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M.Martini EriciGe-
STRINII, MATH. INF.
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Profeiloris publici,
Publice tuert conabitur
DANIEL DANIELIS DA.
Linus L-Gothur*.
HabshiturDiffatatio in JuditorioMajori
dieρ. Februarijhorismatutinis»
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Excudebat Efchillus Matthias
ANHO M. DC. XXXIII.
REVERENDOifö
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JOHANNI ΒΟΊ1
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Norcopenii Gotho. S.S. Thea
ιlogi#acPhilofophiseDo&origraviillmo,Am·
piiifim# Di#ceieosLincopeniis Epiilopodi·
gniilimo,longeq;deiideratiiTimo, Patrono
meomagno, &Maecenati indubitatifli-
mo, propenfiflimo, omni honoris, debit#q; obfervanti# cuitu
honoratiiTimo. ^
Simu! ac
T O T I
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>acTr&ßantißimuM
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Gymnaßarchis ittic folertijßmis, pra-ceptoribu*, tutoribusacpromoto=·
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Kee nen ^
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Honorandis, Do&iilimis, ac
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mis, fautoribusacbenefa&oribus pri
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lutty datqumiemdebuit fa Defeniöxibus Deditus
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DANIELE DANIELIS DALiNO
L-Gotho.
Τ η ε s ι s L
Νantecedenfi
difputdtio*
neGeometriengeneratimegi"
m9'deM agnitudinc,primoξβ adaquato Geometriλ objecto,
ejutfygeneralibut affeélicni-
DUSjquatumMagnitudiniper
f
fe £5?
abfolute conßderata in-unt) tumetdem unicum altera coüatacompetunt}
1Tihuncverd d nobis ordinispoßulat ratio, utadejus
&
Speeles
defeendamut> Mas affedtiones tntu&a*l" Vftur)quaJj>eciatim un'tcuidpmapnitudini ßtntpro*
bria. \
i* 'Tresautem e(JeMagnitudinis
vontinua(j)é~
piesLineam,Supcrficiem & Corpus ortas trt~Irinaem eju<s dimenßone, longitudine, iatitudine
ξβ
^rofunditate,partiminpracedentibusa
nobü eßpro-
1^batumj partimin hac prafenti ex incidenie qua-
. iwt occaßone demonßrabimm>qua cum
lovgiortm
F c on-
contemplationem requirant, quamutUrticadifiuii*
tione comprehendiqueant; ideo^inhocdifcurfu de
Lineaprimaillarum acßmplicijßmafieciatimagc-
mm-,reliquas commodioritempori
refervaturi.
3. ki hacttadpdeLinea theoria> quatvornobii proponimmquafiionej dUucidandas. /. An fit li-
II carealequid& pofitivuminrerumnaturai
2. Quid iit, &an ex indivifibiübus confteo?
3. Qmenain fint primaria? ejus divifiones&
Speeies \ 4, Quotnam principales Linese af-
fediones ? Hüriteexpoßtüßrefolutis,eaquaad
hane mäteriam
cognofeendamJpeBare
videntur>e- njadentmanifefliortL·».4. Dari in rerumnaturaLineam■>βαfe βα
reliquü Mag. jpeciebm diflinBamycontraquosdam
ageometretos,qui ipfamßmulcum PunBo exclu-
dunt-,βreliquas duosdarifolum contendunt> hifee
rationibmGeometricisconfirmareannitemur.
. S. Primo: cumenimCorpora exißantyeoruft muliafuperficievifibili claudantur-> 2. def.1.11. el.
Eucl. 2. e. 1.22. G. R; Ergo etiamfuperficies
terminabiturvelfeipsd>velnegatione>vel aliquo
alio inh&renti , cujm natura ab ea eB difiinBtu.
Nonfeipfdycum omne terminatum terminum dfe
natura dißinBum poßulet> quemadmodum docet Arift.1. 6.ph.c.2. Eucl. 1.1. el. def.3. 6.131.11«
d.2. Ram.1.1. Geom.e. 5.1.2. e.3.1.5.e. 3. \m-
e. 2: nedpnegatione feualterim continuationispri-
vatione·>cumqu&libet fuperficiesformaliterßtfini¬
ta·, ββquandofy infinitampofiulatGeometrat
idfi-
lumfit
difeiplina
causdξβquantafatüßt,utdocet
Arift.1, ^Phyf c.6, Ergo aliquo also
mharents
al
in
1<
R
fio
vci( ra
Ii
fll
ni 13.
en
rii
βκ
au 1di
lfa
pa
m
,cu
(3
tn
'.de g*.
ibii li-
•a?
•Λ
k af*
ad
t*
'iA< al·eadiftinBo\ iüudautemid eB>qusdLineam
ejß
intettigimu* ξβappellamus, Eucl. 1.1,e.d. 5.Ram«
i5.G.e.3<DaturergoLineas.
