longs;&longs;e conftans e&longs;t omnium a&longs;&longs;ertio? quarum prior anno 1572. in con
&longs;tellatione Ca&longs;&longs;iopeæ apparuit. Secunda anno 1600. in Cygno, quæ nec dum
extinguitur. Tertia anno 1604. inter Sagittarij &longs;tellas vi&longs;a e&longs;t, de quibus vi
de P. Clauium in &longs;phæra breuiter de illis tractantem: aut &longs;i mauis, & vacat,
vide quoad primam primum volumen progymna&longs;matum Tychonis Brahe,
vbi etiam aliorum a&longs;tronomorum de eadem certi&longs;&longs;imas commentationes
reperies. con&longs;ule etiam de reliquis duabus Ioannis Kepleri Cæ&longs;areæ Maie
&longs;tatis Mathematici commentaria; & coactus libenter fateberis noui ali
quid cœlo aduenire po&longs;&longs;e.
Po&longs;tremò tandem po&longs;&longs;et qui&longs;piam in hunc
&longs;tet
e&longs;&longs;e pariter cœle&longs;tes, Huic memorati A&longs;tro
nomi &longs;ic re&longs;ponderent; id quidem mathematica, & infallibili ratione non
colligi, imò aliquot parum infra Lunam extiti&longs;&longs;e, non omninò negandum
videri: at verò in &longs;uperiori aeris plaga, in tam fluxa, ac in&longs;tabili mundi par
te, cometas vnquam efful&longs;i&longs;&longs;e, nemo &longs;ibi ob allatas rationes meritò per&longs;ua
dere po&longs;&longs;e.
Summæ 2. cap.
5.
ca Astrologiam &longs;peculationibus, Solis magnitudo maior e&longs;t quàm terræ; & diftax
tia multò maior a&longs;trorum ad terram quàm So is; &longs;icut Solis ad terram quàm Lu
næ; non
vmbraterræ, quæ vocatur nox, erit apud astra; &longs;ed ne
cun&longs;picere, & nulli
& ex figura ibi de&longs;cripta, facilè e&longs;t intelligere præ&longs;entem locum; nam cum
Sol &longs;it multò maior terra, vt ibi probatur, ac minus di&longs;ter à terra quàm fixæ
&longs;tellæ, magis tamen quàm Luna, vt patet ex &longs;olari eclyp&longs;i, &longs;equitur nece&longs;&longs;a
riò vmbram terræ, quæ nox e&longs;t ip&longs;a, effici turbinatam, & valdè procul à ter
ra acumen coni vmbræ a&longs;cendet, &longs;ed paulò &longs;upra Lunam conus hic vmbræ
permittet radios Solis &longs;e ip&longs;um ambientes iterum &longs;imul committi, quod il
lis verbis
tur vmbra apud Lunam &longs;it &longs;atis gracilis, breui &longs;upra Lunam de&longs;inet, neque
vllo pacto ad affixa &longs;ydera protendetur, quod
etiam experientia confirmat, cum nunquam a&longs;tra illa, quæ Soli opponuntur, quare &longs;ine vllo ter
ræ impedimento Sol pote&longs;t af&longs;ixa omuia &longs;ydera perlu&longs;lrare. Exactiores ha
rum rerum demon&longs;trationes &longs;unt alterius loci.
Eodem cap.
po&longs;&longs;ibile autem e&longs;t & hoc, &longs;i enim videns quieuerit & &longs;peculum, & quod videtur
omne in eodem puncto &longs;peculi eadem apparebit
ueatur &longs;peculum, & quod videtur, in eadem quidem di&longs;tantia ad videns, & quie
&longs;cens; ad inuicem autem
po&longs;&longs;ibile eandem imaginem in eadem e&longs;&longs;e parte &longs;peculi. Quæ autem in lactis circu
lo feruntur a&longs;tra, & Sol, ad quem fit reflexio, mouentur manentibus nobis, & &longs;i
militer, & æqualiter ad nos di&longs;tantia; à &longs;e ip&longs;is autem non æqualiter: aliquando
enim medijs noctibus Delphin oritur, aliquando verò diluculo. partes autem lactis
eædem manent in vnoquoque; atqui non oportebat, &longs;i erat imago, &longs;ed non in ei&longs;dem
adhuc e&longs;&longs;et hæc pa&longs;&longs;io locis)
xiam apparere per quandam reflexionem vi&longs;us no&longs;tri ab illa parte c&ecedil;li, ceu,
ex quodam &longs;peculo ad Solem: probat autem hoc e&longs;&longs;e impo &longs;&longs;ibile ratione
de&longs;umpta ex parte Optices, quæ dicitur Catoptrica, &longs;iue &longs;pecularia, quia
tractat de vi&longs;ione reflexa, quæ fit mediante &longs;peculo, quam quidem rationem
&longs;i vellem mathematicè explicare, longa nimis, ac præter in&longs;titutum fieret
tractatio. Pauca tamen addam, quæ Ari&longs;totelis
reddant. &longs;i igitur inquit, Galaxia nihil aliud e&longs;&longs;et quàm reflexio no&longs;tri vi&longs;us
ex illa cœli parte, in qua ip&longs;a apparet tanquam ex &longs;peculo ad Solem, ita vt
nihil aliud ip&longs;a e&longs;&longs;et, quàm Sol vi&longs;us per reflexionem exilla cœli parte tan
quam &longs;peculo; &longs;equeretur eam non &longs;emper in eadem cœli parte apparere,
&longs;ed modo in vna, modo in alia, ita vt &longs;patio vnius anni totum cœlum perua
garetur: quod tamen non accidit. quod autem illud con&longs;equatur manife
&longs;tum e&longs;&longs;e pote&longs;t ex ob&longs;eruatione eorum, quæ ex &longs;peculis videntur: tunc enim
res per &longs;pe culum vi&longs;a in eadem &longs;peculi parte apparet, quando & videns, &
&longs;peculum, & obiectum immota manent: quod &longs;i & &longs;peculum, & obiectum ad
inuicem accedant, vel recedant, &longs;eruata tamen eadem ab in&longs;pectore di&longs;tan
tia, nullo modo fieri pote&longs;t, vt eadem imago, in eadem &longs;peculi parte &longs;pe
ctanti videatur, ni&longs;i obiectum &longs;peculo per eandem lineam accedat, &longs;ecun
dum quam illi incidebat. At verò partibus illis lactei circuli, &longs;iue a&longs;tris, quæ
in eo fulgent, Sol perpetuò accedit, vel recedit,
Delphini con&longs;tellatio, qui in ip&longs;o ferè lacte exi&longs;tit,
aliquando verò mane, aliquando etiam ve&longs;peri oritur; quod inde accidit,
quia illi Sol modò appropinquat, modò coniungitur, modò ab eo recedit,
quare nece&longs;&longs;e e&longs;&longs;et, vt lacteus orbis, non &longs;emper in ij&longs;dem locis, &longs;ed perpe
tuò in alijs, ex qui
bus con&longs;tat fal&longs;am omninò e&longs;&longs;e eorum &longs;ententiam, qui Galaxiam per huiu&longs;
modi re&longs;lexionem fieri opinabantur. Quæ dicta &longs;unt de &longs;peculo, & obiecto
&longs;atius e&longs;t a&longs;&longs;umpto aliquo &longs;peculo experiri, quàm ea pluribus ob&longs;curare: qua
etiam experientia Ari&longs;t.
ratio confirmabiaur.
Ibidem
xio, mouentur mancntibus nobis, & &longs;imiliter, & æqualiter ad nos di&longs;t antia à &longs;e
ip&longs;is autem non æqualiter)
vera propter apogæum, ac porigæum Solis, quæ quidem duo ab omnibus
a&longs;tronomis a&longs;&longs;eruatur: quando igitur Sol e&longs;t in apogæo, maiori multo in
con&longs;tat diametris terræ duobus, & quadraginta, hoc e&longs;t milliarijs 208000.
ferè, ide&longs;t octonis millibus &longs;upra ducenta millia. quæ differentia facit vt Sol
manife&longs;tè appareat nobis minor apogæus, quàm perigæus. Sol præterea &longs;i
militer ip&longs;is inerrantibus &longs;tellis fit tantumdem modo remotior, modo pro
pinquior: &longs;ed fortè Ari&longs;t.
i&longs;ta non occurrerunt, vel tunc temporis nondum
per&longs;pecta erant.
Ibidem
xiam non &longs;emper &longs;eruare à Sole di&longs;tantiam eandem, accipit tanquam huius
rei &longs;ignum, manife&longs;tum, quod Delphini con&longs;tellatio aliquando medijs no
ctibus oriatur &longs;upra horizontem, aliquando verò diluculo; non ideò tamen
putes hanc rationem &longs;upponere Delphinum e&longs;&longs;e in ip&longs;o lacteo circulo, quod
tamen verum non e&longs;t, non enim e&longs;t in Galaxia, &longs;ed tamen illi proximus, vt
noctu videre e&longs;t in cœlo, vel etiam &longs;i mauis in globo a&longs;tronomico: non ta
men ob id Ari&longs;t.
ratio minus valida redditur, cum Delphinus &longs;emper Gala
xiæ eodem modo &longs;it proximus,
Summæ 2. cap.
6. Sunt qui velint Ari&longs;t. Galaxiam nihil aliud e&longs;&longs;e, quàm
quandam refractionem lucis &longs;tellarum illarum, quæ &longs;unt in ætherea Gala
xia, quæ inquam refractio fiat circa &longs;upremam aeris regionem ex occur&longs;u
exhalationum, quæ ibi perpetuò con&longs;eruantur, & vi earumdem &longs;tellarum
&longs;ur&longs;um &longs;emper attrahuntur, quæ refractio fiat ad eum modum, quo halo cir
ca Solem, & Lunam. & quemadmodum halo, &longs;iue area omnibus
a&longs;picientibus &longs;emper videntur in eodem cœli loco, hoc e&longs;t è regione Solis,
vel Lunæ; &longs;imiliter Galaxia in aere omnibus
reat in eadem cœli parte, ide&longs;t ex aduersò eorumdem &longs;yderum, quæ cœle
&longs;tem lacteam viam conficiunt. Porrò qui &longs;ic mentem Ari&longs;t.
exponunt, nul
lo modo po&longs;&longs;unt à Mathematicis redargui per rationem de&longs;umptam à di
uer&longs;itate a&longs;pectus (quam po&longs;tea explicabo) quamuis phy&longs;icis rationibus re
fellantur. Alij &longs;unt, quorum &longs;ententia magis videtur
Ari&longs;t. &longs;ummum Philo&longs;ophum pueriliter in a&longs;tronomia lap&longs;um fateri cogan
tur. Exi&longs;timant hi Galaxiam hanc Ari&longs;totelicam nihil aliud e&longs;&longs;e, quàm ip
&longs;as tenues exhalationes in aere &longs;ubuectas, directèque infra &longs;tellas illas la
cteum circulum in cœlo con&longs;tituentes nobis obiectas. qui præter innumera,
ac magna ab&longs;urda è naturali Philo&longs;ophia petita, vnum maximum ex A&longs;tro
nomia, nempè ex diuer&longs;itate a&longs;pectus de&longs;umptum, nullo modo vitare po&longs;
&longs;unt;
nibus,
è diuer&longs;is, & præcipuè ab inuicem valde di&longs;&longs;itis, circa diuer&longs;a a&longs;tra &longs;e &longs;e ocu
lis no&longs;tris obijceret: at te&longs;timonio &longs;en&longs;us con&longs;tat, Galaxiam &longs;emper in eo
dem loco;
viam hanc in aere qua&longs;i pendulam fabricare debemus. rationem hanc di
uer&longs;itatis a&longs;pectus a&longs;tronomicè magis explicatam reperies apud Clauium
in &longs;phæra. Porrò hæc ratio quamuis adeo certa, ac no&longs;tra tempe&longs;tate vul
gata, parum tamen à nonnullis de rebus Meteorologicis commentaria con
farcinantibus intellecta, minimè eos ab&longs;terrere potuit, quin prædictam opi
nionem, non &longs;olum Ari&longs;toteli imponerent, verum etiam ip&longs;i
auxilio Philo&longs;ophiam aggrediuntur.
Eodem cap.
& adhuc &longs;par&longs;is vocatis)
ti&longs;&longs;imam e&longs;&longs;e videmus, &longs;ed præterea eandem &longs;tellarum admodum feracem
appellare licebit, &longs;i quidem &longs;tellæ omnes illæ nouæ, quæ no&longs;tra tempe&longs;tate
apparuerunt, omnes in hac via exortæ &longs;unt. prima enim anno 1572. efful&longs;it
in Ca&longs;&longs;iopea; altera anno 1600. in Cygno. tertia demum anno 1604. in Sa
gittario, quæ omnes con&longs;tellationes intra lacteum circulum continentur. Veri&longs;&longs;imum præterea e&longs;&longs;e hoc idem confirmatur in&longs;trumenti illius mirabi
lis auxilio, quod &longs;uperiori anno in Belgio excogitatum, & po&longs;tea in Italia
à Galilæo perfectius
Latinè verò, & quidem aptè à Græcis mutuato vocabulo alius Tele&longs;copium
appellauit: hoc inquam &longs;pecillo adhibito per&longs;picuum &longs;tatim fit non &longs;olum
in via lactea innumeras &longs;tellas contineri, verum quid ip&longs;a &longs;it, certò certius
con&longs;tat; &longs;ed &longs;atius e&longs;t ip&longs;ius Galilæi verba ex Nuncio &longs;ydereo referre: Quod
tertio inquit, loco à nobis fuit ob&longs;eruatum e&longs;t ip&longs;iu&longs;met lactei circuli e&longs;&longs;en
tia, &longs;en materies, quam Tele&longs;copij beneficio adeò ad &longs;en&longs;um licet intueri,
vt & altercationes omnes, quæ per tot &longs;æcula Philo&longs;ophos excruciarunt ab
oculata certitudine
e&longs;t enim Galaxia nihil aliud, quàm innumerarum &longs;tellarum coaceruatim
con&longs;itarum congeries, in
&longs;tatim &longs;tellarum ingens fre quentia &longs;e &longs;e in con&longs;pectum profert,
plures &longs;atis magnæ, ac valdè con&longs;picuæ videntur; &longs;ed exiguarum multitudo
pror&longs;us inexplorabilis e&longs;t. hæc ille.
Eodem cap.
&longs;criptione)
ex de&longs;criptione alicuius Globi a&longs;tronomici, in quo &longs;olent A&longs;tronomi omnes
con&longs;tellationes, ac &longs;tellas &longs;uis locis reddere,
culum graphicè effingere. huiu&longs;modi globum veteres &longs;ph&ecedil;ram Aratæam di
cebant ab Arato Poeta græco, qui
tus e&longs;t, ac proinde globum hunc ordine expo&longs;uit:
Eodem cap.
recentiores informia appellant, eò quod ad aliorum a&longs;teri&longs;morum formas
minimè reuocentur.
Summa 4. cap.
1.
&longs;tuentes)
natur in A&longs;ia, cum certò certius con&longs;tet, ip&longs;um in Græcia Europæ regione
&longs;itum e&longs;&longs;e. fortè legendum e&longs;t, vt vult Vicomercatus, ex Paropame&longs;&longs;o, non
autem ex Parna&longs;&longs;o, quamuis Græci codices aduer&longs;entur; Paropame&longs;&longs;um
dam montis Cauca&longs;i: Cauca&longs;um autem &longs;upra Pontum orifi, &
canum, & vltra mare per totam A&longs;iam &longs;e proferre, tradunt veteres Geo
graphi. vide The&longs;aurum geographicum Abrahami Ortelij.
Strabo lib.
15.
&longs;ic: Indiam à &longs;eptentrione Tauri extrema terminant, ab Ariana v&longs;que in
orientale mare, quæ extrema indigenæ particulatim nominant Poropami&longs;
vocant.
Ibidem
Arabiam, ac Per&longs;iam alluit,
&longs;cis Geographis Rubrum mare appellatur, cuius alterum Rubrum mare,
quod inter Africam, & Arabiam &longs;e in&longs;inuat, e&longs;t quidam &longs;inus, quem nunc
communiter omnes Rubrum mare appellant. de illo inquam meritò intel
ligit Alexander, non de hoc Aegyptiaco, cum ex a&longs;pectu illius à monte Pa
ropame&longs;&longs;o, &longs;equatur ip&longs;um e&longs;&longs;e editi&longs;&longs;imum, quod non &longs;equeretur ex altero
ob illius propinquitatem. Dixit autem mare, quod e&longs;t extra, ide&longs;t extra
terram habitatam, ad di&longs;tinctionem maris Mediterranei, quod e&longs;t intra
terram habitatam, ac propterea Mediterraneum dictum e&longs;t.
Ibidem
xes. ab hoc autem ab&longs;cinditur Tanais pars exi&longs;tens in Meotidem paludem fluit au
tem, & Indus ex ip&longs;o, omnium fluuiorum fluxio maxima)
& impo&longs;&longs;ibilia; nam cum Bactrus Bactrianam regionem irriget, quæ e&longs;t vl
tra Per&longs;iam, Choa&longs;pes verò Per&longs;iam ip&longs;am, Indus
quì fieri pote&longs;t, vt in Regionibus adeò inuicem di&longs;&longs;itis orti fluuij ab eodem nec minus fal&longs;um e&longs;t illud de Ta
nai, quod &longs;it qua&longs;i ip&longs;ius Araxis ramus quidam, Tanais enim ex Riphæis
Diony&longs;ius Afer &longs;ic cecinit:
verùm huiu&longs;modi errata Ari&longs;t.
donanda &longs;unt, cum nondum Geographia &longs;atis exculta e&longs;&longs;et.
Eod. cap.
lem, & multitudine, & altitudine &longs;igna autem altitudinis quidem, quia
videtur & à vocatis Profundis, & à nauigantibus in Stagnum in&longs;uper il
lu&longs;trantur à Sole ip&longs;ius &longs;ummitates, v&longs;que ad tertiam partem nocte, & ab
aurora, & iterum a ve&longs;pera)
Ca&longs;pium, &longs;upra Cholchidem, & Iberiam regiones, vbi polus eleuatur 47.
circiter grad. ac re&longs;pectu Græciæ, & maris Euxini vergit ad eam mundi pla
gam, vnde illis æ&longs;tiuo tempore Sol oritur. ait Ari&longs;t.
eum e&longs;&longs;e omnium mon
tium illius plagæ alti&longs;&longs;imum, quod probat primò, quia admodum à longè
cernitur,
tis nu&longs;quam ibi fundus reperiatur. & præterca à Nauigantibus in Stagnum,
&longs;iue in Meotidem paludem, quæ quidem loca minimùm di&longs;tant a Cauca&longs;o
560. milliaribus. Secundò, probat il ius altitudinem ex eo, quòd &longs;ummi
tates ip&longs;ius Lo
cum hunc fu
de Comparatione Platonis, & Ari&longs;t.
quo in opere plurima habet ex Mathe
maticis de&longs;umpta, quibus naturalem Philo&longs;ophiam mirificè illu&longs;trat,
fe&longs;tumqueIs igitur &longs;ect.
3. cap.
5. de hoc Ari&longs;t.
loco &longs;ie loquitur:
hic locus diligenter expendendus videtur tum quia difficillimus e&longs;t,
multis an&longs;am dedit reprehendendi Ari&longs;t.
tanquam puerilia effutientem. tex
tus
vt tertia illa pars ad montem referatur, qua&longs;i dicat, quod antequam Sol ima
montis illu&longs;tret, illuminat illius cacumen
&longs;ed hæc Mazonij expo&longs;itio nulla e&longs;t, cuiu&longs;libet enim montis etiam medio
cris altitudinis Sol illu&longs;trat non &longs;olum tertiam partem, &longs;ed & dimidium, &
duas tertias, & ferè totum, antequam ad planam illius ba&longs;im de&longs;cendat. Ego &longs;ic exponendum cen&longs;eo, vt Ari&longs;t.
dicat, mane, ide&longs;t initio Crepu&longs;culi
matutini, & ve&longs;pere, ide&longs;t, in fine Crepu&longs;culi ve&longs;pertini ip&longs;ius tertiam par
tem illuminatam con&longs;pici ab ijs, quorum horizonti tunc incipit, vel de&longs;init
Crepu&longs;culum; ex quibus illi nece&longs;&longs;ariò re&longs;pectu Cauca&longs;i &longs;unt occidentales,
quì manè hoc vident, vti &longs;unt ij, qui in Euxino, &longs;eu Ponto, & Meotide naui
gant, vel loca proxima inhabitant: illi verò, qui in fine Crepu&longs;culi ve&longs;per
tini hoc cernunt, nece&longs;&longs;ariò re&longs;pectu Cauca&longs;i erunt orientales. Alter huius
loci &longs;en&longs;us e&longs;t, ait Mazonius, vt non de tertia montis parte, &longs;ed de tertia
noctis portione loquatur, ita vt manè. v. g. initio tertiæ, & vltimæ noctis
parte, cacumen Cauca&longs;i illuminetur. hæc ille.
vbi animaduertendum expo
&longs;itionem
pus recidat; nam &longs;i dixerimus initio Crepu&longs;culi matutini illuminari ter
tiam partem Cauca&longs;i, tempus hoc coincidit cum initio tertiæ partis noctis,
quantitas enim Crepu&longs;culi in poli eleuatione 47. grad. qualem habet Cau
ca&longs;us, per totam æ&longs;tatem tres horas plus minus continet, vt patet ex tabu
la quantitatis Crepu&longs;culi, quæ e&longs;t apud Nonium, & apud Clauium in &longs;phæ
ra vltimæ editionis; quæ quantitas reperiri geometrico calculo pote&longs;t, vt
docent Nonius, Clauius, & Maginus lib.
10. primi mob. quod quidem trium
circiter horarum tempus e&longs;t tertia ferè noctis pars in ijs regionibus, quibus
polus eleuatur 47. grad. &longs;iue ergo dicamus id contingere initio Crepu&longs;culi,
&longs;iue initio tertiæ partis noctis, erit idem tempus, trium &longs;cilicet horarum. &longs;i ergo, inquit Mazonius, &longs;equamur priorem declarationem, nece&longs;&longs;arium
e&longs;t dicere, quod ea tertia pars montis, quæ initio auroræ Solis lumine per
funditur, &longs;it ea montis altitudo, qua ip&longs;e exuperat illam aeris regionem,
vnde Crepu&longs;culum incipit apparere. quo po&longs;ito aptè, ac &longs;agaciter altitudi
nem Cauca&longs;i inue&longs;tigat hoc pacto. præmittit autem &longs;eptem propo&longs;itiones
apud Mathematicos manife&longs;tas, quas ego mi&longs;&longs;as facio cum non mihi nece&longs;
&longs;ariæ videantur. po&longs;tea &longs;ic di&longs;currit; His ergo ita &longs;e habentibus, dico nos in
uenire po&longs;&longs;e viam, qua &longs;altem rudi Minerua, montis altitudinem comper
tam habeamus. &longs;i enim in principio Crepu&longs;culi v. g. matutini (ita enim, vt
&longs;upra annotaui intelligendus e&longs;t Ari&longs;t.) illuminatur tertia pars, nece&longs;&longs;arium
vidctur tertiam illam partem &longs;upra cam regionem collocari, ex qua Cre
pu&longs;culum in planitie apparere incipit, &longs;ed illa regio ex Alhazino, & Vitell.
de Crepu&longs;culis milliaribus 52. à terra recedit, ergo duæ tertiæ montis par
tes, quæ Solem initio auroræ non vident, &longs;unt 52. milliaria ad perpendicu
lum, & tertia alia pars illuminata e&longs;t ad perpendiculum 26. milliaria: ita
vt totius montis altitudo perpendicularis &longs;it 78. mill. &longs;ed papè in quos acurident enim hoc Ari&longs;t.
dictum Mathematici,
putant enim eum pueriliter lap&longs;um e&longs;&longs;e. Cæterum ego pro præceptoris tu
rela, dico eum &longs;equutum e&longs;&longs;e famam. hæc Mazonius, quorum nonnulla in
digent con&longs;ideratione cuiu&longs;modi, &longs;unt illa, quando dicit, nece&longs;&longs;arium vi
detur, quod ea pars &longs;upra eam regionem attollatur, vnde Crepu&longs;culum in
planitie apparere ineipit. videtur enim his verbis velle dicere, quod quan
do habitantibus planitiem, quæ e&longs;t ad pedem montis Cauca&longs;i, vel horizon
tem eiu&longs;dem, incipit Crepu&longs;culum, ij&longs;dem etiam tunc tertia montis pars
appareat illuminata; in quo &longs;en&longs;u errat po&longs;tea in colligenda montis altitu
dine, quamuis enim verum e&longs;&longs;et partem illuminatam eminere totam &longs;upra
52. milliaria, non tamen &longs;equitur ip&longs;am &longs;olam eminere, &longs;ed alia etiam pars
eminere pote&longs;t, quod &longs;ic geometricè demon&longs;trabo. de&longs;cribatur enim figura
& Clauius, in qua terræ globus e&longs;t F L G E, regiò vaporum, & exhalatio
num M X N T. horizon a&longs;tronomicus O P. phy&longs;icus Q R, tangens terram
in puncto F, vbi etiam ponendus e&longs;t huius horizontis habitator, vnà cum. Cauca&longs;o F V.
Sol A B C, qui initio Crepu&longs;culi infra horizontem O P, depri
mitur gr. 18. vti ab A&longs;tronomis compertum e&longs;t, hoc e&longs;t, arcum D P, e&longs;&longs;e
grad. 18. radius autem C I K, tangens terram, incipit illuminare halitus,
qui &longs;unt ad K, in extremo horizonte &longs;en&longs;ibili F K. quique po&longs;&longs;unt videri ab
oculo in F, ide&longs;t ab huius horizontis habitatore. Cæterùm prædicti autho
res po&longs;t longam ratiocinationem ex calculo planorum
o&longs;tendunt in triangulo H F K, latus H K, continere milliaria 3631. ex quo
detracta H L, &longs;emidiametro terræ, quæ e&longs;t milliar, 3579. reliqua L K, &longs;um
ma halitunm eleuatio relinquatur 52. milliar. quibus ab ip&longs;is demon&longs;tra
tis, &longs;i H F, terræ &longs;emidiameter, quæ continet milliar. 3579. ponatur &longs;inus
totus 100000. & latus F K, ponatur tangens anguli ad H, quem pr&ecedil;dicti au
thores probant e&longs;&longs;e grad. 8. 54. erit F K, tangens partium 15659. fiat igi
tur per 2. pro. trjang.
rectil.
Clauij;
& inueniemus per auream regulam latus F K, continere milliar.
560. quan
ta &longs;cilicet e&longs;t di&longs;tantia ab oculo no&longs;tro ad exhalationes Crepu&longs;culi initium
efficientes. Con&longs;ideremus iam triangulum F K V, vt ip&longs;ius latus F V, quæ
e&longs;t Cauca&longs;i altitudo, in milliaribus innote&longs;cat. iam ip&longs;ius latus F K, inno
tuit, angulus verò ad F, e&longs;t rectus; at angulus ad K, &longs;ic manife&longs;tabitur; in
quadrilatero F K I H, quatuor anguli &longs;unt æquales 4. rectis ex 32. primi. duo
autem F, & I, &longs;unt recti ex 18. 3. ergo reliqui duo H, & K, æquales erunt duo
bus rectis, quorum alter H, e&longs;t gr. 17. 48. vt præditi Mathematici
reliquus igitur ad K, erit gr. 162. 12. vt compleat duos rectos. qui &longs;i detra
hatur à duobus rectis, qui &longs;unt deinceps ad lineam F K, reliquus angulus
F K V, erit gr. 17. 48. &longs;i ergo latus F K, notum ponatur &longs;inus totus 100000.
latus verò F V, tangens anguli noti, erit ip&longs;a 32100. fiat igitur,
180. cuius pars F X, quæ e&longs;t in
fra habituum altitudinem continet milliar. 52. quibus detractis ex 180. re
manent 128. pro tota X V, quæ tota e&longs;t &longs;upra vapores, nondum tamen illu
minata. vnde patet Mazonium erra&longs;&longs;e in colligenda hoc modo Cauca&longs;i al
titudine, ex prima Crepu&longs;culi illuminatione in horizonte Cauca&longs;i facta,
cum ex præmi&longs;&longs;o calculo con&longs;tet partem montis F V, totam tunc temporis
e&longs;&longs;e tenebro&longs;am, quamuis &longs;uperet multò regionem vaporum, contrà quàm
ip&longs;e putabat, &longs;uperat enim eam milliar. 128. quare duæ tertiæ montis erunt
non 52. mill. vt ip&longs;e ait, &longs;ed mill. 180. & proinde tota altitudo erit mill. 270.
quod &longs;anè ridiculum e&longs;t, cum nullius montis altitudo &longs;e&longs;quimilliare tran
&longs;cendat. Quod &longs;i &longs;equamur alteram expo&longs;itionem, vt nimirum Ari&longs;tor. lo
quatur non de tertia montis parte, &longs;ed noctis, ita vt dicat, circa initium
tertiæ partis noctis apicem montis illu&longs;trari, altitudo eius erit tantum
adhuc tamen ab&longs;urda e&longs;t.
Si v
Cauca&longs;i, &longs;ed alterius, cuius habitator in principio &longs;ui Crepu&longs;culi tertiam
Cauca&longs;i partem iam illu&longs;tratam videat, vti accideret &longs;i Cauca&longs;us &longs;tatuere
tur in L K, vbi incipit Crepu&longs;culum habitanti in F. tunc e&longs;&longs;et altitudo tanta,
quanta colligit Mazonius, &longs;i tamen Ari&longs;t.
intelligatur de tertia montis par
te; e&longs;t enim L K, altitudo habituum 52. mill. & duæ tertiæ montis, quare
totus mons erit 78. &longs;i autem intelligatur circa tertiam noctis partem, mon
tis apicem illuminatum videri ab habitatore F, &longs;ic altitudo eins erit tan
tummodo 52. mill. quæ tamen adhuc omnem veritatem nimium &longs;uperat. Cum ergo hinc inde &longs;equantur ab&longs;urda, putat Mazonium excu&longs;andum e&longs;&longs;e
Ari&longs;tot. dicendo eum &longs;equutum e&longs;&longs;e famam, Verumenimuerò &longs;apientiores iudicent num rectè philo&longs;ophus, cuius e&longs;t re
condita,
&longs;equutum e&longs;&longs;e.
Tandem monendus mihi Lector e&longs;t, in demon&longs;tratione Magini, quæ e&longs;t
apud Mazonium &longs;ect. 4. citati operis; a&longs;&longs;umi radium Solis tangentem terræ
globum, qui cum horizonte faciat angulum gr. 18. quod fal&longs;um e&longs;t, &longs;olus
enim radius centralis, qui à centro Solis ad centrum terræ ducitur talem
facit angulum,
no&longs;tra e&longs;t 270. mill. &longs;ua verò 276. vbi etiam, &longs;icut & nos a&longs;&longs;umit horizon
tem Cauca&longs;i.
Aduertendum tandem Mazonium admodum aduer&longs;antia loquutum e&longs;&longs;e,
&longs;ect. enim 3. demon&longs;tratinè concludit altitudinem 76. mill. &longs;ect.
verò 4. &longs;i
mul quæ nimis ab innicem di&longs;crepant, cum tamen
ritas &longs;it vna. At verò cau&longs;a huius di&longs;crepantiæ e&longs;t, quòd &longs;ect.
3. accipit Cre
pu&longs;culum non horizontis Cauca&longs;i, &longs;ed illius, in cuius extremitate orientali,
vbi incipit Crepu&longs;culum, Cauca&longs;us &longs;itus &longs;it, vt &longs;upra o&longs;tendimus.
&longs;ect.
verò 4. accipit horizontem ip&longs;ius Cauca&longs;i, vt ex
figura illic de&longs;cripta videre e&longs;t. ex hac igitur horizontum varia &longs;uppo&longs;itio
ne, varia etiam altitudo colligitur, quamuis
ne
Eodem cap.
ctiaiem in Gallia, flaunt l&longs;ter, & Tarte&longs;&longs;us, iste quidem extra columnas, I&longs;ter au
tem per totam Europam in Pontum Euxmum)
tum, qui falsò tradit I&longs;trum, &longs;ine Dannbium ex Pyreneis de&longs;luere, nam Iu
ce clarius con&longs;tat ip&longs;um ex ijs Alpibus, quæ Heluetiorum montes dicuntur,
propè Ba&longs;ileam ex Adula monte ortum ducere.
quem & Bœtim alij nominant ex Pyreneis de&longs;cendere. Tarte&longs;&longs;um hunc Ma
ginus putat e&longs;&longs;e Tagum, cui fauet vocabulorum quali&longs;cunque &longs;imilitudo. extra tamen columnas Herculis qui&longs;quis &longs;it in Oceanum occidentale illa
bitur. Igno&longs;cenda &longs;unt i&longs;ta Ari&longs;t.
tunc enim Geographia
Ad fin
qui
enim in tota Liguria quidpiam tale reperitur.
Svmma 4. cap.
2. quod e&longs;t de permutatione, & vici&longs;&longs;itudine aquarum,
& continentis. Pergratum Lectori fore exi&longs;timaui, nec alienum ab
in&longs;tituto, &longs;i occa&longs;ione huius permutationis maris, ac terræ, rem ex
po&longs;uero &longs;citu digniffimam, quam pridem ob&longs;eruare cœpi, ac in dies
ob&longs;eruo, præ&longs;ertim cum nullus præteritorum &longs;criptorum, quod &longs;ciam, eam
literis mandauerit: Terræ &longs;cilicet totius molem paulatim reduci ad perfe
ctam &longs;phæricitatem, ita vt aliquando nece&longs;&longs;e &longs;it futurum ip&longs;am à mari inun
dari, Prrmum igitur illud ex &longs;acris lite
ris &longs;tatuendum, orbem terræ in &longs;uo primordio fui&longs;&longs;e ab opifice rerum om
nium, figura &longs;phærica donatum, hoc e&longs;t
vallium depre&longs;&longs;ionibus. quod patet ex eo, quia
ita vt minimè apta e&longs;&longs;et animantibus ad inhabitandum. redditam verò ha
bitabilem, cum ip&longs;ius conditor
dam eminentiorem effeci&longs;&longs;et; transferendo nimirum maximam terræ por
tionem ex vno loco in alium, vnde illic maris concauitas, i&longs;tic verò mon
tium &longs;ublimitas emer&longs;it. quo facto aquæ omnes in loca illa decliuiora &longs;ua
&longs;pontè rece&longs;&longs;erunt, quæ aquarum congregatio Mare appellatum e&longs;t. Hine
nonnulli auctores graui&longs;&longs;imi a&longs;&longs;erere non dubitarunt, montes
&longs;e ex terra illa, quæ locum illum occupabat, quem po&longs;tea maria inua&longs;erunt. quæ cum ita &longs;int.
&longs;equitur terram
propterea in quodam &longs;tatu violento, præ
terea cum terra &longs;it grauior quàm aqua, nulla ratione deberent terræ partes
&longs;uperiores a quæ &longs;uperficiem &longs;uperare, cuius tamen
&longs;uperficies ip&longs;a terræ, & multò magis
uis non parum &longs;uperant; quæ altera violentia terræ, & aquæ ine&longs;t, & ideò
minimè mirum e&longs;t, imò
pri&longs;tinam, ac primigeniam figuram, ex qua con&longs;ectarium erit aquam
&longs;uam pariter illam &longs;ibi primæuam recuperaturam e&longs;&longs;e figuram. cau&longs;am au
tem re&longs;tauratri
fluuiales iamdiù ob&longs;eruauimus, vt ex &longs;equentibus ob&longs;eruationibus patebit.
Primò, videmus flumina quotidie montium radices corrodere, ac qua&longs;i
&longs;uffodere, ita vt pa&longs;&longs;im ex hoc, vel illo monte magnas faciant ruinas, ac pr&ecedil;
cipitia,
Iob cap.
14. allunione paulatim terra con&longs;umitur. humum porrò illam ex
montibus delap&longs;am &longs;emper ad loca humiliora fluuij &longs;ecum detrahunt. Ex
continua etiam hac inter montes corro&longs;ione facta manife&longs;tè apparet, flumi
num alueos in montanis modò e&longs;&longs;e humiliores quàm olim, quamuis contra
rium accidat alueis &longs;luuiorum per plana decurrentium, qui modò altiores
&longs;unt Illud autem liquidò apparet
ex &longs;ignis, &longs;eu &longs;ymbolis, &longs;eu ex &longs;imilitudine terræ, aut lapidis, quæ in alti&longs;&longs;imis
fluminum ripis hinc inde pa&longs;&longs;im
flumen eos ab inuicem &longs;epararet;
illa ambula&longs;&longs;e; quemadmodum in Pyramo Ciliciæ amne ob&longs;eruauit Strabo,
dum libro 12. de illius ripis hæc tradit, mira præterea e&longs;t montis cæ&longs;ura,
per quam alueus ducitur; nam quemadmodum in petris per medium &longs;ci&longs;&longs;is
contingit, alterius partis depre&longs;&longs;ioribus ita conuenire alterius partis emi
nentias, vt coniungi po&longs;&longs;int: &longs;ic videre e&longs;t imminentes flumini petras vtrin
que ferè
&longs;patio concauitates qua&longs;dam eminentijs oppo&longs;itas habere. hæc Strabo de
vno, quod nos in pluribus ob&longs;eruauimus. Pr&ecedil;terea videmus quotidie pluuias
aquas, idem quantum po&longs;&longs;unt efficere, &longs;uperficies montium, eorum maxi
mè, qui coluntur, perpetuò ab&longs;umentes,
tes. hinc videre e&longs;t, montes cæteris duriores, vt &longs;unt lapido&longs;i, cæteris altio
res reman&longs;i&longs;&longs;e; quippe qui magis & pluuijs, & fluuialibus aquis &longs;ua duritie
ob&longs;titerunt. idem montani incolæ omnes confirmant, qui omnes aiunt &longs;ibi
hanc montium demolitionem iampridem innotui&longs;&longs;e, ex eo quod nonnulli
montes olim &longs;ibi impedimento erant, ne arcem, turremuè in vlteriore mon
te &longs;itam con&longs;picerent, quam deinde plures po&longs;t annos intermedio monte
depre&longs;&longs;o, commodè videbant. Ad hæc; antiqua in montium verticibus con
&longs;tituta ædeficia, propterea intercidunt, quia terra hinc, & inde ab aquis
paulatim confumpta,
relinquit; deindé terra etiam ip&longs;a, qua fundamenta innitebatur &longs;en&longs;im de
lap&longs;a, ip&longs;a
ius &longs;igna infinita propemodum videri po&longs;&longs;unt; vnum tamen, quod toti orbi
confpicuum e&longs;t, non ommittam; Capitolium videlicet Romanum, cuius
modo fundamenta tota extant, quæ olim altè &longs;ub terram de&longs;cendebant. vi
de pulcherrimam hac de re tractationem apud Georgium Agricolam lib.
3.
cap.
1. qui amplius addit illud, quod & mihi maximè probatur; flumina ni
mirum producere montes,
extiti&longs;&longs;e tot particulares montes ab inuicem di&longs;cretos, &longs;ed fui&longs;&longs;e perpetua
quædam terræ iuga eminentia quidem, &longs;ed non tot vallibus di&longs;&longs;ecta: v. g.
mons no&longs;ter Apenninus erat iugum, &longs;iue dor&longs;um quoddam terræ eminens
quidem, &longs;ed nullis vallibus in tot particulares colles, aut montes di&longs;&longs;ectum;
&longs;ed po&longs;tquam flumina à &longs;ummitate ip&longs;ius deor&longs;um fluere cœperunt; paula
tim corrodentes humum in dies magis, ac magis effecerunt valles,
ratione in colles, hæc de
montibus &longs;ufficiant, nunc ad plana de&longs;cendamus.
Contrarium igitur omninò accidere videmus in planis, quoniam eædem
aquæ, quæ ex montibus quotidie terram &longs;ecum deducunt, eam ad humilio
ra loca, vt &longs;unt plana, & campe&longs;tria, &longs;iue ibi &longs;int maria, &longs;iue arida, compor
tant, hinc videmus antiqua ædeficia in planis locis
ex&longs;tructa, e&longs;&longs;e iam penè tota &longs;epulta, contra quam in montanis, cuius exem
plum habes etiam Romæ propè ip&longs;um Capitolium, in Arcu triumphali Sep
timij, qui iam ferè totus ruino&longs;a vndique terra obruitur. &longs;ic Pantheon.
&longs;ic
etiam templa Epi&longs;copalia, quæ
infra terram con&longs;piciuntur. Idem affirmant cœmentarij, & architectores
canant, occurrit primò terra quædam, quam ip&longs;i motam appellant, quæ li
gnis, ruderibus, ferramentis, numi&longs;matis, &longs;epulturis,
mixta e&longs;t; qua eruta, reperitur terra alia, quam nunquam fui&longs;&longs;e motam, ap
paret, ex eo quod &longs;olida, ac benè compacta &longs;it, neque vllis externis rebus,
præ&longs;ertim artificiatis admixta, terra illa, quam motam dicunt, variam va
rijs in locis &longs;ortita e&longs;t altitudinem, prout aquæ plurimum, vel minimum
montanæ terræ huc, vel illuc comportarunt: alicubi vt hic Parmæ erit &longs;ex
vlnarum, alibi viginti, vt Mutinæ; alibi triginta, vt Romæ, nonnullis in lo
cis. Comprobatur tandem hæc no&longs;tra ob&longs;eruatio ex arte illa, qua per ea&longs;
dem fluuiales aquas &longs;olent, tam loca depre&longs;&longs;iora per aggerationem paula
tim replere,
ro&longs;ionem deprimere. qua in arte exercitati&longs;&longs;imum P. Augu&longs;tinum Spernac
ciatum no&longs;træ Societatis videmus modo de mandato Summi Pontificis Pa
dum, ac Renum Bononien&longs;em ob aggerationem &longs;tagnantes in mari emitte
re; cui totus hic no&longs;ter di&longs;cur&longs;us maximè probatur. Ex quibus omnibus &longs;e
quitur &longs;uperficiem terræ tam montium, quam planorum quotidie variari. illam nimirum deprimi, hanc attolli.
vnde aliud maximum notandum &longs;e
quitur, videlicet hac tempe&longs;tate non e&longs;&longs;e eandem agrorum &longs;uperficiem, quæ
erat antiquitus, cum in montanis agris &longs;it multò humilior, in campe&longs;tribus
verò altior, quàm antiqua illa, ac primigenia; quapropter mirum videri
non debet, &longs;i quorumdam locorum adeò immutata natura e&longs;t, vt quæ olim
genero&longs;a vina ferebant, vel quouis alio e&longs;&longs;ent prædita munere, adeò dege
nerauerint, vt & vina, & alia nullius modò valoris, vel in par
ferant. Quod verò ad marium aggerationem &longs;pectat, dicimus ij&longs;dem aquis
magnam arenarum copiam perpetuò impertantibus, fieri aggerationem,
hoc e&longs;t littora quotidie magis cre&longs;cere, &longs;eu in mare ingredi, & con&longs;equen
ter mare recedere. quod primò Ari&longs;t.
te&longs;timonio in hoc cap.
comprobatur,
cum quo pariter &longs;entiunt veteres Geographi, & Hi&longs;torici omnes. Ari&longs;t.
igi
tur in comprobationem huius adducit primò magnam Aegypti aggeratio
nem; pars enim illa Aegypti, quæ Delta,
rodoto, ex arenis, & limo, ex Aethyopiæ montibus &longs;imul cum Nilo in mare
delabentibus, e&longs;t conflata,
mare ce&longs;&longs;it;
nas importante &longs;it facta. &longs;ecundum, Ari&longs;t.
exemplum e&longs;t Ammonia Regio,
cuius humiliora loca. f.
maritima, palam e&longs;t, inquit, quod aggeratione facta,
fiunt &longs;tagna, & continens: &longs;uccedente autem tempore, &longs;tagnans aqua ob
nouam aggerationem de&longs;iccata e&longs;t, & iam annihilata. tertium e&longs;t Meotidis
Paludis; At verò, ait, & quæ &longs;unt circa Meotidem Paludem creuerunt allu
uione fluuiorum tantum, vt multò minores magnitudine naues, nunc innare
po&longs;&longs;int, quàm anno ab hinc quare ex hoc facilè e&longs;t ratiocinari,
quod & primò, vt multa &longs;tagnorum, ita & hoc opus e&longs;t fluuiorum, & tan
dem nece&longs;&longs;e e&longs;t totum fieri &longs;iccum.
vnà cum præcedentibus &longs;atius e&longs;t apud ip&longs;um, vel potius apud eius expo&longs;i
torem Vicomercatum videre, vt breuitati con&longs;ulatur. Accedit & Plinij te
&longs;timonium, qui tradit multas terras na&longs;ci, non &longs;olum fluminum inuectu, &longs;
Athenarum verò
bit. Huc facit locus quidam Strabonis ex lib.
12. de Pyramo Ciliciæ fluuio:
&longs;ic; montes verò egre&longs;&longs;us tantum limum in mare deducit, partim ex Ca
taonia, partim ex Ciliciæ campis, vt huiu&longs;modi de co oraculum feratur;
In &longs;acram veniet conge&longs;io litore, Cyprum:
hic enim fluuius è regione Cypri in&longs;ulæ in mari influit, &c.
hæc Strabo.
Verùm recentiora non de&longs;unt exempla.
Rauenna olim erat in extremo
littore &longs;ita, nunc paulatim aggeratione aucto litore, mare multum ab ea
rece&longs;&longs;it. Patauium pariter, vt fertur mare alluebat, quod modo 25. pa&longs;&longs;uum
millibus ab eo di&longs;tat. Aæ&longs;tuarium ip&longs;um Venetum, ob arenas à varijs &longs;lu
minibus in ip&longs;um immi&longs;&longs;as adeò fundum extulit, vt vix amplius nauigatio
ni &longs;it aptum,
fiat terre&longs;tris. demum exemplum &longs;it Bononien&longs;ium Renus, qui quamuis exi
guus &longs;it torrens, paucis tamen annis Padum ip&longs;um, in quem immi&longs;&longs;us fue
rat arena ita repleuit, vt & &longs;ibi, & Pado magno vicinorum agrorum damno
viam in mare ob&longs;truxerit. Cum igitur mare ob hanc adaggerationem co
gatur &longs;e quotidie magis recipere,
coangu&longs;tari, quod
iam
quibus in locis &longs;u
contra maritimas innundationes: quibus antiquitus minimè fui&longs;&longs;e opus hi
&longs;toricorum, ac Hoc igitur modo ter
ra, qua montes,
deportata, cau&longs;a e&longs;t, vt mare &longs;en&longs;im modo hac, modo illac, terræ &longs;uperfi
ciei &longs;uperfundatur,
tabilis reddatur: quod tunc maximè accidct cum aquæ tam fluuiales, quàm
pluuiæ, &longs;uper faciem terræ perpetuò di&longs;currentes, totam illam montanam
terram in pri&longs;tinum locum, vbi ab initio fuerat,
tuerint; tunc terra erit iterum rotunda, & &longs;phærica, hoc e&longs;t &longs;uæ primigeniæ
iterum figuræ re&longs;tituetur: quapropter mare etiam rur&longs;us &longs;icut initio mundi
totam terræ faciem
Hinc nonnulla colligi po&longs;&longs;unt non minus notatu, ac &longs;citu, quàm præceden
tia digni&longs;&longs;ima, quibus Ethnicorum Philo&longs;ophorum error redarguatur, &longs;ides
verò no&longs;tra magis roboretur: mundum nimirum ab æterno neutiquam ex
titi&longs;&longs;e, vel &longs;altem terram ab æterno non fui&longs;&longs;e hac figura præditam, qua nunc
videmus, ncc mundum perpetuò duraturum. nam &longs;i hæc montuo&longs;a illi figu
ra ab æterno ine&longs;&longs;et, iampridem tota illa montium tubero&longs;itas fui&longs;&longs;et ab
aquis exæ&longs;a, & con&longs;umpta:
bauimus, reducetur ad rotunditatem,
inhabitabilis, vnde nece&longs;&longs;ariò mortalium genus interibit. Quapropter ni&longs;i
igne illo, quem &longs;acræ literæ innuunt catacly&longs;mus ille præueniatur, aqua
mundus interiturus e&longs;&longs;et. &longs;ed de his hactenus.
Quoad magnum illud Diluuium, quod Ari&longs;t.
hoc capite exi&longs;timat po&longs;t
multa &longs;ecula reuolui, hoc veritati e&longs;&longs;e con&longs;entaneum argumento &longs;unt, ac
pariter admirationi varia
te, tùm in Alpibus ob&longs;eruaui;
to; præ&longs;ertim in tam immen&longs;a copia,
ta, quæ nulla vis humana illuc contuli&longs;&longs;er, ni&longs;i temporibus catacly&longs;mi ebul
lientibus aquis maris &longs;uper terram facta fui&longs;&longs;et hæc varia rerum maritima
rum cum terre&longs;tribus commixtio: quæ quidem optimè ex Pomponio Mela
comprobantur, qui libro 1. de Numidia &longs;ic narrat: interius, & longè &longs;atis
à litore, &longs;i fides res capit, mirum admodum, &longs;pinæ pi&longs;cium,
rumque
nis, infixæ cautibus anchoræ, neque locus ille Ouid. Met. 15. extra rem:
E&longs;&longs;e fretum, vidi fact as ex æquore terras:
Et procul à Pelago conchæ iacuere marinæ,
Et vetus inuenta e&longs;t in montibus anchora &longs;ummis.
Nos autem Chri&longs;tiani ad Noemi Diluuium i&longs;ta referre debemus.
Cap. 1. ait multa e&longs;&longs;e maria, quæ ad inuicem non communicant.
Eorum rubrum mare vnum e&longs;&longs;e; quod cum Oceano
e&longs;t extra Herculeum fretum ad occidentem parum videtur com
mi&longs;ceri &longs;iue Ari&longs;t.
pro Rubro mari intelligat Oceanum illum, qui
Arabiam, ac Per&longs;iam alluit, &longs;iue illius &longs;inum, qui Arabiam,
interluit, fal&longs;um e&longs;t ip&longs;um parum communicare cum occidentali Oceano,
vt quotidianis Lu&longs;itanorum nauigationibus ad Indos patet. &longs;ed meritò hoc
Ari&longs;tot. condonandum, cum tunc temporis nondum tota Africa e&longs;&longs;et certò
circumlu&longs;trata,
quens, patefactum e&longs;&longs;et.
Summæ 2. cap.
2.
tas)
ortu a&longs;trorum fixorum, aut con&longs;tellationum, quæ &longs;unt in firmamento, vti
e&longs;t Orion (& Canis, de quo po&longs;tea) intelligunt &longs;emper de ortu ip&longs;orum, qui
fit matutino tempore, quando &longs;cilicet vel &longs;imul cum Sole, vel paulò ante
Solem emergunt, ita vt videantur à nobis; qui ortus dicitur Co&longs;micus, tunc
propriè, quando &longs;imul a&longs;trum cum Sole oritur; quando autem incipit appa
rere manc ante Solem, dicitur ortus Heliacus. i.
&longs;olaris, quia oritur quodam
modo ex radijs Solis, &longs;ub quibus antea latebat. A&longs;tra verò incrrantia, &
planetæ Sole tardiores oriuntur nam cùm ip&longs;a Sol, quippe il
lis velocior primum a&longs;&longs;equitur, ea &longs;uo lumine obtegit,
rum heliacus: cum verò eadem præterierit, ac po&longs;t &longs;e reliquerit fit, vt mo
tu diurno toto cœlo conuer&longs;o, mane ante Solem effulgeant, &longs;iue heliacè
oriantur: & cum quotidie magis Sol ab illis recedat, ip&longs;aque magis à Sole
te Solem videantur.
oriantur; tum ante mediam noctem po&longs;tea paulò ante occa&longs;um Solis. de
mum cum fuerint Soli oppo&longs;ita, occidente Sole oriantur, qui ortus dicitur
Ve&longs;pertinus, vel Acronicus. po&longs;tea oriuntur &longs;emper in die ante Solis occa
&longs;um, donec Sol ip&longs;a iterum a&longs;&longs;equatur,
liacè occidere; & mox cum ip&longs;o Sole occumbant, quod Acronicè e&longs;t occi
dere. Totum porrò illud tempus, quo per diem oriuntur, non eorum ortui,
&longs;ed occa&longs;ui deputatur, eò quod non cernuntur oriri, vt &longs;equenti loco expli
cabitur. Quæ omnia adhibito Globo a&longs;tronomico, in quo con&longs;tellationes
omnes depictæ &longs;unt,
Sole &longs;uo loco in Zodiaco, qui paulatim per Zodiacum orientem ver&longs;us gra
diatur, & interim diurno motu globus conuertatur, ad &longs;en&longs;um manife &longs;ta
apparebunt. In &longs;umma auctores intelligunt de ortu, qui mane fit ante So
lem, quia tunc primum po&longs;t diuturnas latebras incipit apparere.
intelligunt de ortu Acronico, quia ante hunc ortum videbatur noctu,
ortu Acronico non fit noua apparitio; ideo de hoc non intelligunt. fit au
tem ortus hic Orionis, heliacus, & matutinus, de quo Ari&longs;t.
hoc loco, & alij
auctores, no&longs;tra hac tempe&longs;tate paulò ante Solis ingre&longs;&longs;um in Cancrum, &longs;i
ue ante &longs;ol&longs;titium æ&longs;tiuum circa 22. Iunij.
Eodem cap.
oriens, quia in tran&longs;mutatione temporis accidit occa&longs;us, & ortus, a&longs;tate, aut hye
me, & propter magnitudinem a&longs;tri dierum &longs;it aliqua pluralitas)
mercatus ex &longs;ententia a&longs;tronomorum occa&longs;um Orionis fieri autumni tem
pore, Sole Scorpionem ob&longs;idente docet, quod & verba Ari&longs;t.
clarè &longs;ignifi
cant, cum dicat ortum ip&longs;ius fieri æ&longs;tate; in tran&longs;mutatione verò temporis,
videlicet in autumno fieri occa&longs;um. Porrò occa&longs;us hic fieri incipit primum
mane oriente Sole,
Orion e&longs;t in occidente, & infra orizontem cadit: deinde paulò ante Solis or
tum, &longs;ed tamen nocturno tempore, ita vt occa&longs;us eius videri po&longs;&longs;it, donec
occidat parum po&longs;t Solis occa&longs;um, & tandem cum Sole ip&longs;o heliacè euane
&longs;cat. Scriptores autem ferè &longs;emper cum loquuntur de occa&longs;u inerrantium
&longs;yderum, de eo, qui noctu videatur, intelligunt: &longs;icuti ortum intelligunt
eum, qui noctu fit, affixa
&longs;es noctu oriuntur,
pus, ortui ip&longs;orum deputamus: Reliquum verò
tur, & idcircò ortus illorum minimè apparet, nulla ratione ortui debuit
a&longs;cribi: totum verò tempus, quo noctu occidunt, & occidere cernuntur, oc
ca&longs;ui illorum meritò attribuitur. &
quo primo de nocte apparere incipiunt, dicitur ab&longs;olutè ortus cuiu&longs;uis &longs;y
deris; &longs;ic etiam initium temporis illius, quo primum per noctem ea occide
re videmus, &longs;impliciter occa&longs;um appellamus.