6 Deindéex13, d. l.i.e. Eucl. &5.e.l.i. G,
R.colligituromnem remterminatamunicddimen*
fionefuum
fupérare
terminums: E.etiamfuperfi-ciesfuum fuperabit, quicumfuperficiesejfenullA
ratiöriequéat3 eritquidinlatitudine ind'vvißbile^i
Wud Lineanomine venit: 6. daturLineas.
Tertioßcontinuipartescopulantur commu-
niterminofeuuniuntur inunaextremitatecommu- ni>Ariil. lib. Cat.c„ deQuarit. Eucl. 1.r.el. d„
13.Ram. Li. G.e.4. E. etiamfuperficieipartesuni-
enturineadem communiab utracpdißinBa3 qua e- rit Lineas* f^.g. Sit
fuperficies A
BDΕ continua, cujuspartesßntEC> FB.Jam vero harum par¬
tiumterminusfeuextre-
mitas communis utpoté FCi aut erit continuλ
autnoncontinna\Sicon- tinua,ejuspartesalidex¬
tremitate unientur in in*
finitum3βnon continua, eritautquantitasdifcreta?.
c c* autquid,juxtahanecopulationemξβunionem, in-
lh j
divißbito*
Difcretanoneß,
cumhocßtcontrathe-[Z° fin; Erit ergo FCextremitas ξβterminus utrim^
n~ partisEC, FB
indivißbile
quid, quodLineam no*minamus. 8. datier Lineas
8. Lineamhancutramqupartemfuperficiei alt-
)oteri- lumina-
ß-
^
cujus conneBentem>non abfurdeconcipereρJ?
tousexierminofuperficiei alieujuspartimimi'
1 ρ ^
partimobumbratA. Limesenimutrittsj, longitu-
doψ -dam eß,adiongitudinemipjutemet luminis φ
·.,mOr.?extmfzyfedomnislatitudinisexpers,cumά
p » u*deßitualur.
o. 'Tandem lineam effe magnitudinemrealen
s nonßrabimusexreali contacluCylindrt βfuper
feieiplana:. Namβ Cylindrarfeucolumna
perfett]
rotunda-,
cadatfuperfuperßeiem
exadleplanum^nd pote\·"? illam tangerentfiinlineos. Si enim in linünontangeret, effetinCylindroaliqua exlenßo plan βfic Cylindra*noneffetCyl'tndru*, quodeflabfiträ ξβcontra22. def. 11. el. Eucl. 11. d.i. 11. Campi
Ii.e.l. 21.G. R. Tangitergoinlineay ξβcumgon
tactusßtrealis, erit ξβLineapoßtiva βrealis.
10. Hincpatetcaufa curCylindru*faciliusmt·
veatur inpiano,quam Cubu* aut aliudquodpid folidum reguläre^. Namcumex contaBu mobik
cumpianofac1Utas βtarditasmotu*ßt aBimanm.
itautquo minor eBcontaBu*, eominus harere(n
tiusfdiveilt faciatyquo veromajor,eodiutius:
β
utmobiletångensplanum inlineafacilius·,infuper ficieverodifßcilimemoveatur, quemadmodumetu
oftendit Arift.q.9. Mech.
11. Vtramfy
hanc
theßn fic düußrabimw'· ζιSit Cylindrut AB, tångensplanum CD iu^ γΛ
contaBu*ßt AB. Dico β contaBum hunc
φι
lineam, βfuperhoc etiamfacilimcCylindrurnmü^
veri. Si enimABnon eBlinea, ßt contaBu*
itte
Λ frliqua extenfio planas, fßuoniam ergo
ABpL
eßextenßoy Cylindru*ΑΒβplanumCD,qua
fe
modo tangunt, adeequabuntur ineo>ιΛ quo
fe
ta&\ni piano enim corpori non
poteB nißplanum
^^ dftvAt
te
&
m V(
f«
Vi ac
qu
no
gunt
im
is
g||
mά
da ipa fett
,MH\
lintr 'am
råI
pjj
fort
mι
iwA
Aquari. Cum ergoCylindrusAB inaliqu.iexten*
ßonetemgatplanumCD'y
necejfiirio
infehaberetex-lenßonem plamm ,quodeH inconveniens. CIan-
I gl*erg°CylindrusAB planumCDin lineas.
mtf 12. Cum ergo Cylindrarplanum tangat in Ii-
neCr,1 neaABfUt oßenjum efl;ßjam rotetur Cylindrm verpcsD adanteriora·> non mancbitfixun inAßt
' I
fedAByinquafitcontaBus,afcendet in Ε F: GΗ r, vero lineadefcendet inlK-, nudd Cylindriß) plant
adveniente ofifenfatione > cum in linea latitudinis
prorfaexpertefiattantumycontaBus-,inquampau-
cioresab
axeinßßuntperpendiculares
motum mo-Λ
/ff rantesξβ impedientes.
13. Solidum vero
aliud^v.g.