Eodem cap.
nes intelligit tropicos, quod & tropici etymon confirmat,
valeat, ac conuer&longs;iuus. circa Canis ortum eadem &longs;unt notanda, quæ &longs;upra
de ortu Orionis annotaui; intelligit enim eum Canis ortum, qui mane fiat
Cum porrò in c&ecedil;lo &longs;it Canis maior, & Canis minor, qui & Procyon, ide&longs;t
Anticanis dicitur, exi&longs;timo Canem maiorem e&longs;&longs;e eum, qui vulgò Canicula
nominatur, de quo etiam
putò Ari&longs;t.
intelligere. eius porrò ortus in no&longs;tra poli eleuatione quadra
ginta quinque graduum, circa diem tertium Augu&longs;ti contingit, Sole autem
10. gradum Leonis occupante. Ex Magini tabulis ante ephemerides.
Eodem cap.
quidem ad &longs;uperiorem polum, qui no&longs;ter e&longs;t; altero ad alterum, & ad meridiem:
ductæ lineæ, & faciunt duos conos, bunc quidem habentem ba&longs;im tropicum, alte
rum autem habentem ba&longs;im circulum &longs;emper manifestum, verticem autem in me
dio terræ. eodem autem modo ad inferiorem polum alij duo coni terræ &longs;egmenta fa
ciunt)
&longs;tot. concipias,
Maior circulus &longs;it cœlum, in quo polus L, articus; M, antarticus, ille eleua
tus &longs;upra no&longs;trum horizontem S N, 45. gradibus, i&longs;te verò totidem infra
depre&longs;&longs;us.
X O, Capricorni, vt vides in figura. Terra &longs;it A B C H G F E D Z K. à cu
ius centro Z, educantur primo duæ lineæ rectæ Z R, Z S. ad circulum &longs;em
per apparentium maximum, quæ in terra tran&longs;eant per puncta B, K. & iun
gatur linea B K: iam vides conum S R Z, cuius ba&longs;is e&longs;t circulus &longs;emper ap
parens S R, vertex autem Z, in centro terræ, vt ait Ari&longs;tot. educantur nunc
duæ aliæ rectæ ad tropicum Cancri Z T, Z Q, quæ in terra faciant puncta
I, C,
&longs;is e&longs;t circulus Cancri, vertex verò centrum terræ Z. con&longs;idera iam figuram
B K I C, inter duas rectas B K, I C, & duos circuli terræ arcus contentam;
hanc Ari&longs;t.
appellat tympanum vnum terræ habitabile, quod e&longs;t ad Vr&longs;am,
ide&longs;t in &longs;eptentrionali plaga, in qua &longs;umus nos: quæ quidem portio &longs;i con&longs;i
deretur vt &longs;olida, & à reliqua terra præci&longs;a, erit corpus rotundum,
tamen duobus planis circulis ad in&longs;tar tympani terminatum: Ductis dein
de &longs;imiliter alijs quattuor lineis à centro Z, ver&longs;us polum antarticum fit al
terum tympanum H D E G, au&longs;tralis terræ habitabilis, vt in figura manife
&longs;tum e&longs;t. fui&longs;&longs;e autem huiu&longs;modi habitabilis terræ &longs;egmenta figuræ tympa
ni &longs;imilia, optimè declarant veteres figuræ geographicæ Ptol&ecedil;mei, & patet
etiam ex longitudine, & latitudine, vt benè ait Ari&longs;t.
quas Geographi por
tioni terræ habitabili attribuebant, longitudinem enim dixerunt eius di
men&longs;ionem ab occa&longs;u ad ortum: latitudinem autem à &longs;eptentrione in meri
diem, eò quòd illa multò hac longior e&longs;&longs;et. Ex quibus apparet habitatam
fui&longs;&longs;e veluti Zonam, terram ab occa&longs;u ad ortum præcingentem. quæ Zona
&longs;i &longs;umatur cum &longs;oliditate, quam ambit, ab Ari&longs;t.
tympano a&longs;&longs;imilatur.
Eodem cap.
bra enim non
&longs;ubdeficiat, aut permutetur vmbra ad meridiem. Quæ autem &longs;ub Vr&longs;a, è frigore
inhabitabilia)
Zona torrida, non po&longs;&longs;e habitari, fal&longs;um e&longs;&longs;e o&longs;tendunt plurimæ regiones
tam veteris, quam noui orbis, &longs;uperiori &longs;eculo patefactæ, in quibus magna
in amœnitate, ac fertilitate, Quoad vmbram il
lam, intellige meridianam. i.
quam Sole circa meridiem exi&longs;tente, nos qui
Boreales &longs;umus, &longs;emper ad Quod &longs;i ad meridiem
perrexerimus, occurret inhabitabilis (vt falsò putat) terra, prius quam. vmbra meridiana in Boream vergens deficiat.
quæ &longs;igna &longs;unt no&longs;tram habi
tationem e&longs;&longs;e citra Zonam torridam, in Boreali parte. Quæ autem &longs;ub Vr
&longs;a, ide&longs;t &longs;ub polo arctico, ob nimium frigus inho&longs;pita omninò habetur, nam
Iupiter vrget.
Verumtamen, quæ &longs;ub
Eodem cap:
per caput e&longs;&longs;e nohis, cum fuerit &longs;ecundum meridianum)
quæ corona Ariadnæ dicitur, hæc cum in cœlo manife&longs;tè &longs;it Borealis,
&longs;troque
cat nos
Eodem cap.
illic autem, ide&longs;t &longs;ub Zona torrida, compertum autem e&longs;t nunc totam ferè
torridam Zonam, & quidem alicubi percommodè habitari, cuius cau&longs;æ &longs;unt
quatuor, quæ ip&longs;um latuerunt. prima
e&longs;t perpetuum æquinoctium, quo Sol tantum &longs;upra, quantum infra terram
immoratur. accedit, quòd Sol nocturno tempore maximè ad imum cœli fe
ratur, atque ob
hanc &longs;olam rationem Campanus in &longs;ua &longs;phæra Zonam hanc putat maximè
e&longs;&longs;e habitabilem: quamuis hæc &longs;ola cau&longs;a, vt quotidiana docet experientia,
non &longs;ufficiat. &longs;ecunda &longs;unt pluuiæ, quæ alicubi quotidie &longs;tata hora decidunt.
tertia venti, qui veluti flabella quædam aerem agitant.
quarta præalti mon
tes perpetuis niuibus ob&longs;iti. quæ quatuor torridam hanc pa&longs;&longs;im refrigerant,
atque habitabilem reddunt.
Summæ 2. cap.
3. de
ne con&longs;iderare)
qua, vt in &longs;equenti con&longs;iderare; &longs;olet enim Ari&longs;t.
figuras, imò demon&longs;tratio
nes ip&longs;as Mathematicorum, de&longs;criptiones appellare, vt &longs;æpius in Logicis
monuimus.
rotundus)
referret,
eodem modo diuidi poterit)
ræ habitatæ partem, quæ quamuis rotunda non &longs;it, poterit tamen, ac &longs;i ro
tunda e&longs;&longs;et in figura circulari repre&longs;entari,
quo circulus &longs;ecatur, &longs;ecari.
&longs;ecundum locum; &longs;icut &longs;ecundum &longs;peciem contraria, plurimum di&longs;tant &longs;ecundum
&longs;peciem. plurimum autem di&longs;tant &longs;ecundum locum, quæ per diametrum opponuntur,
&longs;it igitur vbi A, occidens æquinoctionalis, contrarius autem huic locus vltimus B,
ortus æquinoctionalis)
ip&longs;ius extremitate vbi A. &longs;it occa&longs;us æquinoctialis, qui fit Sole exi&longs;tente in
alterutro æquinoctio; huic igitur per diametrum opponatur ortus æquino
ctialis in B. qui pariter contingit tempore æquinoctiorum: linea autem B A,
refert ip&longs;um æquatorem.
G, &longs;it Vr&longs;a: huic autem contrarium ex oppo&longs;ito illud, in quo H, meridies)
meter erit ip&longs;a linea meridiana. pro Vr&longs;a verò intelligit &longs;eptentrionem,
quod ibi &longs;it Vr&longs;æ con&longs;tellatio.
puncta iunguntur linea F E, quæ refert &longs;ectionem tropici, Cancri cum ho
rizonte: ortus enim, & occa&longs;us æ&longs;tiualis contingunt Sole Cancri tropicum
percurrente.
tem D C, erit &longs;ectio tropici Capricorni, & horizontis; Sole enim hunc tro
picum attingente ortus, & occa&longs;us hybernus fiunt.
di&longs;tantia &longs;ecundum locum, contraria &longs;unt &longs;ecundum locum: plurimum autem di
stantia, quæ &longs;ecundum diametrum; nece&longs;&longs;arium e&longs;t, & flatuum hos inuicem con
trarios e&longs;&longs;e, vocantur autem &longs;ecundum po
&longs;itionem locorum venti &longs;ic; Zephyrus quidem ab A, hoc enim e&longs;t occidens æquino
ctialis. Boreas autem, & Aparetias à G. hic enim Vr&longs;a, contrarius autem huic
Notus ab H. Meridies enim e&longs;t hic, à quo flat, & H, ip&longs;i G, contrarium e&longs;t; &longs;ecun
dum enim diametrum &longs;unt. Ab F, autem Cæcias; hic enim oriens æ&longs;tiuus e&longs;t; cui
contrarius est, non qui flat ab E, &longs;ed qui à C. Libs, i&longs;te enim ab occidente hyemali
flat; Qui verò à D,
Eurus, i&longs;te enim ab horiente hyberno flat, vicinus existens Noto, vnde & &longs;æpè Eu
ronoti flare dicuntur:
vocant, hi quidem Arge&longs;ten, hi autem Olympium, alij verò Scironem; iste enim ab
occidente æ&longs;tiuo flat, & &longs;ecundum diametrum ip&longs;i &longs;olus opponitur. Venti igitur, qui
&longs;ecundum diametrum po&longs;iti &longs;unt, & quibus alij aduer&longs;antur, ij &longs;unt. Alij autem
&longs;unt, &longs;ecundum quos non &longs;unt contrarij venti, ab I, quem vocant Tra&longs;ciam, qui me
dius e&longs;t inter Argesten, & Apparitiam, à K, autem, quem vocant Me&longs;en, Mtdius
enim e&longs;t Cæciæ, & Aparetiæ. Diameter autem K I, iuxta circulum &longs;emper con&longs;pi
cuum e&longs;&longs;e &longs;olet, non tamen exactè)
diametrum circuli omnium &longs;emper apparentium maximi, eo quod &longs;it ferè
&longs;ub diametro illius, in qualibet enim &longs;phæra obliqua, ide&longs;t, in qua polus cle
ximum, quem de&longs;cribunt ex ip&longs;o polo, tanquam centro, & interuailo v&longs;que
ad horizontem, circa ip&longs;um polum: hunc appellant &longs;emper apparentium,
maximum, quia intra hunc alios quamplurimos concipiunt circa eundem
polum, quorum minores &longs;emper &longs;unt polo propinquiores. huius igitur dia
metrum vult Ari&longs;t.
per lineam, quæ à K, in I, duceretur (quamuis non exa
ctè) repre&longs;entari.
ab eo, in quo M. hoc enim illi e&longs;t &longs;ecundum diametrum;
quod punctum per diametrum aduer&longs;um illi e&longs;t, &longs;piraret. Ni&longs;i ab eo veniat, qui ta
men non longè progreditur ventus quidam, quem accolæ Phæniciam vocant. maxi
mè igitur præcipui, & definiti venti hi &longs;unt:
rò omnia ex &longs;equenti figura optimè poterunt intelligi, quam diligenti ope
ra ad mentem Ari&longs;t.
ex græcis codicibus re&longs;tituere conatus &longs;um, cum ani
maduerterem figuras val dè deprauatas pa&longs;&longs;im apud Porrò ad literam M, in figura &longs;crip&longs;i ventum Libonotum, quem Ari&longs;t.
qui
dem non ponit propter ip&longs;ius paruitatem; imò apertè dicit Hele&longs;pontum
non habere contrarium: &longs;ed feci, vt completum ventorum numerum, quem
alij tradunt, haberemus.
Antequam textuum explicationem aggrediar, illud animaduerten
dum e&longs;t,
& refrangere; ibi Vicomercatum in &longs;ua interpretatione meritò,
& propriè v&longs;um e&longs;&longs;e verbis; reflexio, & reflecti: differunt enim
valdè apud Opticos refractio, & reflexio, vt etiam refrangere, & reflectere. propterea optimè hoc loco Olympiodorus di&longs;tinguit inter
diaxladiReflexio enim fit ex repercu&longs;&longs;o, vt
quando lumen Solis incidens in aliquod &longs;peculum, inde re&longs;ilit in oppo&longs;itum
parietem, illud re&longs;ilire e&longs;t propriè per&longs;pectiuis reflecti, vnde reflexio. Re
fractio autem fit ex tran&longs;pectu: vt quando lapis, qui e&longs;t in aqua, emittit
fuam &longs;peciem ad oculum, qui e&longs;t in aere, tunc enim, quia &longs;pecies lapidis re
pre&longs;entatiua non tendit recta ad oculum, &longs;ed in confinio aquæ, & aeris fran
gitur, dicitur fieri refractio, & refrangi, in refractione igitur requiruntur
duo media, per quæ &longs;iat vi&longs;io, quæ &longs;int diuer&longs;æ den&longs;itatis, vt &longs;unt aqua, &
aer: vapor, exhalatio, & aer: vitrum, & aer, &c. quando igitur videmus
Solem, aut Lunam per vapores, aut exhalationes fit refractio, quia den&longs;ior
e&longs;t vapor, & exhalatio, quam aer.
Notandum etiam Aream, de qua mox dicam explicari po&longs;&longs;e tam per re
flexionem, quàm per refractionem: per reflexionem, quia &longs;upponunt Philo
&longs;ophi e&longs;&longs;e in acre rorido innumcra &longs;pecula parua inuicem valdè proxima,
ide&longs;t guttulas, per quas re&longs;lectatur ad oculum no&longs;trum &longs;pecies &longs;yderis. per
re&longs;ractionem verò, vt vult Vitellio, quia &longs;umit totum illum aerem humi
dum magis den&longs;um e&longs;&longs;e aere paro, qui e&longs;t circa oculos no&longs;tros, & hoc modo
con&longs;tituit diucr&longs;a media in den&longs;itate, per quam fiat vi&longs;io; corpus inquam Vicomerca
tus igitur quamuis vtatur voce reflexionis in Halone, non tamen ex prædi
ctis videtur reprchenden dus.
Summæ 2. cap.
2. De Areæ figura
circa Solem, aut Lunam vi&longs;us; quapropter non ex oppo&longs;ito &longs;icut iris, apparet.
diqueab co
dem enim &longs;igno ad idem &longs;ignum æquales frangentur &longs;uper circuli lineam &longs;emper. &longs;it
A C B, & quæ A F B, & quæ A D B, æquales autem
& hæ A C, A F, A D, inuiccm. & quæ ad B, inui
cem &longs;cilicet C B, E B, D B. & protrahatur A E B,
quare trianguli æquales, etenim &longs;uper æqualem, quæ
e&longs;t A E B, ducantur autem
ex angulis; à C, quidem, quæ e&longs;t C E; ab F, autem,
quæ e&longs;t F E; à D, autem, quæ e&longs;t D E, æquales itaque
hæ, in æqualibus enim triăgulis, & in vno plano om
nes, ad rectum emm omnes ei, quæ e&longs;t A E B. & ad
vnum punctum E, copulantur, circulus igitur erit
de&longs;cripta, centrum autem E. &longs;it autem B, quidem Sol,
A, autem vi&longs;us, quæ autem e&longs;t circa C D F, circun
ferentia nubes, à qua refrangitur vi&longs;us ad Solem)
quia &longs;uppono Aream, &longs;iue Halonem fieri per re
fractionem, vt vult etiam Vitellio, propterea
ctatio de refractione innititur; e&longs;t autem huiu&longs;
modi; ea, quæ
aliquo refractionis angulo, manentibus nobis &
a&longs;tro, & medio ij&longs;dem in locis, non po&longs;&longs;unt vide
ri &longs;ub diuer&longs;o angulo à priori, ncc per con&longs;
alibi apparere. v. g. Sol (vt in præ&longs;enti figura)
videatur ab oculo A, media nube C D F, &longs;ub an
gulo refractionis B C A, vel B F A, & alijs &longs;imilibus angulis in eadem nube;
manente igitur oculo A, & a&longs;tro B, necnon nube C D E. eodem in loco, im
po&longs;&longs;ibile e&longs;t Solem videri ab eodem oculo &longs;ub diuer&longs;o angulo à priori, nec
con&longs;equenter alibi apparere, quam in B. Nunc ad textus declarationem, in
quo continetur Geometrica demon&longs;tratio rotunditatis Areæ, quam &longs;ic bre
uiter prius veteres excogitarunt: Viderunt primò Solem in Area apparere
in orbem, & con&longs;imiliter: hinc intulerunt nece&longs;&longs;e e&longs;&longs;e apparere etiam per
con&longs;imiles, &longs;iue æquales refractionis angulos; quia diuer&longs;i anguli, diuer&longs;am
etiam
gulos nece&longs;&longs;e e&longs;t in circulum
ditatis huius, e&longs;t angulorum refractionis æqualitas. Sed iam textum Ari&longs;t.
qui geometricam huius rci continet demon&longs;trationem, explicemus. Suppo
nit igitur primò Ari&longs;t.
lineas vi&longs;uales à &longs;ydere B, ad oculos no&longs;tros A, per
nubem roridam C D F, procedentes, in nube con&longs;imiliter refrangi, ide&longs;t
diquequod etiam
C D F, radij vi&longs;uales tres refracti in nube &longs;int B C A, B D A, B E A, facien
tes con&longs;imilem refractionem, ide&longs;t angulos refractos B C A, B D A, B E A,
æquales in punctis C, D, F: Supponit &longs;ecundò lineas à &longs;ydere ad nubem, v&longs;que exten&longs;as e&longs;&longs;e æquales, vt
&longs;unt B C, B D, B F: &longs;imiliter reliquas tres à nube ad vi&longs;um A. pares e&longs;&longs;e C A,
D A, F A. his &longs;uppo&longs;itis, &longs;i deinde protrahatur recta A B, ab oculo ad &longs;ydus,
exurgunt tria triangula omninò æqualia, & &longs;imilia, cuni duo latera vnius
&longs;int æqualia duobus alterius
ba&longs;is &longs;it communis; ideò per quartam primi &longs;unt omninò æqualia. ducan
tur nunc ex angulis C, D, F, tres perpendiculares ad rectam A B, quæ &longs;int
C E, D E, F E, in figura; quæ tres nece&longs;&longs;ariò erunt æquales, cum &longs;int ductæ
ab angulis æqualibus æqualium triangulorum ad communem ba&longs;im, & di
uident nece&longs;&longs;ariò ba&longs;im in eodem puncto E, cum diuidant triangula æqua
lia proportionaliter;
in nube concipitur ex 5. 11. Quare &longs;i concipiamus &longs;uperficiem, &longs;iue planum
delineari circa E, ad interuallum linearum æqualium C E, D E, F E, de
&longs;criptus erit circulus per 9. tertij, cuius circumferentia C D F. Ex quibus
patet tria illa puncta C, D, E, per quæ Sol tran&longs;paret e&longs;&longs;e in orbem di&longs;po&longs;i
ta. cau&longs;a igitur rotunditatis Areæ, e&longs;t &longs;imilitudo angulorum refractionis,
quibus Sol tran&longs;paret: vel ideo rotunda e&longs;t, quia &longs;imiles anguli nece&longs;&longs;ariò
in orbem con&longs;tituuntur, vt o&longs;ten&longs;um e&longs;t. Eadem ratione omnia alia puncta
eiu&longs;dem
do ad &longs;imilitudinem trium linearum A C B, A D B, A F B, refractarum, in
finitæ
prædicta, aliæ verò in alia periphæria maiori, aliæ etiam in minori, ita vt
ex tota nube fiant refractiones circulares plurimæ, ex quibus in nube area
con&longs;tituatur.
reddunt,
culo portionem, & de alijs accidentibus circa ip&longs;am, ex de&longs;criptione
erit con&longs;iderantibus manife&longs;tum)
de&longs;criptiones intelligere geometricas demon&longs;trationes, quod
etiam hoc loco confirmatur, vbi Geometrica demon&longs;tratione quam de&longs;cri
ptionem appellat, Iridis figuræ accidentia o&longs;tendit; nimirum cur &longs;it quidem
circularis, nunquam tamen circulus integer, imò
maior, &longs;ed tamen &longs;emicirculo minor.
Ibidem
tro autem K, alio autem quodam oriente puncto, in quo G, &longs;i lineæ, quæ à K, &longs;ecun
dum conum excidentes faciant velut axem lineam in qua G K, & à K. ad M, co
pulatæ refrangantur ab hemi&longs;phærio ad G, &longs;uper maiorem angulum, circuli circun
ferentiam incident lineæ, quæ à K, & &longs;i quidem in ortu, aut in occa&longs;u a&longs;tri reflexio
fiat, &longs;emicirculus ab &longs;i autem &longs;upra, minor
cum in meridie fuerit a&longs;trum)
&longs;upra monui, iterum moneo,
tinendam
uis
tur refractio, e&longs;t enim apud om
nes in confe&longs;&longs;o Iridem fieri per
reflexionem. E&longs;t igitur in &longs;upe
riori figura, quam textui, vt par
erat re&longs;titui, horizon G K O. cuius centrum K. in quo e&longs;t vi&longs;us no&longs;ter,
hemi&longs;phærium no&longs;trum in arcu G A M O, repræ&longs;entatum,
da, in qua Iris appareat, vbi M, quod punctum M, nubem referens, in figu
ra ponitur in hemi&longs;phærij ambitu, quod cœlum repræ&longs;entat, cum tamen
nubes parum à terra &longs;ubuchatur; id enim ad demon&longs;trationem ferè perinde
e&longs;t. in oriente G, &longs;it a&longs;trum.
&longs;i ergò lineæ vi&longs;uales à K, ad M, nubem tenden
tes reflectantur &longs;uper maiorem angulum M K G, ad G, erit reflexarum vna
veluti M G. Porro omnes lineæ viluales, quæ adnubem M, incidunt, nece&longs;
&longs;ariò, vt probabo, cadent in ambitum circularem. debemus enim innume
ras lineas im aginari à K, in coni figuram excidentes, cuius vertex &longs;it in K,
& axis G K O, quas omnes repræ&longs;entat vna K M,
cogitemus axem G K O, circa polos G, O, manentes circumuolui,
lineam K M, circumducere. in hac etiam giratione linea K M, tran&longs;ibit per
omnes illas lineas, quas imaginabamur;
formare debebant. In prædicta autem axis volutatione, extremum M, li
neæ K M, nece&longs;&longs;ariò de&longs;cribit circulum, qui e&longs;t circulus Iridis, & e&longs;t ba&longs;is
memorati coni.
Si igitur oriente, vel occidente a&longs;tro fiat iris, Iris erit &longs;emicirculus, ide&longs;t
illa &longs;emi&longs;&longs;is circuli pr&ecedil;dicti (quem horizon bifariam diuidit) quæ &longs;upra ter
ram extabit. &longs;i autem a&longs;trum eleuatum &longs;upra horizontem fuerit, quando fit
iris, erit &longs;emper arcus Iridis &longs;emicirculo minor; h&ecedil;c tria &longs;unt, quæ deinceps
Ibidem
mum vbi G, & refracta &longs;it K M,
ad G, & planum erectum &longs;it in quo
A, à triangulo in quo G K M, cir
culus igitur erit &longs;ectio &longs;phæ
maximus &longs;it in quo A, differet enim
mbil &longs;i quod
G K, &longs;ecundum triangulŭ K M G,
erectum fuerit planum. lineæ igitur
ab ijs, quæ G, K, ductæ in bac ratio
ne non cen&longs;tituentur ad aliud, &
aliud punctum, quàm &longs;emicirculi
in quo A. Quoniam enim puncta
G, K, data &longs;unt, & quæ K M, vtique data erit; & quæ M G, ad M K; datam igiapud autem aliud punctum, quam ip&longs;ius M N, circunferentiæ, ab
ij&longs;dem punctis, eadem ratio in eodem plano non con&longs;i&longs;tit)
demon&longs;tranda &longs;unt, præmittenda &longs;unt duo nece&longs;&longs;aria fundamenta. Primum
e&longs;t; ea, quæ videmus per reflexionem &longs;ub quopiam angulo, manentibus no
bis &longs;peculo, & obiecto ij&longs;dem in locis, non po&longs;&longs;unt videri &longs;ub alio diuer&longs;o
angulo, nec alibi con&longs;equenter apparere. v. g. in &longs;uperiori figura, quam
textui re&longs;tituimus exi&longs;tente Sole in G, oculo in K, & nube in M. ex qua ra
dius Solis G M, re&longs;lectatur ad vi&longs;um in K, per
impo&longs;&longs;ibile e&longs;t manentibus illis, vt dixi, videri Solem in nube M, &longs;ub diuer
&longs;o angulo à priori, nec alibi apparere. Alterum e&longs;t apud Opticos vulga
tum; ea &longs;cilicet, quæ per reflexionem (de quorum numero e&longs;t Iris) viden
tur, videri, tunc &longs;olum, quando angulus incidentiæ fuerit æqualis angulo
reflexionis, quia tunc breui&longs;&longs;imis lineis fit vi&longs;io; quibus &longs;oli, natura (&longs;i fieri
pote&longs;t) vtitur. v. g. in figura præ&longs;enti &longs;it &longs;pe
culum C D E, obiectum A, oculus B, linea in
cidentiæ e&longs;t A D, & angulus pariter inciden
tiæ e&longs;t A D C. linea verò D B, e&longs;t linea refle
xionis, & angulus pariter reflexionis e&longs;t B D
E, qui duo anguli ni&longs;i fuerint æquales, nun
quam videbitur obiectum A, ab oculo B, hinc
e&longs;t, quod aliquando po&longs;ito &longs;peculo, obiectum
quamuis illi aduer&longs;um, à nobis pariter ante
&longs;peculum con&longs;titutis, videri nequit, quia &longs;ci
licet in tali po&longs;itione &longs;peculi, obiecti, & no&longs;tri, nulla linea incidentiæ, ide&longs;t,
quæ ab obiecto in &longs;peculum tendit, facere pote&longs;t angulum cum &longs;peculo, qui
dicitur angulus incidentiæ, æqualem angulo illi, quem facit linea eadem re
flexa à &longs;peculo ad oculum, quem dicunt angulum re&longs;lexionis. Cum ergo in
Iride videamus colorem Solis per reflexionem, tunc &longs;olum apparebit Iris,
quando Sol, nubes, & oculus fuerint in ea con&longs;titutione, qua radius
nubi, & radius à nube repercu&longs;&longs;us faciant pares angulos. Et quia quando
nubes ro&longs;cida perpendiculariter opponitur Soli, & nobis, po&longs;&longs;unt &longs;ieri præ
dicti anguli æquales non in vno loco nubis, &longs;ed in pluribus, con&longs;titutis ta
men in circuli periphæria, hinc fit, quod Solis color reflectatur ex pluribus
locis in orbem con&longs;titutis, quæ reflexio e&longs;t ip&longs;ius Iridis arcus. ex Vitellion
63. 10. Totam autem figuræ Iridis demon&longs;trationem &longs;ic breuiter puto ad
inuentam e&longs;&longs;e. cum Sol in Iride videatur in orbem,
&longs;e e&longs;t id prouenire ex angulis reflexionum con&longs;imilibus, &longs;iue æqualibus: di&longs;
&longs;imiles enim anguli, di&longs;&longs;imilem atqui con
&longs;imiles anguli, &longs;iue æquales, non ni&longs;i in orbem po&longs;&longs;unt con&longs;titui; igitur an
gulorum æqualitas cau&longs;a erit rotundationis arcus. h&ecedil;c e&longs;t &longs;umma totius di
&longs;cur&longs;us, quem pluribus, & nimis ob&longs;curè Ari&longs;t.
explicat.
Inquit igitur Ari&longs;t. &longs;it enim in oriente, &c.
vbi aggreditur probare vnum
ex tribus illis, quæ &longs;upra propo&longs;uit, nimirum tunc Iridem e&longs;&longs;e &longs;emicircu
lum, quando a&longs;trum fuerit in oriente, &longs;iue in horizonte, vbi G. &longs;i igitur per
triangulum G M K, intelligamus
magnum, vt totum &longs;ece
cans hemi&longs;phærium, tran&longs;eat per
tio repræ&longs;entatur in figura, per &longs;emicirculum in quo A, &longs;iue in quo G A M
R O. nihil autem refert quodcunque intelligas planum &longs;uper axem G K O,
tran&longs;iens &longs;iue per triangulum G K M, &longs;iue per aliud illi &longs;imile. Præmitten
dum præterea non po&longs;&longs;e in &longs;emicirculo &longs;uperiori, quod e&longs;t planum, & &longs;ectio
trianguli G K M, poni alias duas lineas. v. g. G R, K R, ad aliud punctum,
vti e&longs;t R, quæ habeant eandem inuicem proportionem, quam habent prio
res duæ G M, K M, quod probatur, quia &longs;i &longs;int vt G M, ad K M, ita G R, ad
K R, cum G R, &longs;it centro K, propinquior quam G M, erit etiam eadem G R,
longior ip&longs;a G M, per 15. 3. & tamen deberet e&longs;&longs;e æqualis illi; quemadmo
dum K M, e&longs;t æqualis alteri K R; nequeunt autem duæ lineæ inæquales inui
cem, habere eandem rationem ad duas inuicem æquales: ergo non habent
eandem rationem G M, & K M, quam habent G R, & K R. quod &longs;i punctum
R, &longs;umatur &longs;upra M, erit &longs;imilis
tent. his po&longs;itis, ait
&longs;unt po&longs;itione, cum notum &longs;it vbi &longs;int. G, enim e&longs;t in ortu.
K, verò in centro
horizontis, &longs;equitur, quod etiam linea G K, cuins ip&longs;a &longs;unt extrema, data
&longs;it, & po&longs;itione, & magnitudine, per 26. Datorum Euclidis. eadem quoque
ratione data erit K M, linea; &longs;iue quia e&longs;t æqualis ip&longs;i G K, &longs;iue quia per
a&longs;trolabium po&longs;&longs;umus ip&longs;ius longitudinem, & po&longs;itionem inue&longs;tigare; qua
re & punctum M, datum erit per 27. Datorum, quare & linea G M, data
erit quoad &longs;itum, & magnitudinem per 26. Datorum. Quare per primam
Datorum erit data proportio linearum G M, M K, punctum
ambitum datum, qui ba&longs;is e&longs;t coni, quem linea K M, de&longs;cribit in reuolutio
ne axis G K O, &longs;uper polis G, O. cum enim data &longs;it K M, po&longs;itu, & magni
tudine,
&longs;is coni e&longs;&longs;e datum per &longs;imilem definitionem 5. definitioni Datorum. &longs;it
tem
non e&longs;t
falsò pingitur in figura; &longs;ed debemus ip&longs;um concipere tanquam erectum ad
angulos rectos cum prædicto &longs;emicirculo, necnon cum horizonte G K O. Iam &longs;i
punctum ip&longs;ius M, de&longs;cribit prædictum ambitum L M N. hunc ambitum
inquit Ariltot. linea K M, attinger,
Erit præterea &longs;ectio circunferentiarum ho
rizontis, & huius amb tus data, cuius extre
ma puncta e&longs;&longs;ent L, & N. &longs;i enim
in figura non &longs;olum horizontis
G K O, &longs;ed etiam circunferentiam (in qua
circunferentia e&longs;&longs;ent duo illa puncta L, & N,
vt in præ&longs;enti de&longs;criptione melius intelli ge
tur, in qua horizon G N O L, & ambitus
prædictus e&longs;t L M N, qui debet intelligi ele
uatus &longs;upra horizontem perpendiculariter)
tunc &longs;ectio ip&longs;ius mutua cum horizonte e&longs;&longs;et
trema lineæ K M, circumlatæ; & quemadmodum dabatur &longs;uperius punctum
M. eadem ratione ex Datis, dabitur punctum N, & L. quare etiam &longs;ectio
N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.
Illud nunc in memoriam
proportionem linearum G M, K M, non po&longs;&longs;e &longs;eruari in alijs lineis, quæ &longs;int
in eodem plano trianguli G M K, &longs;i ducantur ab ij&longs;dem punctis G, K. pote&longs;t
tamen &longs;eruari in alijs duabus, quæ cadant in prædictum ambitum, &longs;iue
cunferentiam
quod tamen tran&longs;eat per axem G K O,
&longs;upra dictum e&longs;t. Verumenimuerò ad quid probatio hæc?
non po&longs;&longs;e duas
alias lineas in eodem plano, &c.? exi&longs;timo Ari&longs;t.
idcircò hoc proba&longs;&longs;e, quia
&longs;i aliæ duæ lineæ habentes eandem rationem, po&longs;&longs;ent collocari in eodem
plano; e&longs;&longs;ent
æquales prioribus G M, M K, per quas videtur Iris, cum enim K R, &longs;it æqua
lis ip&longs;i K M, erit, & G M, æqualis ip&longs;i G R, per 7. 5. & in eius &longs;cholio. qua
re natura ageret tam per lineas breui&longs;&longs;imas cum ergò con&longs;tet non po&longs;&longs;e has
e&longs;&longs;e prioribus proportionales, &longs;ed maiorem, vel minorem, alteram illarum,
quàm &longs;it G M, &longs;equitur, quod non faciunt angulum æqualem angulo G M K,
&longs;ub quo videtur Iris,
habet enim Iris hunc angulum determinatum, ita vt &longs;ub maiori, vel mino
ri videri nequeat; ex 10. Bapti&longs;ta Porta. &longs;i autem punctum R, e&longs;&longs;et infra M,
angulus G R K, e&longs;&longs;et minor angulo Iridis G M K, &longs;i verò &longs;upra e&longs;&longs;et maior
eodem, quod vel ad &longs;en&longs;um patere pote&longs;t in quouis circulo,
longior euadat hæc tractatio. Fortè etiam addi pote&longs;t, quod alibi exi&longs;ten
te puncto R, quàm in M, non po&longs;&longs;ent anguli incidentiæ, & reflexionis e&longs;&longs;e
æquales, quæ cau&longs;a e&longs;&longs;et cur &longs;ub alio angulo, quam prædicto G M K, Iris
non appareret.
Prædicta omnia &longs;unt &longs;ecundum Ari&longs;tot. di&longs;cur&longs;um, & figurationem dicta,
nam &longs;ecundum veritatem po&longs;&longs;
æquales, nec tamen in eodem orbe, &longs;ed vnus &longs;upra
&longs;enti, &longs;i nubes e&longs;&longs;et vbi B D.
oculus in C, Sol in A. e&longs;&longs;ent
duo anguli A B C, A D C, æ
quales per 33. 3. qui tamen
non &longs;unt in gyrum con&longs;tituti,
po&longs;&longs;et igitur, per
que Sol Iridem efficere. atque
animaduer&longs;io h&ecedil;c videtur ma
gni
mon&longs;trationem
cum hinc v&longs;itatæ demon&longs;tra
tiones infringatur. Fortè confu giendum e&longs;t ad illud, quod Maurolycus, &
10. Bapti&longs;ta Porta ob&longs;eruarunt; debere
trum Iridis e&longs;&longs;e æqualem altitudini, &longs;iue &longs;emidiametro Iridis. Ita vt non &longs;o
ritur vt angulus in orbem con&longs;tituatur, ex quo Iris po&longs;&longs;it apparere. hæc à
nemine hactenus animaduer&longs;a placuit addere, vt ex ijs demon&longs;tratio Iridis
omnibus numeris aliquando ab&longs;olui po&longs;&longs;it, quod infra (ni fallor, fauente
Deo) præ&longs;tabimus.
Ibidem
M K, &longs;ic quæ D, ad B, maior autem quæ M G, ea quàm M K, quoniam &longs;uper ma
iorem angulum reflexio coni, maiori enim angulo &longs;ubtenditur trianguli M K G. Maior igitur e&longs;t & ip&longs;a D, ip&longs;a B. addatur igitur ad eam, quæ B, ea in qua F, vt
&longs;it quod D, ad B, quæ B F, ad D. Deinde quod F, ad K G, quæ B, ad aliam fiat,
quæ K P. & à P, ad M, copuletur quæ P M, erit igitur P. polus circuli, ad quem
lineæ, quæ à K, incidunt)
ctum in Iridis periphæriam, pergit deinceps inue&longs;tigare polum, & po&longs;tea
centrum eiu&longs;dem ambitus, vtraque autem exi&longs;tere in horizonte reperit, vt
hinc inferat Iridis portionem illam, quæ oriente Sole &longs;upra horizontem ap
paret, e&longs;&longs;e &longs;emicirculum, vt propo&longs;uerat. Differt autem polus circuli à cen
tro eiu&longs;dem circuli. polus e&longs;t punctum extra planum circuli, ex quo tamen
vt
toris e&longs;t idem, qui polus mundi:
trum æquatoris e&longs;t idem cum centro mundi, cum æquator per illud incedat.
Dicit
riori &longs;ecunda figura numeri 164. quam nunc iterum in&longs;picere opertet; ex
ponatur alia linea recta B D. quæ diui
datur in partes B, & D. proportionales
cum lineis K M, G M, per 10. 6. cum
ergo K M, &longs;it minor quàm G M, per 19.
primi, quia in triangulo G M K, oppo
nitur minori angulo, erit
nea F, ita vt &longs;it tota F B, tertia proportionalis ad duas B, & D, per 11. 6.
hoc ordine, vt F B, ad D. ita D, ad B. Deinde vt &longs;e habet F, ad K G. ita &longs;it
B, ad aliam, quæ &longs;it K P, in eadem figura per 12. 6. & à puncto P, ad M, iun
gatur recta P M. Dico P, e&longs;&longs;e polum circuli, quem dixi Iridis, & in quem li
neæ à K, procedentes turbinis formam effingunt, probatur autem ab Ari&longs;t.
in &longs;equentibus.
Ibidem
non enim &longs;it, &longs;ed aut ad minorem, aut ad maiorem ea, quæ P M, nibil emm differet. &longs;it enim ad P R. eandem ergo rationem G K, & K P, & P R, inuicem habebunt,
quam quæ F, B, D: quæ autem F, B, D, proportionales crant, quod quidem D, ad
B. quæ F B, ad D: quare quod quæ P G, ad P R, quæ P R, ad eam, quæ P K. &longs;i igi
tur ab ijs, quæ K G, quæ G R, & K R, ad R, coniungantur, coniunctæ hæ eandem
habebunt rationem, quam quæ G P, ad eam, quæ P R, circa eundem enim angulum
P, proportion aliter, & quæ trianguli G P R, & eius, qui K R P. quare & quæ G R,
ad eam quæ K R, eandem rationem habebit, quam & quæ G P, ad eam quæ P R,
habet autem & quæ M G, ad M K, eam rationem, quam quæ D, ad eam quæ B,
quare ambæ à punctis G K, non &longs;olum ad circun&longs;erentiam M N, con&longs;tituentur ean
aem habentes rationem, &longs;ed & alibi, quod quidem impo&longs;&longs;ibile)Primò enim &longs;ciendum in præ
mi&longs;&longs;a con&longs;tructione e&longs;&longs;e, vt F, ad G K, & B, ad K P, ita D, ad P M. nam &longs;i non
&longs;it eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiu&longs;dem D,
ad aliam maiorem, vel minorem ipfa P M. &longs;it ad minorem P R. nihil enim
refert &longs;iue dixeris habere eandem rationem ad minorem, &longs;iue ad maiorem,
ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. &longs;ed li
neæ F, B, D, erant proportionales
D, ad B: quare &longs;imiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. &longs;i igi
tur à punctis G, & K, figuræ nu. 164.
erit vt G R, ad K R, ita G P, ad P R. quia orta
quæ habent eundem angulum ad P. & latera proportionalia circa dictum
angulum. e&longs;t etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in
minori, ex con&longs;tructione, quare per 6. 6. erunt illa duo triangula æquian
gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia;
quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita
B, ad D. & ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R,
ad R G, intra eandem circunferentiam, & in eodem plano: quod e&longs;&longs;e im
po&longs;&longs;ibile &longs;upra o&longs;tendimus, hoc autem impo&longs;&longs;ibile, &longs;equitur &longs;i neges e&longs;&longs;e vt
F, ad G K; & B, ad K P, ita D, ad P M.
Ibidem
(&longs;imiliter enim demon&longs;ir abimus) palam e&longs;t, quod ad ip&longs;am
quare erit, quod quæ M P, ad P K, quæ P G, ad M P. Si igitur eo in quo P, polo
vtens, di&longs;tantia autem ea, in qua P M, circulns de&longs;cribatur, omnes angulos attin
get, quos reflexæ faciunt, quæ à K, G. &longs;i autem non, &longs;imiliter o&longs;tendentur eandem
babere rationem, quæ alibi, quam in &longs;emicirculo con&longs;tituuntur; quod quidem erat
impo&longs;&longs;ibile)
iorem quam P M, habet eam rationem, quæ e&longs;t ip&longs;ius F, ad G K, aut ip&longs;ius
B, ad K P. &longs;imiliter enim demon&longs;tratur ab&longs;urdum &longs;equi. palàm e&longs;t, quoniam
erit D, ad P M, vt prædictæ ad prædictas: quare componendo, & permu
tando, erunt tandem vt G P, ad P M, ita P M, ad P K, & ita G M, ad M K,
a&longs;&longs;ump&longs;imus enim in con&longs;tructione e&longs;&longs;e G M, ad M K, ita F B, ad D, & D, ad
B. quare cum &longs;it vt G M, ad M K, ita F B, ad D. & G P, ad P M. & P M, ad
K P; erunt per 11. 5. vt G M, ad M K. ita G P, ad P M. & P M, ad P K. &longs;i quis
igitur vtens puncto P, tanquam polo, & interuallo P M, circulum de&longs;cribat,
omnes angulos reflexionis attinget, quos faciunt lineæ productæ à K, & re
flexæ ab M, ad G. harum enim infinitam multitudinem debemus imaginari
à K, ad infinita puncta M, produci in ambitu illo con&longs;tituta,
&longs;i enim non attingat omnes illos angulos, &longs;equitur, vt &longs;upra, in eodem &longs;emi
circulo
quod e&longs;t impo&longs;&longs;ibile. Porrò &longs;ub angulo G M K, linearum G M, M K, Iris
apparet: quare apparebit etiam &longs;ub alijs omnibus, quæ à punctis G K, duci
po&longs;&longs;unt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum
non de&longs;inant in eundem
punctum imaginamur circumduci. Ex quibus pater P, e&longs;&longs;e polum Iridis, ex
quo per puncta M, vbi &longs;it reflexio, de&longs;cribitur arcus attingens omnes Iridis
reflexiones.
Ibidem
G K P, que à G, K, reflexæ ad id in quo M; in omnibus planis &longs;imiliter &longs;e habebunt,
& æqualem facient angulam, qui K M G, & quem etiam facient angulum, quæ
K P, & P M, &longs;uper eam, quæ G P, &longs;emper æqualis erit. Trianguli igitur &longs;uper eam,
quæ G P, æquales ei, qui G M P. con&longs;i&longs;tunt. horum autem perpendiculares ad idem
&longs;ignum cadent eius, quæ G P, & æquales erunt, cadunt ad
textus parte concludit Iridis portionem &longs;upra horizontem a&longs;tro
&longs;tentem e&longs;&longs;e &longs;emicirculum, hoc modo; &longs;i igitur imaginatione circumducas
&longs;emicirculum, in quo A, circa diametrum horizontis G K P, in hac circum
uolutione duæ lineæ G M, M K, in omnibus planis con&longs;titui po&longs;&longs;ibilibus cir
ca prædictam diametrum, quæ &longs;upra etiam fieri à triangulis infinitis dixi
mus, &longs;ucce&longs;&longs;iuè erunt; &longs;iue percurrent &longs;imiliter omnia illa plana, & facient
vbique angulum Iridis K M G, eundem: pariter duæ lineæ K P, P M, facient
vndique eundem angulum K P M. quare omnia triangula in predictis planis
imaginata, &
&longs;i igitur ab angulis ip&longs;orum, in quibus M, ductæ &longs;int perpendiculares ad la
tus G P, omnes cadent in idem punctum
quæ tamen cæteras omnes repre&longs;entabit,
G K
gulorum æqualium.
erit centrum ip&longs;ius. &longs;imilia dicta &longs;unt in Halone.
Cum ergò ip&longs;ius centrum
&longs;upra horizontem eminet, e&longs;&longs;e &longs;emicirculum, qui in figura notatur lineis
L M N. Atque hoc accidit Sole, vel Luna in horizonte exi&longs;tentibus; quod
erat primo loco demon&longs;trandum.
Porrò &longs;ciendum po&longs;&longs;e nos breuius polum prædictum inuenire, &longs;i nimirum
ad M, ducatur M P, faciens angulum K P M, æqua
lem angulo G M K, per 23. primi, erunt enim duo
triangula
P, e&longs;t communis, angulus verò M K P, e&longs;t æqualis
duobus G, & G M K, per 32. primi, ergo etiam
duobus ad M, &longs;iue toti G M P, & reliquus K M P,
reliquo, quare per 4.6. latera circa angulos æqua
les proportionalia erunt, & omologa G M, ad M K, ita G P, ad P M, quæ
æqualibus augulis &longs;ubtenduntur.
triangula Ari&longs;t.
in figura, de qua paulò ante dicebam. Verba illa
b quam in &longs;emicirculo constituuntur)
tran&longs;lata, nam Græcè &longs;ic,
e&longs;&longs;ent, quæ in alio circuli loco concurrunt.
Ibidem
axis autem &longs;it nunc in quo G P. Alia igitur omnia &longs;imiliter o&longs;tendentur vt & prius.
Polus autem circuli, in quo P, erit &longs;ub horizonte eo, in quo A C, eleuato puncto,
in quo G. in eadem autem & polus, & centrum circuli, & terminantis nunc ortum,
e&longs;t enim i&longs;te, in quo G P. Quoniam autem &longs;upra diametrum, quæ A C, quod K G,
centrum vtique erit &longs;ub horizonte priori eius, in quo A C, in linea K P, in quo
culo, in qua S T, (nam Q S T, &longs;emicir
culus est, nunc autem inter&longs;ectus e&longs;t ab
horizonte A C;
eleuato ip&longs;o Sole)
tionem &longs;ecundam nimirum Sole &longs;upra
horizontem elcuato, ambitum Iridis
e&longs;&longs;e minorem circuli portionem, &longs;iue
&longs;emicirculo minorem. &longs;it igitur in fi
gura &longs;uperiori, quam textui
tem
talis, &longs;upra quam Sol &longs;it eleuatus in
circulo altitudinis in loco G, axis au
rem coni, quem reflexè faciunt &longs;it
G K
pariter o&longs;tendi po&longs;&longs;unt, &longs;cilicet Iridem fieri tantum per lineas proportiona
les, & æquales lineis G M, M K, quia Iris videri nequit, ni&longs;i in tali, ac deter
minata reflexione, & angulo, vt initio &longs;uppo&longs;ui; & quia lineæ illis propor
tionales non po&longs;&longs;unt alibi con&longs;titui, quam in ambitu circulari, & in diuer&longs;is
planis, &longs;equitur, vt &longs;upra Iridem e&longs;&longs;e circularem M N L;
centrum
zontem in K, in hac vltima figura propter eleuationem Solis &longs;upra A C, in
G, &longs;equitur partem axis, in qua & quia (vt pater ex 64. 10. Vitell.) & P, polus, & centrum
crum K, circuli horizontis, cuius &longs;cilicet diameter e&longs;&longs;et A K S, & Sol, &longs;unt
in eadem linea G K
norem circuli portionem, quam &longs;it &longs;emicirculus &longs;upra horizontem eminere,
in qua po&longs;ui literas S L T, nam Q S L T R, e&longs;t &longs;emicirculus, cuius pars con
tenta inter duos arcus Q S, & T R, e&longs;t infra horizontem. debemus autem
hunc &longs;emicirculum, & hanc portionem ip&longs;ius S L T, extantem &longs;upra hori
zontem imaginari erectam e&longs;&longs;e, vt planum ip&longs;ius circuli faciat angulos re
ctos &longs;iue &longs;it perpendiculare cum axe G K P; &
modo fungi vice horizontis. &longs;ic enim &longs;ola portio S L T, appareret nobis,
&longs;etqueEx quibus 2. Ari&longs;t.
propo&longs;itio manife&longs;ta e&longs;t.
Ibidem
rius & polus, & centrum circuli erit)
rum Sole exi&longs;tente in meridie minimam
ratio autem e&longs;t, quia tunc G, &longs;iue Sol, e&longs;t alti&longs;&longs;imus &longs;upra horizontem, &
con&longs;equenter
culi Iridis portio ab&longs;condetur, & proinde minima apparebit, quod erat vl
timo Non me latet has Ari&longs;t.
figurationes e&longs;&longs;e apud Olym
piodorum nonnullis obiectionibus obnoxias, &longs;ed cum facilè dilui po&longs;&longs;int, &
etiam &longs;i non diluantur, &longs;aluetur tamen veritas Ari&longs;totelicæ demon&longs;tratio
nis, breuitati &longs;tudens, con&longs;ultò eas prætermitto.
Aduertendum præterea Vicomercatum inordinatè citare librum Dato
rum Euclidis, & peius verò
plicare conantur, cum manife&longs;tè illo innitantur.
Cæterum &longs;i quis breues, ac dilucidas harum rerum demon&longs;trationes re
quirat, is legat 74. 75. 76. propo&longs;itiones 10. Vitell. vel &longs;equentem no&longs;tram
de Iride additionem. ego enim longiorem hanc,
ctationem in gratiam textus illius, vt in&longs;tituti mei ratio po&longs;tulabat, per&longs;e
quutus &longs;um.
Ibidem
tumnale
noctio altero, ad æquinoctium alterum circa meridiem non fit Iris, can&longs;a est, quia
quæ ad Vr&longs;am &longs;ectiones omnes maiores &longs;unt &longs;emicirculo, & &longs;emper ad matere
autem e&longs;t occultum, paruum: quæ autem ad æquatoris meridiem &longs;ectiones, quæ qui
dem &longs;upra &longs;ectio, parua; quæ autem &longs;ub terra magna, & &longs;emper maiores, quæ lon
gius. quare in ijs, qui ad æ&longs;tiuas ver&longs;iones diebus propter magnitudinem &longs;ectionis,
antequam veniat G, ad medium &longs;ectionis, infra iam pœnitus fit P; propterea quod
longè di&longs;tat à terra meridies propter magnitudinem &longs;ectionis. In ijs autem diebus,
qui ad hyemates ver&longs;iones, quia non multŭ &longs;unt &longs;upra terram &longs;ectiones cir culorum,
contrarium nece&longs;&longs;arium fieri, modicum enim eleuato in quo G, in meridie fit Sol)
quærit cur po&longs;t æquinoctium autumnale v&longs;que ad vernum, hoc e&longs;t hyemali
tempore, Iris appareat etiam Sole meridiem occupante: reliquo autem
tempore æ&longs;tiuo, quod e&longs;t ab æquinoctio verno ad autumnale appareat tan
tum Sole vel in ortu, aut occa&longs;u exi&longs;tente, vel parum &longs;upra terram &longs;ublato. cau&longs;a autem huius refert in &longs;ectiones parallelorum circulorum, quos Sol
diurno motu inter
qui &longs;unt ad Vr&longs;am, ide&longs;t in parte &longs;phæræ Boreali, qui omnes &longs;unt inter æqua
torem, & tropicum Cancri; &longs;ectiones inquam horum circulorum, quæ &longs;unt
&longs;upra horizontem, maiores &longs;unt &longs;ectionibus infra horizontem depre&longs;&longs;is, &
&longs;emper eò maiores, quò propiores &longs;unt Cancro, ita vt magna yaldè &longs;it ea
portio, quæ e&longs;t &longs;upra terram, exigua verò admodum, quæ infra (intelligan
tur hæc in &longs;phæra obliqua, cuius polus eleuetur grad. 45. circiter) quare
quando a&longs;trum G, con&longs;cenderit meridiem, adeò P, polus Iridis, & etiam
centrum eius infra terram deprimitur, vt aut nihil, aut in&longs;en&longs;ibile quid de
Iridis ambitu &longs;upra terram eleuari po&longs;&longs;it, contrarium accidit in parallelis
meridionalibus, quia eorum &longs;ectiones &longs;uperiores &longs;unt &longs;emper inferioribus
minores, quapropter etiam &longs;i a&longs;trum ad meridiem eleuetur, parum tamen
attollitur, & con&longs;equenter centrum
de&longs;cendit, ac propterea etiam in meridie pars ip&longs;ius &longs;atis ma
gna con&longs;picitur. quæ omnia adhibita &longs;phæra materia
li, eaque a&longs;tronomicè ad &longs;uam eleuationem
accommodata, nullo negotio li
cebit intueri.
Cvm &longs;uperior Ari&longs;tot. de Iride tractatio ob&longs;cura, ac tricis pluribus
impedita eua&longs;erit,
ex parte vacilient, vi&longs;um e&longs;t breuiter expeditam,
ip&longs;ius apponere demon&longs;trationem. Cum igitur in cœle&longs;ti arcu
duo poti&longs;&longs;imum &longs;int, quæ &longs;ui admiratione
templationem alliciant, colores, &longs;cilicet, & figura: nos mirabilem illam co
lorum triadem, tanquam alienam, phy&longs;icis relinquentes, de figura ip&longs;ius iu
re mathematico di&longs;&longs;eremus: rotunditatis &longs;cilicet Iridis cau&longs;am opticis ra
tionibus venabimur, cur aliquando &longs;emicirculus, aliquando &longs;emicirculo mi
nor appareat. vt igitur ordine procedamus.
Primo loco aduertendum e&longs;t tria ad Iridis vi&longs;ionem e&longs;&longs;e nece&longs;&longs;aria, So
lem, oculum, & nubem tenuem, ac ro&longs;cidam, quæ &longs;cilicet minutis guttulis
iam &longs;cateat; hac enim ratione guttulæ illæ innumera erunt veluti parua
&longs;pecula, quæ lumen Solis ob paruitatem imperfecto quodam modo repre
&longs;entare po&longs;&longs;int, ex tali enim repre&longs;entatione Iris apparet. quæ tria debent
e&longs;&longs;e ita di&longs;po&longs;ita, vt Sol, oculus, & centrum Iridis &longs;int in eadem recta linea
con&longs;tituta,
vt in prima figura videre e&longs;t, in qua Sol vbi A, oculus in C. nubes verò
G H L E, in qua apparet Iris in arcu E B F, quem debemus concipere e&longs;&longs;e
in rece&longs;&longs;u, vt pictores aiunt, depictum. i.
non in hoc &longs;itu, & ouali figura, &longs;ed
e&longs;&longs;e perfectè &longs;emicircularem,
&longs;it citra chartam eleuata,
linea horizontali A C L, in quo &longs;itu oculo C, totus ex oppo&longs;ito directè &longs;pe
ctaretur, non aliter ac Iridem ip&longs;am con&longs;picere &longs;olemus. Quod autem ne
ce&longs;&longs;aria &longs;it nubes ro&longs;cida, pulcherrima hac experientia
in Sole po&longs;iti ore aquam efflantes leui a&longs;pergine aerem Soli, ac nobis ad
uer&longs;um irroremus, actutum Iridis arcum guttulis illis, quamuis volitanti
bus inhærcntem &longs;umma voluptate &longs;pectabimus. Quod præterea oculus no
&longs;ter, cum Iridem videmus, medius &longs;it inter Solem, & Iridis centrum, expe
rimento diuturno, manife&longs;tum e&longs;t.
Secundò, notandum e&longs;t, arcum per reflexionem fieri: quod quidem pri
mo eadem experientia, qua præcedens conclu&longs;io confirmatur: deinde, quia
Iridem &longs;emper in oppo&longs;ita Soli, ac nobis parte
in eadem figura o&longs;tenditur, quod aliter quàm per reflexionem fieri nequit.
Tertiò, &longs;ciendum e&longs;t ex Maurolyco, & 10. Bapti&longs;ta Porta, tantam e&longs;&longs;e di
&longs;tantiam C D, ab oculo ad centrum arcus, quanta e&longs;t altitudo, &longs;eu &longs;emidia
meter D B, ob&longs;eruarunt enim ip&longs;i angulos D C B, & C B D, e&longs;&longs;e &longs;emirectos,
& proinde æquales, & con&longs;equenter duo latera C D, D B, trianguli C D B,
per 6. 1. æqualia &longs;unt.
Quartò, con&longs;iderandum e&longs;t lineas A B, A D, ob maximam Solis ab Iride
di&longs;tantiam in&longs;en&longs;ibiliter differre; & ideò &longs;upponi po&longs;&longs;unt æquidi&longs;tantes,
quare angulus A B C, qui æqualis e&longs;t alterno B C D, &longs;umi pote&longs;t ab&longs;que vllo
errore pro &longs;emirecto. hic autem angulus A B C, dicitur angulus reflexionis
Iridis, &longs;ub tali enim reflexione lumen Solis occurrens nubi in B, reflectitur
ad oculum C.
Quintò, &longs;equitur ex prædictis arcum videri &longs;emper &longs;ub &longs;tato, ac determi
nato reflexionis angulo, &longs;cilicet &longs;ub &longs;emirecto, quod etiam probari pote&longs;t ex Ari&longs;t.
quia nimirum videmus arcum apparere
con&longs;imiliter in ambitu circulari, ergò nece&longs;&longs;ariò apparebit
lo ambitu per con&longs;imilem reflexionem, &longs;iue per æquales reflexionis angulos,
pro quibus omnibus vnus cernitur in figura angulus A B C.
Sextò, ad Iridis vi&longs;ionem, præter ea, requiri aeris rorantis multiplica
tionem; &longs;icuti enim nebulam videre nequimus, ni&longs;i aer exhalatione illa in
fectus multus &longs;it ante oculum no&longs;trum: &longs;ic etiam exi&longs;timo ad Iridis appari
tionem, opus e&longs;&longs;e plurima nube rore&longs;cente, vt ex multiplicatione guttula
rum, quarum aliæ po&longs;t alias &longs;int, totus tandem Iris appareat. quia paucæ
guttulæ, etiam &longs;i quælibet illarum aliquid Iridis efficeret, ob paruitatem
tamen illarum, nulla arcus figura &longs;pectaretur. Quod &longs;i ante oculum pluri
mæ &longs;int in toto aere aliæ po&longs;t alias, tunc &longs;e mutuò iuuantes, obiectum &longs;atis
&longs;en&longs;ibile, quoc Iris e&longs;t, efficere po&longs;&longs;unt. Adde, quod etiam ex tali guttula
rum multiplicatione, aer opacatur, quæ opacatio plurimum iuuat ad Iri
dem &longs;pectandam.
Septimò, Iridis rotundationis cau&longs;am ex præmi&longs;&longs;is con&longs;tare poti&longs;&longs;imum
ex duabus. primò, ex angulo reflexionis determinato, qui videlicet &longs;it ferè
&longs;emirectus. &longs;ecundò, ex paribus di&longs;tantijs C D, D B, huiu&longs;modi enim plures
anguli, qui ad Iridem &longs;unt nece&longs;&longs;arij (debent enim &longs;ingulæ Iridis partes &longs;ub
huiu&longs;modi angulo repre&longs;entari) non po&longs;&longs;unt aliter quàm in gyrum
cumuerti circa lineam horizontalem A C L, fixam, tanquam circa axem. in
hac enim conuer&longs;ione angulus Iridis B, de&longs;cribet circulum, qui erit Iris, &
pertr an&longs;ibit omnes angulos, qui in tali Solis, oculi, ac nubis &longs;itu, arcum ef
ficere &longs;unt idonei.
Sed contra prædicta de angulo Iridis determinato eadem nobis obijcies,
quæ nos &longs;upra ad finem numeri 164. Ari&longs;t. & alijs obiecimus, plures
rum
qui non &longs;int in eodem orbe con&longs;tituti, in quo &longs;unt omnes anguli B. Iridem
reflectentes,
prædictam Iridis altitudinem non e&longs;&longs;e, vti diximus, determinatam, cum
po&longs;&longs;it angulus B, alios &longs;ibi æquales tam &longs;upra, quàm infra habere, qua ra
tione deberet etiam Iris, & altius, & inferius apparere.
Huic dubitationi re&longs;pondeo, quod quamuis huiu&longs;modi plures anguli
æquales fiant, non tamen Iridis generationi ob&longs;tant, quinimò ad eam valdè
nece&longs;&longs;arij &longs;unt;
tæ figuræ num. 164. quæ modo in&longs;picienda e&longs;t, vt &longs;unt in ea anguli A D C,
A B C; quæ circunferentia ob &longs;ui circuli immen&longs;itatem ad &longs;en&longs;um e&longs;t in&longs;tar
lineæ rectæ, fit vt omnes illi anguli tàm qui &longs;upra B, quàm qui infra &longs;unt,
&longs;int quoad &longs;en&longs;um in eadem recta C D B, ante vi&longs;um proten&longs;a,
apparet in D, & in B, &c. ob medij rorantis multiplicationem vnam
oculo Iridem repre&longs;entet. locus tamen, in quo apparet, & vbi e&longs;t angulus
B, qui propriè Iridis appellatur, e&longs;t in tanta di&longs;tantia à centro arcus, quan
ta e&longs;t ab eodem centro ad oculum, vt &longs;upra dictum e&longs;t.
Quod verò alibi extra circunferentiam illius circuli, poni nequeat angu
lus æqualis angulo B, præ&longs;entis figuræ, qui re&longs;lectat ad C. patet &longs;ic, &longs;it enim
angulus A N O, &longs;emirectas, & ideò æqualis angulo B, erunt ergo B C, N O,
parallelæ, quare non corcurrent
O, quæ propterea oculo in O, eodem modo o&longs;tendi
pote&longs;t,
angulo B, qui oculo C, Iridem valeat o&longs;tendere. Ex quibus &longs;atis patefacta
e&longs;t cau&longs;a rotunditatis arcus, angulus &longs;cilicet determinatus cum di&longs;tantia
rum C D, D B, paritate, necnon cum medij rorantis &longs;ufficienti multiplica
tione. Ex his etiam Iridis definitio in hunc modum concinnari pote&longs;t, Iris
e&longs;t arcus multicolor in nube rorida, ex radiorum Solis, aut Lunæ reflexio
ne &longs;ab &longs;tatuto angulo effulgens.
Octauo loco Problemata nonnulla re&longs;oluemus.
Cur oriente, aut occumbente Sole, Iris &longs;emicirculus e&longs;t?
Cau&longs;a huius hæc e&longs;t; &longs;upra enim dictum e&longs;t, in omni Iridis appari
tione tria hæc, Solem, oculum, & Iridis centrum e&longs;&longs;e in eadem re
cta linea, v. g. in linea A C D, præcedentis figuræ, cum igitur Sol
tam oriens, quam occidens &longs;it in horizonte, v. g. in A, horizontis
puncto, &longs;imiliter oculus &longs;it in C, horizontis centro, con&longs;ectarium e&longs;t, cen
trum etiam Iridis D, e&longs;&longs;e pariter in horizontis &longs;uperficie, quare &longs;ecabitur
ab horizonte per centrum, vnde etiam &longs;equitur ip&longs;ius Iridis portionem
E B F, quæ &longs;upra horizontem extat e&longs;&longs;e &longs;emicirculum. Quod &longs;i horizon non
ob&longs;taret,
An
Maior quidem, imò etiam integer circulus, &longs;ed ab oculo in &longs;ummitate
montis con&longs;tituto,
hac &longs;ecunda figura cernitur, vbi euecto Sole ad locum E, &longs;upra horizontem
A B, poterit oculus in vertice montis C, po&longs;itus Iridem F G H I, comple
tam videre, quia infra lineam E C D, in qua exi&longs;tunt Sol, oculus, & Iridis
centrum, nihil e&longs;t ad partes D, vbi nubes irrorat, quod Iridis apparitioni
&longs;it impedimento.
Cur quanto Sol altior e&longs;t, tanto inferior,
culo minor appareat Iris?
Qvia eleuato Sole ad E, vt in hac tertia figura, nece&longs;&longs;ario centrum Iri
dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D.
Sol E, oculus C,
portionem F G H, &longs;upra horizontem extantem, &longs;emicirculo minorem e&longs;&longs;e.
Cur Iris in&longs;equentes fugit, fugientes verò in&longs;equitur?
Pvlcherrimum i&longs;tud phænomenon primus omnium Philippus Mendæus
Platonis di&longs;cipulus, ob&longs;eruauit; Cuius ratio e&longs;t, quia arcus non ni&longs;i &longs;ub
determinato angulo, di&longs;tantijs etiam illis paribus, ac tandem idone a a&longs;per
gino&longs;æ nubis multiplicatione &longs;pectatur; quapropter &longs;i quis per aerem to
tum
fuerint Iris apparebit: quod &longs;i in aperta planitie obequitans arcu con&longs;pe
cto, additis equo calcaribus citatum cur&longs;um ad eum direxerit, fugientem
ante &longs;e Iridem &longs;umma cum iucunditate mirabitur.
Ex dictis pr&ecedil;tere a patet, &longs;impliciter nimis eos hallucinari, qui exi&longs;timant
in plana, aut concaua nubis &longs;uperficie Iridem tantummodo apparere po&longs;&longs;e.
Curlunares Irides fiunt rariores?
Qvoniam iuxta plenilunia tantum, cum &longs;cilicet Luna plurimo lumine
abundat, quod Iridem efficere debet, contingunt: præterea quia cum
lunare lumen debile &longs;it, ni&longs;i aliæ cau&longs;æ perfectæ admodum concur
rant, quod rarò accidit, Iris nullo modo effulgere valet. Hactenus de Iri
dis figura &longs;it &longs;atis.
Textus
ca&longs;u, & nec &longs;upra Solem, nec infra, &longs;ed ex lateribus, nec propè admo
dum, nec procul omninò. propinquam enim concretionem Sol di&longs;&longs;oluit:
&longs;i autem procul ab&longs;it, a&longs;pectus non reflectetur, &longs;i enim à paruo &longs;peculo
procul protenditur imbecillus fit. quare, & Coronæ è regione Solis non fiunt.
&longs;i igi
tur &longs;upra fuerit, & proxima; eam Sol di&longs;&longs;oluet: &longs;i v
quam vt reflecti po&longs;&longs;it in Solem non-incidet; à latere autem fieri pote&longs;t, vt &longs;pecu
lum ita distet à Sole, vt non &longs;oluatur, & a&longs;pectus totus ad eum perueniat, eo quod
ad terram dum fertur, qua&longs;i per immen&longs;um feratur, peruenire nequeat. &longs;ub Sole
verò non fit, quia cum ad terram propius acce&longs;&longs;erit à Sole di&longs;&longs;oluitur, cum medium
cœli tenuerit a&longs;pectus di&longs;trahitur. omninò ne à latere quidem, Sole medium cœli
tenente, efficitur, quia a&longs; pectus &longs;ub terram non fertur, quare exiguus ad &longs;peculum
producitur, & qui reftectitur pror&longs;us imbecillis redditur)
concretionem Sol di&longs;&longs;oluit)
cter loquar) admodum debiles. præ&longs;ens ea e&longs;t, vt Parelium non fiat propè
Solem, quia illa nubis concretio, quæ Parelio nece&longs;&longs;aria e&longs;t, nequit adeo So
li propinqua e&longs;&longs;e, quia nimirum Sol ob propinquitatem eam di&longs;&longs;olueret; &longs;ed
quis non videt eam nubem, quam vulgò exi&longs;timamus e&longs;&longs;e Soli propinquam,
&longs;eu qua&longs;i inter nos, & Solem tantum, imò etiam minus aliquando à Sole ve
rè di&longs;tare, quàm alia, quàm vulgò remotiorem à Sole putabimus? præte
rea omnes nubes no&longs;tri horizontis re vera æquidi&longs;tare à Sole certum e&longs;t, ob
maximam enim Solis di&longs;tantiam totus no&longs;ter horizon phy&longs;icus e&longs;t in&longs;en&longs;i
bilis quantitatis ad Solem, & vnius puncti vicem gerit.
Ibi verò
appareat in nube à Sole valde remota &longs;ecundum vulgarem æ&longs;timationem,
vnde vulgarem etiam rationem affert, ait enim, nubem illam e&longs;&longs;e veluti &longs;pe
culum Solis repre&longs;entatiuum, &longs;peculum autem tàm longè à Sole po&longs;itum,
reddi debile, & proptereá non po&longs;&longs;e Solis imaginem referre: Verùm ratio
hæc nulla e&longs;&longs;e videtur, quis enim ignorat non propterea e&longs;&longs;e remotius à So
le, quamuis maiorem habere videatur à Sole lateralem di&longs;tantiam, vt pau
lò ante dixi? Eandem rationem illi dubitationi accommodat, cur
deatur &longs;upra Solem, quamuis non ei quadret, pote&longs;t enim aliqua nubes vi
gignitur. Ait po&longs;tea
à latere vicina, in di&longs;tantiam à Sole refert: &longs;ed quæ dudum dicta &longs;unt, i&longs;tud Verba illa
feratur, peruenire nequeat)
&longs;unt reliqua, præ&longs;ertim quæ ibi
pius acce&longs;&longs;erit)
inanis e&longs;t reddit; nunquid enim non po&longs;&longs;umus tam infra Solem, quàm &longs;upra
ita &longs;peculum accommodare, vt Solem no&longs;tris vi&longs;ibus remittat? huic certè
Optice tota repugnat. Cum igitur Mathematica ratione hæ rationes non
con&longs;i&longs;tant, alias alij excogitent. Mirum tamen e&longs;t, omnes, quos viderim
commentatores, eas tanquam optimas admittere.
Tex. 11.
ad cogno&longs;cendas cau&longs;as accidentium &longs;ub&longs;tantijs: &longs;icut in Mathemati
cis quid rectum, & quid obliqaum, aut quid linea, & planum, ad co
gno&longs; cendum quot rectis, trianguli anguli &longs;unt æquales)
quodque
mentarijs ip&longs;arum; quamuis autem ibi non definiatur
in genere, definitur tamen linea recta, & obliqua, & plana &longs;uperficies, &longs;iue
planum, ex quibus facilè definitio recti, & obliqui colligi pote&longs;t: quæ defi
nitiones nece&longs;&longs;ariæ &longs;unt ad cogno&longs;cendum quot rectis angulis æquales &longs;int
tres anguli cuiufuis trianguli. vide quæ de hac æqualitate &longs;crip&longs;i lib, primo
Priorum, &longs;ecto 3. cap.
1.
Tex. 13.
que ip&longs;am &longs;eparari: &longs;i verò nulla e&longs;t propria ip&longs;ius non vtique erit &longs;eparabilis. &longs;ed
&longs;icut recto in quantum rectum multa accidunt, vt tangere æ
dum punctum, non tamen tanget hoc, rectum ip&longs;um &longs;eparatum: in&longs;eparabile enim,
&longs;i quidem cum corpore quodam &longs;emper e&longs;t)
neam rectam, duo quælibet puncta
pinqua in circuli ambitu a&longs;&longs;umpta coniungentem
cadere intra circulum. v. g. puncta A B, quantum
uis &longs;ibi inuicem propinqua fnerint, attamen &longs;i line a
A B, ea coniungat, ip&longs;a cadet intra circulum, &
veluti chorda &longs;ubtendet arcum A B, quantulum
cunque. ex qua demon&longs;tratione colligitur in corol
lario eius lineam rectam tangentem circulum ip
&longs;um in vnico puncto tangere. v. g. rectam C D, tan
gere circulum in puncto E. &longs;i enim dixeris tangere
in duobus admodum propinquis, vt in E F, tunc non erit amplius tangens,
&longs;ed &longs;ecans, quia vt modo dixi, pars lineæ rectæ, quæ
quia contra hypothe&longs;im, cum &longs;upponamus illam &longs;olùm tangere, non autem
&longs;ecare circulum. Ex hac Euclidis doctrina Theodo&longs;ius primo &longs;phæricorum,
propo&longs;itione 3. probat planum, &longs;iue &longs;uperficiem planam tangere &longs;phæram
in vnico puncto, vt hoc loco innuit Philo&longs;ophus. probat autem hac ferè ra
tione. &longs;it &longs;phæra A B C, quæ tangat quodpiam planum
in duobus punctis A, B, &longs;i fieri pote&longs;t. per quæ duo pun
cta intelligatur ducta recta linea A B, intelligatur
circulus A B C, qui &longs;ecet &longs;phæram per centrum C. &
per puncta A, B, ergo ex demon&longs;tratis ab Euclide li
nea A B, quæ coniungit puncta A B, cadet intra prædi
ctum circulum; &longs;ed linea hæc e&longs;t in plano tangente ex
&longs;uppo&longs;itione, circulus verò in &longs;phæra; ergò cum linea
cadat intra circulum, cadet etiam nece&longs;&longs;ariò planum
in quo e&longs;t linea, & cum linea cadat intra circulum, cadet etiam nece&longs;&longs;ariò
intra &longs;phæram;
planum &longs;ecat &longs;phæram, non autem tangit, quod e&longs;t ab&longs;urdum, quia contra
hypothe&longs;im, &longs;upponunt autem Mathematici, entia hæc mathematica e&longs;&longs;e
perfecta, qualia in &longs;ublunaribus fortè non reperiuntur; ænea enim &longs;phæra
nulla erit perfectè rotunda, vel planum aliquod perfectè complanatum, vt
ip&longs;i &longs;upponunt, eò quod materiæ imperfectio, ac ruditas id nequaquam pa
tiatur. quare cum huiu&longs;inodi entia non reperiantur ab&longs;tracta ab impura hac
materia, nullum erit inquit Ari&longs;t.
ab&longs;tractum planum, quod po&longs;&longs;it mathe
maticè,
nece&longs;&longs;aria &longs;unt mathematica ad huius loci ex quibus ea etiam,
quæ ad phy&longs;icum &longs;pectant manife&longs;ta fiunt, nimirum &longs;icut entia mathemati
ca à materia non exi&longs;tunt &longs;eparata, quia &longs;ic nullam haberent operationem;
ita etiam anima, &longs;i nullam habet propriam operationem non exi&longs;tet à cor
pore &longs;eparata.
Tex. 12.
oftendere, &longs;icut plures definitionum dicunt, &longs;ed & cau&longs;am ine&longs;&longs;e, & ap
parere. nunc autem, vt conclu&longs;iones rationes definitionum &longs;unt, vt quid
tetragoni&longs;mus? æquale altera parte longiori rectangulum æquilaterum
e&longs;&longs;e, talis autem definitio ratio conclu&longs;ionis. dicens autem, quod tetragoni&longs;mus e&longs;t
medij inuentio rei cau&longs;am dicit)
mittit duplicem e&longs;&longs;e definitionem, alteram &longs;cilicet, quæ explicat &longs;olum rei
e&longs;&longs;entiam, quam dicunt formalem definitionem; alteram verò, quæ præte
rea explicat etiam rei cau&longs;am, quam dicunt cau&longs;alem definitionem: vtram
que autem exemplo Geometrico explicat.
In cap.
igitur de relatione plura &longs;crip&longs;i de tetragoni&longs;mo, &longs;eu qua dratio
ne circuli, quæ huc &longs;pectant. propterea nunc tantum propria huius loci
clarandaloquitur igitur hic Philo &longs;ophus non de quadratione circuli,
&longs;ens figura A B C D, cuius quadrandæ ratio e&longs;t huiu&longs;modi. per 13. 6. inue
niatur recta linea media proportionalis inter
duo latera figuræ A B, B C,
quenti figura. e&longs;&longs;e autem mediam proportio
nalem nihil aliud e&longs;t quam ita e&longs;&longs;e A B, ad B D,
&longs;icut B D, ad B C.
nalis, quia in hac habitudine medium locum obtinet. quadratum autem li
neæ B D, æquale e&longs;t rectangulo dato A B C D, per 17.6. Inuentio porrò hu
ius mediæ proportionalis, quia facilis e&longs;t, & &longs;citu iucunda, eam &longs;ic habeto.
accipe duo latera A B, & B C,
guli,
ctam con&longs;tituant A C, vt apparet in figura; de
inde diui&longs;a tota A C, bifariam in E, facto cen
tro in E, de&longs;cribe &longs;emicirculum &longs;uper lineam
A C, demum à puncto B, in quo duo latera con
iunguntur, erigatur linea perpendicularis
ad periphæriam, quæ &longs;it B D. hæc enim B D, e&longs;t media proportionalis inter
latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo
que obtinet B D, ad B C. Quadratum igitur huius B D, hoc e&longs;t quadratum,
cuius quatuor latera &longs;iut æqualia lineæ B D, quale e&longs;t præ&longs;ens, æquale erit
dato &longs;uperiori rectangulo A B C D,
acta erit quadratio, &longs;eu tetragoni&longs;mus dati quadrilateri
A B C D. Vides igitur, qua ratione quadratum con&longs;ti
tuatur æquale dato quadrilatero; & qua rationem inuen
tio illius mediæ proportionalis &longs;it cau&longs;a quadraturæ re
ctanguli, & proinde &longs;i quis dicat quadrationem hanc e&longs;&longs;e
effectionem rectanguli æquilateri, ide&longs;t quadrati, æqualis dato quadrilate
ro, hic definitionem formalem &longs;olum afferet: quæ definitio, vt dixit in Lo
gicis, e&longs;t in&longs;tar conclu&longs;ionis. &longs;i quis verò dicat tetragoni&longs;mum hunc quadri
lateri dati e&longs;&longs;e mediæ prædictæ inuentionem cau&longs;alem afferet definitionem,
cum rei cau&longs;am dicat. Aduerte 10. Grammaticum immeritò accu&longs;are Ale
xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor
tionalis tradi in 2. Elem.
nam verè in 14. 2. traditur talis inuentio, quam
uis enim ibi nulla fiat expre&longs;&longs;a mentio huiu&longs;modi mediæ, in ip&longs;a tamen e
reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura
14. prædictæ, quæ eadem e&longs;t cum figura 13. 6. qua docemur prædictam in
uentionem.
Tex. 86.
multo parùm; non igitur velox e&longs;t acutum, graue autem tardum, fed fit illius qui
dem propter velocitatem motus huiu&longs;modi, huius autem propter tarditatem)
quæ de hac re primo topic. cap.
13. dicta &longs;unt, illa enim omnia in hunc lo
cum quadrant. Verum occurrit illa dubitatio; quod cum Ari&longs;t.
ibi dicat
tia dicere videtur. cui dubitationi &longs;ic occurrendum; vt dicamus ibi Philo
&longs;ophum dicere vocem acutam e&longs;&longs;e velocem, quatenus acumen vocis oritur hic verò di&longs;tinguere acutum à ve
loci, tanquam effectum à cau&longs;a.
Tex. 159.
habet, vt apparet &longs;ol vnius pedis, per&longs;ua&longs;um autem e&longs;t, eum maiorem e&longs;&longs;e babitata)
habitata, ide&longs;t terra habitata. Vide, quæ cap.
3. &longs;ummæ 1. primi Meteor.
Item capite 5. &longs;ummæ 2. de Solis magnitudine &longs;crip&longs;i, ea enim huic loco
abundè &longs;atisfaciunt.
Tex. 21.
men&longs;uratione diametri, & co&longs;tæ &longs;cripta &longs;unt lib.
1. Priorum, cap.
23. vnde &longs;atis huic loco fieri pote&longs;t.
Tex 25.
&longs;tratur &longs;i cut priuatio)
terioris lineæ
etiam, & &longs;uperficiem, nam quemadmodum punctus oritur ex diui&longs;ione li
neæ, ita linea ex diui&longs;ione &longs;uperficiei, & &longs;uperficies ex di& quamuis punctum, linea, &longs;uperficies, &longs;int indiui&longs;ibilia, mon&longs;trantur ta
men quatenus &longs;unt priuationes, &longs;eu negationes, illud vlterioris lineæ, i&longs;ta
vlterioris &longs;uperficiei, hæc tandem vlterioris corporis.
Tex. 32.
re & p
A, ad C, ita B, ad D, hunc argumentandi modum à permutata proportio
ne explicaui in primo Po&longs;ter. cap.
5. tex. 13. dicitur etiam alterna ratio;
& definitur ab Euclide definitione 12, 5.
Cap, 6.
omnem cantum, di&longs;t antia enim eius ad extremos &longs;onos latet)
Die&longs;is apud Mu&longs;icos explicatum e&longs;t primo Po&longs;ter. tex. 38. cum
autem Die&longs;is &longs;it minima di&longs;tantia, &longs;eu vt loquuntur Mu&longs;ici, mini
mum
tremos &longs;onos non exaudiatur, quemadmodum nec minima particula alicu
ius magni corporis à longè vi&longs;i
Cap. 8.
vinum non temperatum, quàm temperatum; & mel, & colorem, & neten &longs;olam. quàm in diapa&longs;on, quia ob&longs;curant &longs;e inu
in mu&longs;icis in&longs;trumentis omnium chordarum acuti&longs;&longs;ima, cuiu&longs;modi apud
nos e&longs;t, quam vulgò canto appellant. Hypate verò e
graui&longs;&longs;ima, qualis e&longs;t ea, quam modo Ba&longs;&longs;o vocant. hæ duæ &longs;imul pul&longs;atæ
edebant conionantiam, quæ Diapa&longs;on dicitur, & vulgò octaua. ex quibus
&longs;en&longs;us verberum Ari&longs;t.
manife&longs;tus e&longs;t.
Eodem cap.
&longs;on, explicaui in primo Po&longs;ter. tex. 1. Diapente verò e&longs;t con&longs;onantia ex duo
bus &longs;onis coale&longs;cens, quorum proportio &longs;it vt
3. ad 2. quæ dicitur &longs;e&longs;quialtera. v. g. &longs;int duæ
chordæ æqualis cra&longs;fitiei,
tamen habeat ad alteram proportionem &longs;e&longs;
quialteram, vt in figura apparet; &longs;i &longs;imul pul
&longs;entur, edent con&longs;onantiam Diapente. vulgò autem quint
Cap. 1.
&longs;ic meminit tres angulos cuiu&longs;uis trianguli &longs;imul &longs;umptos æqua
les e&longs;&longs;e duobus angulis rectis &longs;imul &longs;umptis. lege annotata primo
Po&longs;ter. &longs;ecto 3. cap.
1.
Cap. 3.
vt mathemata)
petuam de mon&longs;trationum connexionem, qua Geometræ omnes, & præci
puè Euclides opera &longs;ua ab initio ad finem v&longs;que, diuino planè ingenij acu
mine deduxerunt.
Cap. 2.
mouetur apparent quælibet, &longs;ed etiam cum &longs;en&longs;us ip&longs;e mouetur, &longs;i eodem
modo moueatur, quemadmodum à &longs;en&longs;ibili. dico autem velut terra vi
detur nauigantibus moueri, dummodo vi&longs;us ab alio)
cur nauigantibus videatur terra ip&longs;a moueri, ac retrocedere, non autem
ip&longs;i nauigantes, quin potius ip&longs;i fibi &longs;tare videantur. cau&longs;am igitur eam e&longs;
&longs;e ait, quia ex motu nauis, terra ip&longs;a manente, accidit, vt eodem modo im
mutetur &longs;en&longs;us vi&longs;us, ac &longs;i terra ip&longs;a moueretur, vi&longs;us verò quie&longs;ceret. At cur eodem modo afficitur &longs;en&longs;us?
Per&longs;pectiuirationem e&longs;&longs;e dicunt, quia
ea, quæ circa oculum &longs;unt, vt nauis, & ea, quæ in naui &longs;unt, non mutant &longs;i
tum re&longs;pectu oculi, quemadmodum facerent, &longs;i nos ip&longs;i &longs;ine naui progrede
remur. arbores autem, & reliqua, quæ in terra &longs;unt, variant &longs;itum re&longs;pectu
oculi, non &longs;ecus, ac &longs;i ip&longs;æ arbores retro deferrentur. propterea igitur vi&longs;us
tunc arbores remeare iudicat, quia quæ circa oculum &longs;unt re&longs;pectu ip&longs;ius
oculi non mouentur, &longs;iue non variant &longs;itum ad ip&longs;um; ex variatione enim
&longs;itus rei re&longs;pectu oculi, percipimus cuiu&longs;uis rei localem motum.
Cap. 3.
&longs;olum app trebit, &longs;ed etiam putabitur duo, quod e&longs;t vnum. Si verò non lateat appa
rebit quidem, non putabitur tamen)
cum aliquod obiectum intuentes, interim digito alterum oculum &longs;ur&longs;um
pellimus, ita vt oculi propterea varient &longs;itum re&longs;pectu obiecti, &longs;iue non eo
non amplius concurrunt &longs;imul in rem vi&longs;am. Vnde &longs;equitur &longs;peciem rei in
tentionalem oculis vario &longs;itu affectis imprimi, ac proinde eam eundem &longs;i
tum in vtroque oculo minimè obtinere, &longs;ed ea, quæ oculo à &longs;uo naturali
&longs;latu dimoto accidit ab altera alterius oculi differt; quapropter vario
ctiam modo, duplici nimirum, obiectum repre&longs;entant. atque hæc
ip&longs;a cau&longs;a e&longs;t, cur illud, quod vnum tantum e&longs;t, duo tamen
emoto oculorum altero, videatur. Vide Alhaze
num lib.
3. propo&longs;it. 11. & 12. & infra
Problem. 7. &longs;ectionis 31.
EX PRIMO
METAPHYSICAE.
Capite 1.
enim gens Sacerdotum vacare permittitur)
bilis Mathematicarum origo, cum ab Aegyptiorum Sacerdoti
bus te&longs;te Philo&longs;opho fuerint adinuentæ, quibus occa&longs;ionem præ
buit anniuer&longs;aria agrorum ob Nili innundationem, diui&longs;io: cum enim iam
perplures dimetiendorum agrorum rationes repertæ fui&longs;&longs;ent, Sacerdotes
ip&longs;i, quibus per otium licebat, illarum praxium demon&longs;tr ationes cœperunt
perue&longs;tigare,
quæ deinde ij&longs;dem ad res a&longs;tronomicas per&longs;crutandas
ratione reliquas etiam in mathematicas inciderunt.
Cap. 2.
&longs;unt cau&longs;am
Automata. erant autem Automata apud veteres Gr&ecedil;cos machinæ qu&ecedil;dam,
quæ à Mathematicis Mechanicæ artis occultis quibu&longs;dam ingenijs, ea arte
con&longs;truebantur, vt à &longs;eip&longs;is de loco ad locum, ac &longs;i viuæ e&longs;&longs;ent &longs;pontè pro
grederentur; vnde, & automata, qua&longs;i &longs;pontanca dicebantur. Extat adhuc
de huiu&longs;modi machinis liber Heronis Alexandrini, quem nuper ex græco
latinum reddidit docti&longs;&longs;imus Abbas Gua&longs;tallenfis. de huiu&longs;modi artificio&longs;is
operibus, quibus &longs;æpè pri&longs;ci ita admirationi fuere, vt præ&longs;tigia quædam ar
tificium ignorantibus, viderentur, intelligit hoc loco Ari&longs;t.
Cap. 3. (
in
eleuationem eius, aut depre&longs;&longs;ionem meridianam, videatur moras trahere,
quamuis no&longs;trum &longs;it explicare, ob rei tamen facilitatem omittantur. Hoc
tantum &longs;cias velim &longs;ol&longs;titiorum cau&longs;am e&longs;&longs;e Zodiaci ad Tropicos longio
rem adhæ&longs;ionem, ide&longs;t, quòd Zodiacus propè contactum tropicorum ab ijs
parum recedat, cum ergo Sol motu proprio &longs;emper per Zodiacum inam
bulet, fit vt ip&longs;e
mum &longs;ecus illos incedat, ita vt eo tempore, quo ad eos paulatim accedit,
aut ab eis paulatim recedit, qua&longs;i &longs;tare, &longs;iue quie&longs;cere apud eo&longs;dem videa
tur:
& noua elcuatio, aut depre&longs;&longs;io Solis &longs;upra horizontem nuila ferè appareat.
Ibidem (
detur, &longs;i quid, cum non &longs;it minimum non men&longs;uretur, decet autem in contrarium,
& in melius &longs;ecundum prouerbium con&longs;umare, quemadmodŭ in his fit, cum di&longs;cant,
nihil enim magis vir Geometricus admiraretur, quàm &longs;i diamcter commen&longs;urabi
lis &longs;ieret1.
cap.
1. Videtur inquit mirum à principio Geometriam aggredienti diame
trum, & latus eiu&longs;dem quadrati non commen&longs;urari, cum in neutro eorum
detur minimum, &longs;eu indiui&longs;ibile, videtur enim omne diui&longs;ibile po&longs;&longs;e men&longs;upo&longs;tea tamen cum in Geometria ver&longs;atus fuerit, maximè admirare
tur, &longs;i audiret diametrum e&longs;&longs;e lateri commen&longs;urabilem.
Summa 2. cap.
3. (
nebant, & in cis enutriti, eorum principia, entium
principia
rò multò melius & &longs;ibi, & Philo&longs;ophiæ con&longs;ulerent. At verò non &longs;ine ma
gno artium,
&longs;tate de&longs;pectui habentur; &longs;ed quid mirum cum quas &longs;cientiarum omnium
alumni Pythagorei omnibus &longs;cientijs anteferebant; eas no&longs;tri &longs;eculi quam
plures omnibus alijs facultatibus po&longs;thabeant.
Tex. 47. (
nes, po&longs;tulata, axiomata, quæ &longs;unt tria principiorum genera, ex quibus to
ta Geometria deducitur.
Tex. 14. (
bus fabulo&longs;a, ac puerilia plus po&longs;&longs;unt propter con&longs;uetudinem, quàm &longs;i
ea cogno&longs;ceremus
res Mu&longs;ici leges appellabant, eò quòd eas &longs;olas, cæteris abroga
tis liceret lata lege decantari. Vide declarationem problematis 15. & 28.
&longs;ect. 19.
Tex. 3. Verba huius textus, cum &longs;atis per&longs;picua &longs;int, ac parum ma
thematicis indigeant, omittenda duxi. Quod ad mathematicas
attinet, ait, eas non demon&longs;trare, nec per cau&longs;am finalem, nec
per efficientem (quod intelligendum e&longs;t de Mathematicis puris,
& &longs;peculatiuis nam mathematicæ practicæ reliquas etiam cau&longs;as, efficien
tem, & finalem nece&longs;&longs;ariò habere debent, quapropter &longs;ophi&longs;ta quidam no
mine Ari&longs;tippus, eas irridebat,
bus po&longs;thabebat, quæ cau&longs;am efficientem, quia &longs;cilicet operantur, & fina
lem &longs;cilicet quæ&longs;tum &longs;ibi proponunt. fuit autem i&longs;te ex Plutarcho, & Laer
tio primus, qui pacto pretio doceret,
efficerent; & finali, nihil lucrarentur. videas igitur quales &longs;int pulcherrima
rum facultatum contemptores, ij nimirum, qui philo&longs;ophiæ, aut lucri, aut
ambitionis cau&longs;a dant operam. Quod autem Mathematicæ nihil efficiant,
enim plures mathematicæ practicæ, quæ innumera,
opera, huiu&longs;modi &longs;unt Geometria
practica, qua men&longs;urationes omnes vel &longs;olo vi&longs;u perficiuntur. Arithmeti
ca, cuius v&longs;us quàm latè patet? Mu&longs;ica practica, qua quotidie ip&longs;i oblecta
Mechanica pra
ctica, cuius op
commouentur. Per&longs;pectiua, quæ Pictoribus, & Architectoribus adeo in&longs;er
uit, vt A&longs;tronomia tandem, &longs;i in praxim de
ducatur, ex vna &longs;olum eclyp&longs;ium prædictione, quantam vniuer&longs;o orbi ad
mirationem parit? mitto hanc &longs;olam dierum, men&longs;ium, & annorum di&longs;tri
butionem, ac temporum emendationem exhibere, rem adeò Reipublicæ
Chri&longs;tianæ nece&longs;&longs;ariam.
Eodem tex. 3. (
quorum &longs;unt demonstrationes, cum quid e&longs;t &longs;ciamus, vt puta quid tetragoni&longs;mus,
quòd inuentio mediæ
fu&longs;ius explicata.
Tex. 8. (
eorum e&longs;t, quæ &longs;entimus, illa verò non &longs;en&longs;ibilium e&longs;t
metriæ practicæ, ea &longs;cilicet, quæ circa diui&longs;ionem &longs;uperficierum ver&longs;atur. audi Pedia&longs;mum de men&longs;uratione: Terræ inquit men&longs;uratio in duas partes
diuiditur, Geometriam &longs;cilicet, & Geodæ&longs;iam: Areæ
tem men&longs;uratio, & terræ men&longs;uratio e&longs;t, & meritò Geometria vocatur. Vnius verò, & eiu&longs;dem areæ, &longs;eu loci diui&longs;io inter diuer&longs;as per&longs;onas, parti
tio quædam e&longs;t terræ, & iure optimo Geodæ&longs;ia appellatur. hæcille.
dicitur
autem Geodæ&longs;ia à Vocabulum tamen i&longs;tud Geo
dæ&longs;iæ fuit po&longs;tea ad latiorem tran&longs;latum &longs;ignificationem: extat enim Geo
dæ&longs;ia Heronis Mechanici antiqui &longs;criptoris, quampridem Baroccius lati
nitate donauit, quæ quidem ars e&longs;t eadem cum Geometria practica, cum
non &longs;olum diui&longs;iones, &longs;ed men&longs;urationes omnes etiam per dioptricam f
cultatem, &longs;eu per lineas vi&longs;uales doceat inue&longs;tigare.
Tex. 4.
bæc enim habet partes: ac prima quædam, & &longs;ecunda &longs;cientia e&longs;t: cæ
teræ
mæ &longs;cientiæ &longs;unt Geometria, & Arithmetica, quia ip&longs;æ à cæteris
nulla ratione dependent; imò cæteræ ip&longs;is innituntur, quæ &longs;ecundæ hoc lo
co appellantur, hæ &longs;unt Per&longs;pectiua, Mu&longs;ica, Mechanica, A&longs;tronomia. illas
duas recentiores &longs;ubalternantes, has verò &longs;ecundas &longs;ubalternatas vocant. Exempla &longs;ubalternationum varia attuli in Logicis tex. 20. & 23. primi Po
&longs;ter. vbi clarè licet intueri quid &longs;it &longs;ubalternatio, vnde etiam præ&longs;ens lo
cus illu&longs;tratur.
Tex. 28. (
primo Priorum, &longs;ecto 1. cap.
23. de hac commen&longs;urabilitate, & incommen
&longs;urabilitate tractata &longs;unt.
Tex. 2. (
&longs;ius quid erat e&longs;&longs;e, & borum genera, vt ip&longs;ius Diapa&longs;on duo ad vnum,
& &longs;impliciter numerus, & partes, quæ in ra ione &longs;unt
plum cau&longs;æ formalis ex Mu&longs;ica petitum;
illius con&longs;onantiæ, quæ Diapa&longs;on dicitur,
duplam proportionem, ide&longs;t, quæ e&longs;t inter duo, & vnum, id, quod omnes
Mu&longs;ici quod vtinelius intelligas, repete, quæ in 2. Po&longs;ter. ad tex. 1.
&longs;cripta &longs;unt: necnon quæ in libro de Sen&longs;u in cap.
8. Amplius inquit can&longs;am
formalem genericam eiu&longs;dem Diapa&longs;on e&longs;&longs;e numerum, & partes numeri,
&longs;ub numero enim continentur & duo, & vnum. Occurrit hoc loco vnum
magnopere notandum, videlicet tam con&longs;onantias, quam di&longs;&longs;onantias ha
bere proportiones numerorum, hoc tamen di&longs;crimine, quod con&longs;onantiæ
habent &longs;olùm proportiones numerorum eorum, qui quaternario continen
tur, ex veterum præ&longs;ertim Pythagoreorum &longs;ententia, qui propterea vltra
quaternarium progredi vetabant. Recentiores tamen y&longs;que ad &longs;enarium
procedunt, quippe, qui omnes vocum con&longs;onantias admittunt, quæ pro
portionibus numerorum &longs;enario contentorum præditæ &longs;int. Di&longs;&longs;onantiæ
verò &longs;eoundum pri&longs;cos habent proportiones numerorum extra quaterna
rium progredientium, iuxta no&longs;tros autem extra &longs;enarium. qua de re pluri
bus Zarlinus colloquio 2. definit. 3.
Tex. 3.
materiali. libuit locum hunc annotare in gratiam Geometricarum demon
&longs;trationum, quorum media &longs;æpè &longs;unt ex cau&longs;a materiali &longs;umpta, quod ta
men non ita ab omnibus ob&longs;eruatur,
nem quampiam de aliquo &longs;ubiecto, ex eo, quod &longs;ubiectum illud &longs;it, vel di
midium alicuius, vel duplum, vel reliquum, vel tertia pars, & his &longs;imilia,
erit talis ratio in genere cau&longs;æ materialis.
naturalibus &longs;cientijs a&longs;&longs;ueti, negent huiu&longs;modi materiam veram e&longs;&longs;emate
riam, ac proinde neq, Geometricas demon&longs;trationes veras e&longs;&longs;e demon&longs;tra
tiones; dicendum enim talem quidem materiam non e&longs;&longs;e veram materiam
phy&longs;icam, & proinde illas demon&longs;trationes
&longs;irationes, e&longs;&longs;e tamen veram materiam intelligibilem, quæ Geometriæ &longs;u
bijcitur, & proinde demon&longs;trationes illas veras e&longs;&longs;e demon&longs;trationes Geo
metricas; id quod Ari&longs;t. &longs;æpius in libris Po&longs;ter, apertè &longs;ignificat, tum a&longs;&longs;er
tionibus, tum exemplis quamplurimis. Quapropter cauendum e&longs;t illis, ne
ingrati animi notam incurrant, dum pulcherrimam artem re&longs;olutoriam,
quam Ari&longs;t. à Mathematicis acceptam omnibus &longs;cientijs accommodauit
(vt initio Priorum o&longs;ten&longs;um e&longs;t) eam ip&longs;i ita alijs facultatibus adaptent, vt
Mathematicis ip&longs;is, ex quibus orta, & &longs;ub quibus adoleuit, pulla ratione
conuenire poi&longs;it. De hac materia fu&longs;ius infra in additamento de natura Ma
thematicarum.
Tex. 3. (
&longs;unt, quemadmodum &longs;upra tex. 2. huius cap.
explicatum e&longs;t.
Tex. 4. (
demon&longs;trationum primæ enim demon&longs;trationes, quæ in pluribus demonstr ationbus
in&longs;unt, hæc elementa demon&longs;trationum dicuntur
&longs;criptionum
e&longs;t, præ&longs;ertim in Logicis, & ex hoc loco pariter confirmatur. Ex hoc por
rò loco illud innote&longs;cit dignum, quod præcipuè à Mathematico non igno
retur, quæ nam &longs;int demon&longs;trationes illæ, quæ nomine
appelllari, necnon cau&longs;a cur Euclides &longs;uum opus elementa nuncupauerit,
&longs;unt enim illæ, quæ in pluribus demon&longs;trationibus in&longs;unt, ide&longs;t, quæ &longs;æpius
in alijs demon&longs;trationibus citantur, vti &longs;unt præcipuè &longs;ex priores libri Eu
clidis:
Tex. 12.
tem idem in cunct is generibus vnum, &longs;ed hic quidem die&longs;is, hic verò vocalis, aut
muta)
&longs;is, quæ e&longs;t minima vox, aut &longs;onus, qui &longs;ub Mu&longs;ici con&longs;iderationem cadat. Porrò ad tex. 38. primi Po&longs;ter. de die&longs;i plura &longs;unt dicta.
Tex. 17.
&longs;itionem inuenies 1. Priorum, &longs;ecto 1. cap.
23.
Tex. eodem
tentia dicitur)
quadratum illius, ide&longs;t quadratum &longs;uper ip&longs;am con&longs;tru
ctum. v. g. quadratum in quo C, dicitur potentia lineæ
D B, quia &longs;uper illam con&longs;tructum e&longs;t.
Tex. 34. (
notata 1. Priorum, fecto 1. cap.
23.
Tex. 35. (
re tres angulos æquales duobus rectis angulis. Vide declarationem huius
lib.
primo Priornm, &longs;ecto 3. cap.
1.
Tex. 1. (
notanda &longs;unt hæc aduer&longs;us quo&longs;dam, qui negant in Mathemati
cis cau&longs;as reperiri, vt hinc enim
uerò apertè patet eos falli ex toto hoc Ari&longs;t.
di&longs;cur&longs;u.
rabitur
tionem horum reperies 1. Priorum, &longs;ecto 1. cap.
23.
Tex. 20. (
inuenirent, quod &longs;i diui&longs;æ e&longs;&longs;ent, manife&longs;i è e&longs;&longs;ent, nunc autem in&longs;unt potentia, cur
triangulus duo recti? quia qui circa vnum punctum anguli duobus rectis æquales
ptiones, vel figurationes, vel de&longs;ignationes intelligendas e&longs;&longs;e demon&longs;tra
tiones Geometricas &longs;æpius &longs;upra dictum e&longs;t, & pariter ex hoc loco com
probatur. Dicit igitur, quod demon&longs;trationes &longs;uas Geometræ inueniunt,
reducendo ad actum ea, quæ erant in potentia, diuidentes enim educunt in
actum, figuras, angulos, lineas, & cætera huiu&longs;inodi, quæ prius &longs;olùm erat
in potentia, ex quibus po&longs;tea &longs;uas demon&longs;trationes perficiunt (
lus duo recti
tras demon&longs;trare producendo ad actum entia quædam Mathematica, quod
exemplum, vt intelligas ijs opus habes, quæ primo Priorum, &longs;ecto 3. cap.
1.
con&longs;cripta &longs;unt (
angulos æquales duobus rectis angulis (
duobus rectis angulis æquales &longs;unt
fal&longs;um e&longs;t, nam omnes anguli, qui circa vnum
punctum, v. g. A, &longs;unt con&longs;tituti, æquales &longs;unt
non duobus, vt e&longs;t in textu, &longs;ed quatuor rectis,
vt patet ex corollario 2. 15. primi Elem.
quot
quot enim anguli con&longs;tituantur ad punctum A,
omnes &longs;imul erunt æquales quatuor rectis, quos
faciunt præ&longs;entes lineæ B C, D E. vniuer&longs;i enim
illi congruent his quatuor rectis: &longs;ed Ari&longs;t. &longs;en
&longs;us e&longs;t omnes angulos ad ea&longs;dem partes con&longs;ti
tutos, v. g. ad partes &longs;uperiores lineæ B C, e&longs;&longs;e
æquales duobus rectis B A D, D A C, vt o&longs;tenditur in 13. primi, necnon
etiam patere pote&longs;t ex corollario 2. 15. eiu&longs;dem. tales &longs;unt quatuor anguli
ad &longs;uperiores partes lineæ B C, & ad punctum A, con&longs;tituti, qui, vt patet,
&longs;unt æquales duobus rectis B A D, D A C,
tales etiam &longs;unt in hac &longs;ecunda figura tres
anguli B C A, A C D, D C E, qui quidem
æquales &longs;unt duobus rectis angulis. hoc
&longs;en&longs;i&longs;&longs;e Ari&longs;t.
patet ex demon&longs;tratione 32.
primi, quæ demon&longs;trat
&longs;tot. trianguli affectionem, & ad quam
propterea ip&longs;e &longs;pectabat, cuius figura e&longs;t
eadem cum hac &longs;ecunda, in qua Euclides o&longs;tendit prædictos tres angulos
æquari duobus rectis. &longs;ubdit po&longs;tea, &longs;i igitur linea C D, quæ ad latus A B,
parallela e&longs;t in potentia, educeretur in actum, videnti mox e&longs;&longs;et manife&longs;tum
tres angulos trianguli A B C, e&longs;&longs;e pares duobus rectis. ducta enim C D, pa
rallela lateri B A, apparet &longs;tatim angulus A, æqualis angulo A C D, & an
gulus B, angulo D C E; cum reliquus verò
prædictos duos ad idem punctum C, con&longs;tit utus;
rectis æquentur, mox in&longs;picienti talem figur ationem manife&longs;tum fit tres an
gulos illius trianguli e&longs;&longs;e duobus rectis æqu ales.
Ibidem (
quia &longs;i tres æquales, & quæ
ba&longs;is e&longs;t duo, & quæ ex medio &longs;upra stat recta, videnti manifestum erit ei, qui illud
&longs;ciatnunc &longs;olùm
hæc addenda &longs;unt. Re&longs;pondet Ari&longs;t.
quæ
&longs;ito pr&ecedil;cedenti, cur &longs;cilicet angulus in &longs;e
micirculo &longs;it rectus, qualis e&longs;t in figura
angulus A C B,
in figura tres lineæ &longs;unt æquales, duæ ni
mirum, in quas ba&longs;is B A, diuiditur, quæ
&longs;unt B D, D A, & tertia, quæ ex medio
ba&longs;is erigitur,
&longs;emidia metri ciu&longs;dem circuli. educta
&longs;i cuipiam trium harum linearum æqualitas innote&longs;cat, continuò ei etiam
manife&longs;tum erit angulum A C B, in &longs;emicirculo, e&longs;&longs;e rectum. quia &longs;tatim ap
parent duo i&longs;o&longs;celia B D C, A D C, quorum anguli ad ba&longs;es B C, A C, &longs;unt
æquales inuicem; & anguli duo ad D, &longs;unt dupli duorum
D C B, ex quibus conflatur totus angulus A C B, ergo duo anguli ad D, &longs;unt
dupli anguli B C A, &longs;ed duo anguli ad D, &longs;unt æquales duobus rectis, ergo
duo recti &longs;unt dupli anguli A C B, ergo angulus B C A, e&longs;t dimidium duo
rum rectorum. cum autem omnes recti &longs;int æquales, con&longs;ectarium e&longs;t dimi
dium duorum rectorum e&longs;&longs;e angulum rectum. patet igitur, qua ratione ex
ductu linearum prædictarum actu, manife&longs;tum fiat angulum in &longs;emicirculo
A C B, e&longs;&longs;e rectum. ne mireris &longs;i vulgatam tran&longs;lationem antiquam non
&longs;um &longs;equutus, indigebat enim correctione, quam iuxta græcum exem
plar adhibui.
Tex. 22. (
rectos habere, modo non, mutaretur enim
tiam per demon&longs;trationem 32. primi Elementorum. quomodo autem tri
angulus habeat duos rectos, ide&longs;t tres angulos æquales duobus rectis angu
lis, explicatum e&longs;t primo Priorum, &longs;ecto 3. cap.
1.
Ibidem (
primum nullum e&longs;&longs;e; aut quo&longs;dam quidem, quo&longs;dam verò non
7. Elem.
&longs;ic numerus ille, qui à Mathematicis dicitur primus, definitur, pri
mus numerus e&longs;t, quem vnitas &longs;ola metitur, vnde patet inter numeros pa
res &longs;olum binarium e&longs;&longs;e primum, cum ip&longs;um &longs;ola vnitas bis replicata men
&longs;uraret. quaternarium autem, &longs;enarium, &c.
pares, non e&longs;&longs;e primos, cum
eos non &longs;ola vnitas, &longs;ed alius numerus metiatur: quaternarium enim bina
rius bis replicatus men&longs;urat: &longs;enarium men&longs;urat & binarius, & ternarius:
quare verum erit exi&longs;timare inter pares numeros aliquos e&longs;&longs;e primos, ide&longs;t
binarium, aliquos verò non, ide&longs;t cæteros pares vltra binarium.
Tex. 4. (
nimum enim tempus hic habet. quapropter in A&longs;trologia tale vnŭ prin
cipium, & men&longs;ura e&longs;t. motum
ponunt, ad quem cæteros tudicant
primo cœlo, &longs;eu mobili a&longs;eribunt, hic enim veloci&longs;&longs;imus e&longs;t omnium reli
laris, & propterea minimum habet tempus, ide&longs;t tempus vnius diei natura
lis, quo tempore totum primum mobile circulationem integram perficit. per minimum tempus, po&longs;&longs;uut etiam intelligi partes diei, quæ &longs;unt horæ, &
horarum partes. con&longs;iderant hunc motum in circulo æquàtoris, quia æqua
tor motu primi mobilis, &longs;eu diurno vniformiter, ae maximè regulatiter
mouetur: hac de cau&longs;a hunc motum tanquam reliquorum men&longs;uram, ac
normam meritò a&longs;&longs;ump&longs;erunt.
Ibidem
nimum interuallum, quod à Mu&longs;icis con&longs;ideretur, e&longs;t men&longs;ura maiorum in
teruallorum. ad tex. 38. primi Po&longs;ter. &longs;atis dictum e&longs;t de Die&longs;i, quæ videas.
Eodem tex. &longs;ed cap.
3.
aliquando plura, vt puta die&longs;es duæ, non quidem &longs;ecundum cuditum, &longs;ed in ratio
nibus, & voces plures, quibus men&longs;uramus, & diameter duobus men&longs;uratur, & la
tus, & omnes magnitudines)quid die&longs;is
dictum &longs;it ad tex. 38. primi Po&longs;ter. quando autem ait
ide&longs;t duæ die&longs;es &longs;unt men&longs;ura vnius interualli mu&longs;ici, qui tonus appellatur:
quæ quidem duæ die&longs;es non &longs;unt men&longs;ura &longs;en&longs;ibilis, quæ &longs;cilicet auribus per
cipiatur, &longs;ed tantummodò exi&longs;tunt in numerorum proportionibus, ibi per
intellectum excogitatis, quando ait
quando vtimur eodem interuallo, &longs;iue eadem voce ad cantus men&longs;uram,
tunc &longs;unt plures men&longs;uræ numero, quamuis vna tantam &longs;pe
meter duobus men&longs;uratur)& latus pariter quadrati, duobus.
v. g. pedibus me
do reliquæ omnes magnitudines po&longs;&longs;unt ab ea
ta men&longs;urari.
Eodem tex.
que magnitudo, & &longs;ecundum vnumquedque,
manife&longs;ta apparet, cur Geometræ practici me
quam longitudinem, vt puta per vlnam, digitum, vnciam, &c. &longs;uperficies
etiam per aliquam &longs;uperficiem, &longs;ed quæ &longs;it quadrata, vt puta per vlnam qua
dratam, palmum quadratum, &c. corpora
&longs;it cubus, vt per vlnam cubicam, palmum cubicum, vnciam cubicam, &c.