Cubus undfuper-_ f* · ' v
j j '"""V -i
ΤΛquotßmtpmnBain j
fuperficienixu
e ö
no coniunfta■>quafefemutuo tinquam vir^s
t0\
ßabiliunt, βfulciunt,nefaciPe deijciantur. ttux®
ergopianotantb magisinbaret Cubustquanto/►
quA·'
F 3 -
S( lh
ne
rij
m
di
nu
ni
pa
tit al
ne
ni
ru
in
bus ab eopunBis tangiturj J^uane, difflcilimeme-
«vetur,contraquaminCylindro, ubicumpauciom
reBapiano infiflantyfacilima
ab
illopiano fleride·
jecliopotefl.Majorergo
inharentia
majus eflimpe-dimentums. Patet ergoutrumq^.
14. Exhifce demonflrattsmanifeflum efjepil¬
tarnasLinearareverddaritåinrerumnaturaex-
?+ Quid
*ftw> jam
veroquid illa Ht expendamu*,
i$Linea.» if. fijuamvis enim variedvarijsAuthorihvÅ
*deflribatur Linea,utvidereeflm lib.io. expet.&
fug. cap. 3. & 4. Greg. Vallie; Itenu Com
ment. Clavij ad2. def. 1. Eucl. nobistarnen ill
arridet,quamadfert
Euclides
L1. el.d.2.flcilicel quodLinea «·33λΛ7ϊί,Longitudo latitudinisexpersiflve abfq;latitudino.
16. InhacEuclideadeftnitione duonobiscow
fiderandaproponuntur, j. q
uid
fit Lineal dein-
^quidnonfio?
17. Quid fit LineaexplicatEuclidesperW cabulum μηκ®». fljuoniam enim Linea natura
naturampunclilonge
excedit,
itautmintl·sfimpkx Ψ
fitpunBojquodomnisprorßtsdimenßonis
eft
expemi.def.1.1, el. Eucl;ideo%·, ut
dpunBo Linear»dp eJJ
ftmguat, aiteam
ejfe longitudinernfeu quanti- S1
tatemdivifionem admittenterru. Etcumh$\ e longitudototalem lineanaturam nondeclaret?
quip* Φ
pequatå fitperficiei tåfolido infit;ideoej
addi flin* Ψ
Sionemfuperficiei tåcorporisy removet
ab illa la- ηι
tit udinem,itautlonga quidemfit%fed ,
latanonfio. Hinc ergopatet
Euclidem hdcfus | \
definitioneLineamtumaffirmando tumnegan-
rt
do
definijjfLsi U
de
ta, tu
m
iorei de npe-
ϊ8. Perperdm igiturfacitRamus,
qui lib.
3.Schol.Math. addef.i.&z.EucLdefmitionem hane
linea Euclideamtam
anxiereprehendit,
quafipure negativa effet,quanegatioddefinitionibus
,tefe
A- rifl, 6.Top. c. 3. omnind excludendcu*; non ani-,pu,
madvertens eandem β affirmative per longitu-
tex, dinemab Euclide definiri,ßmalq, inter principia
numeräri,quibusnegativaorationes,juxtdΡarme-
ibtvi n*disfententiam>pracipue
conveniunLquodprinci-
t β palis poflremacp caufa,perfolas
tradatur abnega-
)m, tiones. Omne
fiquidem
principiumab eis
quafunl
tk Φ™)**** aliend
conflat effentid
>ξβ hor
umnegatio-
(icel nesΜ*#* nobispatefaciuntproprietatenu>.
jfuod
e-nimhorumeH caufa-, nihilautemhorum eH, quo-
rum eHcaufa-,ex eanegatimeejus naturautcundp inclarefcit. Greg,valla lib.10. exp.&fug.c.4.
19. NecpeHcurpraferatRamus
Joco cit.fuam
definitionem Linea definitioni nominaliEuclidis
^ tanquam
affirmantior
em,dum Lineamper magni-
^Jtudinem
tantumlongam definitl.
2.G.
e. 2.lex 1uafi tantum longum afßrmaret
ξβ
longutruI
abiq;
latitudinenegaret,nonanimadvertens hac
di-
eife
Synonyma^. Tantum enimlongum & lon-
ltj, gumabfq;latitudine
idem denotqre,
quis nonvi-htutf
Exclufivam
ergoparticulamRdmi tantum>
,'ip.
qualatitudinemdlineaexcludit,
nonmagis nqgare
in' %u<*mEuclidü
particulam abfq; cuilibet facile in-
la- n°tefcit.
;7inj 20, Hane
longitudinem abfq. latitudine in
re-fuLfa*naturalibu* ξβ artifcialibm fenfu
objeHis
repe-Eirenulld rationefas eHy licet
nonnulla adeo lati-
ff«ιtudinisfintexpertia >ut vifumpene
fugiantjäuale
F 4 i
Μ
<8Γ0»
ein
Jt
.fitenuiffimumfilumarancαaut bombycis,lineail¬
la,quam
Anettes
tertiocoloreduxitinProtogenis ab- igntts.tabella,prioresduas aß ξβProtegenecertatim duclasβcans>nullumrelinquensampl/usßubtilitatilocum_s. Lines? enim Geometriea?>noncolore fed animoimaginantur&concipiuntun..