Tex. 11.
tiam &longs;ubiectam in d. fferentia &longs;ecundum formam eadem &longs;it: quemadmodum quadra
tum maius minori &longs;imile e&longs;t, & lineæ inæquales, hæ enim &longs;imiles quidem, verŭ non
cædem &longs;impliciter &longs;unt)
quæ &longs;unt æquiangulæ inuicem, & quæ habent latera proportionalia circa
æquales angulos. cum ergò quadratum maius, & minus &longs;int æquiangula,
quia habent omnes angulos rectos; & præterea habeant latera circa æqua
les angulos proportionalia, &longs;icut enim latera maioris quadrati circa vnum
angulum rectum &longs;unt in proportione æqualitatis; ita
circa vnum angulum rectum &longs;unt illis proportionalia, cum &longs;int inuicem pa
riter in proportione æqualitatis, erunt nece&longs;&longs;ariò &longs;imilia hæc duo quadrata. duæ ctiam, exempli gratia, lineæ rectæ &longs;unt inuicem &longs;imiles, quamuis vna
&longs;it maior altera.
Eodem tex.
dum diuer&longs;i, ponit in entibus Mathematicis, &longs;icut enim po&longs;uit idem e&longs;&longs;e in
Mathematicis, quando duæ figuræ &longs;unt &longs;imiles, & æquales: ita ex oppo&longs;ito
diuer&longs;um erit in Mathematicis, quando duæ figuræ fuerint di&longs;&longs;imiles, & in
æquales,
e&longs;t in præcedenti expo&longs;itione.
Svmma r.
cap.
2.
primas &longs;uperficies principia e&longs;&longs;e ponat. bæc non &longs;unt &longs;ub&longs;tantiæ &longs;eparabiles,
verùm &longs;ectiones, & diui&longs;iones, illæ quidem in &longs;uperficierum, hæc verò cor
porum, puncta verò linearum &longs;unt, & etiam ip&longs;arum earumdem termini;
hæc autem omnia in alijs &longs;unt, & nihil &longs;eparabile e&longs;t)
ne lineæ, quamuis &longs;int etiam termini illius; lineas verò oriri ex diui&longs;ione
&longs;uperficierum, quamuis &longs;int etiam termini illarum. &longs;uperficies
ex diui&longs;ione corporum, quamuis &longs;int etiam termini, illorum. Hæc placuit
annotare propter
Summa 3. cap.
2.
læ&longs;tis, &longs;eu Caniculæ. Vide quæ libro &longs;ecundo Meteororum, &longs;umma 2. cap.
2.
de hac &longs;tella &longs;crip&longs;imus.
Tex. 44.
Mathematicarum &longs;cientiarum, videlicet ex A&longs;tronomia con&longs;iderandum
est: hæc enim de &longs;ub&longs;tantia &longs;en&longs;ibili quidem, ac &longs;empiterna &longs;peculatur)
pluralitatem nimirum cœle&longs;tium motuum petendam e&longs;&longs;e a&longs;&longs;erit
ex præcipua totius Philo&longs;ophiæ parte, quam ait e&longs;&longs;e A&longs;tronomiam. dignum
porrò con&longs;ideratione e&longs;t, quanti faciat Ari&longs;t. Mathematicas di&longs;ciplinas, ac
præcipuè &longs;yderalem &longs;cientiam.
Tex. 45.
quorum primus quidem e&longs;&longs;et, qui inerrantium &longs;iellarum; &longs;ecundus verò &longs;ecunduns
id, quod per medium Zodiacum; tertius tandem, &longs;ecundum quem qui in latitudine
Zodiaci obliquatur. in maiori autem latitudine obliquari eum &longs;ecundum quem Lu
na, quàm eum &longs;ecundum quem Sol &longs;ertur)
culta fuerat A&longs;tronomia, vt propterea minimè mirandum &longs;it, eum hoc lo
co imperfecta admodum circa c&ecedil;le&longs;tia tradere. omittit enim in Sole orbem
motum augis conficientem; necnon duos eccentricos, qui &longs;olis anomaliam, attribuit præterea Soli motum
quendam in latitudinem, quod fal&longs;um e&longs;t omninò, cum Sol perpetuò directè
&longs;ub eclyptica incedat. In Luna pariter plures nece&longs;&longs;arios illi orbes ad motus
ip&longs;ius &longs;aluandos prætermittit. Ex &longs;ententia tamen Tychonis Brahe hos or
bes, ac circulos tanquam ab inuicem di&longs;tinctos abrogare debemus.
Tex. 46.
primam quidem, & &longs;ecundam eandem illis e&longs;&longs;e: etenim, quæ fix arum eft eam illam
e&longs;&longs;e, quæomnes fert: at cam, quæ &longs;ub ip&longs;a ordinata e&longs;t, ac quæ &longs;ecuxdum Zodiacum
lationem habet, communem omnibus e&longs;&longs;e. Tertiæ verò omnium polos in eo, quod
per medium Zodiacum e&longs;t. Quartæ autem lationem &longs;ecundum eum, qui obliquatus
ad medusm eius e&longs;t; e&longs;&longs;e verò tertiæ &longs;phæræ polos aliarum quidem proprios, Veneris
autem, & Mercurij eo&longs;dem)
Sol autem, & Luna hoc nomine non e&longs;t complexus, eo quod ip&longs;a mereantur
potius duo mundi luminaria appellari, quàm cum c&ecedil;teris &longs;tellis in ordinem
redigi. Reliquis igitur
bat, quarum prima, & &longs;ecunda eodem modo &longs;e habebant, ac in Sole, & Lu
na, etenim octaua &longs;phæra, &longs;eu firmamentum, quod affixa &longs;ibi &longs;ydera differt
communicabat, &longs;ecundum ip&longs;um reliquis inferioribus &longs;phæris motum &longs;uum
peculiarem, videlicet diurnum, quo ab oriente in occidentem tota c&ecedil;li ma
china conuertebatur. fecundam eam facit, quæ Planetas omnes &longs;ecundum
Zodiaci longitudinem ab occidente in
modo &longs;e habet in &longs;ingulis. Tertiam verò eam confinxit, cuius poli e&longs;&longs;ent in
eclyptica, in quibus cita, ab eclyptica vltrò, Quartam
demum po&longs;uit, quæ tertiam bifariam &longs;ecaret,
eclyptica plus iu&longs;to ver&longs;us mundi polos exorbitaret. porrò in reliquis vo
luit polos tertij orbis e&longs;&longs;e peculiares, Veneri autem, & Mercurio eo&longs;dem
e&longs;&longs;e, ide&longs;t e&longs;&longs;e in eadem linea. Ex mente igitur Eudoxi cœle&longs;tes orbes in
vniuer&longs;um 27. numerantur, in Sole &longs;imul, ac Luna 6. in reliquis quinque er
rantibus 20.
ob ratas po&longs;teriorum a&longs;tronomorum ob&longs;eruationes non &longs;ub&longs;i&longs;tere. at verò
hic non ip&longs;ius placita, &longs;ed præcipuè textus intelligentiam per&longs;equor.
Tex. 47.
e&longs;t di&longs;tantiarum ordinem. pluralitatem autem &longs;tellæ quidem Iouis, ac Saturni ean
dem illi attribuebat. Solis verò, & Lunæ duas adbuc putabat &longs;phæras addendas
e&longs;&longs;e, &longs;i quis eorum, quæ &longs;en&longs;ibilitcr apparent, can&longs;as a&longs;&longs;ignare debeat. Cæteris ve
rò errantium vnicuique vnam. nece&longs;&longs;e verò e&longs;&longs;e, &longs;i debent omnes &longs;imul po&longs;itæ, quæ
apparent reddere, &longs;ecundam
res e&longs;&longs;e, quæ reuoluant, & ad idem po&longs;itione &longs;emper primam eius astri &longs;phæram,
quod inferius ordinatum e&longs;t, con&longs;tituant. boc enim modo &longs;olùm contingit errantium
lationem omnia facere. Cùmigitur, in quibus ip&longs;a quidem feruntur &longs;phæris, hæ
quidem octo, bæverò horum &longs;ane non oportet illas &longs;olas reuo
lai, in quiòus fertur, quod infimè ordinatum e&longs;t. quæ quidem duarum &longs;phærarum
primas reuoluant, &longs;ex erunt. quæ verò pe&longs;teriorum quatuor, &longs;exdecim.
cunctarum
verò numerus, tùm earum quæ ferunt, tùm quæ reuoluunt eas, quinquaginta quin
que. quòd &longs;i Lunæ, & Soli, non addat aliquis quos diximus motus, omnes &longs;phæræ
erunt &longs;eptem, & quadraginta. pluralitas
per paraphra&longs;im &longs;ic explico; Calippus igitur eundem quidem ordinem, at
que di&longs;tantiam &longs;phærarum cum Eudoxo ponebat:
orbium mouentium Saturnum, ac Jouem; quatuor &longs;ed putabat &longs;oli duas addendas, ac Lunæ &longs;imiliter, &longs;i quis eorum
cæteris verò errantium, Marti, Veneri, & Mercurio
vnam. nece&longs;&longs;e præterea exi&longs;timabat e&longs;&longs;e, vt prædictæ omnes &longs;phæræ &longs;imul
apparentias omnes excu&longs;arent, addendas e&longs;&longs;e alias &longs;ingulis planetis toti
dem &longs;phæras vna minus, quas Reuoluentes appellabat; ita vt qui quatuor
Mouentes &longs;phæras habuif&longs;et, tribus præterea reuoluentibus opus haberet:
quæ &longs;phæræ reuoluentes id præ&longs;tabant, vt qua&longs;i priores Mouentes ita in of
ficio continerent, vt priori po&longs;itioni a&longs;trum, quod interiori orbi affigebur
&longs;uo tempore re&longs;tituerent, vt Alexander exponit. hoc enim &longs;olummodo po&longs;
&longs;ibile putabat omnes errantium lationes nos imitari po&longs;&longs;e. Cum igitur mo
uentes &longs;phæræ illæ quidem Saturni, ac Iouis &longs;int octo; reliquorum verò vi
gintiquinque, nam reliqui Planetæ
habent, quæ omnes &longs;imul numerum
&longs;olæ inferiores, quibus a&longs;trum affixum volebat, non indigebant reuoluente,
&longs;equitur duorum &longs;uperiorum Saturni, & Iouis, quorum octo erant mouen
tes, &longs;ex debere e&longs;&longs;e reuoluentes. Inferiorum verò quatuor planetarum re
uoluentes erunt &longs;exdecim: &longs;ed hoc loco Ari&longs;t.
memoria fallit, deberet enim
dicere, reliquorum
planetæ &longs;eptem, quorum Saturno, ac Ioui &longs;upremis &longs;ex reuoluentes attri
buit habita ratione &longs;phæratum mouentium; reliquis igitur
habita ratione &longs;uorum orbium mouentium, 25. cum &longs;inguli habeant
mouentes, habebunt ex prælcripto Calippi &longs;inguli 4. reuoluentes; ac pro
inde 20. in vniuer&longs;um erunt reuoluentes. Omnium igitur &longs;phærarum tam
mouentium, quàm reuoluentium &longs;ummam ait, &longs;ed perperam, e&longs;&longs;e quinqua
gintaquinque; cum enim mouentes Saturni, & Iouis &longs;int 8. reliquorum au
tem 25. reuoluentes verò Saturni, & Iouis &longs;int 6. reliquorum autem, vt ip
&longs;e memoria fal&longs;us ponit, &longs;exdecim, conflant quidem &longs;ummam prædictam,
&longs;ed illi in memoria reuocandus e&longs;t, planeta ille, quem oblitus e&longs;t, cuius &longs;unt
quatuor reuoluentes, qui prioribus additi &longs;phærarum errantium numerum
quinquaginta nouem con&longs;tituent: quibus etiam addenda e&longs;t octaua &longs;phæra,
&longs;eu firmamentum, quod inerrantium &longs;edes e&longs;t, non enim &longs;olum errantium,
&longs;ed omnium cœle&longs;tium orbium numerum inue&longs;tigare volebat,
omnes &longs;ecundum Calippum &longs;ph&ecedil;ræ &longs;exaginta. Quod &longs;i Lunæ, & Soli non ad
dantur &longs;ingulis duo mouentes, vt facit Calippus,
illis debiti reuoluentes non erunt omnes, 55. verùm, detractis octo prædi
ctis, erunt tantum 47. &longs;eu vt melius loquatur non erunt in vniuer&longs;um, 60. &longs;ed
52. tantum. Hactenus de numero cœlorum.
Svmma 1. cap.
3.
pulchro dicere, fal&longs;um dicunt. dicunt.
n.
& maximè
nam & &longs;i non
nominant, quia tamen opera, & rationes ostendunt, non ne dicunt de eis? pulchra
maximè à Mathematicis &longs;cientijs o&longs;tenduntur, &c.)
carum commendationem, ac defen&longs;ionem apponere, cum non de&longs;int hac
no&longs;tra tempe&longs;tate ageometrcti complures, qui cas libenter &longs;ugillare &longs;olent.
Qvidquid Mathematicum in his quæ&longs;tionibus occurret, illud, vt
plurimum per paraphra&longs;im exponemus, ita tamen, vt tex. Ari&longs;t.
& figuræ textui re&longs;pondentes per eam, quantum fieri poterit re
&longs;tituantur, & &longs;i quæ &longs;e offerent difficilia, pro viribus &longs;oluantur. E&longs;t autem nonnullis in locis textus tam græcus, quàm latinus adeò corrup
tus, ac deprauatus, vt nullo modo emendari queat.
Quæ &longs;it artis Mechanicæ facultas.
Eorum, quæ miraculo &longs;unt, alia quidem natura contingunt,
quorum ignorantur cau&longs;æ: alia verò &longs;unt, quæ præter naturam per
artificium aliquod ad hominum vtilitatem perficiuntur, in multis
inde oritur, quia natura eundem &longs;emper, ac &longs;implicem &longs;eruat modum: quod
autem nobis vtile e&longs;t, plurimas &longs;ubit varietates. quando igitur quippiam
præter naturam facere opportuerit, illud, quod faciendum e&longs;t, difficultate
&longs;ua nos remoratur, quamobrem eam artis
partem, quæ huiu&longs;modi &longs;uccurrit difficultatibus, Mechanicam appellamus. Cæterùm optimè Antiphon Poeta in hunc modum cecinit;
Quemadmodum accidit, cum minora &longs;uperant maiora, & quæcunque exi
guam vim habentia, magna tamen mouent pondera, & omnia ferè illa, quæ
&longs;ub ea cadunt problemata, quæ mechanica nuncupari &longs;unt autem hæc
mathematicarum contemplationum, Po&longs;tea in
græcis codicibus hæc &longs;equuntur (
de pevi\o\, di\a tw_n fuszxw_n
exi&longs;tant, manife&longs;tum e&longs;t per Mathematica: illud verò circa quod ver&longs;antur,
hoc e&longs;t obiectum, de quo pertractant Mechanicæ quæ&longs;tiones per &longs;cientias
phy&longs;icas habetur, ide&longs;t res naturalis e&longs;t; e&longs;t enim pondus, & vis, aut poten
tia pondus ip&longs;um mouens, quatenus quanta &longs;unt; &longs;iue dixeris e&longs;t quantitas
ponderum, Mathematicæ enim mediæ, de quorum nu
mero e&longs;t facultas Mechanica, con&longs;iderant quantitatem rei alicuius
determinatæ, &longs;ic A&longs;tronomia circa cœle&longs;tium corporum,
tuumque
lium; Mu&longs;ica circa &longs;onorum quantitates ver
&longs;antur. quæ placuit annotare, vt &longs;cien
tiæ huius naturam per&longs;pectam
haberemus.
De dignitatibus,
Cvm vellet Ari&longs;t.
mirabilium effectuum, quos in Mechanicis admi
ramur, cau&longs;am referre in circulum: meritò ante omnia de admi
randa ip&longs;ius circuli natura di&longs;&longs;erit, quo minus mirum deinde vi
deatur prædictas mirabiles-operationes exip&longs;o procedere. quan
doquidem exadmiranda cau&longs;a admirabiles effectus prodire debeant. qua
lia &longs;unt ea, quæ circa yectem, cum magna videmus enim exiguam pror&longs;us vim ingens pondus, quod
mè mouere po&longs;&longs;et, addito etiam ip&longs;ius vectis pondere, facilè
luerit propellere. quod quidem auditu ab&longs;urdum foret, ni&longs;i vi&longs;u con&longs;taret.
omnium autem huiu&longs;modi cau&longs;æ principium circulus obtinet: & hoc qui
dem meritò, ex admirabili enim, quippiam mirandum accidere rationi
omninò con&longs;entaneum eft.
Primò igitur maximè admirandum e&longs;t contraria &longs;imul fieri, aut exi&longs;tere
circulus tamen ex contrarijs e&longs;t con&longs;titutus, oritur enim circulus ex com
moto, & manente, quæ quidem naturaliter &longs;untinuicem contraria. &longs;it au
tem circulus ex commoto, & manente, quia oritur ex circumuolutione
vnius rectæ lineæ, cuius alterum extremum fixum manet, alterum verò cir
cumagitur; quamobrem i&longs;thæc cernentes minus admirari
quæ in ip&longs;o &longs;unt contrarietates. cuiu&longs;modi e&longs;t hæc, quod cum linea, quæ cir
culi orbem complectitur,
latitudinem, ei tamen contraria quodammodo in&longs;unt, concauum &longs;cilicet,
& curuum; quæ quidem eo modo &longs;unt contraria, quo etiam magnum, & pa
ruum, horum enim medium e&longs;t æquale; illorum verò rectum. & &longs;icuti quan
do magnum, & paruum inuicem commutantur, ita vt quod magnum e&longs;t fiat
paruum, quod verò paruum fiat magnum, nece&longs;&longs;e e&longs;t, vt perueniant ad
æquale priu&longs;quam ad extremum alterutrum; ita linea curua antequam fiat
concaua, debet prius fieri recta: & ex concaua, vt tran&longs;eat ad conuexam,
& circularem, debet &longs;imiliter prius e&longs;&longs;e recta.
Alterum contrarium, quod circulo ine&longs;t, e&longs;t &longs;imul
neri: &longs;imul enim ad anteriorem mouetur locum, & ad po&longs;teriotem. & eo
dem modo linea illa, quæ ex vno extremo manens, ex altero verò circum
lata circulum de&longs;cribit, &longs;e habet; contraria enim &longs;imul contipet, primum
&longs;cilicet, & extremum. Ex quo enim primo loco circumagi incipit ad eun
dem rur&longs;us po&longs;tremò reuertitur, ita, vt primum ip&longs;ius, & po&longs;tremum idem
&longs;int; quapropter, vt prius dicebamus non e&longs;t inconueniens, ip&longs;um circulum
miraculorum omnium e&longs;&longs;e principium. Admiranda igitur ea, quæ circa li
bram &longs;iunt, ad circulum
ad ip&longs;am libram: alia autem ferè omnia, quæ circa mechanicas contingunt
motiones, ad vectem reducuntur.
Præter prædicta aliud tandem mirum ip&longs;i ine&longs;t, quia nimirum cum innu
mera &longs;int puncta in vna
tur, inæquali veiocitate mouentur; Nam punctum illud &longs;emper velocius
mouetur, quod remotius e&longs;t à centro circuli, &longs;eu à manente &longs;emidiametri
termino, & proinde illud tardius, quod centro proximius e&longs;t.
mira circuli proprietate,
bus, vt in &longs;equentibus quæ&longs;tionibus manife&longs;tum erit.
Quoniam autem &longs;ecundum contrarias &longs;imul motiones mouetur circulus,
& alterum quidem diametri extremum vbi A, in figura præ&longs;enti antror&longs;um
mouetur; alterum verò vbi B, retror
&longs;um, efficiunt nonnulli, vt ab vnica mo
tione multi contrariò &longs;imul mouean
tur denticulati circuli: vt &longs;unt ij, quos
in locis proponunt &longs;acris, quorum alij
&longs;unt ænei, alij ferrei. &longs;i enim circulus
A B, alterum circulum C D, contige
rit, mota diametro A B, ita vt A, an
tror&longs;um eat, commouebit alteram dia
metrum C D, ita vt C, retror&longs;um, hoc e&longs;t in contrarium ip&longs;i A, veniat, in
contrarium igitur mouebitur &longs;ecundus circulus C D, ad circulum A B, &
rur&longs;us circulus E F. in contrarium ip&longs;i C D, commouebitur ab ip&longs;o C D, ob
eandem rationem. eodem etiam modo &longs;i plures fuerint, idem facient vno
&longs;olo tanquam primo motore hanc igitur circuli naturam animad
uertentes Architecti, in&longs;trumentum artificiosè
occultantes, vt machinæ &longs;olù manife&longs;tum &longs;it illud, quod admirationem
parit, cau&longs;a verò lateat: quod genus machinarum Automata dicebantur,
quia &longs;pontè à &longs;e ip&longs;is mouebantur.
In primis igitur, quæ circa libram accidunt, dubitare faciunt, quamnam
ob cau&longs;am maiores libræ minoribus &longs;int exactiores: huius autem rei prin
cipium e&longs;t illud, quod &longs;upra innuimus, quod &longs;cilicet, quæ à centro plus di
&longs;tat linea, &longs;iue quæ longior e&longs;t, eadem vi commota citius fertur, quam illa,
quæ minus à centro di&longs;tat, &longs;eu quæ minor e&longs;t. Porrò citius bifariam dicitur;
&longs;iue enim in minori tempore æquale pertran&longs;it &longs;patium: &longs;iue in æquali tem
pore, maius conficit interuallum; citius feci&longs;&longs;e dicitur. &longs;i autem duæ lineæ
circa idem centrum moueantur vna maior, & altera minor in æquali tem
pore; maior maiorem circulum de&longs;cribet, quam minor; quia circulus à ma
iori de&longs;criptus, alterum à minori delineatum circumple ctetur,
continebit; maius autem e&longs;t continens, quàm horum autem cau
&longs;a, quoniam quæ circulum de&longs;cribit linea, duabus fertur lationibus, quæ nul
lam inuicem obtinent analogiam: quod antequam probemus, &longs;ciendum
e&longs;t, quod, quidquid duobus motibus inuicem proportionatis, mouetur, ne
ce&longs;&longs;e e&longs;t, quod motu exillis mixto progrediatur per lineam rectam, quæ dia
meter e&longs;t quadrilateri, cuius latera habeant illam proportionem, quam
duo illi motus. &longs;it enim in figura proportio lateris A B, ad latus A C, quam
ctiam habent duo motus, &longs;ecundum quos latum quodpiam feratur,
tum illud A, & feratur motu vno ver&longs;us B, per lineam A B, altero verò mo
tu feratur ver&longs;us C. quod fiet &longs;i cogitemus latus A B,
ip&longs;i æquidi&longs;tanter, dum punctum A, mouetnr
ad B. his duabus lationibus A, latum. nece&longs;&longs;a
niò motu mixto progre dietur per diametrum
A M, quod &longs;ic probari pote&longs;t; &longs;it iam A, mo
tum primo motu
cundo motu &longs;it in G F E, quo motu punctum
A, quod erat in D,
ctum e&longs;t in diametro A M, quoniam enim mo
uetur duobus motibus, cum lineis A B, A C, proportionalibus, motus au
tem
compleatur rectangulum A D F E, erunt &longs;imiliter proportionalia F E, D E,
cum &longs;int æqualia duobus D A, A E, quare per 26. 6. cum quadrilaterum
paruum A D F E, &longs;it &longs;imile toti A B M C, erit A M,
punctum F, in quo e&longs;t A, e&longs;t in diametro A M. eodem modo, de quouis pun
cto in linea A B, ad quod A, perueniat, probabitur ab altero motu de&longs;cen
di&longs;&longs;e v&longs;que ad diametrum. &longs;emper ergò latum A, per rectam A M, diame
trum quadrilateri, cum illis motibus proportionalibus progreditur, quod
probandum erat. è conuersò manife&longs;tum etiam e&longs;t, quod &longs;i quid &longs;ecundum
diametrum duabus fertur lationibus, eas lationes e&longs;&longs;e proportionales late
ribus quadrilateri, cuius e&longs;t illa diameter, &longs;i enim illæ lationes non &longs;unt la
teribus proportionales, latum illud non feretur &longs;ecundum diametrum il
lam, &longs;ed &longs;ecundum aliam alterius quadrilateri.
Quod &longs;i quid duabus lationibus nullam habentibus proportionem per
petuò ferratur, impo&longs;&longs;ibile e&longs;t ip&longs;um motu mixto lineam rectam de&longs;cribere. &longs;i enim dixeris illud po&longs;&longs;e de&longs;cribere rectam lineam, tunc circa rectam il
lam tanquam diametrum de&longs;cribam quadrilaterum, & po&longs;tea o&longs;tendam, vt
proximè o&longs;ten&longs;um e&longs;t, illud latum e&longs;&longs;e &longs;ecundum laterum illius proportio
nem, quare impo&longs;&longs;ibile e&longs;t id, quod mouetur duabus lationibus nullam in
uicem rationem habentibus, ferri per lineam rectam: quapropter
e&longs;t hoc modo Quod autemea, quæ de&longs;cribit circulum linea, dum altero eins manente
extremo circumagitur, duabus &longs;imul feratur lationibus, ex quibus motus
orbicularis oriatur, manife&longs;tum e&longs;t ex &longs;uperioribus, quia & antror&longs;um, &
retror&longs;um impellitur; tùm etiam, quia &longs;i rectà tenderet recta
culum, nunquam ad diametri perpendiculum
perueniret, &longs;ed tamen peruenit, ita vt &longs;it ip&longs;a
à centro perpendicularis diametro. &longs;it circuli
figura A B C D, in qua extremum diametri
B, feratur ad alterum extremum vbi D, per
ip&longs;ius diametri B D, circumuolutionem circa
centrum F, nece&longs;&longs;e e&longs;t aliquando B, perueniat
ad C. &longs;i igitur B, feretur duabus lationibus
aliquo modo proportionatis, v. g. vt e&longs;t pro
portio lateris B E, ad E C, latus, &longs;equeretur
ex demon&longs;tratis ip&longs;um B, ferri per
quæ diameter e&longs;&longs;et quadrilateri B E C G. &longs;ed
progreditur ad C, ita vt ip&longs;a diameter B D, in po&longs;itione A C, fiat perpendi
cularis priori diametro B D. ex quibus &longs;equitur eam moueri duobus moti
bus nullam rationem habentibus; quod erat intentum.
Hoc modo Ari&longs;t.
probare conatur, lineam circulum de&longs;cribentem, dua
bus ferri lationibus, quæ nullam habeant analogiam: Yerùm, vt liberè fa
tear nullo modo mihi videtur intentum a&longs;&longs;equi, nam
ip&longs;am duobus motibus ferri, quibus opus e&longs;&longs;et: neque patet eos (quamuis
concedantur) nullam inuicem habere analogiam: qui enim fieri pote&longs;t, vt
duo motus reperiantur, quì nulla &longs;e mutuò habitudine re&longs;piciant? Præte
rea &longs;i B, ferretur illis motibus, non &longs;equitur debere moueri per lineam cir
cularem, cum præter lineam rectam &longs;int plures curuæ, quæ tamen non &longs;unt
circulares, vt &longs;unt &longs;ectiones parabolicæ, & lineæ &longs;pirales. Deinde pergit.
Vt autem cau&longs;a appareat, cur ea, quæ à centro longior e&longs;t linea velocius
moueatur, &longs;iue quod in eadem &longs;emidiametro remotiora puncta à
locius moueantur, vt &longs;upra dictum e&longs;t, &longs;ciendum e&longs;t, Quod &longs;i duo mouean
tur ab eadem potentia, quorum vnum à quopiam alio mouente plus repel
latur à motu priori, alterum verò minus, rationi
moueri id, quod plus, eo quod minus impeditur; quod videtur accidere
maiori, & minori illarum, quæ à centro egre&longs;&longs;æ circulos delineant. quoniam
enim propius e&longs;t manenti eius, quæ minor e&longs;t extremum, quàm extremum
maioris, propterea plus à centro, cui propius e&longs;t, retrahitur à priori mo
tu,
fit, vt extremum illud de&longs;cribat lineam circularem quidem, &longs;ed tamen
curuiorem quam de&longs;cribat extremum longioris lineæ, quæ circulum minus
curuum, &longs;eu magis ad rectam lineam accedentem delineat. omni quidem
igitur lineæ circulum de&longs;cribenti i&longs;tud accidit, vt duobus feratur motioni
bus; vna quidem, quæ illi naturalis, ac &longs;ecundum circunferentiam, qua re
ctà tenderet ni&longs;i impediretur: altera verò, quæ illi innaturalis, qua in tran&longs;
uer&longs;um agitur, &longs;eu &longs;ecus centrum, ob quam cogitur in gyrum duci, minor
autem linea &longs;ecundum hanc motionem innaturalem plus fertur, quàm ma
ior, ide&longs;t plus ip&longs;ius progre&longs;&longs;io inflectitur in orbem; quia enim e&longs;t centro
vicinior, quod quodammodo retra
hit à motu naturali, propterea ma
gis vincitur, quàm remotior. Quod
ex his erit &longs;it circulus vbi
B C E D, & alter in eo minor, vbi
N M O P, circa idem centrum A. &
proijciantur diametri in magno qui
dem C D, B E, in minori verò M O,
N P. & altera parte longius quadri
laterum compleatur D K R C. &longs;i igi
tur &longs;emidiameter A B, circumacta
de&longs;cribit circulum maiorem, reuer
titur tandem ad locum B A, vnde di
gre&longs;&longs;a e&longs;t. &longs;imiliter M A, circumuoluta
Tardius autem fertur M A, quàm
B A, vt dictum e&longs;t, quia maior illi fit retractio à recta progre&longs;&longs;ione. Sit igi
tur linea A B, mota v&longs;que ad locum A L F, & à puncto L, ducatur L Q, per
pendicularis ip&longs;i A B, in minori circulo. & rur&longs;us ducatur L S, parallela ei
dem A B, & à puncto S, in maiori circulo ducatur S T, perpendicularis ei
dem B A, necnon F X. erunt igitur S T, L Q, latera rectanguli T L, æqualia
per 34. primi. erit po&longs;tea B T, minor quam M Q, quia æquales rectæ S T,
L Q, ductæ à circunferentia ad diametrum perpendiculares in circulis in
æqualibus, ea quæ e&longs;t in maiori circulo minorem re&longs;ecat diametri portio
nem, quàm quæ in minori.
In quanto autem tempore ip&longs;a A L, lata e&longs;t per circunferentiam M L, in
tanto temporis &longs;patio in maiori circulo B, extremum ip&longs;ius B A, latum erit
per maiorem arcum quàm &longs;it B S; iam con&longs;ideraudum e&longs;t motus vtriu&longs;que
lineæ in hoc ca&longs;u æquales e&longs;&longs;e, &longs;unt enim de&longs;cripti per lineas æquales T S,
Q L, quæ &longs;unt rectæ; tam enim linea B A, quàm M A, naturali motu recta
tenderet, vt dictum e&longs;t, Verum lationes innaturales &longs;unt impares, latio enim B T, breuior e&longs;t M
quantitate autem B T, retracta e&longs;t B A, à motu &longs;ibi naturali, & recto: quan
titate verò M Q, retracta e&longs;t M A, vnde apparet motu hoc violento magis
retractam e&longs;&longs;e minorem M A, quàm maiorem B A, quod erat primo de
clarandum.
Quod autem ob id A B, maior c&ecedil;lerius mota &longs;it motu naturali, quàm mi
nor M A, palàm fiet. quia enim oportet
norem eadem vi motam, confeci&longs;&longs;e binos illos motus proportionales, ide&longs;t
ita &longs;e debet habere motus naturalis maioris ad motum innaturalem eiu&longs;
dem, quemadmodum &longs;e habet motus naturalis minoris ad motum innatu
ralem eiu&longs;dem: Oportet ergo, vt &longs;i A B, & A M, &longs;unt eadem vi commotæ,
vt &longs;it eadem ratio T S, ad Q L, quæ e&longs;t B T, ad M Q, non e&longs;t autem, vt o&longs;ten
&longs;um e&longs;t; ergo linea A B, eadem vi commota, ac M A, conficit plu&longs;quam
B S, &longs;ed nece&longs;&longs;ariò peruenit ad F. hoc enim in puncto erunt prædicti motus
proportionales, vt oportet, erit enim motus naturalis in maiori perpendi
cularis F X, & innaturalis B X, in minori verò naturalis L Q, innaturalis
M
gula B X F, M Q L, & erunt per 4. 6. vt F X, ad B X. ita L Q, ad M Q, &
permutando erunt etiam vt F X, ad L Q, ita B X, ad M
naturalis ad naturalem, ita innaturalis ad innaturalem. In alio autem lo
co præter F, non erunt eædem proportiones.
Ex quibus patere &longs;atis pote&longs;t, cur A B, longior à centro velocius mouea
tur quàm minor M A, &longs;eu cur puncta eiu&longs;dem B A, velocius vertuntur, quo
longius ab&longs;unt à centro A, ide&longs;t maiorem arcum B F, peractum e&longs;&longs;e à B,
quàm &longs;it arcus M L, peractus ab M, quod erat o&longs;tendendum.
Atque hic e&longs;t di&longs;cur&longs;us ille Ari&longs;t.
quo putat &longs;e cau&longs;am aperui&longs;&longs;e, cur lon
gior &longs;emidiameter velocius moueatur: quod num rectè attigerit, non puto
operæpretium e&longs;&longs;e hoc loco di&longs;cutere, præ&longs;ertim cum ad naturalem Philo
&longs;ophum &longs;pectet.
Mihi tamen maximè con&longs;iderandum videtur hoc ip&longs;um quod a&longs;&longs;eruit, &
ueri, quàm viciniores; ex hac enim maiori velocitate &longs;equitur maiore etiam
vi moueri, vnde & potentiæ mouenti in extremo eius vis augebitur, & plus
poterit quam &longs;ola &longs;ine vecte, e&longs;t enim vectis duæ &longs;emidiametri altera alte
ram longior; ex quibus fortè apparet vnde vectis vires oriantur.
His igitur tanquam huius Mechanicæ facultatis principijs po&longs;itis, ad va
rias Quæ&longs;tiones di&longs;cutiendas accedit.
Cvr autem maiores libræ minoribus &longs;int exactiores, palàm e&longs;t ex
præmi&longs;&longs;is principijs. con&longs;iderare enim oportet, quod in motu li
bræ de&longs;cribitur quidam circulus, cuius diameter &longs;unt ip&longs;a libræ
brachia, centrum verò e&longs;t fpartum, &longs;iue trutina; hoc enim pun
ctum in motu libræ manet: duo verò brachia &longs;unt veluti duæ &longs;emidiametri
à centro exeuntes, vt in figura cerne
re e&longs;t, in qua centrum, &longs;iue &longs;partum
e&longs;t vbi C, reliqua &longs;unt manife&longs;ta. In
eadem porrò figura libra maior &longs;it
A B. minor verò circa idem &longs;partum
C, &longs;it F G. Iam vt præmi&longs;&longs;um e&longs;t, ea
dem vi, vel eodem onere in lance B,
po&longs;ito, mouebitur velocius brachium
libræ maioris, quàm minoris &longs;it ma
ior tran&longs;lata ad
mota e&longs;t per arcum B E, vel A D. Minor autem libra acta e&longs;&longs;et per mino
rem arcum G I, vel F H, melius autem apparet arcus B E, maior, quam mi
nor G I,
quòd nonnulla pondera in minimis libris adeò paruam brachiorum aper
tionem faciant, vt ægrè percipi po&longs;&longs;it; in magnis verò propter brachiorum
longitudinem valdè &longs;en&longs;ibilem efficiant. quædam verò benè, & in magnis,
& in paruis apparent, &longs;ed tamen &longs;emper melius in magnis ob dictam ratio
nem. Quamobrem machinantur ij, qui purpuram vendunt, vt pendendo
defraudent, tum in medio libræ non ponentes &longs;partum, vt hoc modo bra
chium ex vna parte longius factum facilius moueatur, & proinde à minori
purpuræ pondere; tum etiam
merces imponitur, vel partem illam lancis, quam magis grauitare cu
piunt ex ligno radici proximo, vel ex nodo&longs;o facientes: lignum
enim, quod radici proximum e&longs;t, graue admodum e&longs;t,
quemadmodum etiam nodus; quia nodus e&longs;t,
quædam radix.
mæ quæ&longs;tionis paraphra&longs;is.
piam id amouet, rur&longs;um a&longs;cendit libra? Si autem deor&longs;um constitutum
fuerit, non a&longs;cendit, &longs;ed manet? An quia &longs;ur&longs;um &longs;parto quidem exi&longs;ten
te plus libræ extra perpendiculum fit, &longs;partum enim e&longs;t perpendiculum,
quare nece&longs;&longs;e est deor&longs;um ferri id, quod plus est, donec a&longs;cendat, quæ bifariam li
bram diuidit ad ip&longs;um perpendiculŭ, cum onus incumbat ad libræ partem tractam.
A D: hoc igitur &longs;ur&longs;um erecto, perpendi
culum erit vbi A D M. &longs;i igitur in ip&longs;o B,
ponatur onus, B, quidem de&longs;cendet vbi E;
C, autem a&longs;cendat vbi H, quamobrem ea,
quæ bifariam libram &longs;ecat, primò quidem
erit D M, ip&longs;ius perpendiculi; incumben
te autem onere erit D G, quare libræ ip
&longs;ius vbi E H, quod extra A M, perpendi
culum e&longs;t, vbi e&longs;t D D H, maius e&longs;t dimidio. &longs;i igitur amoueatur onus ab ip&longs;o E, ne
ce&longs;&longs;e e&longs;t H, deor&longs;um ferri, minus enim e&longs;t ip&longs;um E D. &longs;i quidem igitur &longs;ur&longs;um ha
buerit &longs;partum, propter hoc a&longs;cendit libra. &longs;i autem deor&longs;um fuerit, id quod &longs;ub
stat, contrarium facit; plus enim dimidio fit libræ, quæ deor&longs;um e&longs;t, pars, quàm
quod perpendiculum &longs;ecet; quapropter non a&longs;cendit, pars enim eleuata leuior e&longs;t.
K L M, bifariam igitur &longs;ecatur N G. im
po&longs;ito autem onere in ip&longs;o N, erit quidem
N, vbi O, ip&longs;um autem G, vbi R; K L, au
tem vbi K P, quare maius e&longs;t L P O, quàm
L R, ip&longs;o P L. Ablato igitur onere, ne
ce&longs;&longs;e e&longs;t manere; incumbit enim, ceu onus
exce&longs;&longs;us medietatis in quo P L.)
te textum græcum e&longs;&longs;e mendo&longs;um, la
tinum vero mendo&longs;i&longs;&longs;imum. Ego partim ex certa rei intelligentia, vti vi
des re&longs;titui. Porrò quoniam Piccolomineus,
& &longs;i plurimum, vt ip&longs;e fatetur, in&longs;udauerit, non tamen &longs;olutionem huius
quæ&longs;tionis e&longs;t a&longs;&longs;ecutus, eam tibi ex Mechanicis Guidibaldi tradam. Ari&longs;t
igitur ponit duas libræ &longs;pecies, &longs;iue potius duas eiu&longs;dem libræ po&longs;itiones,
vnam, quæ haber &longs;partum, &longs;iue perpendiculum &longs;upra; alteram, quæ infra.
vt in præ&longs;enti figura, &longs;it libra B C, cuius
&longs;partum, &longs;iue perpendiculum A D, &longs;it &longs;ur
&longs;um, ita vt in puncto A, &longs;it affixum perpen
diculum, & circa idem punctum A, tan
quam circa centrum tota libra circum
uertatur. hæc e&longs;t prima libræ collocatio.
&longs;it deinde libra B C, cuius &longs;partum,
&longs;iue perpendiculum A D, &longs;it deor&longs;um, vt in altera figura,
ctum A, tanquam circa
ita fixum, vt ip&longs;i libræ conuer&longs;io innita
tur, quæ e&longs;t altera libræ po&longs;itio. Quærit
igitur, cur &longs;i in libra &longs;ur&longs;um
pendiculum, & centrum, ponatur ex vna
parte onus quodpiam, v. g. in parte B, vt in prima textus figura factum e&longs;t,
libra de primo &longs;itu B C, mouetur ad &longs;itum E H, &longs;ed tamen ablato pondere
reuertitur &longs;ua &longs;pontè ad pri&longs;tinum &longs;itum B C. &longs;i autem in libra, cuius per
pendiculum, ac centrum deor&longs;um &longs;it, vt in &longs;ecunda figura textus, pondus
imponatur, ip&longs;a quidem à &longs;itu B C, ad &longs;itum O R, transferretur; verumta
men ablato onere,
Huic quæ&longs;tioni, vt re&longs;pondeat, tacitè &longs;upponit omne graue tendere de
or&longs;um, hoc pacto, vt centrum grauitatis ip&longs;ius tendat per lineam rectam
ad mundi centrum ab ip&longs;o grauitatis centro protractam, quam lineam Di
rectionis Recentiores appellant. &longs;ciendum autem centrum grauitatis e&longs;&longs;e
punctum quoddam in quolibet graui, ex quo &longs;i graue illud &longs;u&longs;pendatur, &longs;em
per manet in æquilibrio, nec vnquam po&longs;itionem re&longs;pectu &longs;uarum partium
mutat, quamuis ita &longs;u&longs;pen&longs;um huc illuc transferatur. Ita Pappus Alexan
drinus initio octaui libri Mathematicarum collectionum. Totius igitur li
bræ ab&longs;que onere centrum grauitatis e&longs;&longs;et circa punctum D, quod e&longs;&longs;et di
&longs;tinctum à centro circumuolutionis A. quod grauitatis centrum, &longs;emper
quantum fieri pote&longs;t, &longs;i nihil ob&longs;tet, centro mundi appropinquat; & propte
rea facit, vt prior libra &longs;ine onere &longs;u&longs;pen&longs;a in A, in æquilibrio, atque hori
zonti parallela permaneat, &longs;tante enim D, centro mundi maximè propin
quo, &longs;iue in loco humillimo, erit inter punctum A, & centrum mundi, ac
con&longs;equenter in linea directionis. quæ linea directionis in prima figura
textus e&longs;&longs;et eadem cum perpendiculo A D M, manente libra &longs;ine pondere
horizonti parallela; in
perpendiculo K L M, antequam libra ob impo&longs;itum onus ab æquilibrio di
moueretur. per hanc enim lineam centrum grauitatis libræ, quod e&longs;t propè
puncta D, & L, tenderet ad mundi centrum, &longs;i libra liberè ad centrum mun
di dilaberetur. his præmi&longs;&longs;is &longs;ic quæ&longs;tioni &longs;atisfacit, & primò primæ parti,
quando nimirum &longs;partum &longs;upernè collocatum e&longs;t. Ratio igitur, cur tunc li
bra amoto pondere ad horizontis æquilibrium reuertatur e&longs;t, quia pondus
libræ impo&longs;itum in altera tantum libræ parte, grauitando impellit libram
ad alium &longs;itum E H, ita vt maior pars libræ con&longs;tituatur ex altera parte li
neæ directionis prioris A D M, in qua etiam parte exi&longs;tit centrum granita
tis libræ ip&longs;ius, e&longs;t enim circa D, quod centrum vi ponderis incumbentis in
E, cogitur paulùm a&longs;cendere,
mundi centro recedere, vt &longs;i in libra B C, appendatur onus in B, vt in pri
ma textus figura; B, de&longs;cendet ad E, & C, a&longs;cendet ad H, & centrum graui
tatis D, paulùm a&longs;cendet à centro mundi, & linca A D M, quæ libram bi
fariam &longs;ecabat modo tran&longs;lato perpendiculo in A D G, non amplius cam
bifariam &longs;ecabit; &longs;ed libræ E H, maior pars erit vltra perpendiculum A D
M, quæ maior pars e&longs;t D D H.
Si igitur nunc onus amoueatur libræ E H, centrum grauitatis, quod e&longs;t
plius illi æ que ponderat, grauitabit, & quia libra cùm affixa &longs;it ad A, nequit
deor&longs;um recta tendere, circumferretur circa A, trahente ip&longs;am grauitatis
centro, cum nihil ob&longs;it, donec iterum perpendiculum A D G, priori &longs;itui
A D M, congruat: hac enim ratione centrum grauitatis, quantum pote&longs;t,
iuxta naturam &longs;uam de&longs;cendet,
re&longs;tituetur. Si autem deor&longs;um fuerit &longs;partum in &longs;ecunda figura textus, im
po&longs;ito pondere contrarium accidit, quia maior pars libræ, & in qua cen
trum grauitatis e&longs;t, in tali motu de&longs;cendit: altera autem pars minor, ac læ
uior &longs;ur&longs;um tollitur. & quia graue natura &longs;ua nequit a&longs;cendere, propterea
ablato pondere non reuertitur ad æquilibrium B C, cum centrum grauita
tis a&longs;cendere ne queat, quod tunc oporteret.
Sit libra N G, in &longs;ecunda figura, cuius perpendiculum,
nis linea &longs;it K L M, quæ libram in prima po&longs;itione diuidit bifariam; impofi
to autem onere in N. N, trahetur ad O, & G, ad R, & K L, vbi K P. quare
maior e&longs;t O L, in quo
&longs;uperat enim O L, ip&longs;am L R, exce&longs;&longs;u duplæ P L, quod facilè apparet &longs;i po
natur tota O R, 10. & dimidia O P, & O R, 5. & P L, ponatur 2. erit enim
tunc O L, 7. & L R, 3. quæ hanc &longs;uperat 4. duplo &longs;cilicet ip&longs;ius P L, 2. qua
re ne&longs;cio cur Ari&longs;t.
dicat, ip&longs;am O L, &longs;uperare ip&longs;am L R, &longs;olùm quantitate
P L. Quapropter etiam &longs;i onus auferatur, nece&longs;&longs;e e&longs;t ibi libram manere,
quia maior, & grauior ip&longs;ius pars deor&longs;um e&longs;t, nec pote&longs;t natura &longs;ua læui
tare, vel a&longs;cendere, vt oporteret, &longs;i ad pri&longs;tinum &longs;itum N G, re&longs;titui debe
ret. remanebit igitur in O R.
Ex his, quæ&longs;tionis &longs;olutionem, textus explicationem, ac re&longs;titutio
nem habeto.
Aduertendum quoad &longs;ecundam libram, ne &longs;imul cum 10. Bapti&longs;ta Bene
dicto in libro &longs;peculationum immeritò Ari&longs;t.
erroris arguamus: ip&longs;e enim,
quia libram hanc non agnouit, au&longs;us e&longs;t affirmare, Ari&longs;tot. hoc loco fal&longs;um
pror&longs;us dixi&longs;&longs;e, cum dixit libram &longs;patto infimè collocato, non redire ad
pri&longs;tinam po&longs;itionem.
Cvm textus tam græci, quàm latini mendis &longs;cateant,
maioris &longs;int momenti, eos per paraphra&longs;im explicabo, in qua ta
men totus textus continebitur, Cur exiguæ
vires (quemadmodum à principio dictum e&longs;t) adhibito vecte, ma
iora mouent pondera, quam contrarium enim videtur debere
fieri, nam mouenti additur grauitas vectis, & ideò pondus augetur, ergò
difficilius ip&longs;um cum vecte, quàm &longs;ine eo mouere deberet.
Vectis porrò e&longs;t in&longs;trumentum oblongum, quo ad &longs;ubleuandum graue
quodpiam vtuntur opifices, quod innititur cuidam fulcimento, quod græcè
hypomoclion dicitur: hypomoclion antem oncri leuando, quanrum &longs;ieri
mouentis. vt plurimum verò fulcimentum e&longs;t inter pondus, & potentiam:
aliquando etiam e&longs;t ex altero vectis extremo, ita vt onus &longs;it inter fulturam,
& potentiam; aliquando potentia e&longs;t inter vtrunque, vnde tres vectis &longs;pe
cies exi&longs;tunt. vt in &longs;ubiectis figuris apparet.
In prima, vectis e&longs;t A B, fultu
ra E, onus C. potentia autem &longs;eu vis,
&longs;eu aliud pondus
deor&longs;um in D, præmens eleuabit &longs;ur
&longs;um ex altera parte onus C. & vectis
circa fulturam E, tanquam centrum
conuertetur. In altera figura pondus
e&longs;t inter fulturam, & potentiam, ful
tura autem in altera extremitate, vt
patet in figura, hic autem potentia
non præmit deor&longs;um in D: &longs;ed &longs;ur&longs;um
vectem eleuando pondus C, attollitur. In tertia tandem figura potentia, e&longs;t
inter vtrunque, e&longs;t enim in D, ibique
&longs;ur&longs;um vrget. verum tamen e&longs;t hunc vectem artificibus e&longs;&longs;e inutilem, quip
pe qui nullo modo iuuet potentiam, imò verò pondus ip&longs;um grauius reddit:
Re&longs;pondet igitur dubitationi, dicens rationem huius incrementi poten
tiæ motricis, quod fit a&longs;&longs;umpto vecte fortè inde oriri, quod vectis &longs;it quæ
dam libra, cuius alterum brachium &longs;it altero longius; in prima autem quæ
&longs;tione explicatum e&longs;t, cur libra maior, maiorem vim habeat, eam ad cir
culum reducendo; vectis autem fit libra, hypomoclion enim e&longs;t loco &longs;parti,
tam enim &longs;partum, quam hypomoclion veluti centra manent. quoniam ve
rò ab eodem pondere, c&ecedil;lerius, &longs;iue maiori vi mouetur linea, quantò lon
gior à centro fuerit, vt dictum e&longs;t de admiranda circuli natura; hinc fit, vt
cum duæ &longs;int in vecte potentiæ, &longs;iue duo pondera, mouens, & motum, illud
facilius ac maiore vi moueat, &longs;iue vires ex vecte acquirat, quod longiorem
vectis partem pre&longs;&longs;erit. quemadmodum igitur pars vectis longior, quæ &longs;pe
ctabat ad mouentem potentiam, &longs;uperat minorem partem, in qua e&longs;t mo
tum; ita etiam maius e&longs;t pondus &longs;emper autem quan
to ab hypomoclio magis di&longs;tabit potentia, tantò facilius mouebit, cuius
cau&longs;a &longs;upra reddita e&longs;t, quoniam nimirum, quæ plus à centro elongatur ma
iorem de&longs;cribit circulum, qui magis ad lineam rectam accedit: quare ab
eadem potentia adhibito vecte, tantò facilius pars vectis mouens dimoue
bitur, quantò magis à fulcimento di&longs;tabit. Exempli gratia &longs;it in &longs;uperiori
prima figura vectis A B, pondus C, mouens D, hypomoclion E, in qua præ
dicta poteris contemplari. vltima illa textus verba
uens, vbi F, motum autem vbi C, pondus in G,)
mendosè addita.
In hac quæ&longs;tione re&longs;pexit Ari&longs;t. &longs;olùm ad primam vectis &longs;peciem.
Illud
demum, quod dixit eandem habere rationem potentiam ad pondus, quàm
partes vectis inuicem demon&longs;tratum e&longs;t po&longs;tea acuti&longs;&longs;imè ab Archimede
uis modo, & vnica demon&longs;tratione à Guido Vbaldo in &longs;uis Mechanicis pro
po&longs;itione 1. de Vecte, quæ e&longs;t huiu&longs;modi; Potentia &longs;u&longs;tinens pondus vecti
appen&longs;um, eandem ad ip&longs;um pondus proportionem habet, quam vectis di
&longs;tantia inter fulcimentum, ac ponderis &longs;u&longs;pen&longs;ionem, ad di&longs;tantiam, à fulci
mento ad potentiam interiectam. quod de omni vecte ab eo demon&longs;tratur,
cuius propo&longs;itionis &longs;en&longs;us e&longs;t hic; in &longs;uperiori prima figura &longs;i pars vectis
E B, fuerit, v.g. qua drupla partis A E; etiam pondus C, erit quadruplo ma
ius pondere, &longs;eu vi in D, quæ ip &longs;um C, ope vectis &longs;u&longs;tinet. quod etiam trans
ferre debes ad &longs;ecundam figuram.
EI, qui &longs;uperiora intellexerit &longs;atis clara videtur.
Illud tamen non
omittendum, &longs;cilicet dicendum potius Remum e&longs;&longs;e vectem &longs;ecundi
generis, quàm primi, quod fortè Ari&longs;t.
non animaduertit, nec Pic
colomineus, nam mare e&longs;t hypomoclion, re&longs;pectu enim nauis non
mouetur, &longs;ed manet, &longs;calmus autem &longs;imul cum tota naui e&longs;t pondus motum;
verè enim nauis ip&longs;a mouetur. mouens e&longs;t ip&longs;e remex.
Reliqua in textu
&longs;unt clara.
Qvemadmodum in præcedenti quæ&longs;tione Ari&longs;t.
vectem &longs;ecundi ge
neris ad &longs;olutionem non adhibuit, vt par erat, & propterea ob&longs;cu
rior eua&longs;it, ita etiam in præ&longs;enti, qu&ecedil;&longs;tionem ad ve ctem primi ge
neris reducit, quæ ad alterum reducendà erat:
tas, atque prolixitas &longs;olutionis manauit. E&longs;t enim propriè Temo, &longs;iue gu
bernaculum nauis, vectis &longs;ecundi generis, vt mox explicabo, e&longs;t enim temo
in&longs;trumentum in extrema nanis par
te, &longs;eu puppi affixum, vt in figura prç
&longs;enti vides tabellam, in qua B C D,
cuius manubrium A B, intra nauim
recipitur, quæ tabella, &longs;eu temo in
duobus cardinibus, vbi C, & D, cir
cumuertitur à Nauis gubernatore,
manubrium vbi A, tractante; ex qua
conuer&longs;ione nauigium, quò vult ip&longs;e
gubernator facilè dirigit, ip&longs;umque
nauigium huc illuc quamuis adeò magnum ip&longs;e &longs;olus impellit, & agitat. e&longs;t
enim temo vectis, cuius auxilio vires mirum in modum augentur, nam to
ta A B, e&longs;t ip&longs;a Vectis longitudo, cuius hypomoclion e&longs;t mare, cui contra
bus C D, mouenti re&longs;i&longs;tit, & quod præcipuè mouere gubernator intendit. cum igitur motum onus &longs;it intra vectis extrema, hypomoclion in extremo
ad B E, vbi in motu temonis tabella mare vrget, quod minimè cedit,
in hoc motu ferè maneat, & fiat qua&longs;i centrum, circa quod totus temo cir
cumducitur, patet temonem e&longs;&longs;e vectem &longs;ecundæ &longs;peciei, vt dicebam. quod
etiam hinc patere pote&longs;t, quia temo e&longs;t veluti remus, cuius &longs;calmus &longs;int car
dines C, D. &longs;icut ergo remus e&longs;t vectis &longs;ecundi generis, cuius pondus e&longs;t
&longs;ealmus, & mare hypomoclion; ita temo erit vectis eiu&longs;dem generis, cuius
pondus erit vbi cardines, fultura verò mare.