21. Fttarnenaliquam
intelligentium
ξβcognl·tionemLineλhabeamus>imaginarioportetpunBun
ini.d.i. i. el. Eucl. definitum e loco in locumßu-
er e,ζβexhocßuxu vefiigium quodddrelinqui>quod,
cum longumßt omniprorfus latitudine carens , Ii-
neaefformabit. Ftfipun-β
tium Acogitetur moveri
, rx ΑinΒ, ξβ inde in C,
efficietur ex hoc motu i- maginariovefiigiumA B
Cquodlineaappellabitur,
fil
fåquod'punBumAiomni
carensdimenfione,tale vefiigium defiripfirit. Hinc faBumesiut nonnulliLiηeåmfluxum &rveo-
luticneimpundtinonincongrué dixerint.
22. Ex hifiefiquiturprirnoLmcam quam- cunq;. iinitam , utrinq;pun6lis ceuterminis*
claudi. 3,def.l. 1.eL Eucl. Νam cum lineafh-
xupunBi imaginarioefiformetur-,utdiftum·)fit ut uhi lineaincipitveletiamefß definitfiibiflatimpun"
Bi$ terminetur, utpatet infuperiori linea ABC>
fujun
extremaJuntpunBa AξβC.sompona- 23' Secundd Lmeam non ex pundtis tan-
u>t vx in qnam partibusintegrantibns,fedlineisperpc-
f;vlJb,|i·■ tqis in infinitum dividuü
cömponij cumimpojji-
hdefit * ut
indivifibile
βnonq%antttm> qualeeB j.
pun~
punclum>remquantamcofiituat»ut ofienditArifL
1.6; ph. c.i.
24. EtenimfiLineaconfiaret expurtBis tan- quampartibus .fequeretur PrimoCirculumlineam
tanaerepofjeinduobusvelpluribu<spunBisquauno,
quodest contracorol. prop.2.1.3. el. Eucl. Vtfit
circulm CF tångens lineam reBamA B confian-
ternexpunBis, quorum
unumfitin extremoC, linetA G,alterumvero
in Η initio linet HB:
Circulus CF
tängere poterit in G termino/\
communiutriusdp linety
hocft modo tanget utrumq;punBumG ξ£Η, quod eBper corol. prop. 2.1.3.el. Eucl. impoffibile_>„
Sequitur ergo ne%illaduopunBaGH,ne%aliaejus-
dem linetABfemutuotängere-,undenelineamex
hujusmodipunBispofjecompont.
zs.
Deindéfequeretur
circulum minorem t-qualem effe majorifibiconcentrico. Namβa een-
tro communi ducerentur reBt adfingula punBa
circulimajorts, necefjarid
tranfirentper
omniapun¬Bacirculiminoris·,cumduttranfireperunupunBu
) nonpojfint, aliquinvelfierentuna,velnonreBtU*
26. Demonftrabimm:Sint duocirculiconcen-
triciB Cmajor,DEminor> dquorumcommunicen- troA
adfingula
majorispunBaeducanturreBt,Di~! co illastranfireperfingula minorispunBa, atfyita
minoremmajoriejfe tqualemβlineaexpunBiscom- poneretur. Sienim dut linet admajoremperve-
l Af utien-
wientesi tranßrenl
per unum idemop punBum minoris■>
nonin idemma¬
jorüdeßnerent, ut funtADB) ξβAD
C) tuncdu# reBa DB, DChaberent idem communefe-
gmenlum 9 quod
eß
contratd
9quod demonßrat
Claviusex Procio ad 10. ax. Clavianum 1. ι,
el. Eucl. ejusdemreB&ADBplura
effent
extre¬maj utpoteΒ C> ßc reBanon ex aqualifuain-
teriaceret extrema)contradef.4.1. i.el.-Eucl,Jguod fidicaturunareBa^ltera
obiiqua;
nonerunligitur äquales,qu&dcentroejusdem
circuli ad eandemperi- ι jfåeriam dueuntur contra 15,d. 1.i.elc Eucl.; cumj
duo latera ÄD> DCtrianguliADCreliqno ACßnt
|
majoraper20. ρ.1. 1. el.
Eucl.
W7. e.1.6.G,R,Batet ergolineamex
hujutmodipunBis
noneße
con- flatami quemadmodumhocipfumfußusGeometriee oßendimus in diß. de Pun&o th.14.&15.· &diip.deconft.Geom:th.45. &46.