Quærit igitur Ari&longs;t.
vnde nam tantas vires paruus nauis temo guberna
tori &longs;uggerat,
turam obtineat, cuius inquit onus e&longs;t mare, melius autem, vt dixi, dixi&longs;&longs;et
onus e&longs;&longs;e nauim, mare autem hypomoclion, mouens autem e&longs;t gubernator. Differunt autem remus, & temo, quamuis
&longs;ecundum latitudinem nauis, &longs;eu ad latera nauis mari obnititur. temo au
tem in directum ferè nauigij con&longs;titutus mare &longs;cindit. hinc fit, vt remus ad
nauem antror&longs;um rectà agitandam, gubernaculum verò ad eam in latera,
& obliquè contor quendam idoneum &longs;it. quoniam enim mare e&longs;t hypomo
clion, fit vt dum gubernator mouet an&longs;am temonis in A, &longs;eu ad dextram,
&longs;eu ad &longs;ini&longs;tram &longs;ecum ad eandem partem trahat nauigium, quod temoni
e&longs;t connexum; ad
impingit.
Po&longs;thæc &longs;equuntur huiu&longs;modi verba
iacet, quoniam
celerrimè fertur, quoniam quemadmodum in ijs, quœ feruntur in fine deficit latio,
&longs;ic ip&longs;ius continui in fine imbecili&longs;&longs;ima e&longs;t latio, imbecili&longs;&longs;ima autem ad
est facilis, propter hœc igitur in puppi gubernaculum ponitur)
videtur difficilis,
&longs;unt. Piccolominæus quidem plura quàm Ari&longs;t.
fatur, &longs;ed non clariora.
dif
ficultas e&longs;t in verbis illis
&longs;ius continui in fine imbecili&longs;&longs;ima eft latio)
continuum aliquod proiectum fertur per aera, pars ip&longs;ius anterior ea e&longs;t,
quæ præ cæteris partibus principaliter mouetur, & ad cuius motum reliquæ
po&longs;teriores tanquam &longs;ub&longs;e quentes moueantur; qua&longs;i dicat tota vis lationis
e&longs;t in anteriori parte: &longs;iue ip&longs;i impetus maior ine&longs;t: videmus enim proiecta,
quorum vna pars e&longs;t cæteris grauior, quia ei parti melius imprimitur mo
tus, eam etiam fieri anteriorem in latione, quamuis initio fuerit po&longs;terior. &longs;ic etiam quando graue fertur deor&longs;um, dicimus ip&longs;um ferri &longs;ecundum cen
trum grauitatis ip&longs;ius,
ctis partem anteriorem dicere po&longs;&longs;umus e&longs;&longs;e, &longs;ecundum quam totum conti
nuum fertur:
ri impetu,
uis priorem æqua velocitate con&longs;equatur, non tamen tanto impetu, cum ip
&longs;a ad alterius impetum moueatur, & propterca latio ip&longs;ius e&longs;t admodum
imbecillis.
Si quis &longs;agittam per aerem latam à &longs;uo motu vellet deflectere, eam faci
lius in po&longs;teriore parte à &longs;uo cur&longs;u deuiaret, quàm in anteriore. hunc con
cinui corporis motum continuo proiectorum motui a&longs;&longs;imilat: quemadmo
dum enim motus proiectorum in fine debilior lente&longs;cit: &longs;ic totum conti
nuum in po&longs;trema parte &longs;egnius impellitur. Quia igitur nauis e&longs;t
quod vi remorum recta antror&longs;um fertur, & propterea maiore vi prora,
quàm puppis, facilius e&longs;t à &longs;uo directo cur&longs;u nauem deflectere, eam in pup
pi, quàm in prora commouendo. hac igitur de cau&longs;a, gubernaculum puppi
affigitur. quæ quidem ratio, & quantum valcat, & an naui quadret, & num
benè &longs;it explicata, phy&longs;icorum e&longs;t iudicare.
Ego tamen aliam huius rationem video, quia nimirum &longs;i temo in priori
parte e&longs;&longs;et, quando à rectitudine ip&longs;ius nauis ad dextram, aut ad &longs;ini&longs;tram
e&longs;&longs;et inclinandus, tunc quia aqua in vnam tantum ipfius partem, &longs;eu faciem
tota impingeret, in eam &longs;cilicet, quæ antror&longs;um re&longs;piceret, eam aqua re
tror&longs;um &longs;imul cum tota naui auerteret,
vt prora, cui adhæreret temo extrema fieret. impetus igitur aquæ, & naui
gij temonati, cogit temonem e&longs;&longs;e po&longs;tremum non primum, nec medium. &longs;ubdit po&longs;tea aliam
eiu&longs;dem rationem, quia nimirum parua motione facta in puppi multo ma
ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ
&longs;unt longiores, eò maiorem &longs;ubten&longs;am &longs;ibi lineam re&longs;picit, quod facilè in
ad&longs;cripta figura intueri licet; in qua duæ
lineæ A B, A C, continent angulum A, cui
angulo &longs;ubtenduntur tres lineæ parallelæ
F G, D E, B C, quarum B C, maxima e&longs;t,
quia ibi maiores, &longs;iue remotiores &longs;unt ab
angulo A, duæ rectæ A B, A C, ip&longs;um con
tinentes, quod Geometricè per 4. 6. pro
bari pote&longs;t. &longs;ic etiam facta motione, vel
parua in puppi, tota nauis transfertur ad
alium &longs;itum, ita vt prora multum aliò transferatur, quod non accideret, &longs;i
eadem motio fieret ad medium nauigij. propterea igitur apti&longs;&longs;imè puppi
gubernaculum connectitur.
Ex ij&longs;dem etiam rationibus mathematicis patet, cur magis antror&longs;um
procedit nauigium, quàm remi ip&longs;ius palmula retror&longs;um: eadem enim ma
gnitudo, ij&longs;dem mota viribus in aere plus, quàm in aqua progreditur. Sit igitur A B, remus, G, verò
A, autem in nauigio &longs;it remi initium.
B, verò in mari palmula.
&longs;i igitur A, vbi D, transferatur, per totum &longs;pa
tium A D, non permeabit tantumdem &longs;patij B, B E, enim ponitur
æqualis ip&longs;i A D, &longs;ed minus interuallum propter re&longs;i&longs;tentiam aquæ ex &longs;up
po&longs;itione percurret, quale e&longs;t B F, quod minus e&longs;t quàm A D, quare etiam li
nea B G, abbreuiabitur,
quæ facta e&longs;t D Y, propter duo &longs;indlia triangula D Y A, B Y F, &longs;imilia au
tem triangula &longs;unt ea, quorum anguli vnius &longs;unt æquales angulis alterius,
quo po&longs;ito &longs;unt etiam latera vnius proportionalia lateribus alterius, vt pa
tet ex prima definitione 6. necnon ex quarta eiu&longs;dem demon&longs;tratione. hæc
quidem duo triangula &longs;unt &longs;imi
lia, & rectè concluditur F Y, mi
nus e&longs;&longs;e quàm D Y, &longs;ed tamen
non videntur i&longs;ta propo&longs;itum
o&longs;tendere, quod erat, plus nauim
procedere, quàm palmulam re
trocedere. Fateor quidem tex
tum hunc e&longs;&longs;e ob&longs;curi&longs;&longs;imum,
cteres, qui corrigendi &longs;unt vti nos facimus. ne&longs;cio qua ratione Piccolomi
neus videatur &longs;ibi locum hunc explica&longs;&longs;e. For&longs;itan addenda &longs;unt nonnulla
hoc pacto; cum initio remigationis ponamus remum in &longs;itu A B, in fine ve
rò primæ impul&longs;ionis in D F, &longs;calmum verò circa medium remi in G, pri
mo; vltimo erit etiam circa medium D F, vbi H, quare &longs;calmus tran&longs;latus
e&longs;t à G, ad H,
probare e&longs;&longs;e maiorem ip&longs;a B F, quam palmula obiuit, & con&longs;equenter pro
ba&longs;&longs;et nauigium plus proce&longs;&longs;i&longs;&longs;e, quàm palmula rece&longs;&longs;erit: quod propo&longs;ue
rat. Verum hoc non demon&longs;trat;
po&longs;tea
&longs;ubdit
e&longs;t, extremo B, procedit, vbi extremum in nauigio e&longs;t A, non procederet autcm
vbi est D, ni&longs;i commoueretur nauigiŭ, & eò transferretur vbi e&longs;t remi principium)
vbi in textu mendosè legitur C, pro G.
Sen&longs;us porrò horum verborum e&longs;t hic; &longs;i remus cirea &longs;calmum G, verte
retur, & tamen nauis ab eo non propelleretur, &longs;ed &longs;taret, tunc medium na
uis maneret vbi G, per motum enim remi impellitur in contrarias partes
ip&longs;i palmulæ B, quæ e&longs;t in mari, quia &longs;equitur motum alterius extremi A,
manubrij &longs;cilicet remi, qui e&longs;t in naui: quod autem nauigium à remo mo
neatur, &longs;ignum e&longs;t, quia manubrium A, non procederet vbi e&longs;t D, ni&longs;i pari
ter cum remo nauigium illor&longs;um con&longs;equeretur. Hæc quidem Ari&longs;t.
circa
motum nauigij imperfectè admodum ni&longs;i textus corruptionem cau&longs;etur, di
xi&longs;&longs;e videatur. Quapropter operæpretium me facturum exi&longs;timo, &longs;i Petri
Nonij acuti&longs;&longs;imi Mathematici, &longs;ubtili&longs;&longs;imas,
problema anuotationes hoc loco de&longs;crip&longs;ero, ex quibus perfectè, ac ma
thematicè toti huic quæ&longs;tioni fit &longs;atis, quæ &longs;ic &longs;e habent.
ex remis, annotatio Petri Nonij.
Cvm olim di&longs;cipulis no&longs;tris mechanicas Ari&longs;t.
quæ&longs;tiones interpre
taremur, nonnulla circa problema illud annotauimus, cur magis
procedat nauigium, quam remi palmula in contrarium. Ari&longs;tot.
enim ratiocinatio ob&longs;cura e&longs;t; quam nos tamen, vt aliquid lucis
haberet, ad hunc modum explicauimus; & propter materiæ &longs;imilitudinem
hi&longs;ce no&longs;tris libris de nauigandi ratione adiunximus. Supponit autem ip&longs;e
auctor remi palmulam retrocedere, quoties nauigium in anteriora progre
ip&longs;ius remi po&longs;itum e&longs;&longs;e, vt &longs;cilicet tantum di&longs;tet à manubrio, quantum à
palmula. Duæ
in C, puncto medio &longs;e inuicem &longs;ecent, & connectantur A B, & D E: remus
autem in initio vnius remigationis po&longs;itionem habeat rectam lineam A B, Cum igitur A, remi ca
put in fine ip&longs;ius remigationis eò tran&longs;latum fuerit D, non erit B, vbi E; &longs;i
enim ibi fuerit; remus igitur po&longs;itionem
habebit rectam lineam D E; & quoniam
contrapo&longs;iti anguli, qui ad C, æquales &longs;unt,
& duo latera A C, & D C, trianguli A D C,
duobus lateribus B C, & C E, trianguli B
E C, æqualia etiam &longs;unt: reliqui igitur an
guli,
les erunt per 4. propo&longs;itionem primi libri
Euclidis, & propterea tantum &longs;patium per
curret B, quantum A: &longs;calmus verò C, im
motus omninò erit: & nauigium idcircò, in
quo ip&longs;e &longs;calmus, immotum etiam erit con
tra hypothe&longs;im. &longs;upponitur enim in que&longs;tio
ne, quod nauigium illa remigatione in anteriora moueatur, remi verò pal
mula retrocedat. Scalmus porrò quamquam circularis remi motus expers
&longs;it; motu tamen nauigij commouetur. Remus igitur po&longs;itionem habeat in
fine ip&longs;ius remigationis rectam lineam D Z, quæ quidem rectam A B, &longs;ecec
in T, inter B, & C; rectam verò B E, in Z. Et quoniam duo coalterni anguli
C A D, & C B E, æquales
æqualis e&longs;t: duo igitur triangula A T D, & B Z T, æquiangula erunt per 32.
primi, & communem &longs;ententiam. Similia
raqueMaior e&longs;t autem A T, quàm B T: maior igitur D A, quàm B Z, quod etiam
per
Maius
tran&longs;uehetur nauigium, quò remi capulus deportatus fuerit: nauigium igi
tur in diuer&longs;a procedens, plus &longs;patij, quàm remi palmula tran&longs;mittet. Vti
mur aurem tralatione, Aduer
tendum e&longs;t tamen, quod cum remus po&longs;itionem habuerit D Z, remi palmu
la erit infra Z. Nam quoniam
lia po&longs;ita &longs;unc: duo igitur anguli, qui ad D, & A, æquales erunt: angulus
igitur A D T, angulo D A T, maior erit: & idcircò latus A T, trianguli A
T D, latere D T, maius erit per 19. primi. Aæqualis porrò o&longs;ten&longs;us e&longs;t an
guius B Z T, angulo A D T, præterea angulus D A T; angulo T B Z, æqua
lis: angulus igitur B Z T, angulo T B Z, maior erit, & propterea latus B T,
trianguli B T Z, latere T Z, maius erit: tota igitur recta linea A B, tota
D Z, maior erit: & idcircò cum remus po&longs;itionem habuerit rectam lineam
D Z palmula erit vltra Z. E&longs;to igitur in K, & connectantur rectæ lineæ B D,
& B K: &longs;patium igitur decur&longs;um ab ip&longs;a palmula non erit B Z, &longs;ed B K: quod Nam quoniam duo latera
B D, & D K, trianguli B D K, duobus lateribus B D, & D E,
æqualia &longs;unt, &longs;ed minor e&longs;t angulus B D K, angulo B D E: minorigitur erit
ba&longs;is B K, ba&longs;e B E, per 24. primi, quod demon&longs;trandum erat
Præterea, quod Ari&longs;t.
ratiocinando &longs;umit tantum &longs;patium conficere na
uigium, quantum remi manubrium, ambiguum e&longs;t. Nam remi manubrium
duabus fertur motionibus: vna propria,
verò, qua vnà fertur cum ip&longs;o nauigio. &longs;patium igitur, quod omninò decur
&longs;um e&longs;t à remi manubrio, eo quod à nauigio confectum e&longs;t, mains erit. At
&longs;i paria &longs;patia decur&longs;a e&longs;&longs;e intelligat à remi manubrio motu proprio, & à
nauigio, Nam nauiginm interdum maius &longs;pa
tium percurret, interdum minus, iuxta remigum vires, & prout mari remi
palmula immer&longs;a fuerit: remi verò manubrium tamet&longs;i ab exiguis viribus
moueatur haud minorem tamen ambitum de&longs;cribet, quàm &longs;i à multo ma
iore virtute moueretur. Quapropter, vt huiu&longs;modi Ari&longs;t. &longs;ententiam exa
minaremus, Theoremata, quæ &longs;equuntur, demonftrauimus.
Si Remiges nauigium mouere po&longs;&longs;unt, maius &longs;emper &longs;pa
tium remi manubrium percurrit, quàm nauigium.
Sit enim remus A C, manubrium A, &longs;calmus B, qui propter nauigij
motum &longs;patium percurrat à B, in D, in quo loco ip&longs;eremus A C, &longs;i
tum rectitudinis habeat E F. Spatium
itaque, quod A, conficit, curna linea
&longs;it A E, cui recta linea re&longs;pondeat A Z, in re
ctam E F, perpendieularis. Nauigium verò
idem &longs;patium conficiet, quod &longs;ealmus B: aio
igitur ip&longs;am A Z, rectam lineam, recta B D,
maiorem e&longs;&longs;e. &longs;ecet enim recta A C, rectam
E F, in G: æquiangula &longs;unt igitur bina trian
gula A G Z, & B G D, quapropter &longs;icut A G,
ad B G, &longs;ie A Z, ad B D, per. 4. 6. libri Eucli
dis: maior e&longs;t autem A G, ipfa B G, & maior
igitur erit A Z, quam B D. & proinde maius
&longs;patium remi manubrium percurrit, quam
nauigium, quod demon&longs;trandum erat.
Quod &longs;i à puncto B, rectam lineam vtrinque
ducamus H K, ad remi men&longs;uram, rectos facientem angulos cum B D,
ctamque
&longs;tare ex A I, & I Z, quarum prior re&longs;pondet curuæ A H, quæ motu proprio
manubrij de&longs;cripta e&longs;t; po&longs;terior verò æqualis e&longs;t rectæ B D, quæ motu na
uigij decur&longs;a e&longs;t.
Si remi manubrium motu proprio, & nauigium, æqualia
&longs;patia pertran&longs;ierint, fieri non poterit, vt palmula mo
ueatur: &longs;ed veluti centrum immota manebit.
Esto iterum remus A C, manubrium A, &longs;calmus B: tantum autem &longs;pa
tium conficiat nauigium; quantum motu proprio A. Dico, quod C,
remi palmula immota manebit. Nam &longs;i a loco &longs;uo dimota fuerit:
&longs;patium igitur permeet C D, ad po&longs;teriora: quo quidem decur&longs;o,
remus A C, po&longs;itionem rectitudmis habeat F D, &longs;calmus
erit in G. Excitetur autem à puncto B, in
rectos angulos &longs;uper B G, & à
&longs;uper D F: itemque à puncto E, recta C E, &longs;uper
E R; ip&longs;arum verò rectarum linearum E R, &
A H, &longs;ectio &longs;it in K, &longs;ed C F., & D F, &longs;it in Z, & quo
niam A K, id &longs;patium e&longs;t, quod motu proprio re
mi manubrium permeauit, curuilineo enim re
&longs;pondeat A R, recta autem B G, id &longs;patium e&longs;t,
quod nauigium confecit: ip&longs;æ igitur rectæ lineæ
H K, & B G, æquales erunt. Atqui in duobus æqui
angulis triangulis E B C, & B A K, vel per 26.
propo&longs;itionem primi Euclidis, vel 4. 6. æquales
e&longs;&longs;e concludes A K, & E C, rectas lineas: quapro
pter æqualis erit E C, rectæ B G, per communem
&longs;ententiam: eidem autem B G, æqualis e&longs;t E Z,
in parallelogrammo, per 34. propo&longs;itionem ip
&longs;ius primi libri: æqualis igitur erit recta E Z, re
ctæ E C, pars toti, quod e&longs;t impo&longs;&longs;ibile. Et pro
pterea immota manebit palmula C, quod erat à
nobis o&longs;tendendum.
Si remi manubrium motu proprio duplum confecerit &longs;pa
tium, quàm nauigium, tantum prouehetur ea remiga
tione nauigium, quantum palmula retroce&longs;&longs;erit.
Remus enim incipiente motu po&longs;itionem habeat A C, de&longs;inente
verò rectitudinis &longs;itum F G. &longs;calmus igitur B, propter nauigij
motum, &longs;patium con&longs;iciet B D. Excitetur à puncto B, in
partem perpendicularis E Z, in quam veniant a punctis A, & C,
ad rectos angulos rectæ lineæ A E, & C Z: &longs;patium autem A E, à manubrio
decur&longs;um motu proprio &longs;patij B D, duplum
&longs;it: recta verò linea C H, curuæ re&longs;pondeat
C G, quæ à remi palmula de&longs;cripta e&longs;t. Di
co ip&longs;as rectas lineas B D, & C H, æquales
e&longs;&longs;e. Nam in duobus triangulis B A E, &
C B Z, duæ rectæ lineæ A E, & C Z, æqua
les &longs;unt. In parallelogrammo autem B H,
duæ B D, & H Z, æquales, atqui recta A E,
dupla e&longs;t rectæ B D, per hypothe&longs;im; dupla
e&longs;t igitur, & C Z, rectæ H Z, quapropter
C H, & H Z, æquales erunt, Duæ igitur
C H, & B D, æquales per communem &longs;en
tentiam.
Et quia nauigium tantum &longs;patium de
currit &longs;emper, quantum &longs;calmus: &longs;i igitur
remi manubrium motu proprio duplum
confecerit &longs;patium, quàm nauigium, tan
tum prouehetur nauigium, quantum pal
mula retroce&longs;&longs;erit, quod demon&longs;trandum
erat.
Si nauigium minus &longs;patium decurrat, quàm remi manu
brium, &longs;ed &longs;upra dimidium, magis prouehetur, quàm pal
mula retrocedat; &longs;i verò citra dimidium, minus.
In de&longs;cripta enim figura ponatur B D, minor quam A E, &longs;ed eius dimi
dio maior. Dico, quod ip&longs;a B D, maior e&longs;t quàm C H.
Nam B D, &
H Z, æquales &longs;unt: Ad hæc A E, & C Z, æquales &longs;unt rectæ lineæ; ma
ior igitur erit H Z, dimidio ip&longs;ius A E: quapropter reliqua C H, mi
nor dimidio erit eiu&longs;dem A E, & minor igitur erit C H, quàm B D. Spa
tium autem B D, id e&longs;t, quod nauigium conficit, &longs;patium verò C H, remi
palmula in contrarium decurrit; idcircò prior pars Theorematis vera e&longs;t. Po&longs;terior autem &longs;imiliter o&longs;tendetur.
&longs;i enim B D, minor e&longs;t dimidio ip&longs;ius
A E: minor igitur erit, & H Z, dimidio eiu&longs;dem A E; & quoniam A E, &
C Z, æquales &longs;unt: reliqua igitur C H, dimidio eiu&longs;dem A E, maior erit: &
proinde minor erit B D, quàm C H. Nauigium igitur minus &longs;patium de
curret in anteriora, quam remi palmula in contrarium, quod demon&longs;tran
dum &longs;u&longs;cepimus.
Ex hac, & præcedenti infertur, quod &longs;i remi manubrium motu proprio
maius &longs;patium decurrat, quàm nauigium, &longs;iue id &longs;it duplum, &longs;iue mi
anteriora, & quod palmula remi in contrarium &longs;imul iuncta, ei quod ip&longs;um
remi manubrium motu proprio conficit, æqualia erunt. &longs;emper enim B D,
æqualis e&longs;t H Z: tota verò C Z, quæ æqualis e&longs;t A E, ex &longs;uis partibus C H,
& H Z, con&longs;tabit.
Si nauigium longius progrediatur, quàm remi palmula re
trocedat, &longs;patium conficiet plu&longs;quam dimidium eius,
quod motu proprio remi manubrium decurrit:
&longs;i minus, citra dimidium.
Si celerius feratur nauigium, quàm remi manubrium, mo
uebitur palmula in vlteriora,
det,
motum manubrij &longs;uperat.
Habeat enim remus incipiente motu po&longs;itionem A C: de&longs;inente
verò
mus igitur B, propter nauigij
motum tran&longs;latus, erit in D, &longs;it
mi manubrio motu proprio decur&longs;um: &longs;ic
enim celerius dicetur ferri
manubrium. Dico, quòd palmula C, in
vlteriora mouebitur. Nam cum &longs;calmus
B, prouectus fuerit in D: tran&longs;lata erit ip
fa palmula C, vbi G, in rectitudinis &longs;itu,
re&longs;pondet C K: mouebitur igitur palmula
in vlteriora. Nihil autem vnquam retro
cedere, o&longs;tendetur in hunc modum. eadem
enim celeritate mouentur A, in H, & C,
ver&longs;us I, circa &longs;calmum. Atqui per hypo
the&longs;im celerius fertur nauigium, quam A.
in H, celerius igitur ip&longs;um nauigium fer
tur, quàm C, ver&longs;us I. &longs;ed mouetur idem
ad I, quapropter nihil vnquam retrocedet ip&longs;um C, imò verò in vlteriora
progredietur,
I C, ex I K. &longs;i enim remi palmula tota ip&longs;a nauigij celeritate moueretur, vl
tra K, progrederetur, cum B, perueniret ad D: &longs;ed retrahitur interim, pro
pter eum motum, qui fit circa B. Sic igitur palmulæ celeritate, quæ à mo
tu nauigij prouenit retardata, decur&longs;um &longs;patium erit C K. Videtur autem
&longs;olo remorum impul&longs;u hoc fieri non po&longs;&longs;e, &longs;ed alia in&longs;uper virtute impel
lente opus e&longs;&longs;e, vt venti, vel aquæ.
Ex his Theorematis liquet, inquit Nonius, quàm incerta interroget Ari
&longs;toteles, & quàm in&longs;citè re&longs;pondeat. Nam non continuò &longs;i nauigium in an
teriora mouetur, remi palmula retroceder; neque etiam &longs;i retrocedat, mi
nus &longs;patìum tran&longs;mittit in contrarium, quàm nauigium progrediatur. De
mon&longs;trant hoc &longs;ecunda, & tertia propo&longs;itio. Remi verò manubrium motu
proprio, qui circa &longs;calmum fit, & vnà cum nauigij motu maius &longs;patium con
ficit quàm nauigium. &longs;olo autem proprio motu, &longs;i contingat tantum &longs;pa
tium conficere, quantum nauigium, fieri non poterit, vt palmula mouea
tur. fru&longs;tra igitur conatur in vniuer&longs;um demon&longs;tr are remi manubrium ma
ius &longs;patium decurrere, quàm palmulam in contrarium. Præterea quando
nauigium
tium decurrit, quam manubrium: igitur hon æquale. Et proinde con&longs;tat
neque veritatem in propo&longs;ito, neque demon&longs;trationem in ijs, quæ conge
rit, reperiri.
Hucu&longs;que Petrus Nonius:
Reliqua huius textus vtinam quemadmodum &longs;unt clara, ita etiam vera
e&longs;&longs;ent: &longs;ed quia quæ modo dixit de remo, eadem temoni applicat propte
rea ij&longs;dem etiam obnoxia &longs;unt difficultatibus.
Qværit cur quanto Antenna &longs;ublimior fuerit, ij&longs;dem velis, & vento
eodem celerius ferantur nauigia. Re&longs;pondet inde id prouenire,
quia malus, &longs;iue arbor nauis in huiu&longs;modi ventorum impul&longs;u ve
ctis euadit, cuius auxilio idem ventus, qui mouens e&longs;t, maiorem
vim acquirit, quanto longior fuerit pars vectis, quæ inter hypomoclion, &
vim mouentem intercipitur: quando autem altior fuerit antenna, tunc ea
vectis pars longior euadit, & propterea accidit, vt vires ventorum augean
tur. &longs;ed i&longs;ta melius in figura in&longs;piciamus.
&longs;it nauis A B, cuius arbor C D E,
antenna F C G, velum F G H, vectis e&longs;t arbor, cuius fultura e&longs;t in E, extre
mo mali in fundo nauis, onus autem in D, vbi malus exit è carina. mouens
potentia e&longs;t ventus, qui mouet in antenna F C G. quanto igitur &longs;ublimior
e&longs;t antenna, tanto longior euadit vectis E C,
vires. dixi autem onus e&longs;&longs;e in D, quia &longs;i nauis vento ob&longs;i&longs;teret, ip&longs;a inuerte
retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente
veluti centro parte E. quia ve
rò ob maris liquiditatem na
uis minimè obfi&longs;tit, &longs;ed facilè
cedens à ventis vrgetur, hinc
fit, vt meritò dixerim pondus
nauis e&longs;&longs;e ad D, fulcimentum
verò ad E.
Quæ&longs;tio &longs;eptima, & &longs;atis pec
&longs;e clara e&longs;t;
ci e&longs;t eam exponere.
Cur ex figurarum genere quæcun que rotundæ &longs;unt, & cir
culares facilius mouentur?
Tribus autem modis circulum rotari contingit; aut enim &longs;ecun
dum ap&longs;idem, &longs;iue curuaturam centro &longs;imul moto, quemadmo
dum plau&longs;trorum rotæ vertuntur: aut circa manentem axem,
tanquam centrum veluti rotulæ illæ, ex quibus trochlea compo
nitur; vel quibus ad puteos vtimur, quæ quidem rectæ ad horizontem &longs;o
lent con&longs;titui. aut quem ad modum rota figuli, quæ pariter circa
trum gyratur, &longs;ed qua&longs;i pro&longs;trata horizonti æquidi&longs;tans collocata e&longs;t. Quæ
igitur primo modo mouentur, fortè facilius quam figuræ rectilineæ, vt &longs;unt
triangulares, quadratæ, pentagonæ, &c. mouentur, quia circulares figuræ
parua &longs;ui parte, & qua&longs;i in puncto planum, &longs;eu pauimentum contingunt, vn
de fit, vt
tus e&longs;t angulus, ide&longs;t tali angulo planum contingunt, vt ab eo &longs;tatim rotæ
curuatura à terra eleuari incipiat, & propterea parum terræ hæreat: in fi
guris verò rectilineis, in quadrata. v. g. &longs;ecus accidit, quia ab angulo ad an
gulum linea recta tenditur, vnde in ip&longs;ius volutatione po&longs;t contactum vnius
anguli tota recta linea &longs;equens, plano adaptabitur, & non &longs;emouebitur &longs;ta
tim in altum, & ideò multum offen&longs;abit, & impinget,
uebitur. Præterea circulares etiam, &longs;i cui obulam fiunt corpori, illud &longs;imi
liter &longs;ecundum pu&longs;illum tan gunt: rectilineæ verò figuræ, rectitudine &longs;ua
plani multum contingerent. Ad hæc motor mouens huiu&longs;inodi rotas, eas
mouet, quò nutant: nam quando rota erecta e&longs;t &longs;uper pauimentum, dia
meter ip&longs;ius, quæ à contactu pauimenti ad angulos rectos, ad &longs;upremum
le pondus in æquilibrio con&longs;tituatur, cum ex vna parte tantum &longs;it, quantum
ex altera; ex quo fit, vt vel exigua vis ip&longs;am impellere valeat: quando enim
duo æqualia pondera &longs;unt in æquilibrio, quelibet vis pote&longs;t ea ab æquilibrio
dimouere. quando po&longs;tea rota e&longs;t in motu, vel cum primum ei motus fuerit
à motore inditus, &longs;emper nutat ad partes illas, ad quas primum fuit incita
ta per impre&longs;&longs;am motionem, quapropter nullo negotio ad ea&longs;dem partes,
&longs;eu antror&longs;um mouetur; quò enim
tur: quemadmodum è contrario difficillimum e&longs;t in contrariam nutus &longs;ui
partem vnumquodque pellere. Huc etiam pertinet, quod nonnulli dicunt,
circuli nimirum periphæriam perenni ver&longs;ari motu,
ueri. &longs;icuti etiam dicunt, quod manentia propterea manent, quia contrani
tuntur, & ob&longs;i&longs;tunt mouenti: quod fortè dicebant propter maximam circu
li ad motum aptitudinem. & quia &longs;icut diameter ad diametrum, ita maio
ris circuli periphæria ad minoris periphæriam (vt po&longs;tea o&longs;tendam) & quia
quo
periphæria maioris facilius, quàm minoris moueatur, &longs;iue dixeris, quod an
gulus maioris circuli ad angulum minoris nutum quendam habet; & quia
facilius mouetur angulus maioris, quàm minoris, fit, vt maior rota adhi
beatur ad minorem mouendam: & quia intra maiorem infinitæ circa idem
centrum concipi po&longs;&longs;unt, hinc fit, vt rotæ maiores facilius moueantur, &
motæ moueant cæteras intra &longs;e contentas. quod dictum e&longs;t de nutu anguli
maioris circuli ad angulum minoris ex appo&longs;ita figura facilè patebit, vbi
pro minore angulo intelligendus e&longs;t arcus C B,
pro maiore autem arcus D E, quorum
catur angulus, quoniam angulo A, qui e&longs;t in cen
tro opponuntur. Atque hæc &longs;ufficiant deijs, quæ
primo modo moueutur.
Nunc ad ea, quæ reliquis duobus modis cieri
&longs;olent, quæ &longs;cilicet non mouentur &longs;ecundum ap&longs;i
dem, &longs;ed aut iuxta planitiem, ide&longs;t, quæ æquidi
&longs;tanter pauimento collo
aut quæ in loco à terra eleuato, vt troclearum or
biculi. rotæ hæ facilius ip&longs;æ, & ea etiam, quæ ip&longs;is annectuntur commouen
tur, quam &longs;i rectilinea figura con&longs;tarent; non quia parua &longs;ui portione vel
tangant planum, vel offen&longs;ent, &longs;ed ob aliam inclinationem, de qua initio
huius operis ante quæ&longs;tiones dictum e&longs;t, vbi diximus circulum duas incli
nationes ad motum obtinere, &longs;ecundum quas à motore mouetur; vna e&longs;t,
quam diximus naturalem, qua &longs;olet cieri &longs;ecundum periphæriam, motor
enim &longs;emper mouet circulum in periphæria, & &longs;ecundum hanc inclinatio
nem extremum diametri rectà, non circulariter moueretur: hanc inclina
tionem fortè habet à materia grauitante, & in ip&longs;o circulo con&longs;tituta in
æquilibrio: quæ autem in æquilibrio, facillimè cedunt; & qui talia mouent,
qua&longs;i prius mota mouent, & ideò facillimè. Secundum igitur inclinatio
nem hanc, quæ in obliquum e&longs;t, ide&longs;t, quæ &longs;ecundum circunferentiam &longs;it,
ip&longs;am rotam mouens facillimè mouet. altera latio e&longs;t, &longs;ecundum quam cir
trahit continuò extrema diametri; ne recta &longs;ecundum naturalem lationem
ferantur, &longs;ed in orbem circulariter circa centrum gyrentur. hæc Ari&longs;t.
Re
&longs;tat vt &longs;atisfaciam promi&longs;&longs;is.
Dictum e&longs;t ab Ari&longs;t.
in textu
lus ad maiorem)
rijs, vti expo&longs;ui, manife&longs;tum e&longs;t ex 11. propo&longs;it. 5. Pappi Alexandrini, quæ
talis e&longs;t: Circulorum circunferentiæ inter &longs;e &longs;unt vt diametri. quam etiam
Pater Clauius demon&longs;trat propo&longs;. 2. lib.
8. & propo&longs;.
1. lib.
4. Geom. pract.
&longs;i autem de ip&longs;is circulis intelligerentur fal&longs;a e&longs;&longs;ent, non enim e&longs;t circulus
ad circulum, vt diameter ad diametrum; &longs;ed circuli &longs;unt inter &longs;e, quemad
modum à diametris ip&longs;orum quadrata per &longs;ecundam 12. Elem.
quadrata
autem &longs;unt inter &longs;e in duplicata ratione laterum per 20. 6.
rium; hoc e&longs;t &longs;i fiat, vt latus maioris quadrati ad latus minoris, ita latus mi
noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus
ip&longs;ius ad tertiam illam lineam; non autem vt ad latus minoris. cum ergo
circulus &longs;it ad circulum, vt quadratum diametri ad quadratum diametri,
& quadrata non
&longs;ed illorum duplicatam,
Illud demum non ignorandum, quod Guidus Vbaldus propo&longs;it.
1. de Tro
chlea, demon&longs;trat, quod nimirum potentia &longs;u&longs;tinens pondus per rotulam,
cui funis &longs;upernæ fuerit circumductus, qualis ea e&longs;t, qua ad hauriendam ex
puteis aquam vtimur, talis inquam potentia e&longs;t æqualis ponderi; cuius ra
tio e&longs;t, quia tunc trochlea fit vectis, cuius fulcimentum e&longs;t in medio vectis,
pondus verò, & potentia in extremitatibus &longs;unt, & æquidi&longs;tant ab hypomo
clio, & propterea cum &longs;it eadem proportio ponderis ad potentiam, quæ di
&longs;tantiæ ad di&longs;tantiam, vt &longs;upra qu&ecedil;&longs;t. 3. probatum e&longs;t ex Archimede, & Gui
do Vbaldo, di&longs;tantiæ autem &longs;int æquales, er
æqualia, ide&longs;t, &longs;i pondus e&longs;&longs;et vnius libræ, &longs;u&longs;tineretur à tanta vi,
e&longs;t ad libram vnam &longs;u&longs;tinendam, & non amplius. vt autem clarè appareat
vectis in trochlea, & hypomoclion, & æquales di&longs;tantiæ, &longs;it figura, in qua
pondus D, ductario funi D C B E, alligatum. poten
tia
nam potentia premit rotulam in B, & pondus in C, &
cum rotula &longs;u&longs;tineatur in A, à &longs;u&longs;pen&longs;orio F A. erit
punctum A, hypomoclion, quia in motu vectis eua
dit centrum, æquales autem
di&longs;tantiæ
enim ex centro eodem. ex quibus manife&longs;tum e&longs;t hu
iu&longs;modi rotulam nullam vim mouenti addere, &longs;ed &longs;o
lum illud præ&longs;tat, vt omne tollat impedimentum,
quemadmodum ait Ari&longs;t.
manife&longs;tum etiam e&longs;t ma
iorem vim quamlibet, quam &longs;it ea, quæ &longs;u&longs;tinet, po&longs;&longs;e
idem pondus &longs;ur&longs;um mouere. hæc & præ&longs;enti loco, &
&longs;equentibus lucem afferre po&longs;&longs;unt.
tius mouentur? veluti per maiores trochleas, quàm per minores, & &longs;cy
talas &longs;imiliter? An quanto maior fuerit illa, quæ à centro e&longs;t, in æquali
temporis &longs;patio maius &longs;patium conficit? quamobrem æqualì inexi&longs;tente
onere, idem faciet, &longs;icuti diximus maiores libras minoribus exactiores e&longs;&longs;e; &longs;par
tum enim in illis centrum e&longs;t: partes verò libræ vtrinque à &longs;parto &longs;unt veluti lineæ
ex centro)
legantur, quæ dicta &longs;unt de libra in prima quæ&longs;t. & quæ de rota, & trochlea
in proxima præcedenti, à paraphra&longs;i ip&longs;ius &longs;uper&longs;edebo. Illud tamen, quod
magis nece&longs;&longs;arium e&longs;t, non omittam, vt &longs;cilicet difficultatibus quibu&longs;dam
occurram. Et primo, quod Ari&longs;t.
ait, ea quæ per maiores circulos veluti
trochleas, &longs;eu rotulas trahuntur, facilius trahi, quàm ea, quæ per minores,
non videtur ex omni parte verom. nam &longs;icuti in
&longs;um e&longs;t ex Guido Vbaldo, trochlea &longs;implex, &longs;iue rotula illa &longs;triata, cui funis
&longs;upernè inditur, vt in &longs;uperiori figura; nullas addit vires potentiæ, quia re
ducitur ad vectem, cuius fultura &longs;it in medio ip&longs;ius. &longs;iue igitur rotula illa
magna fuerit, &longs;iue parua, &longs;emper in talem vectem re&longs;oluetur, & propterea,
vt etiam experientia con&longs;tat eodem labore aquam hauriunt, &longs;iue rotula illa
magna fuerit, &longs;ine parua. nec minus vera videtur re&longs;pon&longs;io, cum ait
quanto maior fuerit illa, quæ à centro e&longs;t, in æquali
quæ quidem vera &longs;unt, &longs;i intelligantur hoc modo, nimirum, quod quando
plures
moueri nequeat, tunc quanto maior fuerit diameter, & con&longs;equenter cir
cunferentia, tanto velocius mouebitur. &longs;i autem intelligantur de duobus
circulis ab inuicem &longs;eparatis, quorum vnus
quando vtimur modo rotula magna, modo parua ad aquam hauriendam
non videntur vera, in quo &longs;en&longs;u manife&longs;tè loquitur Ari&longs;t. Quapropter vt &longs;in
cerè loquar, nunc ne&longs;cio, qua ratione Ari&longs;t ab errore excu&longs;are valeam, alijs
fortè occurret.
Secundo loco videndum quid &longs;int &longs;cyntalæ.
Vt autem con&longs;tat ex &longs;equenti
quæ&longs;tione 11. &longs;cyntala erat in&longs;trumentum quoddam vectorium, quod ro
tas, &longs;icut currus, aliter tamen factas, habebat, porrò
inter alia &longs;ignificat
quibus vtimur in &longs;ucculis, vulgò Na&longs;pe; & in axe in peritrochio, vt videre
e&longs;t apud hinc factum e&longs;t, vt apud Lacædemonios &longs;cytala
&longs;ignificaret quoddam genus epi&longs;tolæ, quam &longs;cytalem laconicam dicebant,
quia in charta in&longs;tar zonæ oblonga, & circa &longs;cytalam, hoc e&longs;t circa bacillum
quendam &longs;piratim circumuoluta exarabatur; ita vt yer&longs;us &longs;cripturæ &longs;ecun
dum &longs;urculi longitudinem ducerentur, ex quo &longs;iebat, vt per iuncturas mem
branæ, literæ, ac verba procederent, membranam hanc ex &longs;cytala reuolu
tam, & aliter complicatam Imperatori mittebant, re&longs;olutio autem mem
citra iuncturas, partim vltra: eæquè partes, quæ &longs;imul fuerant &longs;criptæ, &
continuatæ, po&longs;t re&longs;olutionem erant ab innicem valde di&longs;&longs;itæ. quapropter
Imperator commenti totius con&longs;cius, eandem membranam &longs;cytali alteri
priori omninò &longs;imili,
iuncturæ priores redibant, quæ literas, ac verba mutila, & imperfecta in
integrum re&longs;tituebant, vt facilè legi po&longs;&longs;ent. hoc illi vtebantur &longs;ecreto, cum
literas ad Imperatores &longs;uos mi&longs;&longs;as, ho&longs;tibus occultas e&longs;&longs;e volebant.
Ex quibus conijcere licet &longs;cytalam fui&longs;&longs;e lignum oblongum, & teres, &longs;iue
vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus e&longs;&longs;ent
margines duo aliquantulum prominentes, ceu binæ rotæ, cum ip&longs;o tamen
continuæ, & connexæ, vt cum ip&longs;o &longs;imul conuoluerentur; non tamen tan
quam circa axem. cuius hanc accipe fi
guram. Quærit igitur Ari&longs;t.
cur huiu&longs;
modi &longs;cytalæ facilius moueantur, quo
maiores ip&longs;arum &longs;unt rotæ. Cui quæ
&longs;tioni &longs;imul, ij&longs;demque verbis, quibus
quæ&longs;tioni de maioribus rotulis re&longs;pondet, &longs;ed non &longs;atisfacit ob eandem ra
tionem, quam ibi attuli. Crediderim tamen maiores &longs;cytalas, & maiores
curruum rotas, & alia id generis, quæ volutantur, ita vt motu progre&longs;&longs;iuo
mutent locum, facilius moueri, &longs;ed ob aliam cau&longs;am, quia nimirum maio
res rotæ minus &longs;i quid obuiam fiat, offen&longs;ant, quia &longs;ua magnitudine quem
libet obicem facilè &longs;uperare po&longs;&longs;unt; cuius cau&longs;a e&longs;t angulus
quem cum terra facit; at verò exiguæ rotæ, &longs;i cui maiori ob&longs;taculo obuia
rint, ip&longs;um nequeunt &longs;uperare, aut &longs;upera&longs;cendere, quia angulum cum ter
ra faciunt in&longs;to maiorem, vnde facilè ip&longs;orum cur&longs;us inhibetur,
pterea præ maioribus tardiores euadunt. Atque hæc in hanc quæ&longs;tionem
dicta &longs;ufficiant.
Cvr libræ, quæ omni incumbente pondere &longs;unt vacuæ ab impo&longs;ito
pondere facilius mouentur, quàm &longs;i quopiam inexi&longs;tente pondere
aliud rur&longs;us onus &longs;uperaddatur. &longs;imiliter etiam rota, & huiu&longs;modi
quippiam, quod grauius quidem e&longs;t, difficilius commouetur quàm
læue, v. g. rota ferrea difficilius, quàm lignea. &longs;imiliter quæ maiora &longs;unt,
etiam &longs;i ex eadem materia con&longs;tent difficilius mouentur quàm minora, vt
rota maior ferrea, quàm minor etiam ferrea. Habet hæc quæ&longs;tio tres par
tes, quibus Ari&longs;t.
re&longs;pondet dicens, quod graue e&longs;t ægrè moneri non &longs;olum
contra nutum &longs;uum, idc&longs;t &longs;ur&longs;um, &longs;ed etiam in obliquum, &longs;eu ad latera, quia
grauia deor&longs;um
re, quia &longs;unt grauiores, & rota ferrea quàm lignea, & ferrea ctiam maior,
quàm minor grauior e&longs;t, ideò difficilius agitatur.
Contra quam re&longs;pon&longs;ionem &longs;ic fortè obijcies; in præcedenti enim quæ
commoueri, hic autem dicit maiorem rotam difficilius quàm minorem mo
ueri. Hanc obiectionem Piccolomineus di&longs;&longs;imula&longs;&longs;e videtur, cui ego, inge
nuè fateor, me &longs;atisfacere ne&longs;cire, vt enim in præcedenti annotaui, nulla
mihi ratio Ari&longs;t.
excu&longs;andi occurrit, alijs fortè occurret. In præ&longs;enti au
tem benè quidem re&longs;pondet, &longs;ed tamen intimam rei cau&longs;am non attingit.
Sciendum igitur e&longs;t id, quod Guidus Vbaldus in tractatu de libra pluri
bus demon&longs;trauit: quod &longs;i quoduis graue &longs;u&longs;pendatur pror&longs;us in
uitatis, ita vt in perfecto &longs;it æquilibrio, tunc &longs;iue magnum, &longs;iue paruum,
&longs;iue graue, grauiu&longs;uè fuerit, à quauis exigua vi poterit ab æquilibrio dimo
ueri. cur ergo in libris, & rotis grauioribus, aut maioribus
trarium o&longs;tendit? ratio e&longs;t, quia hæc omnia communiter non collocantur,
ita vt circa centrum &longs;uum, quod etiam centrum grauitatis e&longs;t, conuerti
po&longs;&longs;int: verum aptantur circa axem, & quidem iu&longs;to maiorem, laxiu&longs;que
circa ip&longs;um conuertuntur, vnde fit, vt ip&longs;a ob in&longs;itam grauitatem premant
axem in &longs;uperiori parte, vnde quando ab aliquo gyrantur, non propriè gy
rant, &longs;ed in &longs;uperiori axis parte hærentes ip&longs;um atterunt; ex qua attritione
fit, vt retardentur,
Ex his, & textus, & ratio Ari&longs;totelis &longs;atis clara redduntur.
Cvr &longs;uper &longs;cytalas facilius portantur onera quàm &longs;uper currus, cum
tamen currus magnas habeant rotas, &longs;cytalæ verò pu&longs;illas?
Quidnam &longs;cytala e&longs;&longs;et explicatum e&longs;t in 9. quæ&longs;t.
Quo autem
modo per &longs;cytalas onera
talas inuicem æquidi&longs;tantes, & aliquantulum &longs;emotas inuicem &longs;ic di&longs;poni,
vt efficiant in&longs;trumentum vectorium currus in&longs;tar, & fortè veteres vteban
tur his &longs;cytalis eo modo, quo nunc architectores vtuntur duobus illis lignis
longis, ac rotundis, quæ vulgò dicuntur Ruccioli.
Re&longs;pondet igitur id accidere, quia rotæ &longs;cytalarum &longs;imul &longs;unt cum &longs;uo
axe coinpactæ, ita vt &longs;imul cum ip&longs;o rotentur: rotæ autem curruum, quia
&longs;eiunctæ &longs;unt ab earum axe, ita vt &longs;ine illius rotatione ip&longs;æ voluantur, fit vt
illæ firmius incedant, nechuc,
ad ip&longs;um axem offen&longs;ent, quemadmodum i&longs;tæ. addit aliam rationem, quia
currus nimia oneris grauitate premens rotas ip&longs;as ferè &longs;i&longs;tit, quod &longs;cytalis
non accidit, cum rotæ ip&longs;arum vnum, & idem cum &longs;uo &longs;int axe. quæ ratio
quantum valeat, ne&longs;cio, nam quamuis rotæ &longs;cytalarum non premantur ab
axe, premitur tamen axis ip&longs;arum ab onere, à quo &longs;imiliter &longs;i&longs;ti debe
rent &longs;cytalæ.
Crediderim ego facilius portari magna onera per &longs;cytalas, propter ip&longs;a
rum firmitatem, currus enim
maioribus oneribus &longs;ufficiunt. Concludit po&longs;tea quæ&longs;tionem dicens, quia
quàm currus, imò ab ip&longs;o onere iam commoto, ip&longs;æ quoque incitentur, &
præterea à potentia per planum infernè, benè &longs;ub&longs;tratum, & complanatum
trahantur, fit, vt qua&longs;i in duobus locis ip&longs;arum rotæ impellantur ab onere
&longs;upra, & à potentia infra; &longs;icque facilius quam currus ingentia præ&longs;ertim
onera vehunt.
Non videtur declaratione indigere.
Declarandum prius quid &longs;it hoc loco iugum: e&longs;t igitur iugum li
gnum illud cylindricum, quod vulgò dicitur Subbio. quo
na ponuntur in ea machina textoria, quam vulgò dicunt Telaio,
qua&longs;i telarium, eo quod in ip&longs;a telæ texantur. alteri autem iugo
conuoluitur &longs;tamen: alteri verò contexta iam tela &longs;ubinde cum opus e&longs;t cir
cumponitur: quæ duo textores faciunt ip&longs;a iuga conuertendo. quæ vt faci
lius conuertant, iugis vtrinque in&longs;erunt per bina foramina binos collopes. qui collopes &longs;unt duo ligna oblonga &longs;atis gracilia vnius vlnæ ferè in longi
tudinem; quibus quanto autem
collopes &longs;unt longiores, facilius iugum circumagitur. cuius cau&longs;a e&longs;t, quia
collops ad vectem reducitur, cuius fultura e&longs;t circa medium iugi, pondus
verò e&longs;t extima iugi &longs;uperficies è qua telæ, aut &longs;taminis pondus pendet: in
altera verò extremitate collopis, quæ extra iugum multum prominet, e&longs;t
potentia: ibi enim textoris manus premit, vel trahit. quando ergò longior
e&longs;t collops, ea pars, quæ e&longs;t inter fulturam, & vim, augetur; altera non mu
tata; quia &longs;emper inter fulturam, &longs;eu centrum iugi, & vltimam iugi &longs;uper
ficiem continetur; quanto autem illa hanc &longs;uperar, tantum virium po
tentiæ addi.
Secundò, videndum quid &longs;it &longs;uccula: hanc vulgò Na&longs;pa appellant, ni fal
lor à verbo græco cum
quo, & voce, & &longs;ignificatione conuenit; e&longs;t enim in&longs;trumentum, quo &longs;æpius
architectores in extrahendis &longs;ur&longs;um ruderibus effo&longs;&longs;is vtuntur. e&longs;t autem
compago quædam cylindrica non admodum longa, cuiex vna parte poti&longs;
&longs;imum prominent plures collopes non mobiles, vt in iugo, verum &longs;tabiles,
ac cum ip&longs;a &longs;uccula compacti, quibus manu appræhen&longs;is &longs;uccula &longs;upra bi
nos polos ver&longs;atur,
&longs;ur&longs;um pondus educit. cuius imaginem
quærit igitur,
cur quanto gracilius fucrit corpus &longs;ucculæ A B, tanto facilius vertitur. Ratio e&longs;t, quia collops, quemadmodum etiam iugum, reducitur ad vectem,
cuius hypomoclion e&longs;t in medio
&longs;ucculæ, &longs;iue in axe ip&longs;ius &longs;ucculæ;
potentia verò e&longs;t in &longs;ummitatibus
collopum, vt in C, E, F, D, pon
dus verò e&longs;t vbi funis ductarius
cum onere pendet è &longs;uccula in &longs;u
perficie nimirum, vt vbi L, quare
pars vectis inter axim, & &longs;uperfi
ciem &longs;ucculæ eadem e&longs;t, quæ inter
hypomoclium, & pondus. quanto
igitur &longs;ucculæ corpus gracilius fuerit, tanto hæc pars minuetur; & con&longs;e
quenter altera inter hypomoclium, & potentiam productior euadet: eaque
propter facilius à motore ver&longs;abitur.
Satis per&longs;e clara videtur.
Notandum primò, quæ Græcis
Vmbilicos appellari; de his enim loquitur Cic. 2. de Oratore, vbi
&longs;ic, non audeo dicere de talibus viris, &longs;ed tamen ita narrare &longs;ole
bat Sceuola, conchas, eos, & vmbilicos ad Caietam, & ad Lucri
num legere con&longs;ueui&longs;&longs;e. hos autem vmbilicos exponunt Grammatici e&longs;&longs;e
lapillos paruos, acrotundos, polito&longs;que, de quibus etiam Ari&longs;t.
loquitur. Quare decipitur Piccolomineus dum negat, nos harum crocarum latinum
nomen habere. Cæterùm, & quæ&longs;tio, & re&longs;pon&longs;io, ex &longs;uperioribus &longs;atis
per&longs;picua e&longs;&longs;e videntur.
Ex appo&longs;ita figura totus huius problematis textus, alioquin &longs;atis cla
rus patebit. &longs;int duo ligna oblonga, vnum altero longius, & cra&longs;&longs;ius.
in eleuatione maioris, fulcimentum e&longs;t in B, vbi manus altera ferè
manens appræhendit; in C, verò, vbi altera manus mouens premit
e&longs;t potentia, &longs;iue maius onus. in A, verò onus ip&longs;ius ligni, deor&longs;um tendens
premit, quod nunc e&longs;t in&longs;tar potentiæ motricis, quare A, & C, &longs;unt &longs;ibi in
uicem, & potentiæ, & pondera. In minori autem ligno, onus ligni in D,
fultura manus in E, potentia alterius ma
nus in F. iam inquir Ari&longs;t.
maius lignum
A B C, magis flectitur, quamuis cra&longs;&longs;ius
&longs;it, quàm lignum D E F, quod e&longs;t tenuuius,
&longs;ed multò breuius; quia in maiori onus
ip&longs;ius ligni, quod circa A, deor&longs;um pre
mit
in minori ligno. Ex quo &longs;equitur iuxta
ip&longs;ius principia, vt onus A, facilius lignum mouere, aut inflectere
po&longs;&longs;it.
Cæterùm exi&longs;timo, quod &longs;i maioris ligni longitudo ad eiu&longs;dem
cra&longs;&longs;itiem haberet
dem cra&longs;&longs;itiem,
&longs;um, fore, vt
eandem rationem ad di&longs;tantias ab hypomoclio, oportet igitur vt &longs;int non
analoga, &longs;ed aloga, vt eis præ&longs;ens problema Ari&longs;totelis vnà cum eiu&longs;dem
&longs;olutione competat.
Cvr paruo cuneo magna finduntur onera, & corporum moles,
valida fit impre&longs;&longs;io? fortè, quia cuneus duobus vectibus &longs;ibi inui
cem oppo&longs;itis con&longs;tat; quorum vterque, & potentiam mouentem,
& hypomoclion, & hypomoclion autem illud ip&longs;um
e&longs;&longs;e ait, quod cuneo diuellitur; hoc autem dicit Ari&longs;tot. quia non agnouit
alium, præter primi generis vectem, vt &longs;upra etiam dixi.
Verum &longs;atius e&longs;t cum Guido Vbaldo reducere cuneum ad duos &longs;ecundi
generis vectes, quorum fultura &longs;it in cunei apice extremo, pondus verò in
tra vectem, ea nimirum pars ligni, que à cuneo vrgetur, ac diuellitur. cuneo
præterea vires adduntur ex valida mallei percu&longs;&longs;ione; malleus autem ip&longs;e
magna vi percutit, quia motus mouet, &longs;eu quia mouens malleum, mouet
ip&longs;um etiam dum e&longs;t in ip&longs;a latione, vnde ip&longs;a lationis celeritate malleus
fit valentior:
quàm ip&longs;a vectium magnitudo po&longs;tulet. &longs;it cuneus A B C. lignum autem &longs;cinden
dum D E F G,
B C, quorum commune hypomoclion e&longs;t
in C, onus autem vectis B C, e&longs;t pars li
gni G, hæc enim ip&longs;i contranititur,
ab eo expellitur. potentia verò mouens
vectem e&longs;t in malleo, dum &longs;uperius latus
cunei A B, percutit. alter huic auer&longs;us
vectis e&longs;t latus A C, cuius fultura e&longs;t C,
cadem cum priori, onus propul&longs;atum D, cunei igitur virtus partim ex vectibus, partim ex percu&longs;&longs;ione con&longs;tat.
Hvius quæ&longs;tionis &longs;en&longs;us, ac verba optimè intelligentur ex &longs;equen
tibus. Trochlea, vt patet ex &longs;uperioribus Ari&longs;t.
e&longs;t orbiculus in
periphæria &longs;triatus, vna cum toto loculumento, cui in&longs;eritur:
cuius imaginem ad 8. quæ&longs;t. exhibui.
Apud Architectores verò
trochlea con&longs;tat &longs;altem ex duobus prædictis loculamentis, in quibus &longs;unt
orbiculi; & vnus orbiculus e&longs;t &longs;upernè collocatus, alter verò infernè, vt pa
tebit in &longs;equenti figuratione: quod
à nonnullis dicitur etiam Rechamo. Auxilio huius in&longs;trumenti machinato
res parua vi attollunt ingentia pondera. communiter autem con&longs;tat ex plu
ribus orbiculis, qui partim &longs;uperiori loculamento,
partim infetiori inditi &longs;unt, per quos orbiculos cer
ta lege circumductus e&longs;t ductarius funis, qui deinde
in &longs;ui po&longs;trema parte à potentia tractus omnes illos
orbiculos, per quos tran&longs;it circumuoluens inferius
loculamentum, cui appen&longs;um e&longs;t pondus, vnà cum
ponderc attollit. figuram &longs;implicis trochleæ, con
&longs;tantis &longs;cilicet ex duobus tantum orbiculis, facilita
tis cau&longs;a exhibebo, in hac enim melius apparebit,
qua ratione trochlea ad vectem reducatur. vnde, &
Ari&longs;t. &longs;en&longs;um, quamuis ob&longs;curi&longs;&longs;imum, ac proinde
problematis &longs;olutionem optimè percipere licebit. Sit igitur orbiculus &longs;uperior A, qui in pegmate I K
L D, voluatur circa axem G,
rius fixum, & immobile à clauo H, pendens. Infe
rior orbiculus B, in loculamento O P Q R, circa
axem B, conuoluatur: &longs;itque funis ductarius circa
hos orbiculos hoc modo circumductus. primo ca
put funis religetur clauo D, in &longs;uperiori pegmate
infixo, hinc demi&longs;&longs;us &longs;ubtus inferiorem rotulam per
ip&longs;ius &longs;triam de&longs;cendat per puncta L S, a&longs;cendatque
po&longs;tea per M E N, ad &longs;uperiorem rotulam, &longs;upra
quam a&longs;cendat per punctum T,
inde demittatur ad
tentia in F, traxerit funem F V, deor&longs;um, interim
partes T, N, E, M, &longs;ur&longs;um attrahentur, & locula
mentum inferius &longs;imul cum appen&longs;o pondere eleua
bitur, manente tamen interim fune prope D, vbi
clauo D, e&longs;t religatus, & immobilis. &longs;ed vbinam hic
vectis? con&longs;idera diametrum M L, inferioris orbiculi, hæc enim ea e&longs;t, quæ
huius enim extrema L M, à fune tanguntur, & ab eius medio
B, onus pender, & grauitat; & quia funis in M, &longs;ur&longs;um trahitur,
parte illa &longs;ur&longs;um eieuat diametrum L M, erit potentia mouens, & eleuans
in M. pondus verò intra vectem ad B, medium vectis; quare fulcimentum
erit in reliquo extremo L, vbi funis &longs;u&longs;tinet loculamentum, & vbi diameter,
&longs;eu vectis innititur. quare diameter hæc e&longs;t vectis &longs;ecundi generis expo&longs;iti.
aduerte præterea vectem hunc e&longs;&longs;e mobilem, &longs;imul cum
ex parte M, &longs;ur&longs;um tollitur &longs;imul cum toto orbiculo, ac loculamento, &longs;ub
&longs;equitur etiam alterum extremum L, quod fune fulcitur, & in ip&longs;o fune &longs;ur
&longs;um ver&longs;us D, a&longs;cendit; & hoc modo inferius tignum cum onere tandem ad
&longs;uperius tignum &longs;ublatum erit. hinc verum dixi&longs;&longs;e Ari&longs;t.
con&longs;tat, trochleam
&longs;cilicet idem e&longs;&longs;e, ac vectem. quod tamen de &longs;olo inferiori orbiculo intelli
gi debet, &longs;uperior enim rotula quamuis vectis fiat, non tamen vires vllas
potentiæ tribuit, cum eius hypomoclion &longs;it in medio, quemadmodum &longs;upra
ad 8. quæ&longs;t. expo&longs;ui.
Inferior igitur ille e&longs;t, qui mouenti maximo e&longs;t adiu
mento. quod &longs;i &longs;cire aueas quantum iuuet, re&longs;pondeo ip&longs;um vires potentiæ
duplicare; adeo vt &longs;i quatuor. v. g. homines erant nece&longs;&longs;arij ad pondus tol
lendum, auxilio huius &longs;implicis trochleæ duo tantum &longs;ufficiant. quod &longs;i ad
dantur duo alij orbiculi, vnus &longs;uperior, alter inferior, rur&longs;us vires duplica
buntur, quod &longs;i plures aliæ rotulæ tam
&longs;upernè, quàm infernè addantur, vt&longs;olet in maioribus trochleis, quas ve
teres Poly&longs;pa&longs;tos, ide&longs;t multum trahentes dixerunt, augebuntur vires in in
finitum. quod dixi de virium duplicatione con&longs;tat ex 6. & 7. propo&longs;itione
Archimedis de Aequip. quia enim in vecte no&longs;tro L M, dupla e&longs;t proportio
inter L M, & L B, eadem etiam proportio erit inter pondus, & potentiam,
quare pondus C, duplum erit potentiæ in M, hoc e&longs;t à minore potentia &longs;ibi
&longs;ubdupla &longs;u&longs;tinebitur: & à quauis adhuc
Qui plura de trochlea de&longs;iderat, adeat Guidi Vbaldi, Mechanica, cuius
auxilio fateor me verum &longs;en&longs;um harum Mechanicarum Ari&longs;t. & præ&longs;ertim
huius loci enuclea&longs;&longs;e. quæ &longs;i cum Piccolominei expo&longs;itione contuleris, vide
bis eum nequaquam cognoui&longs;&longs;e, vbi nam vectis in trochlea lateret, eumque
tam &longs;uperiorem, quàm inferiorem
Io. Bapti&longs;ta Benedictus pariter erra&longs;&longs;e videtur in &longs;uis &longs;peculationibus, cum
inferiores tantummodo vice vectium fungantur, vt probatum e&longs;t.
Pattim ex &longs;e, partim ex dictis in 17. quæ&longs;t.
&longs;atis clara e&longs;t.
placet au
tem his, quæ de cuneo, & &longs;ecuri dicta &longs;unt, nonnulla ex Guido Vbal
do loco corollarij adijcere, videlicet. Ad huiu&longs;modi facultatis in
&longs;trumentum ca
&longs;ione, &longs;iue impul&longs;u incidunt, diuidunt, perforant,
munera; vt en&longs;es, gladij, mucrones, &longs;ecures, terebræ, & &longs;imilia: &longs;erra
ad hoc reducitur, dentes enim percutiunt,
Libet etiam huic tractationi de &longs;ecuri nonnulla addere, quæ olim oc
ca&longs;ione ex Proclo accepta in tenebris diu delite&longs;centia in lucem re
&longs;tituimus, &longs;unt autem hæc. Primò, antiquæ &longs;ecuris, necnon bipen
nis figuram re&longs;tituam. Secundò, o&longs;tendam angulum &longs;ecuris, qui
curuilineus e&longs;t, æqualem e&longs;&longs;e angulo trianguli æquilateri, qui rectilineus e&longs;t. Proclus igitur in comm. 23. primi Euclidis, &longs;ic ait: o&longs;ten&longs;um fuit ab anti
quis, &longs;cilicet Geometris, quod angulus figuræ illius, quæ &longs;ecuri &longs;imilis e&longs;t,
æqualis e&longs;t angulo rectilineo, quippe qui duabus tertijs anguli recti æqualis
e&longs;t. hanc anguli &longs;ecuris affectionem, cum nec ille, nec alij, quod &longs;ciam de
mon&longs;trent, ego paulò po&longs;t demon&longs;trabo. deinde &longs;ubdit; fit autem huiu&longs;mo
di &longs;ecuralis figura, quæ pelecoides vocatur duobus circulis per centra &longs;e
mutuò &longs;ecantibus. hæc Proclus.
Ex his autem po&longs;tremis verbis de&longs;criptio
nem antiquæ &longs;ecuris, &longs;ic puto eruendam. Ducatur primo recta A C, quæ
erit in&longs;tar manubrij &longs;ecuris. de
inde ex centro C, interuallo. v. g.
C B, de&longs;cribatur circulus B F; &longs;i
militer eodem interuallo B D, ex
centro D, de&longs;cribatur circulus
B E; tandem ex B, centro, atque
eodem interuallo ducatur alius
circulus D E F C, qui priores duos &longs;ecabit in punctis E F.
reliquis circulorum partibus ommi&longs;&longs;is, curuilineam figuram B E F, quam
e&longs;&longs;e veteris &longs;ecuris formam ex
c
habeat angulos E F, tantos, quantos ip&longs;e tradit, vt mox patebit; linea au
tem A B C, &longs;ecuris manubrium refert.
Quod autem tam angulus E, quàm angulus F, &longs;int æquales duabus tertijs
vnius angulirecti, &longs;iue quod idem e&longs;t angulo trianguli æquilateri, manife
ftum erithoc modo. De&longs;cribatur iterum &longs;ecuralis figura prædicto mode,
C A, quæ con&longs;tituunt trianguium æquilaterum A B C, tria enim ip&longs;ius late
ra &longs;ubtendunt tres arcus æquales A B, B C, C A,
&longs;unt enim tres &longs;extantes æqualium circulorum,
quitur tres ilias circulorum portiones, quas re
ctè cum &longs;uis arcubus con&longs;tituunt e&longs;&longs;e inuicem
æquales, & limiles portiones nimirum A B E,
B C D, C A F. hinc pr&ecedil;terea &longs;equitur angulos ip
&longs;arum e&longs;&longs;e inuicem æquales, angulos, v.g. A B E,
C B D, mixtos e&longs;&longs;e æquales, quod facilè e&longs;t per imaginatiam &longs;uperpo&longs;itio
nem demon&longs;trare. cum igitur prædicti duo anguli &longs;int æquales, &longs;itque intet
eos medius alius angulus E B C, qui pariter mixtus e&longs;t, &longs;i ip&longs;e addatur tanl
angulo C B D, quàm angulo A B E, inuicem æqualibus, erunt duo anguli ille autem e&longs;t angulus
æquilateri, qui æqualis e&longs;t duabus tertijs vnius recti ex corollario 32. primi. hic verò e&longs;t angulus &longs;ecuris.
e&longs;t igitur angulus &longs;ecuris æqualis duabus ter
tijs vnius recti, vt ait Proclus, quod demon&longs;trandum erat. quod etiam ma
nife &longs;tum &longs;ignum e&longs;t &longs;ecuris figuram a me re&longs;titutam e&longs;&longs;e illam veterem, de
qua idem Proclus loquitur.
Re&longs;tat, vt de antiquæ bipennis etiam figura di&longs;&longs;eramus; quæ nihil aliud
erat, quàm duplex &longs;ecuris, &longs;iue &longs;ecuris anceps, qualis e&longs;t præ&longs;ens figura, vt
propterea etiam &longs;æpius
&longs;a &longs;ecuris appelletur. dicitur enim
bipennis, qua&longs;i binis pinnis, quæ &longs;e
cures erant, con&longs;tet, vt & Græcis te&longs;te etiam No
nio, illud bipenne e&longs;t, quod
acutum e&longs;t. collegi autem
hanc bipennis figuram ex Simmiæ
peruetufti poetæ græci quod
epigramma carminibus loco linearum con&longs;tat, quæ in &longs;ecuris formam con
&longs;tituta &longs;unt.
Sciendum namque e&longs;t Simmiam, poeticam hanc &longs;ecurim concinna&longs;&longs;e in
gratiam Epei illius, qui equum Troianum ligneum fuerat architectatus, vt
e&longs;t apud Virg. Et ip&longs;e doli fabricator Epeus.
qui cum &longs;oluendi voti cau&longs;a
vellet &longs;ecurim, &longs;iue bipennem, qua in equi Durij molitione v&longs;us fuerat, Mi
neruæ Deæ, quod &longs;ibi in eo opere faciendo auxilio fui&longs;&longs;et, dedicare,
vt Ari&longs;t.
in libello de admirandis audit. num.
104. narrat, in templo græ
cæ Mineruæ
&longs;u&longs;pendere, a præfato Simmia quæ&longs;iuit, vt epigrammate aliquo dedicatio
nem hanc &longs;uam complecteretur. qui vt illi morem gereret ingenio&longs;æ illius
bipennis dedicationem, vt melius imitaretur, &longs;ecuri hac carminum com
plexus e&longs;t. quæ dedicatio, &longs;iue epigramma, quod adhuc extat, deinceps &longs;e
curis Simmiæ vocitata e&longs;t; ex qua figura bipennis illius, equi Durij fabrica
tricis nobis adhuc magna cum voluptate innotuit. Porrò gratum,
ea, quæ diximus intelligenda vtile Lectori fore arbitrati &longs;umus, ip&longs;am Sim
miæ bipennem ex operibus Theocriti, quibus addi &longs;olet, huc referre; quam
P. Ricardus E&longs;ius de no&longs;tra Societate linguæ græcæ periti&longs;&longs;imus, in hunc
modum tran&longs;tulit. hoc autem ordine legenda e&longs;t: lectio à manubrio
incipiat, deinde legatur carmen; forti&longs;&longs;imæ Deæ, quod &longs;ub&longs;e
quatur; dedit Epeus, & &longs;ic in orbem lectio,
dium circumducatur. hæc &longs;unt, quæ præ&longs;ertim
in gratiam eorum, qui &longs;uaui&longs;&longs;imo an
tiquitatis &longs;tudio tenentur, la
tere nolui.
Bipennis.
Antequam ad textus explicationem accedamus, con&longs;ultius e&longs;&longs;e iu
dico veteris &longs;tateræ figuram, atque de&longs;criptionem præmittere,
quàm ex hoc Ari&longs;t.
loco, magna mihi licuit cum delectatione col
ligere: quod etiam antiquitatis &longs;tudio&longs;is pergratum fore non du
bito:
adulterinas reijcere; erat igitur
&longs;tatera, quantum ex Ari&longs;t.
conijcio
primum ha&longs;ta oblonga, qualis e&longs;t in
præ&longs;enti figura A B, ex cuius altero
extremo B, pendebat appendicu
lum, quod propriè æquipondium
dicitur: ex altera verò extremitate
A, lanx vna pendebat; in qua carnes, aliæuè merces ponderabantur: in me
dia
hac, modo illa, prout pondus emptoris po&longs;tulabat &longs;u&longs;pendebatur,
terim tantum mercis lanci imponebatur, donec æquipondio præpondera
ret in æquilibrio. &longs;ingulæ autem trutinæ ad aliquod determinatum pondus
trutinandum, erant con&longs;titutæ, v. g. vna ad &longs;ex libras, altera ad octo, &c. quam diui&longs;ionem, ac fabricam &longs;tateræ non e&longs;t difficilè exhibere, cum ex Ar
chimede propo&longs;. 6. & 7. de æquip.
eadem &longs;it proportio inter pondus mer
cis, & pondus æquipondij, quæ e&longs;t permutatim inter di&longs;tantias vtrinque ab
a&longs;&longs;umpta trutina, quæ in trutinando hypomoclij vicem gerit: nam &longs;tatera
reducitur ad vectem; pondus erit æquipondium; & merces in lance erit po
tentia mouens: &longs;unt autem in tota &longs;tateræ ha&longs;ta trutinæ plures, hoc enim
modo tota fit vniformis quoad pondus. æquipondium præterea debet ha
bere tantum pondus, quantum e&longs;t in nuda lance, vt &longs;ic tota &longs;tatera &longs;it per &longs;e
&longs;ola æquilibrabilis: & præterea debet habere pondus &longs;tatum, a c legitimum,
v. g. vnius libræ, aut duarum, aut trium, prout magis
neum erit, & hoc erit proprium æquipondij pondus. vt autem ex &longs;ingulis
trutinis &longs;ingula pondera ponderentur. &longs;ingulis nota aliqua &longs;culpenda e&longs;t, vt
facilè mercatores merces ponderent, quod hac ratione fieri pote&longs;t. pona
mus æquipondium e&longs;&longs;e 12. librarum. dico, quod trutina C, dabit in lance
pondus mercis 12. librarum, &longs;i ex ea fiat æquilibrium, e&longs;t enim vt A C, ad
C B, ita permutatim æquipondium 12. ad mercem; &longs;ed A C, ip&longs;i C B, e&longs;t
æqualis, ergò etiam æquipondium 12. erit merci æquale, hoc e&longs;t vtrunque
erit, 12. librarum.
Similiter &longs;i &longs;ieret a quilibrium ex trutina D, e&longs;&longs;et vt A D, 3. ad B D, 9.
ita 12. ad 36. tandem trutina E, æquilibrante, e&longs;&longs;et vt A E, 9. ad E B, 3. ita
12. ad 4. Si igitur trutina C, notetur 12. numero, trutina D, num. 36. tru
tina E, num. 4. & idem de cæteris: &longs;tatim facilè erit quodlibet pondus per
huiu&longs;modi &longs;tateram exhibere. Vnde videas contrario ab illis modo in no
æquipondio trutinam quodammodo per ha&longs;tam moueri.
His præmi&longs;&longs;is ad textus paraphra&longs;im veniamus.
Cur &longs;tatera, qua carnes ponderantur, paruo appendiculo magna truti
nat onera, cum alioquin tota &longs;tatera nihil aliud &longs;it, quàm dimidiata libra,
vbi enim onus mercis imponitur vna lanx pendet, quam vnicam &longs;tatera ha
bet; in altera autem parte, vbi libra habet alteram lancem, &longs;tatera nullam
habet, &longs;ed &longs;ola &longs;ine lance e&longs;t. Cau&longs;a igitur e&longs;t, quia &longs;tatera &longs;imul, & libra e&longs;t,
& vectis. libra e&longs;t, quia &longs;partorum, &longs;iue trutinarum quælibet fit veluti cen
trum libræ,
pondium, quod libræ incumbit,
impo&longs;itum; manife&longs;tum enim e&longs;t, quod æquipondium &longs;tateræ tantumdem
trahit oneris, quantum e&longs;t illud, quod in altera lance e&longs;t. eapropter &longs;tatera
quodammodo tot libras in &longs;e continet, quot trutinas: quarum vna quæque
cum &longs;it intra appendiculum, & lancem, apta e&longs;t e&longs;&longs;e medium, &longs;eu centrum
&longs;tateræ,
parte, ex altera verò pro lance æquipondium. &longs;tatera verò dicitur, quate
nus ex vna parte habet non lancem, &longs;ed perpendiculum. &longs;ed hoc nihil e&longs;t
aliud quàm e&longs;&longs;e plures in vna libras; Cur autem &longs;parta, quæ lanci, &longs;iue ap
pen&longs;o oneri proximiora &longs;unt, maiora &longs;ubleuent onera, cau&longs;a e&longs;t vectis natu
ra, quæ &longs;tateræ ine&longs;t. e&longs;t enim &longs;tatera vectis, quamuis quodammodo inuer
fus, e&longs;t enim ip&longs;ius fulcimentum trutina ip&longs;a &longs;upernè collocata, pondus ve
rò leuandum e&longs;t ip&longs;a merx, potentia verò appendiculum. quantò autem pro
ductior fuerit pars vectis à fulcimento ad potentiam, tanto facilius poten
tia mouet, vt in præ&longs;entia accidit. mouet autem
modo pars illa productior &longs;tateræ, quæ vergit ad æquipondium, facit, vt
onus &longs;tateræ impo&longs;itum facilè trutinetur.
Cvr Medici facilius dentes extrahunt dentiforcipis onere adiecto,
quàm &longs;i &longs;ola manu vtantur? fortè, quia ex manu facilius dens ela
bitur propter &longs;ui ip&longs;ius lubricitatem, quàm ex forcipe. Vel etiam,
quia digiti propter carnis mollitiem cedentem nequeunt dentem
&longs;
dens, melius dentem comprçhendit. Aut tandem, quia forceps hæc duos
in &longs;e continet contrarios vectes; quorum, vnum tantum e&longs;t hypomoclion,
corum &longs;cilicet connexio; Virtute igitur
vectis arctius dentem per&longs;tringunt,
adeò obtinent,
commouent. &longs;it dentiforcip is figura, ex
po&longs;ita, cuius alterum extremum, vbi &longs;unt
A, B, e&longs;t illud, quod binis &longs;emicirculis
concurrentibus dentem arctè Vectis vnus e&longs;t A G D, alter B G C, communis fultura e&longs;t G,
vbi e&longs;t ip&longs;orum decu&longs;&longs;ata connexio; dens loco ponderis e&longs;t; vtroque igitur
C, & D, tanquam manubrijs vectium dentem Medici compræhendentes ip
&longs;um facilè commouent: quando autem commotus fuerit, facilius manu,
quàm in&longs;trumento extrahitur.
Tempore Ari&longs;t.
vt colligitur ex hac quæ&longs;tione, ad frangendas nu
ces peculiare in&longs;trumentum ligneum adhibeant, quod erat in&longs;tar
forcipis, ita tamen concinnatum, vt non ad &longs;cindendum, nec ad
extrahendum, &longs;ed ad frangendum per cuius hanc qualemcum que figuram in&longs;pice.
cuius latus inferius A D, fortè
manu tractabatur, vt &longs;ic expeditæ nucium plurima quantitas breui po&longs;&longs;et
confringi. Credibile e&longs;t nucifragam hanc ad capita F E, habui&longs;&longs;e aliquod
impedimentum, ne omninò con&longs;tringeretur, vt nuces
non autem comminuerentur. Cur igitur nuces
tur hi&longs;ce in&longs;trumentis, quæ ad eum fiunt v&longs;um? contrarium
deberet, vtentes enim prædictis in&longs;trumentis, omnibus illis viribus de&longs;ti
tuuntur, quas motio, ac violentia percu&longs;&longs;ionis afferre &longs;olent. præterea cur
ligneo vtuntur, ac proinde leui? non ne aptius e&longs;&longs;et durum,
&longs;um veluti ferreum?
His re&longs;pondendum e&longs;t, nucifragum i&longs;tud in&longs;trumentum reduci ad binos
vectes, quemadmodum etiam dentiforcipem. nuxigitur hoc modo duplici
vecte comprimitur. vecte autem facilè onera quælibet
qui duo vectes vnicum habent hypomoclion ip&longs;am &longs;cilicet connexionem
A. vectes &longs;unt binæ in&longs;trumenti ha&longs;tæ, F A D,
E A C.
tur etiam alia extrema F, E, & impo&longs;ita nuce in
hiatu K, quæuis potentia con&longs;tringendo C, D,
con&longs;tringet &longs;imul F, E,
get. quod igitur cum percu&longs;&longs;ione feci&longs;&longs;et pon
dus mallei, id valentiori vectium virtute efficiunt F A D, E A C. quanto au
tem locus nucis K, propinquior fuerit hypomoclio A, tanto celerius
confringitur, quia partes vectium A C, A D, tunc à centro
A, productiores fiunt, ide&longs;t multò maiores fiunt,
quàm &longs;int di&longs;tantiæ inter nucem, & cen
trum A, quod maximè poten
tiam iuuat.
Ex quibus præ&longs;enti quæ&longs;tioni &longs;atisfactum videtur.
Rhombus ex definitione 23. primi Elem.
e&longs;t figura æquilatera qui
dem, &longs;ed non æquiangula, habet enim
binos oppo&longs;itos angulos acutos, & alies
binos oppo&longs;itos obtu&longs;os, talis e&longs;t præ
&longs;ens figura A B D C. In præ&longs;enti porrò quæ&longs;tione
&longs;upponitur punctum A, quod e&longs;t vnum extremum
in rhombo moueri &longs;uper latus A B, ver&longs;us B, & &longs;i
militer interim æqua velocitate moueri alterum
extremum B, &longs;uper idem latus A B_{2} ver&longs;us A, & in
terim dum hæc duo puncta hoc modo &longs;ibi obuiam
procedunt, moueri latus totum A B, eadem ve
locitate, ver&longs;us latus C D, ita vt &longs;emper ip&longs;i C D,
æquidi&longs;ter,
u&longs;que ip&longs;i C D, congruat.
Horum igitur trium motuum quemadmodum
æquæ &longs;unt celeritates, ita etiam &longs;patia, quibus peraguntur, nam puncta duo
mouentur in latere A B, ip&longs;um verò A B, mouetur in lateribus A C, & B D,
quæ cum priori A B, &longs;unt æqualia.
Aduertendum præterea, quod hac ratione duo puncta A, & B, duabus la
tionibus mouebuntur, &longs;i quidem proprio motu
& quia latus A B, per quod ip&longs;a incedunt eodem tempore mouetur ver&longs;us
C D, &longs;equitur, quod etiam ip&longs;a hoc eodem motu ferantur. erit igitur ip&longs;o
rum motus ex his duobus mixtus; & quidem ip&longs;ius A, latio erit per longio
rem diametrum A D; ip&longs;ius verò B, per breuiorem B C. Quare cum pun
ctum A, peruenerit ad D, etiam punctum B, eadem c&ecedil;leritate acce&longs;&longs;erit ad
C. maius autem e&longs;t &longs;patium A D, quod confecit A, quam &longs;patium B C, con
fectum a C. Quærit igitur primò, cur cùm A, & B, mota fmt æquali celeri
rate in vtra que latione, vnum tamen maiorem lineam, quàm alterum per
tran&longs;iuit? Quærit &longs;ecundò, cur punctum B, confecit lineam B C, quæ mi
nor e&longs;t quam ip&longs;um latus A C, quod in &longs;uo motu conficit latus A B, quando
ad D C, acce&longs;&longs;it. & tamen B, duplici fertur latione; A B, verò vnica; vtrun
que autem in æquali velocitate? Quod autem punctus A, motu illo de&longs;eri
bat lineam A D, punctus verò B. lineam B C, manife&longs;tum erit hoc modo. &longs;it
v. g. punctum A, motu proprio delatum,
ris A B, erit interim totum latus A B, tran&longs;latum vbi e&longs;t F G, hoc e&longs;t, ad &longs;ui
itineris dimidium, quia horum motus ponuntur æquales: hoc autem motu
ip&longs;um punctum A, erit nece&longs;&longs;ariò in K, hoc e&longs;t in linea A D, vt dicebamus. Similiter in fine
extremo &longs;cilicet lineæ A D. &longs;imili ratione o&longs;tendi pote&longs;t de ip&longs;o B, qui cum
æqua velocitate moueatur, ac punctum A, quando A eri
illi occurret in E, proprio motu: &longs;ed alieno à latere B A, pron
lebamus. à quo po&longs;tea di&longs;cedens ver&longs;us C, motu pariter compo&longs;ito &longs;i&longs;titur
tandem in C, extremo lineæ pariter B C. eodem ergo tempore duo rhombi
extrema puncta æquè velocia, &longs;ecundum
la nequaquam æqualia confecerunt, &longs;ed A, maius, nimirum A D; B, verò
minus nimirum B C.
Ex quibus etiam &longs;ecundæ quæ&longs;tionis explicatio, & dubitandi ratio pate
bit: nam cum in rhombo duo &longs;int obtu&longs;i anguli B, & C, & duo acuti A, & D,
punctus ille, qui ab obtu&longs;o angulo B, recedit, fertur duabus lationibus inui
cem ferè contrarijs, propria enim tendit &longs;ur&longs;um ad A, aliena verò deor&longs;um
trahitur ver&longs;us D; cau&longs;a huius contrarietatis &longs;unt lineæ D B, B A, obtu&longs;um
angulum continentes, quæ à prædicto angulo in contrarias partes &longs;eparan
tur: per has autem lineas fiunt prædicti motus, vnde ip&longs;i quoque contrarij
&longs;int nece&longs;&longs;e e&longs;t: & propterea &longs;e mutuò impediunt:
ctum B, motu compofito hinc inhibito minus interuallum B C, pertran&longs;ire. At verò punctum A, quia ab acuto angulo de&longs;cendit,
deor&longs;um, quæ lationes &longs;e mutuò iuuant,
dem tempore, & eadem celeritate peragret &longs;patium A D. nam punctum A,
&longs;ua &longs;pontè
pariter deor&longs;um vehitur. nihil igitur mirum fit, &longs;i A, maius
quam B C, percurrat. cau&longs;a verò huius motuum concordiæ e&longs;t angulus acu
tus A, ob quem latera rhombi magis inuicem approximantur, redduntque
longiorem A D, quàm B C: è contrariò autem, quo obtu&longs;iores &longs;unt anguli
B, C, minorem faciunt ip&longs;am B C, latera enim &longs;emper magis ad rectam li
neam accedunt; donec tandem omni angulo euane&longs;cente in directum con
&longs;tituantur; quo ca&longs;u congruerent cum linea A D,
nullus e&longs;&longs;et.
Ex his igitur &longs;equitur, quod punctum A, ab angulo A, acuto di&longs;cedens,
duobus feratur motibus &longs;imilibus ad eandem partem tendentibus, & quò
acutiores &longs;unt anguli, eò magis tendent ad eandem partem; & melius &longs;e
mutuò inuabunt. B, autem vice ver&longs;a, quoniam quanto obtu&longs;ior e&longs;t angulus
B, tanto magis latera illius diuaricantur; duæ etiam motiones, quibus B,
progreditur in diuer&longs;a
etiam magis &longs;ibi contrariæ erunt; & propterea punctum B, minus interuallum, quale e&longs;t B C, percurret, quan
do A, maius A D, percurrit.
Ad &longs;ecundam verò quæ&longs;tionis partem, re&longs;pondeo con&longs;iderandum e&longs;&longs;e
latus B A, moueri vnico motu ad D C, quare à nullo impedi
tur, vnde nihil mirum videri debet, quòd ip&longs;um vnica
latione maius conficiat &longs;pacium quàm B, quod
quamuis duplici pellatur motu, vnus
tamen ab altero inhibetur.
Vnde e&longs;t, quod &longs;i duo circuli, vnus altero maior, cirea idem cen
trum po&longs;iti, volutentur, ita vt etiam centrum feratur, eo &longs;cilicet
modo, quo plau&longs;trorum rotæ &longs;olent, &longs;ecundum æqualem lineam
conuoluuntur, &longs;iue æquale &longs;patium conficiunt: &longs;i verò &longs;eor&longs;um
&longs;eparati quilibet eodem modo volutetur, non æquale
&longs;ed maior maiorem lineam, quàm minor;
cem eorum circunferentiæ obtinent, cum in hac veluti rotæ conuolutione,
circunferentia tota &longs;ucce&longs;&longs;iuè decur&longs;o &longs;patio adaptetur, ita vt tanta &longs;it de
cur&longs;a linea, quanta e&longs;t rotæ circunferentia? Quin etiam eodem exi&longs;tente
tum minor circulus &longs;olus, &longs;ecundum &longs;uam periphæriam reuolutus perfeci&longs;
&longs;et; Quod autem maior
&longs;olus in &longs;ua reuolutione maiorem lineam de&longs;cribat, manife&longs;tum e&longs;t hinc,
quia &longs;en&longs;u patet maiorem circunferentiam in maiori circulo &longs;ubtendere
angulum, qui fit à diametris in centro; minorem verò circunferentiam
&longs;ubtendere eundem angulum in minori orbe, vt etiam in 8. quæ&longs;t.
eandem igitur, vt proximè dixi habebunt etiam proportionem illæ lineæ,
quæ à &longs;ingulis &longs;eor&longs;um orbibus reuolutis de&longs;ignabuntur. Quod præterea &longs;e
cundum æqualem conuoluuntur, quando circa idem po&longs;iti fuerint centrum,
manife&longs;tum e&longs;t, ita tamen, vt aliquando ambæ æquales &longs;int ei, &longs;ecundum
quam &longs;olus maior conuolueretur; aliquando verò &longs;ecundum quam minor.
&longs;it enim circulus maior quidem vbi
D F C, minor verò vbi E G B,
autem centrum A, linea, &longs;ecundum
quam quadrans F C, maioris per &longs;e
rotaretur, &longs;it F L. linea verò, &longs;ecun
dum quam
iuncti à maiori, volutaretur &longs;it G K,
quæ æqualis e&longs;t dicto quadranti G B,
&longs;icut etiam F I, æqualis e&longs;t quadran
ti F C. &longs;i quis igitur impellat mino
rem orbem mouens &longs;imul commune
centrum A, cui maior e&longs;t circumpo
&longs;itus, donec diameter A B, perpendicularis &longs;it lineæ G K, in puncto K. tunc
pariter diameter maioris A C, erit perpendicularis lineæ F L, in puncto L. G K, autem, & F L, nece&longs;&longs;ariò erunt æquales per 34. primi, æquales igitur
lineas hoc modo peragrarunt inæquales circunferentiæ, &longs;iue quadrantes
G B, F C. &longs;i autem quadrantes hoc præ&longs;tant, manife&longs;tum e&longs;t, quod & toti
ambitus idem efficiunt, quare quando tota periphæria G B E G, fuerit re
uoluta etiam tota F C D F, &longs;uum orbem &longs;imiliter &longs;i ma
iorem quis mouerit, cui minor &longs;it annexus eodem exi&longs;tente centro, &longs;imul ac
pendicularis ip&longs;i G M, in M; &longs;unt autem G M, & F I, æquales, quare quan
do F C, quadrans maioris pertran&longs;iuerit rectam F C, etiam C B, quadrans
minoris tran&longs;actam habebit illi parem G M. hoc autem accidit nulla inter
cedente mora in vllo ip&longs;orum: quando enim mouetur maior, nihil ce&longs;&longs;at
minor: & quando minor agitur, maior nunquam quie&longs;cit. quod &longs;i hoc acci
dit quartæ parti circulorum, idem, & totis accidit periphærijs. vbi in&longs;uper
illud etiam mirum, centrum nimirum ip&longs;orum eadem celeritate motum,
ac vnica &longs;emper exi&longs;tenti latione, modo maius, modo minus &longs;patium per
ficere; idem verò eadem velocitate latum, æquale &longs;emper deberet interual
lum tran&longs;ilire. & tamen in præ&longs;entia vtrouis modo moueas eadem pernici
tate, modò maius, modò minus &longs;patium pertran&longs;ibit.
Huius quæ&longs;tionis enodandæ cau&longs;a, &longs;upponendum primò e&longs;t, quod eadem,
&longs;eu æqualis potentia, hanc quidem magnitudinem tardius, illam verò citius
mouere pote&longs;t. &longs;i enim fuerit quippiam, quod à &longs;eip&longs;o moueri minimè ap
tum &longs;it; & aliud, quod à &longs;e ip&longs;o moueri aptum &longs;it; qui hoc &longs;imul cum illo
coniunctum mouerit, tardius mouebit, quàm &longs;i ip&longs;um &longs;olum moueret. & &longs;i
quid moueatur, quod aptum &longs;it ex &longs;e moueri, verumtamen in eo motu nihil
ex &longs;e moueatur, perinde e&longs;t, ac &longs;i minimè aptum &longs;it ad motum, & proinde
tardius mouebitur; nec fieri poterit, vt plu&longs;quam mouens moueatur, cum
nihil innata motione vtatur. Si quis igitur minorem circulum, quem mo
do B, appello, mouerit &longs;upra &longs;uam circunferentiam, cui annexus &longs;it maior,
quem modo appello A, &longs;ic quidem maior mouebitur, non autem ex &longs;e, &longs;ed
&longs;olum quatenus à minori feretur, vnde tantum pertran&longs;ibit de recta F L,
quantum à minori fuerit impul&longs;us; tantum autem e&longs;t impul&longs;us, quantum
minor e&longs;t motus; quare æqualem cum illo viam confecit. &longs;i igitur minor fe
cit pedalem G K, maior confecit etiam pedalem F L, quia maior nihil de
proprio motu addidit, &longs;ed &longs;olum motione minoris e&longs;t tran&longs;latus. &longs;imiliter
&longs;i quis rotet maiorem &longs;upra &longs;uam circunferentiam annexo minori, tantum
minor mouebitur, quantum à maiori deportabitur, quia nihil ex &longs;e impel
litur. Verum &longs;i &longs;eor&longs;um ambo ex &longs;e &longs;ecundum &longs;uos ambitus moueantur, &longs;iue
citò, &longs;iue tardè, eadem etiam velocitate perficiant integram &longs;uæ periphæ
riæ volutationem, maior maius, minor verò minus conficiet &longs;patium.
Sed fortè augebitur difficultas con&longs;ideranti, quod prædicti circuli &longs;unz
circa idem centrum, & circa illud mouentur. moueri autem circulum cir
ca &longs;uum centrum, e&longs;t moueri &longs;ecundum &longs;uum naturalem motum, ad quem
circuli ex &longs;e &longs;unt apti. &longs;i verò vnus moueretur circa &longs;uum centrum, alter ve
rò non, vt quando alter alteri non e&longs;t circa idem centrum compactus, & ab
altero mouetur, vbi manife&longs;tè apparet, quod fertur omninò ab illo, & in il
la latione non circumuertitur circ
mirum e&longs;t, &longs;i
portetur, cui quoquo modo adiacet, aut appen&longs;us e&longs;t extra illius centrum.
Huic obiectioni
centrici, nihilominus non mouentur ambo &longs;uamet motione, &longs;ed ille, qui ab
alio fertur mouetur &longs;ecundum motionem illam, tanquam &longs;i nullam ad eam
haberet aptitudinem; quamuis enim po&longs;lit moueri circa centrum illud A,
propterea motus debet moueri, quantum mouens, nec plus, nec minus.
Quòd autem &longs;pectat ad id, quod initio dicebatur de eodem centro, & de
mouente eadem velocitate, & de æquali ab inæqualibus orbibus pertran&longs;i
ta linea, &longs;ubeft huic dubitationi paralogi&longs;mus: quamuis enim &longs;it idem am
borum centrum, e&longs;t tamen vnius centrum per &longs;e in motione, alteri verò per
accidens, veluti per accidens e&longs;t eundem virum e&longs;&longs;e mu&longs;icum, & album. ille
enim circulus, qui mouet alterum, obtinet illud centrum per &longs;e, & ex natu
ra &longs;ua; alter verò, qui mouetur, habet illud idem per accidens, quia non
vtitur illo tanquam centro. non igitur circa idem &longs;impliciter centrum fit
horum motus, &longs;ed alio modo vnus, alio modo alter, vnde & reliquis dubi
tationibus facilè &longs;atisfiet.
Cvr lectulorum &longs;pondas faciunt &longs;ecundum duplam proportionem,
hoc e&longs;t longiorem &longs;pondam duplo longiorem, quàm &longs;it altera: il
lam enim &longs;ex pedum, vel paulò plus, hanc verò trium? præterea
cur re&longs;tes, quibus culcitræ &longs;u&longs;tinentur non extendunt per diame
trum, &longs;ed per tran&longs;uer&longs;um?
Ad primum re&longs;pondetur ideò facere &longs;pondas in dup la ratione, vt &longs;int hu
mano corpori proportionatæ, &longs;ic enim lecti longitudinem habebunt qua
tuor cubitorum, latitudinem verò duorum, in tali enim &longs;patio commo
dè cubamus.
Ad &longs;ecundum verò dicendum extendi illos funes non per diametrum, &longs;ed
ex oppo&longs;ito, quia hoc modo ligna ip&longs;ius lecti minus di&longs;trahuntur: facilè
enim ex natura &longs;ua ligna hæc ab inuicem &longs;ecundum longum &longs;eparantur; ar
ctius autem ductis funibus per tran&longs;uer&longs;um, quàm per diametrum inuicem
con&longs;tringuntur: præterea, quia &longs;ic etiam funes minus laborant, cum &longs;int eo
rum ductus breuiores; & quia debent &longs;u&longs;tinere onus ftragulorum,
cìtrarum, &longs;ic certè ex hoc onere minus laborabunt &longs;i tran&longs;uer&longs;im, quàm &longs;i
diametraliter &longs;ubtendantur.
Tertia demum ratio e&longs;t, quia hac ratione minus re&longs;tium ab&longs;umitur, quæ
vt benè intelligatur, de&longs;eriba
tur lectuli figura A F G K, &
bifariam diuidatur latus F G,
in B. & quia tota F G, dupla
e&longs;t ip&longs;ius A F, erit dimidium
F B, æquale ip&longs;i A F. & propte
rea tot erunt foramina, quibus
funes immittuntur in F B, quot
in A F.
hoc modo incipiunt ab A, &
ducunt ad B, po&longs;tea per C, re
de&longs;cendunt ad M, à quo recta tendunt in F, hinc per 2. deducunt ad 3. à quo
foramine, per foramen 4. reflexum faciunt ad 5. à quo iterum per B, de&longs;cen
dunt ad angulum K,
anguli A, & K, re&longs;tis habent capita, & re&longs;tes exten&longs;æ &longs;unt non diametrali
ter, &longs;ed tran&longs;uer&longs;im.
Notandum autem, quod re&longs;tes æquales &longs;unt cum &longs;uis curuaturis.
v. g. re
&longs;tis A B, cum &longs;ua curuatura B C, æqualis e&longs;t re&longs;ti C D, vnà cum eius curua
tura D H, & aliæ eodem modo &longs;e habent, quia eadem demon&longs;tratio omni
bus accommodari pote&longs;t: quia enim figura A B G M, parallelogrammum
e&longs;t, æqualia enim &longs;unt latera B G, A M, & quot foramina &longs;unt in vno, tot
etiam &longs;unt in altero,
parallelas, & æquales, per 33, primi. ex qua etiam &longs;cquitur prædictas cu
ruaturas, B C, D H, E G, e&longs;&longs;e æquales. quare manife&longs;tum e&longs;t in dimidio le
ctulo tot e&longs;&longs;e re&longs;tes æquales re&longs;ti A B, quot &longs;unt foramina in dimidio latere
B G, vel in dimidio F B, hoc e&longs;t e&longs;&longs;e quatuor. porrò oportet quantitatem
harum omnium re&longs;tium per&longs;crutari, vt eam cum quantitate re&longs;tium diame
traliter exten&longs;arum conferamus, quod geometricè hoc modo a&longs;&longs;eque mur:
triangulum enim B G K, rectangulum e&longs;t, ergò per 47. primi, quadrata la
terum B G, G K, æqualia &longs;unt quadrato lineæ B K: latus B G, e&longs;t trium pe
dum, quemadmodum etiam latus G K quadratus autem numerus ternarij
e&longs;t 9. ergo duo quadrati numeri 9. &longs;iue 18. æquales &longs;unt quadrato lineæ B K,
ergò linea B K, e&longs;t radix quadrata numeri 18. quæ radix non pote&longs;t exactè
in numeris repræ&longs;entari, e&longs;t enim, vt aiunt, radix &longs;urda. verumtamen per
radicum extractionem,
quatuor pedum cum vna quarta. cum igitur in toto lecto &longs;int huiu&longs;modi
octo re&longs;tes, erit omnium &longs;umma pedum 34. ferè. &longs;i autem &longs;eeundum diame
trum extendantur re&longs;tes, vti factum e&longs;t in lectulo A B C D, neutiquam re
&longs;tes omnes &longs;imul &longs;uperiori quantitati adæquabuntur, &longs;ed illam longè &longs;upe
rabunt. Sit igitur lectus A B
C D, in quo diametraliter du
ctæ &longs;int re&longs;tes B D, E H, & re
liquæ, vt in figura. harû quan
titas &longs;i per 47. primi, & per ra
dicis quadratæ extractionem
inueniatur, erit &longs;umma earum
pedum quadraginta cum dimi
dio; quæ quantitas præcedenti
maior e&longs;t &longs;ex pedibus cum di
midio.
cis, quàm in latinis codicibus corruptionem, totus re&longs;titui nequiuerit.
Cvr difficilius e&longs;t
&longs;ecundum medium, cùm tamen
dus? An quia dum fertur lignum &longs;uper humeros ab altero extre
mo, alterum extremum vibratur, & agitatur, quæ agitario ip&longs;ius
lationem impedit? An quia licet nihil inflectatur ob agitationem,
gnam habeat longitudinem, difficilius tamen ab extremo fertur, quoniam
facilius ex medio eleuatur, quàm ab extremo, & quia latio e&longs;t qua&longs;i quæ
dam continua eleuatio, propterea etiam difficilius &longs;ic portatur? cau&longs;a au
tem cur facilius ex medio eleuetur e&longs;t, quia hoc modo totum lignum fit ve
ctis, cuius hypomoclion e&longs;t in medio, vbi is, qui eleuat, tenet aut fert: ex
trema autem &longs;ibi mutuò
ta vi, quantum e&longs;t totum ligni pondus &longs;u&longs;tineatur; quod &longs;i ab extremo ele
uetur non &longs;ufficit amplius prædicta vis, &longs;ed opus erit maiori, quia non &longs;o
lum oportebit illud eleuare, &longs;ed præterea etiam illud in æquihbrio con&longs;ti
tuere, & con&longs;eruare. pondus enim totius ligni vergit ferè ad alteram ligni
medietatem, quæ ab hypomoclio productior cuadit, quapropter ad onus
i&longs;tud æquilibrandum, opus e&longs;t alia potentia in altero extremo. &longs;it lignum
A B, &longs;u&longs;pen&longs;um ex medio C.
hoc modo lignum ponderi
bus libratum &longs;uis manet in
æquilibrio, pote&longs;tque à &longs;ola
potentia illud eleuante etiam deferri: quia A, & B, extrema &longs;e mutuò &longs;u&longs;ti
nent. quod &longs;i non ex medio eleuaretur,
&longs;ed ab extremo, vt in &longs;ecunda figura,
eleuans potentia ex C, æqualis oportet,
vt &longs;it præcedenti; &longs;ed præterea opus e&longs;t
alia vi, quæ in B, æquiponderet alteri
extremo A, quod magis grauitat, quo ab C, longius fuerit; & hoc modo in
æquilibrio con&longs;titutum, & con&longs;eruatum poterit non &longs;olum eleuari, &longs;ed
etiam circumferri.
Cvr &longs;i valdè procerum &longs;uerit idem pondus difficilius &longs;uper humeros
ge&longs;tatur, etiam &longs;i ex medio illud feratur, quàm &longs;i breuius &longs;it? quod
enim dudum dictum e&longs;t cau&longs;a non e&longs;t, &longs;ed vibratio, & &longs;uccu&longs;&longs;atio
ligni nunc e&longs;t: quando enim ab humero productius fuerit, magis
vibrantur extrema, quam ob rem contingit portantem difficilius ge&longs;tare. vibrationis autem cau&longs;a e&longs;t, quoniam ab eadem vi moueute magis extrema
fit di&longs;tantià à centro, &longs;eu hypomoclio, quod modo e&longs;t humerus ip&longs;e. &longs;it vt
in prima præcedentis quæ&longs;tionis figura, humerus vbi A. di&longs;tantiæ autem ab
ip&longs;o centro &longs;unt A B, A C, quod autem maior di&longs;tantia; faciliorem reddat
motum o&longs;ten&longs;um e&longs;t initio huius operis.
In&longs;trumentum i&longs;tud, quod græca voce Leonicus interpres Celonia vo
cat, latinis dicitur Tolleno, à tollendo; quod etiam manife&longs;tum e&longs;t
ex Fe&longs;to, qui ait, Tolleno e&longs;t genus machinæ, quo hauritur aqua in al
teram partem prægrauante pondere; quæ tollenonis de&longs;criptio om
ninò machinæ præ&longs;entis quæ&longs;tionis competit. Hi&longs;pani Telonam fortè a tol
lenone nuncupant. E&longs;t autem tolleno
idoneum, quo ru&longs;tici pa&longs;&longs;im vtuntur:
con&longs;truunt, quale à figura &longs;equenti refertur. vbi puteus F, tolleno con&longs;tat
erecto tigno D C, & tran&longs;
uer&longs;a ha&longs;ta A C B, vnà cum
fune B E, & hydria E. ap
ponitur præterea onus &longs;a
tis graue ad
e&longs;t G. ha&longs;ta porrò A B, ve
luti vectis circa
tanquam hypomoclion,
tia funem B E, trahente. &longs;ed iam textus exponatur.
Cur iuxta puteos tolle
nones faciunt eo, quo vi
&longs;untur modo, ligno enim
tran&longs;uer&longs;o A B, adiungunt
onus plumbi G, cum alio
quin vas ip&longs;um E, & vacuum, & plenum pondus habeat: cur inquam, vt fa
cilius moueant tollenonem, tollenonis oneri onus addunt G? An quoniam
cùm opus hauriendi diuidatur in duo, in intingendi nimirum, & &longs;ur&longs;um tra
hendi tempora: accidit quidem
tunc vas e&longs;t vacuum: at verò &longs;ur&longs;um vas deinde plenum trahere, laborio
&longs;ius erit. &longs;i verò addatur onus G, tunc quidem paulò difficilius intingemus,
&longs;ed tamen vas plenum po&longs;tea multò facilius, quod opus, & labor e&longs;t, &longs;ur&longs;um
educemus: operæpretium igitur e&longs;t, onus illud plumbi, aut lapidis adiun
gere in extremo A, quia &longs;ic pondus illud tanquam quædam potentia vecte
A B, vtens &longs;ur&longs;um hydriam plenam rapiet,
bit,
Cvr quando &longs;uper ligno, aut huiu&longs;modi quopiam duo portauerint
homines æquale pondus, non &longs;imiliter grauantur, ni&longs;i quando pon
dus in medio eorum fuerit; &longs;ed magis ille premitur, cui onus vici
nius fuerit? An quia lignum illud vectis efficitur, cuius hypomo
clion e&longs;t vbi pondus ge&longs;tatum &longs;u&longs;penditur; ge&longs;tantium autem oneri proxi
mior gerit vicem illius, quod vecte mouetur, remotior verò e&longs;t potentia
vecte mouens. quanto igitur plus di&longs;tat ab hypomoclio, &longs;eu ge&longs;tato ponde
re, tanto facilius mouet, hoc e&longs;t, alterum magis deor&longs;um premit, contra
nitente nimirum ge&longs;tato onere &longs;i autem in medio fue
rit pondus, nihilo magis alter ge&longs;tantium fit id, quod vecte mouetur, quàm
alter;
Cæterum &longs;ciendum huiu&longs;modi lignum, quo tran&longs;uer&longs;o onera
dici à latinis phalangam, vnde etiam verbum phalangare deducitur, quod
huiu&longs;modi ge&longs;tationem &longs;ignificat;
ait, capream vnam &longs;emilaceram quaterni &longs;imul phalangabant.
Cvm &longs;edemus, præcipuè &longs;i commodè &longs;edeamus, &longs;olemus duos angu
los rectos facere, vnum quidem, quem facit thorax cum femore;
alterum quem facit femur cum crure, vt in figura thorax &longs;it A B,
femur B C, crus C D, anguli duo recti &longs;unt B,
& C. Quærit igitur, cur quando &longs;urgere volumus angu
los ho&longs;ce rectos in acutos commutamus, nam crus re
trahimus &longs;ub femur ad acutum angulum, v. g. ad po&longs;itio
nem C F.
femori aptamus ad acutum angulum E B C, alioquin &longs;ur
gere non valemus? An quia id, quod æquale e&longs;t, quietis
tatis,
&longs;tant, vt quadratum, vt cubus, quieti, ac &longs;tationi &longs;unt
idonea, vt propterea Pytagorei dicerent terram e&longs;&longs;e cubicam, propten ip
&longs;ius immobilitatem. e&longs;t autem angulus rectus, angulus æqualitatis, quia
omnes anguli recti &longs;unt inuicem æquales, vel quia linea illa, quæ angulum
rectum facit e&longs;t perpendicularis alteri lineæ, cui incumbit,
&longs;uper &longs;uperficiem terræ ad angulos rectos non cadant, &longs;ed racta maneant. pariter
cum illo faciant angulos rectos, &longs;ed etiam, quia &longs;imul faciunt cum &longs;uperficie An quia qui &longs;urgit fit rectus; rectus autem manens,
oporter, vt &longs;it &longs;uperficiei terræ perpendicularis. debet igitur e&longs;&longs;e &longs;ecundum
eandem rectitudinem, ide&longs;t caput &longs;upra thoracem, thorax verò &longs;upra femo
ra, femora verò &longs;upra crura in eadem rectitudine, quæ horizonti perpendi
culariter in&longs;i&longs;tat: quando autem &longs;edemus thorax, & crura, non &longs;unt in ea
dem linea horizonti perpendiculariter erecta, quapropter nece&longs;&longs;e e&longs;t pedes
retrahere, caput autem reclinare, vt &longs;ic in eadem recta linea horizonti per
pendiculariter con&longs;tituantur,
Reliquæ quæ&longs;tiones ad Phy&longs;icum &longs;pectant.
In 33. aperit propriam &longs;en
tentiam de motu proiectorum.
In 35. & vltima de vortice quamuis vide atur mathematicam &longs;apere, e&longs;t
tamen phy&longs;ica. Eius autem re&longs;olutiones tres ab Ari&longs;t.
allatas, fal&longs;as e&longs;&longs;e
&longs;u&longs;picor; experientia enim docet, quod &longs;i quippiam ponatur &longs;upra rotam
figuli, id non ad centrum, &longs;ed extra rotam proijcitur. &longs;ed cau&longs;a e&longs;t, quia in
vortice aqua ip &longs;a &longs;piratim circumcurrens tandem in centrum, vbi demer
gitur de&longs;cendit; nece&longs;&longs;e igitur e&longs;t, vt etiam ea, quæ in ip&longs;a &longs;unt, &longs;imul cum
illa ad centrum per plures conuolutiones deducantur. Cæterum &longs;i quis ve
lit Mechanicam facultatem &longs;eriò aggredi, nequaquam paucis his ab Ari&longs;t.
traditis,
de Aquæponderantibus, Commandinum, ac Lucam Valerium de centro
grauitatis &longs;olidorum, ac tandem Guidi Vbaldi Mechanica adeat, vbi hu
ius &longs;cientiæ admiranda plurima,
AD ALEXANDRVM.
Cvm plures libellum hunc Ari&longs;t.
attribuant, cogor loca ip&longs;ius ma
thematica ex in&longs;tituto exponere.