27. Sed ambigat qutspiamξβ dicat duo
eße
in quantitateindivißbiliügenera:ρrimumeorumqua Werl·indivifibiliafunt ξ£nullo modoquanta^ualeeß
punBumy^tQmm quorunda indivifibilium quidemsfed
tarnenquantorum* cujmmodifuntquaaamadeo
- minimalinea,quaatomafunt^tß nuüam
divißonem
admittunt) exquibus> (licetnon exqunBis)lineas
totales ac dividuas>tanquampartibuscomponi? ac proir.dequamlibetlineam in
mßnUumfecari nonpoß
p>
fe,fedtandem
continud quadam divifione ad illas li~
neosindividuasdeveniendumejftj,.
28. Piimoqutadantur linea
commenfurabileSy
Ergo aliqua erit communis
menfura omniu
>quamneceffe eB ejfeatomam-%
Namfi illa div
idu
aefiet,
pofet
femperfecari tåfubfecari in infinitum
per10.ρ.1.1,el.Eucl. quarecumpartes
hujusmodi fint
to-ticommenfurabiles,fequitur aliamtåaltamininfi-<
nitumexifieremenfuram, qua omneshapartes,ac
proinde
totalinea commenfurentur,quodferi nequit,
cumunafit
longitudo
omniumcommenfurabilium li·
nearumcommunismenfuraindividua,exquaomnes linea
commenfurabiles,quapradilla
communimen*furafint
äquales
,componuntur,20. Secundo;Sijam e
B
communismenfur
aatomaomniumlinearum, ejusfiguraplana, reliqua-
rumfigurarumplanarum,
quafitper linets Ulis
com-menfurabilibus confiituuntur>communis
menfura
e-riti tåcumlineaillafit
individua,
erittåfiltra
pa-riterindividuas. „
jo. Tertiö; Si communis menfuraomniumli-
nearumfiatuereturdividua,
effet nulla amplius linea
rationalisnetirrationalis,quodeflcontra
demonfira«
tionesüb.ίο. el.Eucl. Namfi communis
menfura
ejfetdividua,tolleretur
eade
rerumnatura,tå fc li¬
nearis
etiamfymmetria,tå
propterhane rationalitas,
quaexßmmetria
oritur
; quarenedp data
>ad
quamcatera reläta rationales velirrationalesdicuntur,
rationalis eritι E. irrationalesnullaerunt;E. nec illairrationalis erit, quam vocant
Apotomen
exBinomio,five
exduob. nom'm ibus, de
quaEuclides
Prop. 74,1,10. £1.
tåfequentibus fufepcrtraBat,
jr.SedJ/. Sed huictripliciArgumentamied refionfio-
neoccttrremui·, negando omnes lineas commenfwA-
biles unicdquadam
eadernq^
determinatd menfuvA tmenfurari oporterefedomneslmeas->quAadinvicemfunt commenfurabilesy undeademå.menfurd com-
menfurart-,fednon unicdacdeterminatd> cumpof fint
effe
plures e&dmdp menfura communesplurium quantitatum commenfurabilium, utpatetin tribut hifcelinetsA4.Bö.CS-pait.quarumcommunismen*Ό Α Β- C-
-6»
δ.
fura eHD 2. part.Binariut enim tres numeros 4, 6. 8.eXaSlemetitur. 'OndefilineaD2.part.bifa- riamfecelur3eritdimidiumDE1.part.quaparitet
eritcommunismenfara triumpraaiSiarum linearis
cumunitas omnium numerorumfitcommunismen-
furas.
32. Hifceadaüajta argumentafic rejJ>onfis> Geo¬
metriee nosconfirniabimutlineamtotalemexatomis linetsnon confiare,fedquamlibetin infinitaspartes dividuasfecari-,hifcefequentibus rationibut.
33- L Exprop.22)1.1, el. Eucl.patetextri- bus datis re£tis iineis,quarumquAibet dua reit-
quafint majores, pofle Triangulurnconftitui.
Gfuarcfiexlineü
individuis componanturlineato¬tales,poterit extribut iflis atomis lineis conftitui
riunguiumtilluddpAquilatrum^cumomnesindivi-
dua lineafint Äquales. Siergoperpendicularts ex
verticeinbafin ducatur3 cade11Hainmediumbafitr,
qua-
<\:m
quart & in medium individuA Urses,, Utft ex tributStorni»itntU TrUnguiumaquilaterumJBCa
Sierg6 ab a»gulo C ad bafm/,tßdu■«
c»tur perpendicularitCD,fecabit bafin 4Bbtfariam,percoroll. p.3*
1.3.ei.& theor.1. Clavij adprop.
Vg26.1. i.eL Euch Eritfy UmsaBa-
D torn»fabtlisquodeftabfurdum#
34. IL St denturbujmmodi linee indtviduA,
eruneomneslineacommenfurabitistquodeflcontra
demonfirata ab EueIidein1. 10. eh quart& idt,
cjutpotentUβiumfunt commenfitrahtles,trunt&
longttudinecommenfurabües.Jamverolineaindi-
vidu£cumfint invicemÄquales,erunt & ipfkcom-
mtnfurahles longttudine,quart&potentid,omnes
cnim longitudine commerfurabiles, funt &
potentiå commenfurabiles5per 9. prop. 1.10.
ci. Euch undé &quadrataearumomni»comwen-
furabilid,ergo & dividua,cumfintfuperfictes Ion- gitudtnem (f Utitudinem babtntes , qua diverβ funt dimenfiones; ergopoffunt fecundum utramfe dividi, exqua divifione neseßariö lateratpforum,
h:linea iÜA individuA dividentur$qutdefi inten·
vcntens,
31. III. St quadratum ex 4,tineis individuts
tonftituatur,cujusdiagonalt perpetsdumarüexal-
tirutro angulo infiflai, erie qmdvatum (übtenfit,
quadrati» femidisgonalia tf perpmdtmlarüfimui Äquale per 47. Li. eh Euch quart tum
edlemu-
trumq. Uterum ad rt&um fitmww·» iüqwßnsrit
pttmrrtaiη<"£ eeiaw»dttpmmeritquadratumdiame*
%rtiüm rnsdfåth quoédejeribituramtntms: a*
qsali
ffuali tnim aIUta„ reliqus erit individuaminoft fuodeftåbfitrdurtLs,
30. Sit quadratum
ABCDtxquatvorIi*
neisindividuü compo-
ftum, cujw diagonali
A C inßßat ptrpendt·
tularis BE· Britqua
dratum lines BC sqmU qusdrstis
linearurru
BE, ΕC» per 47. p. 1« 1. el.
Euch Del
5. e.1.
ii*G. R.ablatk igiturquadratisBS, EC,erunt
retts
tam BE, quamECminores ipfa BC ex5.ax.L i
eh
Eucl. fguareBC mneritmimma, citat tarnen (it
individua1,quod eflabfurdunu,
37. Conßttuatur deindefuperACdiagonaltqué*
dratum AGECper46, ρ.1.1.eh Euch & erit hoc quairatu duplum quadraU ABCD,per th. Clavi)
ad prop#47«h1.el. Eucl*q$are &diagonalis aC,
major lattreBC. Todatur erpo ex ipfa AC aqualis ipfiBCper 2. prop.1.1. eh
Euch
refidua aut erit aqualisipfiBCaut minor„ Non &qualu,qutatuntdiagonalisföret duplalateris CB,&quadratumdia- gonalis A C, quadruplum quadrati lateris CB pef Xeth.Clavijadp.4.1.%» el Eucl. quodeßabfur·
dum» cum repugnct prop. 47. 1.1.el. Eucl«ntc minor,quiatuntexißeret lineaqusdamminormi·
nimd,quoditemabfurdum*
Tandemfiponantur bujfumodtlintåsto-
m± ieruntSuperficiesab bißtprctrtatåatomsi i-
carporaafuptrficiehm defcriptaatomaXum
tnim ad divifiontm corporis necejfe fic dividi
fts*
ptrfichmι ad bujm
dmfoutm
dividelintam,
qua
quiipfamttrminat,
AriiL
1.u.Μet c.z.Ram.L5«Geom.3. omneq.corpus latituitner» &craßiticm
habtat,per 1.def. l.ii. el. Eucl. nuäumeritcorpus
quoddtvidi nequtat% B, ntclineaertttndividua«
Jollendsergofuntdeverumnaturslineaatoma.
Cumergolinea totalesnetfcexpunBü,ne^
ex indit'sduü linetsconftent,uteßdemenßratum, nect/Jtrtd (ondudendumloidetur, iüas ex perpe- tuis lineis in iniinitum diviilbilibus coaleicere;
quod• tumex·ta£tu re£tarum>/»wZI, exmotucircula-
numfefetangnetium
ofltndi&confirmaripotcfl»
40. Sine dus, Ittsea AB,C B coeuntes in B,in quasingrediaturreBaCA Atautprimumftt in CA firu,deindéin UN,terübin KLtquar16inHl,tum inFG, demum inDE, (fparirattonein omnihus alijs huitu Tringuli locis idens
eveniet,utfeilicet infinit£par¬
teslinearumcoeuniium, quäles funt DBE,FDBEG, HFD
BEGI, KHFDBEGIL, cadantinterinfinitasparteslu
nee tngredtentis, utfuntDEw
,A FG, Hl, KL,MN, CA, at&fic tnanifeftumeil, tdmtotamre*>
&amingredhntent,quamtotasconcurrentesinin*
finita*partesfeeari,itautnuüaparsingredientisp üullafycoeuntiu(uperfit,qu£fefetnutuonottdividut·
^nihiltam mingredientequamineaneurrentibtu
ömanet,quodnondivtdttur. Confiantergolinea Üftaexlineisperpetuis in infimtum4tvtduit,
41,Sint
4t» Sint
itemduocir- .juli A BCMt BBFΜaqua- lest quorum
priormanest " "——
Immobilie>pofteriorverό moveatur (f difcedat
ab A B C Μ circulo manente;βstimnamfapars
egrcflaBFMentmajor (emicirculo, fiet% major
acmajordonec ipfum totumpercurrerst,in<fetall
motuomnespartesegredientis c.irculi,dlvidentuf
ab omnibuspartibus circulimanentis. Lindepatct nihilt[feineorumperipheres, cfuod non dtvtda-
tur:B.nullumin eiseftindividum·
42, Atjjita quid fit Ö* quomodo con- ftituaturLineavidtmus;nuncverbnrarcipuas
ejusdivifioncsacfpecies centemplabimuryqua·
!