In 2. cap.
recen&longs;et Planetarum ordinem, iuxta antiqui&longs;&longs;imorum
A&longs;tronomorum traditiones, qui ob paucas,
&longs;eruationes muita ignorarunt,
tarum &longs;tatuendo, fal&longs;i &longs;unt: A&longs;tronomi enim po&longs;teriores, & maximè Ptolæ
meus, vnà cum recentioribus no&longs;tri &longs;eculi alium ordinem exactioribus ob
&longs;eruationibus,
errores patefecerunt. E&longs;t autem verus ordo, vt Luna &longs;it omnium terris pro
xima, deinde Mercurius, tùm Venus, po&longs;tea Sol, Mars, Iupiter,
à terris alti&longs;&longs;imus, quos omnes &longs;tellarum affixarum &longs;phæra, quæ etiam fir
mamentum dicitur, complectitur. non me latet huius no&longs;tri &longs;eculi di
ligenti&longs;&longs;imos a&longs;tronomos nouam mundani &longs;y&longs;tematis hy
pothe&longs;im inducere; &longs;ed ea prædicto Planctarum
ordini parum, aut nihil repugnat.
In 3. cap.
dam temporibus cum Luna circumag:)
xum attingit, qui quia ex motu præcipuè Lunæ pender, non videtur
alienum hoc loco eum fu&longs;ius explicare,
occurrere, quibus recentiores nonnulli nimis implicautur. Ae&longs;tus maris
e&longs;t quædam maris ebullitio, ob quam vt &longs;olet in ebullientibus aquis, mare
intume&longs;cit: fiunt autem in toto mundo duobus tantum in locis ex hoc æ&longs;tu
tumores duo, quorum vnus &longs;emper directè Lunæ &longs;ubiacet, alter verò in
auer&longs;a terræ parte, &longs;iue huic antipoda, & diametraliter oppo&longs;ita.
Ex his Marium tumoribus fit vt aquœ, quæ naturæ &longs;ua decliuiora petunt,
qua&longs;i exundantes ad littora flaant. atque hic aquarum cur&longs;us fluxus maris
appellatur. decre&longs;cente deinde maris æ&longs;tu, & tumore ex rece&longs;&longs;
iterum ad medium mare refluunt: Cum au
tem in toto die &longs;int 24. horæ & &longs;emper &longs;int &longs;imul in mundo duo æ&longs;tus, & tu
mores, fit vt &longs;int pariter &longs;emper in mundo duo fluxus, qui tumores illos co
mitantur; necnon duo refluxus, qui eo&longs;dem &longs;ub&longs;equantur; hinc fit vt
cuique
&longs;ex verò fluxui, & &longs;ex tandem refluxui Lunæ auer&longs;is, quæ totam Lunæ circa
mundum periodum 25. horarum expleant. Cau&longs;am autem cur mare hoc
modo &longs;tatis horis, paulò tamen &longs;erius ob Lunæ tardiorem ortum &longs;emper
cre&longs;cat, & decre&longs;cat antiqui omnes in Lunam retulerunt, vt primus omnium
Ari&longs;t.
hoc loco, deinde Strabo, Pomponius Mela, Plinius, Solinus, & alij
plures idem &longs;en&longs;erunt. Lunam &longs;cilicet eam habere vim in mare, vt pars il
la, quæ Lunæ &longs;irbiacet, &longs;iue quam Luna radijs ferit, æftuet, & turgeat; non
aliter pars maris huic antipoda, & auer&longs;a, quamuis tota terræ moles inter
ip&longs;am, & Lunam interpona
tur, æ&longs;tuat, fluxumque, ac re
fluxum quamuis priori mi
norem, efficit. quæ omnia
melius in figura cernentur;
vbi infra Lunam vides tumo
rem A, ex quo fluxus deriua
tur. & in parte huic auer&longs;a
tumorem B, ex quo alter flu
xus deriuatur. & quia in alijs
duobus mundi lateribus non
mare ob refrigerationem
&longs;ub&longs;idet, ibi fiunt duo reflu
xus C, & D, ita vt &longs;emper &longs;int
in mari præ&longs;ertim Oceano
quatuor prædicti effectus, qui
&longs;imul, vt ait hic Ari&longs;t. & exhoe e&longs;t
&longs;i Luna, quæ modo e&longs;t in &longs;uperiori parte meridionali, venerit ad locum E,
occidentalem, eam fluxus A, &longs;uble quitur,
dentem E, vnde, & fluxus B, promouebitur ad orientem, ita vt punctum F,
orientalem a&longs;piciat.
Alij præterea duo refluxus eadem proportione promoti erunt, vbi prius
erant fluxus: quæ
ho&longs;ce fluxus, acrefluxus non aliunde quàm à Luna manare. quod adhuc ma
nife&longs;tius erit, &longs;i con&longs;ideremus, quod quanto tardius quotidie Luna oritur,
tanto etiam maris æ&longs;tus tardius incipit. Porrò vt appareat hanc e&longs;&longs;e vete
rum &longs;ententiam libet hic attexere quædam ex lib.
3. Strabonis, quæ ip&longs;e ex
Po&longs;&longs;idonio acceperat. &longs;ic.
Oceani verò motum ait, &longs;cilicet Po&longs;&longs;idonius, &longs;y
deris &longs;ubire circuitum, quendam quidem diurnum, quendam men&longs;truum,
quendam annuum, vt Lunæ etiam contingit. quo etiam tempore i&longs;ta &longs;uper
horizontem a&longs;cenderit, mare terram a&longs;cendere incipit, &longs;en&longs;u te&longs;te,
ad cœVbi verò declinare &longs;ydus ip&longs;um cœ
perit, &longs;en&longs;im rur&longs;us à terr
occidentis painctum Luna de&longs;cenderit. deinde tanto eadem incon&longs;tantia
tempote inanct, quanto Luna ad iplumoccafum conumgitur, & adhuc tan
to magis, quanto &longs;ub terram mota, &longs;ignnm ab horizonte di&longs;tet. po&longs;tea rur
&longs;us mare a&longs;cendere, quou&longs;que &longs;ub teliurem in medio cœli &longs;it Luna, deinde
mare à littore regredi quoad iterum Luna in orientem procedat, ac &longs;upra
horizontem eleuetur, con&longs;i&longs;ere verò v&longs;que quo fignum &longs;upra terram eleue
tur, & rur&longs;us terras mare a&longs;cendere. Hanc diurnam e&longs;&longs;e circuitionem a&longs;&longs;e
rit Po&longs;&longs;idonius, men&longs;truam verò, &c. vbi pergit explicare, qua ratione, ma
ria etiam alijs motibus men&longs;trno. &longs;cilicet, & annuo cieantur, iuxta Lunæ
periodos men&longs;truam, & annuam. Eadem omninò habet Plinius, & alij ve
teres omnes, quos tu con&longs;ulere poteris vnde mirum videri debeat, cur re
centiores plurimi,
nixi, hanc maris affectionem, à Luna effici negarint.
Verum ip&longs;i duabus poti&longs;&longs;imum rationibus id negant.
Prima e&longs;t, quod vario admodum tempore, & modo in diuer&longs;is fiant ma
ribus, & in nonnullis nihil horum æ&longs;tuum appareat.
Huic re&longs;pondendum e&longs;t, id ex varia marium di&longs;pofitione, tum eriam va
rio fitu, quo Lunam a&longs;piciunt prouenire. hoc modo videmus vario tempo
re, & modo, in toto orbe effici dies, ac noctes, æ&longs;tatem, & hyemem; & ta
men certum e&longs;t Solem i&longs;ta omnia efficere. Sed melius etiam huic dubita
tioni occurremus certa quadam,
te Nautica de&longs;umpta. libri enim nautici ab&longs;que vlla dubitatione Luuæ hæc
cmnia verè a&longs;cribunt, dum qua&longs;dam regulas tradunt, eastamen pro varijs
maribus varias, quibus per ætatem Lunæ, & &longs;itum ip&longs;ius &longs;upra horizonteni
illius maris certò certius horam fluxus; & refluxus, imò eorum etiam ma
gnitudinem præuo&longs;cunt, ac prædicunt. huiu&longs;modi librum vidiego Parmæ,
manu &longs;criptum, auctore Augu&longs;tino Cæ&longs;areo, quem ille olim Sereni&longs;s. Duci
Octauio dono dederat. quod &longs;i hi æ&longs;tus à Luna minimè penderent, nulla ra
tione regulæ illæ effici potui&longs;&longs;ent, quibus per ætatem ipfius, ac &longs;itum &longs;upra
horizontem eos prædicere tuto valerent.
Secunda verò ratio, quæ maximè eos torquet e&longs;t quanam ratione à Luna
effici po&longs;&longs;it &longs;ecundus refluxus B, primò oppo&longs;itus, cum tota terræ moles in
teriecta ob&longs;tare videatur.
Verum huic difficultati optimè ex opticis &longs;atisfacere po&longs;&longs;umus, fi dixe
rimus, æ&longs;tum illum effici quidem à Luna, & Sole, &longs;ed tamen per lumen ex
&longs;yderibus ad partem illam auer&longs;am reflexum; quod vt melius explicetur, &
confirmetur. Illud primò &longs;ciendum non &longs;olam Lunam, verumetiam Solem
adæ&longs;tum maris ciendum concurrere, quamuis primas in hoc Lunæ conce
dat; experientia enim con&longs;tat maiorem fieri fluxum, quando Sol, & Luna
&longs;imul &longs;unt coniuncta, vt in nouilunio accidit, quia lumina, & eorum virtu
tes vnitæ fortius eandem maris partem directis radijs percellunt. &longs;imiliter
maior fit, quando luminaria &longs;unt oppo&longs;ita, vt in plenilunio contingit, quia
tunc radij vnius directi, a&longs;&longs;ociantur cum reflexis alterius radijs,
do duplicati ea&longs;dem terræ partes, & directè, & reflexè feriunt, vt melius in
&longs;equenti figura patebit.
Secundò præmittendum e&longs;t, lumen Solis, & Lunæ reflecti ex den&longs;is, ac per
politis corporibns, vti &longs;unt omnia &longs;ydera.
Tertiò, ex opticis a&longs;&longs;umendum, &longs;i corpora plurima &longs;phærica lumen re
flectentia fuerinuin circulari ambitu con&longs;tituta, quemadmodum &longs;unt &longs;tellæ
affixæ in ambitu firmamenti collocatæ, reflectere
& idem punctum, quod &longs;it inter lumen, & ambitum illum; quod a&longs;&longs;umptum
manife&longs;tum e&longs;t ex Iride, vbi ex plurimis &longs;phæricis guttulis lumen Solis re
flectitur ad oculum; quamuis geometricè, & quidem facilè à Per&longs;pectiuo
demon&longs;trari po&longs;&longs;it.
Quartò, ex opticis, dato corpore lumino&longs;o, & &longs;phærico reflectente, &
puncto quouis, ad quod po&longs;&longs;it reflecti lumen, pote&longs;t inueniri in &longs;phæra refle
ctente punctum reflexionis.
Quintò, quanto radij perpendiculariores incidunt, tanto maiorem
vim habere.
Sit ergò Sol, & Luna &longs;imul, vt in figura
cum innumeris in ea affixis &longs;yderibus. e&longs;&longs;e autem totum cœlum &longs;tellis penè
infinitis, ac con&longs;tipatis refertum &longs;en&longs;ui palam fit, adhibito nouo illo, ac mi
rabili Tele&longs;copij inuento.
Iam, vt patet ex 39.5. Alhazeni, ex &longs;ingulis &longs;tellis Solis, ac Lunæ lumen
reflecti pote&longs;t (ni&longs;i quid ob&longs;tet) ad partem terræ D, luminaribus auer&longs;am,
vt quarto loco &longs;uppo&longs;ui. & præterea ex &longs;tellis circa B, po&longs;itis radij Solis re
percuti po&longs;&longs;unt ad eandem terræ partem D, perpendiculares, qui præ cæte
ris maximam vim obtinent. quemadmodum lineæ in figura reflexæ
o&longs;tendunt, ideò a&longs;&longs;erendum e&longs;t eos, æ&longs;tum D, excitare præcipuè po&longs;&longs;e,
terræ quantitas Solis luci obe&longs;t, cum con&longs;tet vmbram terræ parum &longs;upra
Lunæ cœlum produci. pote&longs;t tamen Lunæ e&longs;&longs;e impedimento quoad hos ra
dios perpendiculares; &longs;ed tamen alios minus perpendiculares, &longs;eu parum
obliquos nullo modo impedire pote&longs;t, quo minus ad D, re&longs;iliant. qui quam
uis &longs;int minus quàm perpendiculares efficaces, obtinent tamen non modi
cam vim. Ex &longs;tellis igitur circa A, & C, reflecti pote&longs;t ex quarto fundamen
to lumen
culariter terræ D, incidat. quamuis autem ex &longs;tellis F, E, lumen aliquod ad
D, tran&longs;mittatur, tamen cum obliquè admodum illi accidat, nihil penè ef
ficere valet. Verumenimuerò qui&longs;piam in hunc modum obijciet: hac ra
tione deberet fieri etiam æ&longs;tus in terræ lateribus H, I, quando quidem etiam
illuc lumen ex quarto fundamento reflecti pote&longs;t.
Cui &longs;ic re&longs;pondendum, po&longs;&longs;e quidem aliquod lumen illuc re&longs;ilire, &longs;ed ta
men exiguum admodum, & proinde nullius penè roboris, quod experientia
de&longs;umpta ex illuminatione Lunæ comprobari pote&longs;t; videmus enim, quod
quanto Luna magis Soli opponitur, & proinde &longs;uam illuminationem magis
ver&longs;us terram obuertit, vt in plenilunio, tanto maiorem eam vim habere
æ&longs;tus excitandi. multo verò minorem, quando e&longs;t in a&longs;pectu Solis quadrato,
quia dimidiam tantum &longs;ui illuminationem nobis reflectit. Idem proportio
naliter de &longs;tellis dicendum, quæ enim luminari maximè opponuntur, vt quæ
&longs;unt circa B, illæ totam illuminationem terræ o&longs;tendunt, vnde, & efficacio
res &longs;unt. cæteræ, quo magis ab illis di&longs;tant minus de &longs;ua illuminatione ter
ræ, &longs;eu mari vnde fit, vt quamuis non
nulli radij etiam perpendiculares ad terræ latera H, I, referri po&longs;&longs;int, tamen
quia pauciores &longs;unt, quàm alibi, propterea nullam ibi æ&longs;tus
obtinent. &longs;ydera porrò illa, quæ &longs;upra Solem exi&longs;tunt, etiam &longs;i ip&longs;orum illu
ficiunt, quia in illa vel obliquè admodum radij incidunt, vel ea tantummo
do tangunt. Verum illuminatione &longs;ua ea&longs;dem maris partes, quæ &longs;unt ad G,
vnà cum Sole, ac Luna percellunt.
Ex quibus apparet duas tantum orbis terræ partes totis, ac plenis a&longs;tro
rum luminibus impeti, in quibus &longs;cilicet duo opoo&longs;iti æ&longs;tus ebulliunt.
Idem po&longs;&longs;umus hoc modo confirmare, quia cum totum firmamentum &longs;it
innumeris penè &longs;yderibus &longs;tipatum, loco concaui, ac &longs;phæriçi &longs;peculi ha
beri pote&longs;t, & proinde illius in&longs;tar amborum luminarium lumen reflectere;
qua ratione patet omnem ferè ad partes prædictas D, emitti reflexionem.
His rationibus manife&longs;tum e&longs;&longs;e patet prædictum æ&longs;tus tumorem lumina
ribus auer&longs;um,
Po&longs;&longs;et etiam qui&longs;piam &longs;ic opponere, &longs;i illuc prædicta luminum reflexio
pertineret, non &longs;olum illam aquarum ebullitionem efficeret, verum etiam
lucem aliquam eòdem afferret, quod tamen &longs;en&longs;u minimè apparet. cui &longs;ic
re&longs;pondendum videtur, nece&longs;&longs;arium non e&longs;&longs;e, vt reflexio illa, quæ hoc modo
mare afficit tanta &longs;it, vt etiam illud luce &longs;olito maiori afficiat; quod
rientia
tempore, fluxum, ac re&longs;luxum priorem parit, cum tamen nullam tunc lu
cem nobis afferat? quamuis enim lumen &longs;tellarum &longs;uperficiem maris non
attingat, attingit tamen &longs;uperficiem vaporum, exhalationum, ac nubium,
quæ terram in &longs;phæræ modum ambiunt, ac parum à terra
tolluntur: quem exhalationum ambitum deinde luminarium virtus facilè
penetrare pote&longs;t. Nullum præterea lumen apparet, quia lumen reflexum
præ&longs;ertim ex conuexis corporibus, vt &longs;unt &longs;tellæ, valde debile e&longs;t, quia
uexum
Tandem quærere quis po&longs;&longs;et, cur æ&longs;tus hic &longs;ecundus minor &longs;it priori.
Cui
re&longs;pondendum, quia ille à directis radijs, hic verò à reflexis progignitur:
radios autem reflexos debiliores e&longs;&longs;e directis optici docent,
tia confirmat.
Porrò quando luminaria &longs;unt oppo&longs;ita, vt &longs;i Luna e&longs;&longs;et in B, Sol verò in K,
tunc maximus fit
dijs reflexis alterius; ita vt
directum &longs;imul, v. g. æ&longs;tus, qui Lunæ &longs;ubiacet fit per radium Lunæ directum,
& quia Sol e&longs;t in oppo&longs;itione cum Luna, &longs;it vt ip&longs;ius radij reflectantur, &
vniantur cum directis Lunæ ad eundem tumorem excitandum. &longs;imiliter in
fra Solem directè alius fit à directis ip&longs;ius radijs; & quia Luna ei opponitur
lumen eius ad
ce ad eundem efficiendum concurrit.
Exi&longs;tentibus demum lumidaribus circa quadratum a&longs;pectum, vt &longs;i Luna
e&longs;&longs;et in F, Sole exi&longs;tente in K. exiguus, ac penè nullus fit fluxus, quia eorum
vires non &longs;unt vnitæ, cùm radij nec incidentes, nec reflexi vniantur imò vi
res corum &longs;eparatæ inaria in contrarias partes di&longs;trahunt, vnde fit, vt neu
tro alteri concedente, apud neutrum victoria con&longs;tet.
quam iamdiu
inuentam,
mum &longs;ent. de creatione mundi tantummodo &longs;ine vlla expo&longs;itione,
firmatione proponit. in eadem pror&longs;us &longs;ententia e&longs;t Rogerius Bachon inter
Opticos probati&longs;&longs;imus, cap.
5. de Speculis Mathematicis.
Aliorum demum opinationes, &longs;iue Angelo cuidam, &longs;iue virtuti totam
terram peruadenti hunc æ&longs;tum a&longs;cribentium, non e&longs;t meum refellere, cum
non phy&longs;icum, &longs;ed mathematicum agere in&longs;tituerim.
Cap. 7.
&longs;tatuas, quæ &longs;pontè mouebantur Græci appellarunt Automata, ide&longs;t &longs;pon
tanea, cuiu&longs;modi &longs;unt automata Heronis, Alexandrini, quæ adhuc extant.
De admirandis auditionibus.
Nvmero 82. Quæ de illa in&longs;ula extra Herculis columnas &longs;ita narrat,
eam putant recentiores Geographi, & quidem meritò nouo orbi
conuenire.
Numero 100. Quæ de I&longs;tro, &longs;iue Dannubio tradit, eum &longs;cilicet
e&longs;&longs;e bifidum,
onerari: &longs;unt contra omnes recentiores Geographos; apparet tamen eam
fui&longs;&longs;e veterum nonnullorum opinionem, quos
à quibus etiam multò po&longs;t fal&longs;i &longs;unt Diodorus, Pomponius, & Solinus, qui
I&longs;trum I&longs;triæ Prouincìæ fluuium faciunt, quem ex I&longs;tro Germaniæ veluti ra
mum contra omnem veritatem deriuant. Verùm hoc illis-condonandum
præ&longs;ertim antiquioribus, cum tunc temporis Geographia parum e&longs;&longs;et
exculta.
Primus Strabo hanc fal&longs;itatem libro 1. redarguit, & po&longs;t ip&longs;um Plinius
I&longs;trum i&longs;tum fabulo&longs;um appellat.
De lineis in&longs;ecabilibus, &longs;iue indiuiduis.
Di&longs;putat libellus hic &longs;anè acuti&longs;&longs;imus, Vtrum quantitas con&longs;tet ex
indiui&longs;ibilibus, quam qu&ecedil;&longs;tionem recentiores agitant in Phy&longs;icis
tractatione de Quantitate;
ta: plura &longs;umpturi ni&longs;i operis ob&longs;curitas, & mathematicarum,
ignoratio hactenus ob&longs;titi&longs;&longs;et.
Sciendum igitur primo loco, nos po&longs;&longs;e duo indiui&longs;ibilium genera in quan
titate concipere. primum eorum, quæ verè indiuidua &longs;unt,
partes, &longs;iue nullo modo &longs;unt quanta; cuiu&longs;inodi e&longs;t
Alterum quorumdam indiui&longs;ibilium quidem, &longs;ed tamen quantorum cu
iu&longs;modi e&longs;&longs;ent, quædam adeò minimæ lineæ, quæ omnem effugiant diui&longs;io
nem: ex quibus antiqui opinabantur lineas totales, ac diniduas componi. atque de hec &longs;ecundo indiuiduorum, quantorum genere videtur opu&longs;culum
& quia partim rationibus phy&longs;icis, partim geometricis vti
Ego igitur,
quæ mathematica &longs;unt, exi &longs;tituto exponere aggrediar.
A lint
&longs;int quantitates commen&longs;urabiles, & quæ in commen&longs;urabiles. quæ prima,
& &longs;ecunda definitione 10. Elem.
explicantur;
ca&longs;ione a&longs;ymetriæ diametri cum co&longs;ta &longs;atis expo&longs;ui: vtrumuis locum vide
ris præ&longs;enti nece&longs;&longs;itati con&longs;ultum erit.
Primus locus Mathematicus e&longs;t hic
maticis imbuti di&longs;ciplinis, quiuis lineam aliquam in&longs;ecabilem e&longs;&longs;e concedet. nam
&longs;i, vt aiunt, illæ commen&longs;urabiles &longs;unt lineæ, quæ eadem men&longs;ura dimetiri queunt,
& nihil impedit, quin omnes commen&longs;urabiles reip&longs;a dimetiantur, extabit profe
ctò longitudo aliqua, qua omnes commen&longs;urabuntur; quæ nece&longs;&longs;ario erit indiuidua,
nam &longs;i dicatur e&longs;&longs;e diuidua, huius
munem habebunt, partes enim toti commen&longs;urabiles &longs;unt ita, vt portio partis il
lius, quæ dimidium totius fuerat, efficiatur dupla alterius; quoniam autem hoc
fieri nequit, atoma debet e&longs;&longs;e men&longs;ura hæc communis.
ea men&longs;ura compo&longs;itæ &longs;unt lineæ, veluti ex atomis conflantur.
Affert rationem quandam ex Mathematicis, qua nonnulli probabant ex
tare lineas atomas, ex quibus cæteræ lineæ tanquam partibus con&longs;tarent:
ac proinde negabant lineas e&longs;&longs;e in infinitum diuiduas, &longs;eu quamlibet lineam
&longs;ecari po&longs;&longs;e, &longs;ed a&longs;&longs;erebant
Præmi&longs;&longs;a igitur, vt monui commen&longs;urabilium, & incommen&longs;urabilium
linearum cognitione in hunc modum, & textum Ari&longs;tot. & rationem ip&longs;o
rum exponam.
Mathematici o&longs;tendunt extare lineas commen&longs;urabiles, quæ &longs;cilicet ea
dem communi men&longs;ura men&longs;urantur: at nihil impedit quin omnes
&longs;urabiles
omnes commen&longs;urabiles dimetiamur. hanc autem uece&longs;&longs;e e&longs;t e&longs;&longs;e atomam,
nam &longs;i diuidua &longs;tatuatur, poterit &longs;emper &longs;ecari, & &longs;ub&longs;ecari bifariam, qua
re cum partes huiu&longs;modi &longs;int toti commen&longs;urabiles, &longs;equetur aliam exi&longs;tere
men&longs;uram, qua omnes hæ partes, & proinde tota linea commen&longs;urentur. Verùm hoc fieri nequit, nam hoc pacto non e&longs;&longs;et vna tantum longitudo om
nium commen&longs;urabilium linearum communis men&longs;ura, verùm plures, &
plures in infinitum, quod e&longs;t contra Mathematicorum placita. dicendum,
itaque, communem illam omnium men&longs;uram e&longs;&longs;e omnis diui&longs;ionis exper
tem; & propterea etiam lineas omnes commen&longs;urabiles ex atomis lineis
componi, quæ nimirum prædictæ communi men&longs;uræ æquales &longs;int.
e&longs;t illarum prima argumentatio.
Secundus locus
libus procreantur: nam omnes huiu&longs;modi figuræ erunt etiam inuicem commen&longs;ura
biles, quare
munem men&longs;uram e&longs;&longs;e pariter indiuiduam.
Sciendum e&longs;t omnes lineas
men&longs;urabiles (vt aiunt Geometræ) potentia, ide&longs;t &longs;ecundum quadrata ea
rum vnciarum, & linea trium vnciarum &longs;unt
commen&longs;urabiles longitudine, & potentia,
quia potentia lineæ duarum vnciarum, &longs;iue
lium: & quadratum lineæ trium vnciarum,
e&longs;t nouem vnciarum quadratarum, vt patet
in figuris, quorum quadratorum communis
men&longs;ura e&longs;t vncia vna quadrata. atque hanc
illi nullo modo diuidi po&longs;&longs;e contendebant.
Tertius locus
ram &longs;aciat diuiduam, non erit amplius in rerum natura linea vlla rationalis, aut
irrationalis, re&longs;pectu expo&longs;itæ, ac determinatæ lineæ; neque aliarum vlla erit, de
quibus modo dictum e&longs;t, veluti quam Apotomen vocant ex duobus nominibus. Ve
rùm neque &longs;ecundum &longs;e aliquam definitam naturam habebunt, &longs;ed collatæ &longs;ibi ip&longs;is
tam rationales, quàm irrationales erunt omnes.
Hæc e&longs;t alia eorumdem ratio ad idem comprobandum: quam, vt benè
percipiamus, nonnulla prius ex definitionibus 10. Elem.
&longs;unt explicanda:
vt quæ nam &longs;int lineæ rationales, quæ irrationales, quæ ex binis nomini
bus, quæ Apotomæ.
Propo&longs;ita igitur linea quapiam, v. g. trium palmorum qualis e&longs;t linea A,
po&longs;&longs;unt inueniri quamplurimæ lineæ, quarum aliæ &longs;int illi longitudine com
men&longs;urabiles, &longs;iue quæ cum expo&longs;ita A, ha
beant communem men&longs;uram. v. g. linea B,
A, quia vtramque communis men&longs;ura vnius
palmi metitur: aliæ verò &longs;int eidem A, lon
gitudine incommen&longs;urabiles, qualis e&longs;&longs;et diameter C D, quadrati lineæ A,
quæ e&longs;t cum latere A, incommen&longs;urabilis ex vltima 10.
Cæterum lineam primò expo&longs;itam, vt e&longs;t in præ
&longs;entia A, quod e&longs;&longs;et notæ quantitatis, Græci appella
runt
eam appellant.
Linearum autem longitudine
cum expo&longs;ita rationali A, aliæ &longs;unt, quæ tamen &longs;unt
commen&longs;urabiles eidem potentia, ide&longs;t con&longs;tituunt
quadrata, quæ &longs;unt commen&longs;urabilia quadrato ra
tionali A, vt linea C D, cum &longs;it diameter quadrati li
neæ A, quadratum exhibet, quod e&longs;t duplum quadrati lineæ A, ex 47. primi,
quadratum autem lineæ A, e&longs;t nouem, igitur quadratum eius duplum erit
octodecim, quadratum &longs;cilicet lineæ C D. octodecim autem, & nouem &longs;unt
men&longs;urabiles poteutia tantum, potentia. n.
lineæ dicuntur
Quæ igitur rationali propo&longs;itæ &longs;unt commen&longs;urabiles aliquo modo, &longs;iue
longitudine, & potentia (
e&longs;t etiam potentia) &longs;iue potentia &longs;olùm, rationales ip&longs;æ quoque dicuntur.
Aliæ verò (quarum permultæ in decimo reperiun
tur) quæ nec longitudine, nec potentia illi &longs;unt
commen&longs;urabiles, irrationales appellantur, qua
lis e&longs;&longs;et media proportionalis E F, inter duas A,
& C D, in præ&longs;enti figura ex 11. 10.
Sciendum præterea ex 37. 10. & &longs;equentibus,
quod ex duabus lineis rationalibus re&longs;pectu rationalis expo&longs;itæ. v. g. A, com
men&longs;urabilibus inuicem tantum potentia, componitur linea, quæ cum ea
dem expo&longs;ita e&longs;t irrationalis,
duobus nominibus, &longs;iue Binomium, vt &longs;i ex
latere A, & diametro C D, componatur li
nea A C D, erit irrationalis cum rationali
A, Amplius ex 74. 10. & &longs;equentibus, &longs;i prædictum
minus nomen, &longs;iue minor linea A, detrahatur ex maiori nomine C D, vt re
linquatur B D linea, erit ip&longs;a reliqua B D, irrationalis, quam po&longs;tea appel
lant Apotomen, &longs;iue latinè Re&longs;iduum.
Po&longs;tremò, & hoc non ignorandum ex 43. 10. lineam, &longs;iue
non po&longs;&longs;e diuidi in alio puncto, præter C, in duas lineas, quæ &longs;int rationales
expo&longs;itæ, & potentia tantum inuicem commen&longs;
His præmi&longs;&longs;is textum, ac rationem illorum explicabo in hunc modum.
Si quis faciat diuiduam lineam illam, quæ e&longs;t communis
commen&longs;urabilium, &longs;equetur hoc ab&longs;urdum contra demon&longs;trationes 10.
quod nulla erit amplius linea rationalis, nec irrationalis, quia &longs;i communis
men&longs;ura diuidatur, tolletur ea de rerum natura; vnde non erit amplius in
ter lineas &longs;ymetria vlla, quare neque vllæ erunt rationales, e&longs;&longs;e enim ratio
nale oritur ex commen&longs;urabilitate. quare
&longs;ita, ad quam cæteræ relatæ dicuntur rationales, vel irrationales: quapro
pter etiam irrationales nullæ erunt,
nec irrationalis illa, quam vocant Apotomen ex Binomio, &longs;iue ex duobus
nominibus, de qua Euclides propo&longs;. 74. 10. & &longs;equentibus pertractat.
Notandum in ver&longs;u illo
ca voce illa
græco, magnum Ari&longs;toteli imponi erratum, cum hac ratione dicat apoto
men ex duobus nominibus e&longs;&longs;e compo&longs;itam, quod fal&longs;i&longs;&longs;imum e&longs;t. Apotome
enim, vt &longs;upra dictum e&longs;t, ne dum ex duobus nominibus con&longs;tat, verum ip
&longs;a e&longs;t re&longs;iduum lineæ maioris, &longs;i minor ab ip&longs;a detrahatur. Verumenimuero
vox illa
ex Geometriæ in&longs;citia addita, tolli debet, ne tantæ in&longs;citiæ Ari&longs;t.
ip&longs;e re
darguatur. hæc in hunc locum &longs;ufficiant.
Quartus locus
nes vua quadam, & eadem men&longs;ura oportere men&longs;urari, fal&longs;um t&longs;i admodum, &
nequaquam Mathematicorum &longs;uppo&longs;itionibus concnon enim ita &longs;upponunt
Geometræ,
men&longs;urabiles e&longs;&longs;e, & omnium commen&longs;urabilium linearam communem men&longs;ura
exi&longs;timare. quamobrem ridiculum e&longs;t eos, qui dicunt &longs;e demonstrare ex Geometra
rum decretis, & ex quibus Mathematici docent in contentio&longs;am pariter, ac falla-nam multis modis im
hecillis e&longs;t eiu&longs;modi ratio, & quouis modo licet euitare, ne aut inu&longs;itata dicere, aut
argui videamur.
Refellit hoc loco &longs;uperiores rationes in tribus locis præmi&longs;&longs;is allatas,
quibus nonnulli probabant quantitatem ex indiuiduis con&longs;tare, & proinde
concedenda e&longs;&longs;e quædam Quanta, omninò atoma; &longs;ic igitur inquit. Quod
verò de commen&longs;urabilibus lineis dicunt, omues videlicet vnica quadam,
contra mathematicorum dogmata, non enim Geometræ hoc a&longs;&longs;erunt, cùm
ip&longs;orum demon&longs;trationibus a duer&longs;etur; &longs;ed
ad inuicem &longs;unt commen&longs;urabiles, commen&longs;urari, vna
&longs;ed non tamen vnica, ide&longs;t non vnica, ac determi
nata. po&longs;&longs;unt enim e&longs;&longs;e plures
communes plurium quantitatum commen&longs;ura
bilium, vt præ&longs;entium trium linearum 4. 6. 8.
communis
numeros 4.6. & 8. men&longs;urat. & &longs;i linea 2. bifariam
&longs;ecetur, erit dimidium eius linea 1. quæ pariter
erit communis men&longs;ura trium prædictarum li
nearum, cûm vnitas &longs;it omnium numerorum communis men&longs;u
rum e&longs;t, quod Geometræ, quando &longs;impliciter loquuntur de huiu&longs;modi com
muni men&longs;ura, intelligunt de ea, quæ inter omnes e&longs;t maxima: vt in prædi
ctis tribus lineis maxima earum communis men&longs;ura e&longs;t linea 2.
bi volunt Geometræ, ex quibus totus hic textus intelligi pote&longs;t.
Quintus locus
in rectum ita diuidere, vt infinitæ circunferentiæ, & interualla totidem inuenian
diuiduas non credere, &c.)
ac &longs;ine ratione, imò contra rationem addidit: tum quia in Græco textu non
extant, tum quia &longs;en&longs;us totius &longs;ententiæ is e&longs;t, vt potius debui&longs;&longs;et affirmati
uè dicere
do recta linea A B, vt in appo&longs;ita figura mo
uetur intrando in &longs;emicirculum C A D B, ita
vt primò &longs;it in &longs;itu A B, &longs;ecundò in E F, tertiò
in G H, & &longs;imiliter in alijs omnibus &longs;emicir
culi locis, nece&longs;&longs;ariò accidit, vt inf
ph&ecedil;riæ, quales
cadant inter infinitas partes lineæ ingredien
tis, vt &longs;unt A B, E F, G H,
ingrediens, quàm totus &longs;emicirculus, diuidatur in partes infinitas, ita vt
nulla pars lineæ rectæ,
diuidantur, ergò nihil tam in linea, quàm in &longs;emicirculo remanet, quod non
&longs;ecetur: tota igitur linea recta, & periphæria illa diuidua e&longs;t, quam ob rem
nullo modo con&longs;tare pote&longs;t ex indiuiduis, ex quibus manife&longs;tum e&longs;t perpe
ram additamentum illud factum e&longs;&longs;e, & &longs;imul ratio, & textus Ari&longs;t.
opera patefacta &longs;unt.
Sextus locus
rum æqualium, nam qui&longs;quis borum moueatur, oportet per maiorem &longs;emicirculum
moueri, & quæcunque alia buiu&longs;modi constituta &longs;unt de lineis, fieri non po&longs;&longs;e, vt
talis vllus motus peragatur, quin prius omnibus, & &longs;ingulis interiAtque bæc Mathematicorum &longs;cita, multò magis ab omnibus conce&longs;&longs;a &longs;unt, quàm
illorum dicta.
Hæc e&longs;t alia ratio, qua probat totam circuli periphæriam e&longs;&longs;e diuiduam.
&longs;int enim duo circuli æquales primum in eo
dem loco,
lus B, moueatur, & di&longs;cedat à circulo A, ma
nente; &longs;tatim
maior &longs;emicirculo, & &longs;emper fiet maior, ac
maior.
dientis circuli &longs;ecantur ab omnibus partibus
circuli manentis. vnde patet nihil e&longs;&longs;e in eo
rum periphærijs, quod non diuidatur. nul
lum igitur in eis e&longs;t indiuiduum. falluntur igitur aduer&longs;arij.
Septimus locus
que probabile, neque nece&longs;&longs;arium e&longs;&longs;e lineas vllas indiuiduas extare, tamen ex ijs
etiam, quæ deinceps &longs;abiungam, multò magis per&longs;picuum & primò quidem
per ea, quæ Mathematici demon&longs;trant, at que addi&longs;cenda proponunt, quæ mutare
non decet, ni&longs;t probabiliores rationes habeamus. Nam neque lineæ, neque rectæ li
neæ definitio cum in&longs;ecabili linea con&longs;entit, vt quæ ne
&longs;it, nec medium vllam babeat.
Idem, &longs;ed paulò mutatis verbis po&longs;tea repetit, quæ fortè ab aliquo per
errorem addita &longs;unt. Verumenimuerò maximè con&longs;iderandum e&longs;t, quan
tum hoc loco Ari&longs;t. Mathematicis demon&longs;trationibus tribuat: quod dixe
rim propter recentiores quo&longs;dam, qui eò audaciæ deuenerunt, vt Euclidis
firmi&longs;&longs;imas,
comprobatas, negare non verentur Demon&longs;trationes.
Cæterùm Ari&longs;t.
iterum opinionem
do confutat: nam &longs;i inquit, lineam illam, quam vocant in&longs;ecabilem, e&longs;t non
&longs;olum linea, &longs;ed etiam linea recta, illi conueniret rectæ lineæ definitio, &longs;ed
nullo modo pote&longs;t ci conuenire, ergò tollendæ &longs;unt de rerum natura huiu&longs;
modi lineæ. Porrò definitio lineæ e&longs;t, vt &longs;it longitudo latitudinis expers, &
&longs;i recta &longs;it ex æquo &longs;ua interiacet puncta extrema, ergò ip&longs;a linea media erit
inter duo indiuidua extrema puncta; at verò linea, quam ip&longs;i volunt e&longs;&longs;e
indiuiduum quoddam, qua ratione medium erit inter alia duo indiuidua? ip&longs;i enim
cederent habere medium, iam po&longs;&longs;e
concederent: patet igitur definitionem lineæ minimè illi conuenire, & pro
pterea
Octauus locus
diuiduis lineis dimetientur,
les. indiu duæ autem lineæ &longs;ibi ip&longs;is co
&longs;iat æquales; quare po
Pergit adhuc nouis rationibus aduer&longs;arios refellere, dicens, &longs;i extarent
huiu&longs;modi indiuiduæ lineæ, &longs;equeretur omnes omninò lineas e&longs;&longs;e commen
&longs;urabiles, quod e&longs;t contra demon&longs;trata in 10. Elem.
quia cum omnes lineæ
munes men&longs;uræ, vnde & illæ, quæ dicuntur potentia tantum commen&longs;ura
biles, vt &longs;upra explicaui, erunt etiam commen&longs;urabiles longitudine. indiui
duæ verò ip&longs;æ, cum &longs;int inuicem æquales, erunt ip&longs;æ
les longitudine, quare & potentia, omnes enim longitudine commen&longs;ura
biles, &longs;unt etiam potentia commen&longs;urabiles, ex 9. 10. vnde &longs;equitur qua
drata earum omnia e&longs;&longs;e
quit, ea e&longs;&longs;e 290.)
vnde &longs;equeretur ip&longs;am
ponenda erat indiuidua.
Nonus Iocus, cuius latinam interpretationem, cum admodum e&longs;&longs;et de
prauata ex græco textu, in hunc modum correxi
rem lstitudinem facit applicata, æquale ei, quod ab indiuidua, & pedali copulatis
circa bipedalem, minorem faciet latitudinem, quàm &longs;it indiuidua: erit minus, quod
circa ind
faciat. v. g. linea minor A B, applicata cum ma
iori B C, vt in figura, ita vt contineant figuram
A B C D. Minor A B, facit latitudinem figuræ,
maior verò B C, facit longitudinem. Iam cum
aduer&longs;arij velint extare huiu&longs;modi lineas ato
mas, con&longs;tituatur figura &longs;ub vna ex illis, quæ &longs;it v. g. A B, & altera maiori,
quæ &longs;it pedalis, v. g. B C, vt in præcedenti figura, &longs;umatur deinde linea bi
pedalis E F, cui per 45. primi ap
plicetur &longs;patium E F G H, æquale
&longs;patio &longs;uperiori A B C D, nece&longs;&longs;a
riò latitudo E H, huius &longs;ecundæ fi
guræ minor erit quàm latitudo il
lius, hoc e&longs;t minor, quàm &longs;it indinidua A B, quod e&longs;t ab&longs;urdum. vel dicere
oportet
e&longs;t contra con&longs;tructionem, & propterea pariter inconueniens, non igitur
huiu&longs;modi lineæ &longs;unt ponendæ.
Decimus locus
in omni autem æquilatero perpendicularis
in mediam ba&longs;im mcidit, quare, & in medium indiuiduæ.
Ex 22. primi Elem.
ex tribus datis lineis, quarum quælibet duæ &longs;int, re
liqua maiores pote&longs;t con&longs;titui triangulum: poterit igitur ex tribus indiui
duis con&longs;titui
diuiduæ lineæ &longs;int æquales. &longs;it igitur ex eis triangulum A B C,
&longs;i igitur ab angulo A, ducatur perpendicularis A D, ad ba&longs;im
B C, eam bif
nea B C, &longs;ecabilis, contra quam aduer&longs;arij opinantur.
Vndecimus locus
protracta, & perpendiculari ducta, quadrati ce&longs;ta potentia neque duplum erit &longs;patium à diame
tro con&longs;urgens illius, quod ab indiuidua procreatur: nans æquali ablato, reliquum
erit minus indiutdua, nam &longs;i æqualis, diameter quadruplum de&longs;criberet, &c.)
ide&longs;t &longs;i per 46 primi quadratum. v.g. A B C D, ex qua
tuor in&longs;ecabilibus componatur, cuius diametro B C,
perpendicularis A E, in&longs;i&longs;tat, erit per 47. primi qua
dratum lineæ A B, æquale quadratis
quare tam E B, quàm A E, minores erunt ip&longs;a A B;
quare ip&longs;a non erit minima cum &longs;it indiuidua, quod e&longs;t
ab&longs;urdum. Præterea ex &longs;cholio 47. primi, quadratum
C B F G, diametri C B, duplum e&longs;t quadrati A B C D,
ergò diameter C B, maior quàm A B. Auferatur igitur ab ip&longs;a, C B, æqua
lis ip&longs;i A B, quæ igitur reliqua erit, vel erit æqualis ip&longs;i A B, vel minor. non
æqualis, quia tunc diameter dupla e&longs;&longs;et lateris A B, & quadratum diametri
quadruplum foret quadrati lateris A B. ex &longs;cholio 4. &longs;ecundi, quod ab&longs;ur
dum e&longs;t, repugnat enim 47. primi. nec minor, quia hoc modo exi&longs;teret linea
quædam minor minima, &longs;cilicet atoma, quod pariter e&longs;t inconueniens.
Duodecimus locus
di pote&longs;t, tùm æquales, tùm inæquales, &longs;eindatur linea in tria fru&longs;ta, quæ non con
&longs;tet ex tribus atomis, &longs;ed vniuer&longs;aliter ex imparibus numero atomis, &longs;ic diui&longs;a erit
linea indiuidua. &longs;imiliter autem &longs;i in duo diuidatur linea, quæ ex imparibus
hoc e&longs;t detur linea quæpiam ab aduer&longs;ario ex lineis
paribus, con&longs;tans. v. g. ex quinque; hæc diuidi pote&longs;t in tres æquas partes
per 10.6. Si igitur diuidatur in tria æqualia, nece&longs;&longs;ariò tres ex atomis illam
integrantibus erunt di&longs;&longs;ectæ, nam tertia quælibet pars continebit indiui
duam vnam cum duabus tertijs alterius partibus. idem accidet fi bifariam
per 10. primi, &longs;ecetur quæuis ex imparibus numero atomis conflata.
Decimustertius locus
quæ &longs;olum ex paribus conflata &longs;it. &longs;i iam in duas partes diui&longs;a, in
pote&longs;t diuideretur, &longs;ic
&longs;ita, per inæqualia &longs;cinderetur)
con&longs;titerit: ea igitur diuidatur primo bifariam. deinde iterum diuidatur
quomodocunque, ide&longs;t & bifariam, & non bifariam, nam hoc etiam pacto
indiuidua diuidetur, quod e&longs;t inconueniens.
Decimusquartus locus
ret enim longitudinem, & latitudinem;
aliquid, bæc autem aliquid aliud; quod &longs;i quadratum diuiduum e&longs;t, & linea, vnde
procreatur, diuidua erit)
&longs;cribi patet ex 46. primi, quadratum igitur de&longs;criptum ab indiuidua, cum
&longs;it &longs;uperficies, latitudinem, ac longitudinem habebit, quæ diuer&longs;æ &longs;unt di
men&longs;iones. poterit ergò &longs;ecundum
&longs;ariò latera ip&longs;ius, hoc e&longs;t lineæ, quas indiuiduas illi ponunt diuidentur,
quod e&longs;t inconueniens, non igitur indiniduæ erunt.
Decimusquintus locus
crit impartibile: vno quippe indi
profunditatem contineat: quare nec linea pote&longs;t e&longs;&longs;e atoma. corpus &longs;iquidem in &longs;u
per&longs;icies, &longs;uperficies verò in lineas &longs;oluitur)
linea per aduer&longs;arium extat indiuidua, &longs;ic & fuperficies ab eadem linea de
&longs;cripta erit atoma, & corpus ab hac &longs;uperficie de&longs;criptum erit impartibile. Sciendum enim, quod ex motu puncti de&longs;cribitur linea: ex motu lineæ de
&longs;cribitur &longs;uperficies: ex motu tandem &longs;uperficiei corpus ortum habet, vt &longs;o
let in horum definitionibus explicari.
Si igitur horum vnum nempè linea &longs;it atoma, & reliqua, quæ ab ip&longs;a ma
nant erunt indiui&longs;a, quia corpus diuiditur per &longs;uperficiem, & &longs;uperficies
per lineam, ide&longs;t ad diui&longs;ionem corporis nece&longs;&longs;e e&longs;t diuidi &longs;uperficiem, & ad
&longs;uperficiei diui&longs;ionem diuidi lineam, quæ ip&longs;am terminat. At cum omne
corpus latitudinem, & profunditatem habeat, nullum poterit extare cor
pus, quod diuidi nequeat; quare neque illud, quod ab atoma linea oriretur. Quare nec linea illa corporis procreatrix erit indiuidua; corpus &longs;iquidem
in &longs;uperficies, & &longs;uperficies in lineas quodammodo re&longs;oluitur: & ex diui
&longs;ione &longs;olidi &longs;uperficies &longs;ecari debet, & demum &longs;uperficiei, &longs;ectionem lineæ
&longs;ectio &longs;ub&longs;equitur. Tollendæ igitur &longs;unt de rerum natura lineæ atomæ.
Decimus&longs;extus locus
bus tanget punctis, punctus enim contactus, quiqué e&longs;t in circulo, quiqué e&longs;t in recta,
&longs;e &longs;e mutuò tangunt. quod &longs;i hoc fieri nequit,
quod &longs;i &longs;e tangere nequeunt,
tangere nece&longs;&longs;arium e&longs;t.
In 2. 3. & corollario eius demon&longs;tratur circuli peripheriam tangere re
ctam lineam in vnico puncto. iam &longs;i linea con&longs;taret ex punctis indiuiduis
tanquam partibus, po&longs;&longs;et circulus
Sit circulus, cuius centrum A, tangens lineam
rectam B C, con&longs;tantem ex punctis, quorum vnus
&longs;it in extremo D, lineæ B D, alterum verò in E,
principio lineæ E C, circulus A, tangere poterit
in F, termino communi vtriu&longs;que lineæ, hocque
modo tanget
impo&longs;&longs;ibile per 2. 3. &longs;equitur igitur
gere, & eadem ratione nulla alia
e&longs;t, impo&longs;&longs;ibile e&longs;&longs;e, lineam ex huiu&longs;modi punctis con&longs;tare po&longs;&longs;e.
Reliqua huius opu&longs;culi, quamuis Mathematica alicui videri po&longs;&longs;int,
non tamen &longs;unt, non enim linearibus indigent demon&longs;trationi
bus, ad Phy&longs;i
cum igitur pertinebunt, cuius e&longs;t di&longs;putare, num
indiuidua exi&longs;tant, & quomodo in quanti
tate,
ex Geometria deductis.
Libellum de cau&longs;is proprietatum Elementorum, quamuis nonnulla
mathematica loca contineat, tamen, quia certò con&longs;tat ex ijs,
quæ in eo de Secta Arabum, de Sclauis, de Dalmatis, qui multis
po&longs;t Ari&longs;totelem &longs;æculis floruerunt, auctorem alium e&longs;&longs;e ab Ari&longs;to
tele con&longs;ultò & meritò omi&longs;i.
Alterum de cau&longs;is libellum pariter prætermi&longs;i, cum is vocibus Arabi
cam barbariem redolentibus &longs;cateat: phra&longs;is præterca, & qu&ecedil;dam
de Deo dicta, planè indicant authorem non e&longs;&longs;e Ari&longs;tctelem; &longs;ed potius
Arabem quempiam.
EX LIBRO NONO
DE HIST. ANIMALIVM
Cap. 39.
tanquam excrementum, vt Democritus ait, &longs;ed ab extrin&longs;eco de &longs;uo cor
pore, veluti cortice; aut more eorum animalium, quæ &longs;uos villos iacu
lantur, vt hystricis)
animum meum illa cupido, vt &longs;cilicet certò &longs;cirem, numiure, an iniuria
Ari&longs;t. Democritum hoc loco reijceret, Araneum fila ab intrin&longs;eco emitte
re a&longs;&longs;erentem: quapropter ad magi&longs;tram rerum experientiam confugi, ac
cepto manu bacillo Araneum quendam ex ijs, qui circulares telas, quas
contexunt, lic adij, vt Araneus pro arbittio &longs;uper bacillum liberè inambu
laret, dum ip&longs;e interim curio&longs;ius illum ob&longs;eruarem, quanam videlicet ex
parte filum foras ederet; cum ecce tibi Araneus experienti mihi vltrò fa
uens &longs;e &longs;e ex baculo demi&longs;it, ita tamen, vt ex filo &longs;uo in aere &longs;u&longs;pen&longs;us re
maneret. cum primum ob&longs;eruo ip&longs;um inuer&longs;um, hoc e&longs;t capite deor&longs;um, &
ventre &longs;ur&longs;um pendere. vt autem acutius cernerem, eum opacæ cuidam rei
oppo&longs;ui, ne præ nimia luce tenui&longs;&longs;imum aranei filum aciem oculorum effu
geret; quo facto in temperata luce illa, clari&longs;&longs;imè videbam filum ex &longs;ece&longs;&longs;u
aranei prodire.
ret, coegi deinde ip&longs;um a&longs;cen
dere, & de&longs;cendere &longs;æpius, donec certò certius, mihi con&longs;titi&longs;&longs;et filum illud
non ab extrin&longs;eco, vt hoc loco Ari&longs;t.
affirmat, &longs;ed ab intrin&longs;eco quippe ex
&longs;ece&longs;&longs;u prodire, ac proinde veri&longs;&longs;imam e&longs;&longs;e quamuis ab Ari&longs;t.
reiectam De
mocriti &longs;ententiam. cum Ari&longs;t.
pariter errauit Vly&longs;&longs;es Aldobrandus in &longs;uo
de in&longs;ectis pulcherrimo,
Verumenimuerò opportunè accidit, vt huius dubitationis &longs;olutio, aliam
mihi alterius quæ&longs;tionis, iam olim &longs;ummis votis expetitam afferret expli
cationem. ea e&longs;t huiu&longs;modi.
&longs;æpius fueram expertus, Araneos quo&longs;dam e&longs;
&longs;e, qui ex vno loco ad alium omninò &longs;ibi inacce&longs;&longs;ibilem, tran&longs;eant, &longs;iue quod
idem e&longs;t, ex eo loco, ad illum fila deducant, vt ex vna arbore ad aliam;
quamuis inter
nantur. quod maximè mane æquitantes experimur, dum nobis fila per vias
tran&longs;uer&longs;a, oculis, atque vultui obuiantia adhærent. Qua ratione id Ara
neus perficeret, neminem, quiliteris manda&longs;&longs;et, reperi, ne ip&longs;um quidem. Vly&longs;&longs;em Aldobrandum, qui in hac eruditorum palæ&longs;tra, maiores no&longs;tros
omnes videtur &longs;upera&longs;&longs;e. Phy&longs;iologi à me hac de re interrogati, varij va
ria, nec con&longs;entientia refpondebant. Alij aiebant Araneum &longs;e demittere,
ac &longs;u&longs;pendere ex vna arbore, & deinde ad aliam à vento perferri, at ego his
minimè a&longs;&longs;entiebar, quia m Araneo nullum e&longs;&longs;et naturale in&longs;trumentum, Alij Araneum ex vna arbo
re de&longs;cendere, & po&longs;tea alteram con&longs;cendere, interim emi&longs;&longs;um retro filum
raptando, ac deinde &longs;ur&longs;um attrahendo attollere, ac prætendere: &longs;ed ho
rum re&longs;pon&longs;ionum ob plurima impedimenta, quæ tenui&longs;&longs;imum filum &longs;æpius
&longs;cidi&longs;&longs;ent, &longs;ubridens refellebam. Alij verò aiebant Araneum qualitate qua
dam præditum e&longs;&longs;e, qua ip&longs;e per aera, non &longs;ecus, ac per aquam pi&longs;ces, &
per aerem volucres, ambulare po&longs;&longs;et. Verum opinatio i&longs;ta, ne ri&longs;u quidem
digna videbatur. Huius igitur quæ&longs;iti &longs;olutionem, quam omnes ad hanc
co auctarij initio promi&longs;&longs;i, nunc per&longs;oluam. accidit ergò, vt dicebam, vt
dum Araneus fugæ cupidus ex bacillo in temperatæ lucis loco, nimirum è
regione alicuius opaci penderet, vt cernerem ex filo illo, ex quo &longs;u&longs;pende
batur plura alia fila hinc inde alternatim prodire, quemadmodum ex alter
nis arundinum nodis folia ena&longs;ci &longs;olent. quæ fila, innata læuitate, per ae
rem quoquo ver&longs;us ceu natantia diffundebantur. factum e&longs;t autem, vt eo
rum vnum quendam arboris cuiu&longs;dam ramum attingeret,
reret; quod illicò Araneus optimè per&longs;en&longs;it, quippe quod filum illud vi&longs;ce
ribus eius ex altero capite affigeretur, atque per filum illud, alijs ommi&longs;&longs;is,
&longs;ubitò, vti egregius funambulus accurrit, &longs;ed tamen pedibus &longs;ur&longs;um, dor &longs;o
autem deor&longs;um, non &longs;upra filum, &longs;ed infra ad ramum illum &longs;e contulit,
me ho&longs;tem &longs;uum fuga &longs;æpius elu&longs;it. Ex qna repetita &longs;æpius ob&longs;eruatione lu
ce clarius comperi Araneum non &longs;implex filum, &longs;ed ramo&longs;um, ac multiplex
emittere,
&longs;u&longs;pendatur, alterum
verò, quod &longs;orte hac,
cui rei occurrat,
hæreat, per quod po
&longs;tea
locum &longs;ibi prius inac
ce&longs;&longs;um, aditum parat. qua inre fures eos per
bellè imitatur, qui
&longs;chalas ex funibus con
textas, ac hamis fer
reis munitas, ad fene
&longs;tras proijciunt, vt per
cas ibi affixas con&longs;cen
dere queant. quæ om
nia ex appo&longs;ita figura
melius percipies, vbi
ex &longs;ini&longs;tra arbore pen
det Araneus A, ex filo
B A, ex quo tanquam
rami alia fila C G, D H,
E I, M O, F L, alterSi ergò filum E I, dextræ
arbori occurrerit,
&longs;cius per filum A E I, a&longs;cendit,
rem transfert;
rum capturam contexere; quales aliquando inter duas arbores admira
ri &longs;olemus.
Quæres fortè, num Araneus filum intus tanquam in glomo, vel fpira con
uolutum contineat? dicam, quod non &longs;ine experientia conijcio, exi&longs;timo
Araneum non continere intra &longs;e filum vllum, verum humorem quendam
vi&longs;co&longs;um, qui in tenui&longs;&longs;ima fila &longs;it ductilis; quemadmodum videmus acci
dere gummi, quæ di&longs;rupta exhibet lentorem quendam, qui &longs;olo attritu ita
digitis hæret, vt amoto &longs;en&longs;im digito, filum tenue, & oblongum valdè de
ducatur, hoc inde conijcio, quia aliquando cum ventrem Araneorum &longs;ecui&longs;
&longs;em nullum intus filum, &longs;ed &longs;olus humor quidam lentus apparuit.
Cùm ex paruulis hi&longs;ce meis ob&longs;eruationibus circa animalculum i&longs;tud
vnum tam præclara cognoui&longs;&longs;em, quæ nullus ad hanc
ob&longs;erua&longs;&longs;et; animaduerti lati&longs;&longs;imum patere campum ad animalium hi&longs;to
riam ampliandam, &longs;i ij, qui huic pulcherrimæ cognitioni dant operam, non
ijs &longs;olum, quæ ab alijs per&longs;cripta &longs;unt contenti e&longs;&longs;ent, verùm etiam certi&longs;
&longs;imis,
Atque hæc de Araneo &longs;atis.
Cap. 7.
rum vici&longs;&longs;im &longs;tant, pondusqué &longs;ustinent, nece&longs;&longs;e habent altero progredien
te, inflectere alterum; æqualia namque longitudine nata &longs;unt habere op
po&longs;ita membra. & quod ponderi &longs;ub&longs;tat rectum e&longs;&longs;e oportet, vt perpen
diculum ad terram. quando autem progreditur, fit hypotenu&longs;a, valens manentem
magnitudinem, & eam, quæ interiacet. quoniam autem æqualia &longs;unt membra, ne
ce&longs;&longs;e e&longs;t inflecti id, quod manet, aut in poplite, aut in conflexione)
in gre&longs;&longs;u nece&longs;&longs;ariam e&longs;&longs;e aliquam flexionem membrorum. verum prius
&longs;ciendum, quod lineam hypotenu&longs;am, quemadmodum etiam Athenæus lib.
10. te&longs;tatur, eam appellant geometræ, quæ in triangulo rectangulo recto
angulo &longs;ubtenditur, vnde & denominata e&longs;t hypotenu&longs;a, ide&longs;t &longs;ubten&longs;a, vt
in triangulo A B C, cuius angulus B, rectus &longs;it, recta
A C, angulo recto B, &longs;ubten&longs;a, hypotenu&longs;a dicitur. Ari&longs;t.
igitur ait, quod antequam animal ambulare in
cipiat, dum &longs;cilicet manet, habet crura, quæ manent
recta, &longs;iue perpendicularia horizonti, cum autem in
cipit progredi nece&longs;&longs;e e&longs;t
rizontem. nam primum crus in ingre&longs;lu prolatum fit
hypotenu&longs;a, quia &longs;cilicet &longs;ubtendit angulum rectum,
quem facit alterum crus adhuc quie&longs;cens, cum hori
A D, quæ manente animali, fui&longs;&longs;ent ambo &longs;imul in &longs;itu A B, perpendicula
ria horizonti; incipiens autem animal ambulare, proferat primo crus A D,
A D, fiet hypotenu&longs;a trianguli A B C, & quia crus hoc A D, factum hypo
tenu&longs;a æquale e&longs;t alteri manenti A B, nequit totius veræ hypotenufæ A C,
officio fungi, quæ æquiualet toti A D, & præterea interiacenti D C, vrea au
tem hypotenu&longs;a debet e&longs;&longs;e maior, quia opponitun
maiori angulo nimirum recto B, quam latus A B,
quod angulo acuto C, opponitur per 19. primi, &
propterea ni&longs;i alterum &longs;ub&longs;equens crus A B, incli
netur, vt in &longs;ecunda figura, non pote&longs;t hypotenu&longs;a
A D, terram attingere,
e&longs;t, vt initio gre&longs;&longs;us
pendiculare erat, inclinetur; inclinato igitur crure
A B, antror&longs;um tunc prolatum crus A C, terram
contingit,
Eodem loco
&longs;i quis enim iuxta
parietem per terram ambulet, quæ de&longs;ignatur linea non e&longs;t recta, &longs;ed obtorta, quo
niam minorem quidem flectentis fieri de&longs;criptam nece&longs;&longs;e e&longs;t; &longs;tantis autem, & ere
cti maiorem)
deprimitur, &longs;ignum hoc affert, quia &longs;i quis &longs;ecus parietem per terram am
bulet, linea quam vertex capitis in pariete de&longs;ignat non e&longs;t recta, &longs;eb obtor
ta: quæ linea optimè de&longs;ignatur, &longs;i ambulantis vmbra in pariete apparens
&longs;imul, cum ip&longs;o in pariete ambulet; videmus enim vmbram illam modo al
tiorem fieri, modo breuiorem; quod &longs;ignum e&longs;t ambulantem modo incli
nari, quando &longs;cilicet crus alterum profert, &longs;eu crura dilatat; modo erigi,
cum crus &longs;ub&longs;equens præce denti coniungit, tune enim incedens fit horizon
ti perpendicularis.
Eodem cap.
cto, vel non progredietur: &longs;i enim altero crure recto progreditur alterum, maius
cr
&longs;am, nece&longs;&longs;e igitur e&longs;t, & inflectere id, quod procurrit, & inflexum &longs;imul alterum
extendere, membra enim triangulorum æquilaterorum efficiuntur,
rius, vbi perpendiculum fuerit, in quo firmatum e&longs;t)
ijs, quæ in primo huius capitis loco dicta &longs;unt. proinde ea cum duabus illis
triangulorum figuris repetenda &longs;unt, vt breuius quæ nunc re&longs;tant explicen
tur. quoniam igitur animal antequam gradiatur, maximè homo, &longs;tat hori
zonti perpendicularis, nece&longs;&longs;e e&longs;t ad progrediendum, vt fiat aliqua mem
brorum inflexio, &longs;i enim homo &longs;ine vlla &longs;ui corporis flexura inclinet &longs;e ad
horizontem, ita vt cum horizonte faciat ex anteriori parte. v. g. angulum
recto minorem, &longs;iue acutum, vel concidet, vel non poterit progredi; &longs;i enim
alterum crus præmitteretur, altero manente perpendiculari,
deretur qui&longs;piam, &longs;equeretur crus prolatum, quale e&longs;t A D, iu priori trian
gulo, debere fieri maius altero crure A B, manente, quia fieret tota hypo
tenu&longs;a A C, &longs;ie enim terram attingeret; at non pote&longs;t fieri illo maius, quia
e&longs;t illi æquale, ergò hac ratione ince&longs;&longs;us fieri nequit. nece&longs;&longs;e igitur refle
poris flexuram, vel nodum, vt circa genu, aut alia. crura enim in gre&longs;&longs;u fiunt
latera &longs;uperiora trianguli i&longs;o&longs;celis, vt in &longs;ecunda figura patuit, cuius ba&longs;is
e&longs;t pa&longs;&longs;us. & tunc caput ambulantis fit inferius, quàm antequam gradere
tur; quia tunc ambo crura erant horizonti perpen
dicularia. quando autem caput fuerit in linea
pendiculari
alibi, vt in pr&ecedil;&longs;enti figura, linea
guli huius i&longs;o&longs;celis e&longs;t linea A E, quia ba&longs;i B C, per
pendicularis incidit; quando igitur caput ambulan
tis. v. g. D, fuerit in hac linea,
in quauis alia gre&longs;&longs;us parte: quia tunc crura A B,
A C, &longs;unt maximè diuaricata, & proinde angulus A,
& &longs;imul punctum D, maximè demi&longs;&longs;a.
Cap. 1.
particularum quie&longs;cere ahquam, & propter hoc, & flexus animalibus
in&longs;unt: tanquam enim centro vtuntur flex: bus & fit tota pars, in qua
e&longs;t flexus & vna, & duæ; & recta, & flexa, quæ permutatur potentia,
& actu, propter flexum. cum autem flectitur, & mouetur, hoc quidem &longs;ignum mo-
metri, quæ quidem A D, maneat, quæ cutem B, moueatur, &
fiat A C, &longs;ed hic quidem videtur, &longs;ecundum omnem modum in
diui&longs;ibile e&longs;&longs;e centrum. etenim moueri, vt aiunt, fingunt in ip&longs;is,
non enim mouetur mathematicorum aliquid.
Intendit probare nece&longs;&longs;e e&longs;&longs;e ad motum animalium, vt
vna pars quie&longs;cat, dum altera mouetur. propter hoc enim inquit flexus ani
malibus in&longs;unt, vbi in græco pro voce flexus legitur
cat nodum, articulum, &
quam enim centro quodam vtuntur flexibus, ide&longs;t nodis, &longs;eu iuncturæ &longs;unt
in motu membrorum in&longs;tar centri. v. g. nodus cubiti fit centrum, cum bra
chij parte, quæ e&longs;t inter humerum, & cubitum manente, reliquum brachij
circumducimus; &longs;ic manente genu tanquam centro, crus huc illud agita
mus, & fit tota pars. v. g. totum brachium, in quo e&longs;t cubiti iunctura, & vna
tota pars, quando manet rectum; & duæ
infle ctitur; & fit tota hæc longitudo recta prius, po&longs;tea flexa: quæ propter
flexuram modo vna e&longs;t actu, &longs;ed duæ potentia. modo duæ in actu, &longs;ed vna in
potentia. cum autem flectitur, & mouetur brachium, vnum quidem fignum,
&longs;iue punctum, quod e&longs;t extremum partis manentis, manet; alterum verò &longs;i
gnum, &longs;iue punctum, quod e&longs;t extremum partis motæ
tiguum mouetur &longs;imul cum tota parte mota. quemadmodum, &longs;i diametri
&longs;uperioris figuræ, pars D A, maneat, pars autem A B, moueatur ad A C,
erit huius flexuræ centrum A, quod vt extremum lineæ D A, manentis, maquamuis in mathematicis hæc
quidem duorum centrorum di&longs;tinctio nulla &longs;it, quia centrum mathemati
cum omninò indiuiduum e&longs;t: neque in mathematicis e&longs;t propriè motus,
quamuis enim aliquando Mathematici dicant, &longs;i linea, vel &longs;i punctum mo
ueretur, vel moueatur, & &longs;imilia, huiu&longs;modi tamen motus &longs;unt rebus ma
thematicis extrin&longs;eci, nec quatenus hoc modo mouentur con&longs;iderantur:
patet igitur, qua ratione Ari&longs;tot. partem manentem in motu nece&longs;&longs;ariam
e&longs;&longs;e velit.
Cap. 5.
tanea i&longs;ta erant machinæ, quæ à &longs;eip&longs;is mouebantur, quas Græci automata
dixerunt, cuin&longs;modi &longs;unt Automata Heronis Alexandrini, quæ adhuc
Cap. 8. E&longs;t ibi quoddam triangulum cum elementis more geometrarum
depictum, vnde locus ille videri po&longs;&longs;it mathematicus, verumtamen nullo
modo geometriæ auxilio indiget.
Lib. 2. cap.
1.
chinas illas miro artificio confictas, quæ à &longs;e ip&longs;is intrin&longs;eco prin
cipio mouebantur, quas Græci veteres Automata, ide&longs;t &longs;pontanea,
vel &longs;pontina, vt vertit Interpres vocabant, cuiu&longs;modi &longs;unt Auto
mata Heronis Alexandrini, quæ adhuc extant græca,
&longs;tallen&longs;i in Italicum &longs;unt conuer&longs;a. Automata hodie &longs;unt Horologia, quæ ex
multis dentatis rotis Germani con&longs;truunt.
Lib. 2. cap.
4.
vide quæ de hac re &longs;crip&longs;i lib.
1. Priorum, &longs;ecto 3. cap.
1.
Ibidem
tamen cau&longs;a eorum aliqua & demon&longs;tratio e&longs;t.
Quælibro 1. Priorum, &longs;ecto 1. cap.
23. de hac re annotata &longs;unt, abundè
huic etiam loco &longs;atisfaciunt.
Lib. 1. cap.
7.
ritatis &longs;peculator &longs;it, quid, & qualis &longs;it, indagat)
confirmatur ex eo, quod Fabri omnes vtuntur amu&longs;&longs;i, &longs;eu norma,
quæ nihil aliud e&longs;t quàm angulus rectus, quæ vulgò
&longs;quadra dicitur, vt eius auxilio angulum ip&longs;um re
ctum in opus conferant,
ctu &longs;ua ip&longs;i opera ad angulos rectos, ide&longs;t quadrata,
conficiunt. Geometra verò con&longs;iderat eundem an
gulum, quatenus fit à linea &longs;uper lineam aliam per
pendiculariter in&longs;i&longs;tente, vt e&longs;t in definit. 10. primi,
vt faciat angulos hinc inde æqualis A B D, A B C, prædictos inquam duos
angulos con&longs;iderat e. &longs;e rectos.
contemplatur præterea Geometra omnes
angulos rectos e&longs;&longs;e inter &longs;e æquales, vt in 12. axiomate primi Elem.
ponitur,
& &longs;imilia plura alia, quorum con&longs;iderationem Faber omninò negligit.
Libro 2. capite 6.
Arithmetica ratio, fiue proportio ea e&longs;t, cuius termini cre&longs;cunt per æqua
les exce&longs;&longs;us, vt 2. 6. 10. 14. horum enim terminorum exce&longs;&longs;us æquales &longs;unt,
cum &longs;int omnes quaternarij. &longs;imiliter inter hos terminos 3. 6. 9. 12. e&longs;t arith
metica analogia, cùm omnes ternario numero &longs;uperent præcedentes, & à
&longs;equentibus &longs;uperentur. Porrò apud Mathematicos tria &longs;unt genera pro
portionum, &longs;iue medietatum, Arithmetica quam modo &longs;uppo&longs;ui; Geome
trica, & Harmonica, quas inferius oblata occa&longs;ione opportunius explicabo.
Lib. 2. cap.
9.
cuiu&longs;libet, &longs;ed
&longs;iue centrum dati circuli docet Euclides propo&longs;itio
ne prima 3. hoc modo. in datocirculo ducatur vt
cunque recta B C, quæ per 10. primi diuidatur bifa
riam in F, & per F, ducatur
quæ &longs;ecetur bifariam in E,
lum ip&longs;ius lineæ medium; &longs;ed etiam totius circuli
centrum, quemadm odum ibi demon&longs;trat Euclides.
Lib. 3. cap.
3.
de mundo, aut diametro, & latere, quod nulla inter &longs;e
æquabilitate conueniant)
nulia æquabilitate, ide&longs;t nulla communi men&longs;ura inter &longs;e conueniant, fusè
explicatum e&longs;t libro Priorum, &longs;ecto 1. cap.
23.
Eodem cap.
quemadmodum de&longs;ignationes)
cas demon&longs;trationes &longs;æpius dictum e&longs;t in logicis textibus, quod pariter ex
hoc loco confirmatur. quando autem ait
modum de&longs;ignationes)
ctum e&longs;t in explicatione tituli librorum Re&longs;olutoriorum; quam expo&longs;ui, ni
hil aliud e&longs;&longs;e, quam medij inqui&longs;itionem ad id, quod propo&longs;itum fuerit de
mon&longs;trandum. veram autem,
nem, hoc loco Ari&longs;t.
ip &longs;e confirmat, cum hanc re&longs;olutionem dicat e&longs;&longs;e &longs;imi
lem con&longs;ultationi, &longs;iue inqui&longs;itioni mediorum ad finem in rebus practicis
con&longs;equendum; ip &longs;a verò e&longs;t inqui&longs;itio mediorum ad id, quod in rebus &longs;pe
culatiuis propo&longs;itum e&longs;t, demon&longs;trandum. con&longs;ultatio igitur e&longs;t in rebus
practicis, quod in &longs;peculatiuis e&longs;t re&longs;olutio.
Lib. 5. cap.
3.
quàm numero in vniuer&longs;um proprium e&longs;t)
numerus ex vnitatibus ab&longs;tractis con&longs;litus, ide&longs;t, cuius vnitates non &longs;int res
phy&longs;icæ, &longs;ed à naturalibus ab&longs;tractæ, qualis con&longs;iderat Arithmeticus: omni
tamen numero &longs;iue ab&longs;tracto, &longs;iue non, connenit proportiones &longs;u&longs;cipere,
id e&longs;t & numero, & rebus numeratis.
Ibidem
co intelligenda e&longs;t illa, quam nunc appellant proportionalitatem, quæ e&longs;t
duarum rationum, &longs;eu proportionum &longs;imilitudo, &longs;iue æqualitas, vt manife
&longs;tum e&longs;t ex 4. definit. 5. Elem.
v. g. cum &longs;it eadem ratio 9. ad 6. quæ e&longs;t 6. ad
4. propterea hæc rationum &longs;imilitudo, vel æqualitas dicitur ip&longs;a proportio,
&longs;eu di&longs;tinctionis gratia Proportionalitas.
Ibidem
per&longs;picuum e&longs;t: &longs;ed & continentem nibilominus, vno enim hæc perinde, ae duobus
vtitur, bi&longs;que id accipit in bunc modum, qualis primi re&longs;pectus e&longs;t ad &longs;ecundum,
talis &longs;ecundi ad tertium; bis enim bic, &longs;ecundum dictum e&longs;t, quare &longs;i &longs;ecundum bis
po&longs;itum &longs;it, quatuor erunt ea, quæ con&longs;tant proportione)
cuntur de&longs;umpta &longs;unt, partim ex definit. 6. 5. partim ex 9. definit.
eiu&longs;dem.
breuiter autem &longs;ic &longs;e habent.
Ad con&longs;tituendam proportionalitatem ne
ce&longs;&longs;arij &longs;unt omninò quatuor termini, quod quidem primum per&longs;picuum
e&longs;t in ea proportionalitate, quam Difiunctam vocant, quæ e&longs;t huiu&longs;modi,
vt 9. ad 6. ita 3. ad 2. deinde
quæ talis e&longs;t, vt 9. ad 6. ita 6. ad 4. quæ in tribus quidem terminis 9. 6. 4.
con&longs;i&longs;tit, &longs;ed tamen, qnia medius 6.
duorum gerit, ac proinde e&longs;t, ac &longs;i hoc modo termini di&longs;ponantur 9. 6. 6. 4.
vbi 6. bis ponitur, hinc Ari&longs;t.
textum &longs;atis intelligere poteris.
Eodem cap.
ad quartum; igitur etiam alterna vice, &longs;icut primus ad tertium, ita &longs;ecundus ad
quartum. quare etiam totum ad totum, quod di&longs;tributio binatim copulat.
quæ &longs;i
etiam ita compo&longs;ita fuerint, iustè copulat)
modum, quem Geometræ alternam rationem vocant, 12.
5. exponunt, vt eam rebus ip&longs;is accommodet,
autem huiu&longs;modi, &longs;int primum quatuor termini proportionales, ide&longs;t, vt
primus ad &longs;ecundum, ita tertius ad quartum. v. g. vt 9. ad 6. ita 3. ad 2.
valet con&longs;equentia hæc, ergò etiam alternatim erit, vt primus ad tertium,
ita &longs;ecundus ad quartum, v. g. in allato exemplo, ita erit 9. ad 3. vt 6. ad 2.
quam &longs;equelam e&longs;&longs;e validam probat deinde Euclides propo&longs;it. 16. 5. hinc
aliam deducit con&longs;equentiam, quam Euclides propo&longs;it. 12. 5. demon&longs;trat,
dum ait, quare etiam totum ad totum erit. v. g. quia conclu&longs;um e&longs;t ita e&longs;&longs;e
9. ad 3. quemadmodum 6. ad 2. ita etiam erit totum ad totum, ide&longs;t ita
etiam erunt antecedentes termini &longs;imul ad con&longs;equentes &longs;imul, v. g. ita erit
etiam totum 15. quod e&longs;t totum ex antecedentibus terminis 9. & 6. ad to
tum 5. conflatum ex con&longs;equentibus terminis 3. & 2. In &longs;umma igitur &longs;i fue
rit vt 9. ad 3. ita 6. ad 2. ita etiam erit 15. ad 5. quod verum e&longs;&longs;e apparet in
his numeris, cum tam 9. ad 3. quà 6. ad 2. & 15. ad 5. habeant triplam
proportionem.
Horum exemplum in rebus practicis &longs;it hoc: &longs;it vt Plato ad Proclum, ita
mille aurei ad quingentos aureos, ergò alternatim ita erit Plato ad 1000.
aureos, &longs;icuti Proclus ad 500. quare ita etiam totum erit ad totum, &longs;cilicet
Plato, & Proclus &longs;imul ad 1000. & 500. &longs;imul, quæ duo tota, di&longs;tributio mo
ralis, ac practica diuidit, & binatim copulat, hoc modo dicens, vt Plato ad
iuxta
modis argumentandi ab Euclide comprobatis, nitatur.
Ibidem
propterea quod in Geometrica euenit, vt eandem totum ad totum rationem habeat,
quam habet alterutrum, ad alterutrum)
rationum &longs;imilitudinem Mathematici proportionalitatem Geometricam
appellant, propterea quod in hac duarum rationum geometricarum &longs;imili
tudine accidit, vt &longs;it totum ad totum, quemadmodum etiam partes toto
rum, vt &longs;upra explicatum e&longs;t; quod non accidit in duarum proportionum
arithmeticarum &longs;imilitudine; &longs;i enim ponamus has duas rationes arithme
ticas &longs;imiles, vt 10. ad 8. ita 6. ad 4. quæ &longs;unt &longs;imiles, propter &longs;imiles exce&longs;
&longs;us primorum, & &longs;ecundorum terminorum, cum non erit tamen totum 16. ad totum 12. in eadem ratione cum diui&longs;is ter
minis, cum ibi &longs;it exce&longs;&longs;us binarij, hic verò quaternarij. hæc videtur e&longs;&longs;e
Ari&longs;t.
ratio; quam adhuc melius declara&longs;&longs;e libet. Geometrica igitur pro
portionalitas ita dicta e&longs;t, quia quælibet proportio pote&longs;t in materia Geo
metrica, lineis, &longs;uperficiebus, & corporibus continuari in quatuor termi
nis, ita vt proportionalitas, &longs;eu &longs;imilitudo rationum exurgat, quod in nu
meris fieri &longs;emper nequit, cum plures fint proportiones, quæ numeris ex
primi nequeunt, vt &longs;unt eæ, quas irrationales appellant, cuiu&longs;modi e&longs;t inter
diametrum, & co&longs;tam eiu&longs;dem quadrati, cuius nec proportio, nec propor
tionalitas in numeris reperiri pote&longs;t, quæ tamen in lineis, &longs;uperficiebus, ac
corporibus e&longs;&longs;e po&longs;&longs;unt: e&longs;t enim vt diameter vnius quadrati ad latus eiu&longs;
dem, ita idem latus ad aliam lineam inuentam per 11. 6. vel vt diameter ad
co&longs;tam, ita quælibet alia linea ad aliam inuentam, per 12. 6. omnis igitur
proportionalitas rebus Geometricis ine&longs;&longs;e pote&longs;t; non autem numeris, in
quibus &longs;olum po&longs;&longs;unt e&longs;&longs;e rationes rationales, &longs;eu
latius igitur patet Geometrica hæc &longs;imilitudo, quàm Arithmetica, cùm
Geometrica complectatur tam rationales, quàm irrationales. meritò igi
tur talis proportionalitas appellari debuit à rebus Geometricis, in quibus
&longs;emper reperitur, non autem ab Arithmeticis, cum quibus &longs;æpius reperiri
nequit. Vide Campanum in explicatione definitionis 3. 5. Elemen.
Ibidem
minus efficitur, & cui, & quod)
practicas, non e&longs;t continens, ide&longs;t, quæ con&longs;i&longs;tat in tribus tantum terminis,
quorum medius e&longs;t, ad quem refertur primus, & is qui refertur ad ter
tium; &longs;ed e&longs;t di&longs;iuncta, quia con&longs;tat &longs;emper quatuor terminis, quorum duo
&longs;unt per&longs;onæ aliquæ, reliqui verò duo &longs;unt res, quæ per&longs;onis debentur, vt &longs;i
&longs;int Plato, & Proclus, quibus iuxta meritorum quantitatem debeant diuidi
1500. aurei, debent diuidi aurei in duas partes, quæ habeant eam propor
tionem, quam habet Plato ad Proclum. quod &longs;i Plato duplum m&ecedil;ruit quàm
Proclus, erit vt Plato ad Proclum, ita 1000. ad 500.
Ex quibus patet hanc analogiam in rebus agendis non ni&longs;i in quatuor
terminis con&longs;i&longs;tere po&longs;&longs;e, & ideo non e&longs;&longs;e continuam, &longs;ed di&longs;iunctam, vt vo
lebat Ari&longs;tot.
Lib. 5. cap.
6.
nalitate Geometrica, vel Arithmetica; quæ autem &longs;it proportionalitas
Geometrica, dictum e&longs;t paulò ante in prioribus locis Mathematicis huius
quinti libri; quæ verò &longs;it proportionalitas Arithmetica dictum e&longs;t &longs;uperius
lib.
2. cap.
6. Verum hæc Arithmetica proportionalitas, meritò ab Ari&longs;tot.
hic contradi&longs;tincta e&longs;t à proportionalitate Geometrica: quia Arithmetica
hæc analogia attenditur &longs;olum, iuxta eundem exce&longs;&longs;um numerorum, non,
autem iuxta proportionem, &longs;eu habitudinem terminorum ad inuicem, quod
maximè in Geometrica &longs;pectatur. propterea Mathematici
candam e&longs;&longs;e potius medietatem Arithmeticam, quam proportionalita
tem, cum quibus nunc Ari&longs;t.
con&longs;entit.
Lib. 6. cap.
5.
bere, velnon habere)
ctionem expo&longs;ui.
Lib. 6. cap.
8.
quid &longs;it, quod
puer fieri Mathematicus pote&longs;t, &longs;apiens autem naturalis non pote&longs;t. An quia illa
per ab&longs;tractionem &longs;unt, horum autem principia ab experientia &longs;umuntur)
loco manife&longs;tè apparet Ari&longs;t.
exi&longs;timare principia Mathematica nullo mo
do nobis per experientiam innote&longs;cere, quod nonnulli negant.
Lib. 7. cap.
8.
&longs;icut in Mathematicis &longs;uppo&longs;itiones; nam neque illic ratio e&longs;t, quæ doctrinam tra
dat principiorum, neque hic
corum tria &longs;unt genera, definitiones, po&longs;tulata, axiomata, quæ in ip&longs;o primi
Elementorum ve&longs;tibulo proponuntur: &longs;olaque terminorum explicatione
Cap. 1. (
Pythagoras. Porrò definit.
8. 7. &longs;ic habetur: Pariter par nume
rus e&longs;t, quem par numerus per numerum parem, ide&longs;t paribus vi
cibus, metitur, cuiu&longs;modi e&longs;t numerus 24. quem numerus 6. me
titur per numerum parem, nimirum per 4. quia &longs;cilicet numerus 6. paribus
vicibus, quippe per 4. &longs;iue quater ip&longs;um numerum 24. men&longs;urat, quia to
ties in ip&longs;o adæquatè continetur.
Cap. 2. (
les habere angulos, &longs;umere principinm huiu&longs;modi, anima immortalis est
quæ de hac trianguli proprietate fusè &longs;crip&longs;i lib.
1. Priorum, &longs;ect. 3. cap.
1.
quam affectionem debet Geometra demon&longs;trare ex Geometriæ principijs,
quemadmodum facit Euclides in 32. primi, non autem ex principijs extrin
&longs;ecis, vt quod anima &longs;it immortalis.
Cap. 10.
Per&longs;picuè autem licet hoc in Geometria magis intueri, vbi cum aliqua &longs;ump&longs;eris
principia, vt ea habuerint, ita etiam, quæ ip&longs;a con&longs;equuntur
duobus rectis æquales habet angulos, quadratum & &longs;i tria gulum &longs;ecus, ita etiam, & quadratum commutabitur,
ex altera parte enim ei re&longs;pondet. & &longs;i quadratum quatuor angulis rectis æquales,
non habuerit angulos ne quidem triangulum duobus rectis habebit æquales)
trianguli affectionem, habere &longs;cilicet, &longs;uos tres angulos æquales duobus re
ctis angulis abundè explicaui libro 1. Priorum, &longs;ecto 3. cap.
1. quam Eucli
des propo&longs;it. 32. primi demon&longs;trauit, ex qua demon&longs;tratione, tanquam ex
Geometrico principio &longs;equitur omne
angulos æquales quatuor rectis angulis; omne
tentia duo triangula, cum diuidatur ducta ip&longs;ius diametro in duo quod &longs;i triangulus proprietatem illam non haberet,
conueniret. & &longs;i quadrangulum non haberet quatuor angulos æquales qua
tuor rectis angulis, neque triangulum habere po&longs;&longs;et tres angulos æqua
les duobus rectis, cum nihil &longs;it aliud triangulum, quàm dimidiatum qua
drangulum.
Cap. 16.
æquales habere, & percunctatur propter quid, occurrit, quia etiam triangulŭ duo
bus rectis æquales habet. in his igitur ex determinato &longs;ibi principio propter quid
a&longs;&longs;ump&longs;erunt)
nia huc etiam pertinent. hoc &longs;olum addendum ad illorum verborum (
terminato &longs;ibi principio propter quid a&longs;&longs;ump&longs;erunt
conclu&longs;ione demon&longs;trata, tanquam principio alia demon&longs;trant; quod rectè
fieri Ari&longs;t.
in primo Po&longs;ter. docet.
Cap. 31. (
admodum A, ad B, ita C, ad D.
con&longs;i&longs;tit, quemadmodum pluribus &longs;upra lib.
5. cap.
3. Ethycorum explica
tum e&longs;t: quò nunc Lectorem ablego.
Cap. 5 (
muniter prædicatum &longs;eparari, quippe, quod duplo prius e&longs;t
portionum genera vnum e&longs;t, quod dicitur multiplex, quod &longs;ub &longs;e
infinitas &longs;pecies continet, vt Duplum, Triplum, Quadruplum, & c.
in infinitum. vbi vides, cur Ari&longs;t.
dixerit duplum e&longs;&longs;e primum inter multi
plicia, cum verè naturali ordine numerorum ip&longs;i primus debeatur locus. Vides etiam cur non liceat, Multiplex ip&longs;um genus commune prædicatum
omnibus &longs;peciebus veluti Idæam &longs;eparari; tunc enim ait, ip&longs;um mul
tiplex ab&longs;tractum e&longs;&longs;et prius ordine ip&longs;o primo multiplici, &longs;ci
licet duplo; & Duplum non e&longs;&longs;et primum inter mul
tiplicia, quæ
illud tanquam Idæam licet &longs;epa
ratum ponere.
Cap. 7.
tuor rectis con&longs;tare, manife&longs;tum e&longs;t, quod trigonus duos rectos habens
cau&longs;a eius exi&longs;tat. Verùm &longs;i quid in trigono mutaris, nece&longs;&longs;arium e&longs;t, &
in tetragono mutes, vt &longs;i tres habuerunt, &longs;ex; & &longs;i quatuor, octo; &longs;in
verò non mutes, vt illud, ita hoc
lib.
1. Magnor. moral. cap.
10. &longs;crip&longs;i, ex quibus po&longs;tea &longs;ic locum hunc in
terpretaberis, &longs;i triangulum habet tres angulos æquales duobus rectis an
gulis, nece&longs;&longs;e e&longs;t quodcunque quadrilaterum habere &longs;uos quatuor angulos
æquales quatuor rectis, quia omne quadrangulum continet duo triangula;
& &longs;i natura trianguli fuerit immutata ita, vt habeat tres angulos æquales
non duobus, &longs;ed tribus rectis, tunc nece&longs;&longs;e erit tetragonum
e&longs;&longs;e, quia nece&longs;&longs;ariò habebit &longs;uos angulos æquales non quatuor tantum re
ctis, &longs;ed &longs;ex: pariter &longs;i triangulum habeat tres angulos quatuor rectis pa
res, quadrangulum &longs;uos habebit angulos, octo rectis æquiualentes. His igi
tur ex Geometria &longs;atisfactum &longs;it.
Cap. 10.
trum commen&longs;urabilem e&longs;&longs;e)
tria diametri, & co&longs;tæ eiu&longs;dem quadrati allata &longs;unt, &longs;atis huic etiam loco
facere po&longs;&longs;unt.
Eodem cap.
ra deliberamus: nam illa ad nos non &longs;pectant, hoc verò fieri nequit)
culi quadratio, & qua ratione eam antiqui inue&longs;tigauerint in Prædicamen
to Relationis, & alibi in Logicis, pluribus explicatum e&longs;t. An verò po&longs;&longs;i
bilis &longs;it circuli quadratura, re&longs;pondendum e&longs;t cum di&longs;tinctione, nam theo
rematicè quidem facta e&longs;t ab Archimede, cum ip&longs;e probauerit circulum,
quemuis æqualem e&longs;&longs;e triangulo, cuius vnum latus circa angulum rectum
&longs;it citculi &longs;emidiameter, alterum verò circunferentia. Problematicè verò,
ide&longs;t, vt opere ip&longs;o efficiamus
actum e&longs;t: & propterea problema hoc difficile admodum cen&longs;endum e&longs;t,
præ&longs;ertim cum tota Geometrarum antiquitas,
&longs;tra
dum e&longs;t. quo &longs;en&longs;u locutum e&longs;&longs;e Ari&longs;t.
hoc loco crediderim, dum ait, illud
fieri non po&longs;&longs;e. ab&longs;olutè tamen a&longs;&longs;erere non debemus e&longs;&longs;e impo&longs;&longs;ibilem, cum
nulla id demon&longs;tratione certum &longs;it, imò ego &longs;impliciter, vt aiunt, credo e&longs;
&longs;e po&longs;&longs;ibilem, cum alia theoremata,
illud celebre, quod 47. locum in primo Elemen. occupat, & pro cuius adin
uentione Pythagoras Mu&longs;is Hecatombas &longs;acri&longs;icauit) olim fuerint diù à
multis inca&longs;&longs;um quæ&longs;ita,
perta &longs;unt.
Cap. 12. (
lum habet tres angulos æquales duobus rectis, nece&longs;&longs;e e&longs;t con&longs;equi, vt &longs;upe
rius &longs;epius dixit, quod quadrilaterum habeat quatuor angulos æquales qua
tuor rectis, lib.
1. Magn. moral. cap.
10. &longs;atis de hac re dictum e&longs;t.
Cap. 12. (
e&longs;t omnium maxima oppo&longs;itio, ita vt quæ diametraliter oppo&longs;ita
&longs;unt, amplius di&longs;tare nequeant, quia diameter e&longs;t maxima om
nium di&longs;tantia, &longs;iue fit diameter quadrilateræ figuræ, &longs;iue circuli.
Cap. 2. (
Doricam, modò Phrygiam vocitamus
giam veteres Mu&longs;ici, vt Ari&longs;toxenes, Euclides, Ptolæmeus vocant
Tonos, & Modos, Dorium &longs;cilicet, & Phrygium. per mu&longs;icum au
tem modum intelligebant quandam vocum con&longs;titutionem, &longs;eu rithmum,
quem nos hodie vulgò ariam vocamus, vt doctè explicat Io&longs;ephus Zarlinus
in 4. parte In&longs;titut. Mu&longs;icalium, necnon in lib.
6. &longs;upplem. Denominati au
tem fuerunt prædicti,
ximè in v&longs;u erant, vt Dorius à Dorien&longs;ibus; Phrygius à Phrygijs; Lydius à
Lydijs. Porrò præter prædictos modos alij plures à veteribus Mu&longs;icis com
memorantur; variè tamen, alij enim tres, alij &longs;eptem, alij quindecim, vel
&longs;eptemdecim etiam connumerarunt; Tres tamen præcipui, & ad quos reli
qui reuocabantur, fuerunt Dorius, Phrygius, & Lydius. quorum hæ fuerunt
proprietates. Dorius erat grauis, &longs;euerus, & bellico&longs;us.
vnde pri&longs;ci exi&longs;ti
marunt ip&longs;um in hominum animos prudentiam, ca&longs;titatem,
inducere. Phrygius verò erat hilaris, lætus, placidus, ac propterea fe&longs;tis,
& choreis idoneus. vnde prouerbium illud vetus ortum habuit, à Dorio ad
Phrygium, ide&longs;t à rebus alti&longs;&longs;imis, & &longs;erijs ad humiles, & iucundas. Hos
ambos &longs;olos Plato, & Ari&longs;t.
in Rempublicam admi&longs;erunt. Lydius demum
modus erat horribilis, mœ&longs;tus, ac tri&longs;tis,
crymis aptus. Hoc in funeribus mortuos lamentantes vtebantur, ita vt prç
&longs;entibus lacrymas cierent,
Recentiores Mu&longs;ici &longs;uos modos vocant Tonos, in quibus vtinam anti
quos imitarentur, illi enim &longs;uis rithmis, modi&longs;uè auditorum animos varijs
pro illorum varietate motibus mirè afficiebant: &longs;ed no&longs;tri, rithmos in &longs;uis
cantilenis negligunt, nec illis curæ e&longs;t, vt per rithmos hominum affectiones
percellant, cum tamen Plato a&longs;&longs;erat Mu&longs;ici officium e&longs;&longs;e rithmos adinueni
re; præterea quod animis ciendis valdè ob&longs;tat, cantilenæ verba, ac &longs;en&longs;um
pror&longs;us per &longs;uos, quos vocant, contrapuntos, omninò offu&longs;cant, vt nihil
præter magnum quendam vocum &longs;trepitum concordem exaudiatur:
rithmis imitari hominum mores deberent, mimicis quibu&longs;dam adinuentis
id præ&longs;tare conantur.
Verùm hac de re legantur eruditi&longs;&longs;imi Dialogi de Mu&longs;ica Vincentij Ga
lilæi, cuius præcipuas rationes in fine huius operis, & chronologiæ videre
poteris. Cæterum, qui plura de modis tam antiquis, quàm nouis de&longs;iderat,
lib.
6. &longs;upplemen. virum vatia eruditione refertum,
optimè meritum.
Cap. 3. (
po&longs;uerunt duas &longs;pecies, vnam Doricam, alteram Phrygiam: cæteras
verò omnes vel ad Doricam, vel ad Phrygiam referri.
Vide proximè in præcedenti loco dicta, quæ omnia ita etiam
huic loco quadrant, vt præterea nihil de&longs;ideretur.
Cap. 1. (
e&longs;t &longs;ecundum dignitatem.
Arithmeticam medietatem &longs;upra explicaui lib.
2. cap.
6. Ethi
corum. per eam deinde, quæ e&longs;t &longs;ecundum dignitatem, intelligit
Geometricam, quam &longs;upra lib.
5. cap.
3. Ethicorum expo&longs;ui. vtimur enim
ea, quando opus e&longs;t di&longs;tribuere præmia non omnibus æqualiter, &longs;ed habita
ratione meritorum vniu&longs;cuiu&longs;que. quando autem non &longs;ecundum per&longs;ona
rum dignitatem, &longs;ed omnibus æqualiter fit di&longs;tributio, illa dicitur Arithme
tica proportionalitas, quia &longs;eruat eandem
Cap. 12. & vlt. (
tur, nec tamen rectè. illius enim Reip. quæ e&longs;t optima,
propria a&longs;&longs;ignatur. inquit enim cau&longs;am e&longs;&longs;e mutationis, quia &longs;ic natura compara
tum &longs;it, vt nihil permaneat, &longs;ed in ambitu quodam temporis, mutationem recipiat. e&longs;&longs;e verò principium borum, inquit, quorŭ &longs;e&longs;quitertia radix quinario iuncta, duas
exhibet harmonias. inquiens quando numerus huius diagrammatis efficiatur &longs;oli
dus
tunt (&longs;e&longs;quitertius cubus) &longs;ed qua id ratione ignoro. græcum verbum e&longs;t
præterea &longs;en&longs;ui radix, non autem cubus quadrare pote&longs;t.
Porrò &longs;ciendum
Ari&longs;t.
locum hunc ex Platonis lib.
8. de Rep. accepi&longs;&longs;e, loco Mathematico
ob&longs;curi&longs;&longs;imo, vbi ille de Reip. &longs;eu Gubernation is mutatione, ac duratione
pertractat. quì locus adeò &longs;emper ob&longs;curus habitus e&longs;t, vt Cicero ip&longs;e cum
rem pœnitus ob&longs;curam &longs;ignificare vellet, dicere &longs;olitus e&longs;&longs;et, numero Plato
nis ob&longs;curius. quam ob cau&longs;am Theon Smyrnæus, qui Mathematica Plato
nis loca commentarijs illu&longs;trauit, hi&longs;ce tenebris lucem nullam afferre au&longs;us
e&longs;t, verùm eas di&longs;&longs;imulans cautè declinauit. cùm igitur præ&longs;ens Ari&longs;t.
locus
&longs;it illius Platonici particula quædam, quid mirum, &longs;i non mmori ob&longs;curita
te, ac difficultate impeditus &longs;it? vnde etiam &longs;i quitor huius explicationem,
ab illius explicatione pLocum illum Platonis fu&longs;i &longs;imè expli
cat Mar&longs;ilius Ficinus to 2. operum &longs;uorum pag. 1413. vbi pag.
1421. cap.
12.
uiter, ac dilucidè declaratur. quæ explanatio, quoniam mihi præ cæteris ar
ridet, eam hoc loco, explicatiorem tamen, referam. Illud autem præ&longs;cien
dum e&longs;t, hæc quæ a Socrate lib.
8. de Repub. recen&longs;entur, confingi à Mu&longs;is,
tanquam oraculum quoddam ob&longs;curi&longs;&longs;imum effata; quo arcana quædam
my&longs;teria de Rerump. durationibus, ac mutationibus continerentur.
Aiebatigitur Socrates, Mu&longs;arum &longs;piritu afflatus, optimam Politiam, op
timis &longs;cilicet legibus, ac moribus con&longs;titutam, &longs;ua natura omninò immu
tabilem, Verumtamen mutationi obno
xiam e&longs;&longs;e, quoniam &longs;ie natura comparatum e&longs;t, vt cuncta, quæ naturæ fin
continentur, certa quadam annorum, vel &longs;æculorum periodo exacta, mu
tationem &longs;ubire fatali lege, cogantur. tunc autem harum vice&longs;&longs;itudinum
principium contingere, fatidicæ Mu&longs;æ &longs;ignificare voluerunt, cùm is anno
rum, vel &longs;æculorum numerus ab illius Reip. exordio elap&longs;us fuerit, qui &longs;it
numerus &longs;olidus, & cubus, eius numeri, in quo optima Reipub. con&longs;titutio
con&longs;i&longs;tit. hic porrò numerus, in quo Reip. perfectio &longs;tatuitur, e&longs;t Duodena
rius, quem multis in locis, varias ob rationes extulit Plato, præcipuè verò,
quoniam in &longs;e ip&longs;o duas continet harmonias, &longs;iue duas proportiones har
monicas, quæ &longs;imul iunctæ, perfecti&longs;&longs;imam omnium conflant harmoniam,
quæ Diapa&longs;on dicitur. duæ autem illæ rationes harmonicæ &longs;unt Se&longs;quiter
tia, & Se&longs;quialtera. Se&longs;quitertia reperitur primò inter hos numeros 4. 3.
cùm enim ea inter duas voces, aut &longs;onos reperitur, ij edunt harmoniam,
&longs;eu con&longs;onantiam illam, quæ Diate&longs;&longs;aron appellatur. &longs;imul autem ijdem ad
diti efficiunt 7. qui numerus propterea in textu dicitur radix Epitrite, &longs;iue
Se&longs;quitertia, quoniam vt vidimus
rationem habentibus. Se&longs;quialtera verò ratio reperitur primò inter hos
numeros 3. 2. cùm enim duo &longs;oni in earum fuerint ratione &longs;uauem edent
narium efficiunt; cui quinario &longs;e&longs;quitertia radix adiuncta, quæ e&longs;t 7. Duo
denarium componunt: qui propterea duas exhibet harmonias. Præterea
hæ duæ harmoniæ &longs;imul copulatæ conflant &longs;uaui&longs;&longs;imam Diapa&longs;on con&longs;onan
tiam, nam iunctæ &longs;imul prædictæ duæ rationes &longs;e&longs;quialtera, & &longs;e&longs;quitertia,
eo modo quo tradunt Mu&longs;ici, hoc &longs;cilicet modo 4. 3. 2. oritur inter extre
mos numeros dupla ratio, quæ ip&longs;ius Diapa&longs;on e&longs;t forma. nam ratio 4.ad 3.
e&longs;t &longs;e&longs;quitertia; ratio 3. ad 2. e&longs;t &longs;e&longs;quialtera; ratio verò 4. ad 2. quæ ex il
lis componitur, e&longs;t dupla. quòd &longs;i duo &longs;oni duplam hanc rationem nacti fue
rint, con&longs;onantiam Diapa&longs;on &longs;uaui&longs;&longs;mam re&longs;onabunt. Cùm igitur nume
rus 12. harmonias ha&longs;ce complectatur, per eum Mu&longs;æ optimum Reip. ini
tium, ac &longs;tatum &longs;ignificare voluerunt. Vernmenimuerò cum numerus hu
ius diagrammatis, ide&longs;t huiu&longs;cemodi conditionis, qui e&longs;t 12. factus fuerit
&longs;olidus, hoc e&longs;t, quando Re&longs;p. benè con&longs;tituta ad eam annorum, vel &longs;æculo
rum periodum peruenerit, qui &longs;it numerus &longs;olidus numeri 12. tunc fatali
ordine, mutationem pati incipiet, atque in peius, cùm optimi mutatio &longs;it
pe&longs;sima, prolabi. poriò numerus &longs;olidus ip&longs;ius 12. e&longs;t 1728. vti mox expli
cabo. vult igitur Socrates ibi my&longs;ticè &longs;ignificare po&longs;t tot annorum, aut &longs;æ
culorum numerum Remp. omnem quamuis optimam, in deterius prolpp&longs;u
do, & cubico &longs;ignificatur, &longs;i vlterius progre&longs;sura &longs;it, nece&longs;&longs;ariò &longs;ummam
perfectionem præteribit, ac derelinquet. Quòd autem numerus 1728. &longs;it
numerus &longs;olidus, & cubus ip&longs;ius Duodenarij &longs;ic pal
ea repetiueris, quæ &longs;upra in primo Po&longs;ter. num. 33. marginali, de numero
Quadrato, & Cubo dicta &longs;unt: e&longs;t autem cubus numerus is, qui ex gemina
to ductu alicuius numeri in &longs;e ip&longs;um, producitur. multiplica igitur primò
12. in 12. & producetur numerus 144. qui quadratus, & planus e&longs;t. rur&longs;us
duc 12. in hunc 144.
&longs;olidus e&longs;t, vt loco citato explicauimus.
explicatio &longs;ufficiat.
Cap. 5. (
&longs;iue cum melodia
cat, modulatione, Gr&ecedil;cus tamen textus habet per Mu&longs;icam nudam, forte Ari&longs;toteles intelligit eam,
quæ &longs;olis &longs;onis
quam Io&longs;ephus Zarlinus in 2. parte &longs;uarum In&longs;titutionum Mu&longs;icalium defi
nit, quæ e&longs;t concentus plurium vocum harmonicus cum rithmo, & oratio
ne, ide&longs;t, qua canitur oratio aliqua &longs;ub aliquo rithmo, aut modo, &longs;iue vt
nunc loquimur, con qualche aria.
Ex quibus liquet no&longs;tros contrapuntiftas toto cœlo aberrare, dum &longs;uas
cantilenas, ab&longs;que vlla verborum intelligentia, atque ab&longs;que vllo rithmo
di&longs;perdunt.
Eodem capite propè finem meminit harmoniæ Lydiæ, Mixtæ, Doricæ,
Phrygiæ. de quibus &longs;upra 3. lib. Polit. cap.
2. tractaui,
tes, quas hic Ari&longs;t.
recen&longs;et ibi connumeraui.
Ibidem meminit etiam Rithmorum, & Harmonie.
Quid Rithmus dictum
e&longs;t &longs;uperius lib.
3. Politic. e&longs;&longs;e quem nunc vulgò ariam cantores, ac tibici
nes appellant.
Harmonia e&longs;t plurium vocum ex acuto, & graui concors modulatio.
Verùm de his fu&longs;ius in Problematibus Mu&longs;icis, &longs;ect.
19.
Cap. 7 (
Vide quæ &longs;upra lib.
3. Politic. cap.
2. annotaui.
Sectione 1. num.
3.
Arcturi, Virgiliarum, Caniculæ, qui flatus, imbresqué excitant, qui &longs;ereni
tates, frigora, tepore&longs;uè &longs;olent afferre)
fit, quando a&longs;trum &longs;imul cum Sole oritur: quem ortum abundè in 2.
Meter. &longs;umma 2. cap.
2. explicatum inuenies. Vt autem intelligas, quænam
&longs;int Orionis, Arcturi, Virgiliarum, & Caniculæ con&longs;tellationes, & in qua
cœli parte &longs;int collocatæ, &longs;atius e&longs;t globum aliquem a&longs;tronomicum, in quo
a&longs;teri&longs;mi omnes clarè depicti &longs;int, intueri, quàm hoc loco pluribus verbis
rem per &longs;e claram, ob&longs;curare. De Orione plura dicta &longs;unt 2. Meter. præ
&longs;ertim quo tempore oriatur. Arcturus verò, &longs;iue Bootes, primæ magnitu
dinis &longs;tella, mane vnà cùm Sole in no&longs;tro climate ex Magini Tabulis, circa
28. diem Septembris oritur. De Virgilijs tamen illud exi&longs;timo
quod in Tauri afteri&longs;mo, duæ aliæ partiales con&longs;tellationes continentur; in
capite enim ip&longs;ius, &longs;unt illæ, quæ & Hiades, & Atlantides, & Succulæ nuncu
pantur. in dor&longs;o autem &longs;unt illæ, quæ Pleiades, & Virgiliæ &longs;unt appellatæ,
Vt cecinit Ouidius, quandoquidem &longs;eptima ferè nunquam apparet.
has vul
gus Gallinellam vocat. quod &longs;i eas per Tele&longs;copium in&longs;piciamus, mirum di
ctu, ea&longs;dem plures e&longs;&longs;e, quàm quadraginta &longs;tellas minimas clarè videbimus,
vt optimè primus omnium Galilæus ob&longs;eruauit. Porrò con&longs;tellatio Tauri in
no&longs;tris regionibus oriri cum Sole, incipit vno circiter &longs;e&longs;quimen&longs;e po&longs;t ver
num æquinoctium. De Canicula &longs;atis dixi in 2. Meteor. &longs;umma 2. cap.
2.
Eadem &longs;ect.
num.
17.
poti&longs;&longs;imum pereant, qui morbo longo laborant, & &longs;enes, quam iuuenes potius?)
Lege, quæ &longs;crip&longs;i de occa&longs;u &longs;yderum lib.
2. Meteor. &longs;umma 2. cap.
2. dein
de, quæ in &longs;uperiore proximè loco de Virgilijs: quibus hæc pauca cum intelligat de co&longs;mico Virgiliarum occa&longs;u, qui noctu apparet,
tunc primum, quando oriente Sole, ip&longs;e occumbunt, nece&longs;&longs;e e&longs;t occa&longs;um
hunc incipere po&longs;t autumnale
nibus; cum enim Virgiliæ &longs;int in Tauro, nece&longs;&longs;e e&longs;t occidente Tauro initio
dixi, vt Sol &longs;it in oppo&longs;ito &longs;igno, videlicet in Scorpione; in quo a&longs;teri&longs;mo
Sol reperitur po&longs;t prædictum æquinoctium vno ferè men&longs;e cum dimidio. de
hac re plura Plinius lib.
18. cap.
31.
Nvm. 1.
gur as rectiline as bifariam &longs;ecant, diameter vocata e&longs;t? An quod dia
meter, vt nomen ip&longs;um de&longs;ignat, duas in partes figuram æquè dimetien
do diuidit, nibil dimen&longs;æ figuræ defiruens? igitur hæc, quæ per commi&longs;
&longs;uras, hoc est, per angulos figaram diuidit, appellanda e&longs;t diameter, quoniam hæcquemadmodum faciunt; qui
partiuntur. At cæteræ lineæ, quæ per lineas compo&longs;itam figuram &longs;ecant, eam cor
rumpunt: committitur enim rectilmea figura in argulis, vel &longs;ecundum angulos)
Vt rectè problema hoc percipiamus, proponenda e&longs;t figura rectilinea, &
vna ex ijs, quæ paralle logramma dicuntur, vt &longs;unt Quadratum, Quadrila
terum, Rhombus, Rhomboides, cuiu&longs;mo
di e&longs;t præ&longs;ens, aliter verba Ari&longs;t.
illi non
&longs;emper quadrarent, quia illarum diameter
illas &longs;emper bifariam non &longs;e caret. quemad
modum videre e&longs;t in trapezio. & pentagono
etiam æquilatero. Quærit igitur, cur ex om
nibus lineis, quæ quadrilaterum A B C D,
bifariam diuidunt, quales &longs;unt E F, G H, &
D B. &longs;ola D B, quæ ab angulo ad angulum
ducta e&longs;t, mœruit appellari diameter. Re&longs;pondet autem, eam fortè appel
lationem hanc præ cæteris inde promerui&longs;&longs;e, quòd, quamuis aliæ omnes
æquè parallelogrammum dimetiantur, &longs;ola tamen ip&longs;a D B, ip&longs;um non de
&longs;truit, nec &longs;cindit, cùm ei nouam aliquam diui&longs;ionem non inferat, &longs;ed id per
angulos &longs;ecet, vbi prius laterum commi&longs;&longs;uræ
uas figuræ &longs;ectiones inferunt, cùm eius latera in punctis E, F, G, H,
vbi nulla prius erat diui&longs;io; quapropter ip&longs;am quodammodo de&longs;truunt, Aduertè valgatam verfionem latinam hanc
constant, quæ rectis lineis continentur)
euqugrammon kata\ tas gwni/as,
ponitur enim rectilineum iuxta angulos; quæ interpretatio vera e&longs;t, quia
anguli &longs;unt laterum commi&longs;&longs;uræ, vt dictum e&longs;t.
Eadem &longs;ect.
num.
2.
Vtrum quoniam &longs;ola bi
partitò figuram diuidat? An quod &longs;ola figuram &longs;ecat per partes, &lo