rumprimatfi> qua dtvidttur Linea in fimpli-
ccna ö'nfixtÄ!«.
l,Quae- 4j, Linea
fimplixe/qu«
totofuoflaxune*dT# ueiwmis» tflß vel reßa Wcircolaris.
Tiiioncsw· 44. Re£U linea eft, qua ex aquali fua punåia interiacet«4,dftf.L1. el« Eucl.
4/. Ut reBalineadefinitio k circülari%qud ttiamexaquo (uosterminos interjacere videtur9 fecernatur, fcitndum efl ex aequali interia-
cerelic ideme(feyac stqualitcrcxtcnfumefie
inter fuaextrema, itautreHa dieatur lvneay qua«qualiterexteηfa eft inter fuaextrema
j: i pun£ta>itautnulltbipartesejusajjurgat elattusy
nullibi fubfideant bumilius, (ed aquabilem ac
vnifortqm
interpunfta extremaßtumtsneant^talis
Tallseß linea in hocfibemateJCB, in qua cum
nuüuintermedia pun«
ffumfur(umaufdeor»
fhmübextremis A&B
deßeffat,feeusquamin AFB,AEB% ADBtfed ßngula aquabilem ac
uniformeminceffumoc
A.
eupent,erttjuxtahanc definitionem Linea ACE reffsL#·
46· Confißitproinde linet reffλ offentiα in cqualiintcrfuumprineipiu fiaemq;? utita äicam, interiacentiaquo Santaeil, quanta eß dtßantiaquainterejus terminos intereβ9 qua£
tarnfola mettatur, ut ait Proclus. Siclinea Α Ct, reffa eßtqutainterfitumprineipiumA &fi-
nemB itaaqualiterinterfacet, ut iüudyquod in¬
tereosterminoseß intervallum exaffe aquet (f folaemetiatur,reliquaTterolineaAFBg ABB,AD Btcum iffo fpacio fint malöres,reffα non erunt.
Quapropter rc&a linaa borum terminorum Intervalle & iymoaetra eß & congrua, &
ideofymmetria
&
congruentiaipfipropria«47. Plato ut notat TbemiftiusadIIb. f»
peft.Anal, ö*ipfeEadides in 1.def.Catopt,
hanc fuamdeclaranstreffamlineamdefinierunf, anjas na«dia obumbranc extrems. Utines-
dem lineaACB,ppunffumCauf aliquod aliud
intermedtum interA& B yirtmtem baberet ob*
umbrandi\neaculoeonßitutoinΑ,punffü Biäw
minatumappareret(fcontra6 vtfit pofito|JnB*
G alter
sittrumAlucidumtion lider11ur,Brit IiηesACB
refl49ftcus fil inlineisABB, AFB, ADB utejl perfpicuum.
Speeit· Strtitur éutem linea reda rstionc
aex* Ll#eru»
&aecidf ntium
våris nomtna,#/a utdi-
ri/wTarminata , Nontermniaca , Diego.
naiis,Diameter»Axis,Latus, cujusg>ecush*~
ÜS)Hypothenuia,C hörda,Tangens,Secan»
49. Terminata ieufinitadicitur, quando utfsfypunfls extrems ftu ctrta quantttsam es cenftderstur. Talis eil illa , quam Eucl. io, prop.lib, 1, el.
bifariam
fecarcjubet:
talis iüafuper qua r. prop. 1. i.t\. Trianqulum a-gquilaterumcenflitui
poflulst.
Νamft
Lineahi
proponerenturinfnits,non
pojfunt
neg, bifartamfecari,nctfr
fuper
jjs Triangulum ullum tmftitui,,
tum bifeflio (f trtangult squilateri conßitutto, finitistantiemcompetant.
f§. Nontcrminatafeuinfinitadicitur,eu·
jusarta nenponiturqusntttsa, fedtants aGeo-
metrafumitur, qusntum pro(uo ufuopushabet, refiduonegleflo. Talis
efiiäa,ad
quamå
pun¬cto,quod inea non
eil, perpendiculareoo
, redarn deducere Eucl. jubet 1.1. el. prop,
12. Νsm β iäs finitapropontretur falfa föret propo(itio} cum non
poflit
femper a tältpunflo
perpendicularisad eam deduci. Utfilinta
ftmtu
effet^AB, &punflumextraipfsminfitu C> ma·ntfeftumeftptrpendicu-
Urem in tpfam cadere
nonpoffe,cum exCde» >
firibs circulus, fecans j , j
Μ
■
v , . d
KjéBinduebuspun8is,non licest,quem
adducen*
dam perpendiculanm eam& demonftrandam p
omnino necejjarium requiri EueIides affirmat,
Hacigiturdeceufa oportetreftamdatarn effein·
finttarn,h: nonhaberedeterminatam magnitudi-
mm,ut(altem ad ipfam produÜam,perpendicu·
lariipoßttdeduci.
ft, Diagonioi /^»Dia¬
gonalis ed,qua abangulo in angulum oppofitü dufla,paral·
lelogrammuminduotriangula difpefttt. Taliseft hicLineaAD
Haue lineam bifariårofecare
parallelogrammura , apo~
di&ice confirmat Enclides p. 34· 1· ι. ·1·
$2. Diameterfeu Dimetiens eß re&<L*l
qua per centrumcireuli
quoquover[um
dufla, &exutra% partein eircumferentiamcirculi termi-
nata circulum bifartarn fecat. 17. d. 1+i.eL
£ucl. Taliseß lineadB, trenßtns per centrumD
&terminatainperiphe¬
res Λ Β. Hujus pars dt-
midiaDB,velDA,vtl DCSemidiameterfeu
Radiosappellatur,
Si* Axis eß tüareBa quiefeeus linea.
dreh
quam velSphäre, Ttel triangulum vel
PareBslo-
grammumVolvitur15.19.ss.dd. 1. n.el.
Euch
InSpbaraomnie AxiseßfimulDiameter
(f eodem
Gi *9*
refbeßaanteusy uoncontra, cum DUmetrs pofint) ejfe infinit*.
f4% Latera, in omni figurs reWtlmta ,
dicuntur illt reß* line*, qu* unguium aliquem figur*cotnprqbcnduntytruntjjtot,quotsnguli.Sie
paraßelogrammi Angulicumquatvorfint, totidem
erunt latera angulos illos includ<ntta\ Haec,
quse cx adverfo aequalia eile & parallela,
dernonftrat Eucl. prop. 54,1. 1. cl, Sican*
galtpentagonis funt: Ergd (f latera ί Hexsgoni
(ex,Ergo Ö* totidem latera eruni 0c.
//·. InTrisngulis veroRißtUneis, fi totum inttßigiturTrtangulum,omnestres reß* apptl-
Unturlatera» veliß*fo'um reßt dequa fiorfim oft(ermo : SiverodecertoTriangult anguloagi*
turyreß* iß* du* unguium includentes latera
dicunturytertiaveroBafis,· ißa vero qu* αηξοίο , Jublaterthtu ißis comprqfienfo opponitur veljub*
tenditur,fivein imo loco fit fitafive in fuperipre,
dtettur,prsfertimfiangulo reßa opponitur, Ηy-
pothenufa feu
fubtenfa.
v. g. in hü Irtan*angulisreßangulis,fifermofit de angulis reßis
Βinpriori, VelCpo- ßeriari Latera funt fi
JByBC, &DCyCE: r
Hypoeheauia vero vel fubteofa LsdCf i
tfDE:fi dt angulisB
' ^
adA&DSitfermo, laterafunt BA,CA& CD,
ED, Bafes BC, CΕ: Si de angulis adC & Et
Latera funtκΛ€>Β€>&CΕ, DE,& Baie·
JB$
&
tå DO Singuhautemduo latera
quomodo
cunq; allupta tertio eile
majorayoßendit
EucLp.20.1.1, el.illudveromaximum, quodma¬
ximum angulum in Triangulo
fubtendit,
demonftratprop. \p.1.1,el.j6. Chordaefi illareBalinea infegmento cir*
cul'hquaper
i pherU 9 fetsarcuifuo , inßar nervi fubtenditur*
qualiseßBD^'
in fegmento
circuliBADmaiore,vel BCD minoriveletiam
AB infemcirculo ACB.
S7. Tangens eftilla
reBa,
qua cumcircu-
lum
tangatyßproducatur eundemnonfecat.Secans
verb qua cum circulum tangat>
β producatur,
eundemfecat.2. d. 3.el.
Euch 'Vt reBa A By β
circulum FDB ita tan¬
gatin Bjut
produBa
ad ^Ceundemnonfecet>fed
tota extraipfum mane-
at * Tangens dicitur.
Sivero reBa EG* cir¬
culumitatangatinF>ut produBa fecet eundem»
tåinträcadat, dicetur EG reBa iecans. Et
itafe habentLinea reBa. Sedanteqnam
ad
circ*~larem
explicandam deveniamuiy
LineareBa im-e~ginemprotinus