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Dl SPFTATIONFM GEO.

METRICAR VM TERT lA* &

Z>o '

PRIMA Μ A G ΝIT U-

DINIS SPEC IE

LIN Ε A,

fauanu

Diviftdauxiliante

gratia^

SUB P R £Sl D I ö

M.Martini EriciGe-

STRINII, MATH. INF.

In

Regia Academia Upfalienii

Profeiloris publici,

Publice tuert conabitur

DANIEL DANIELIS DA.

Linus L-Gothur*.

HabshiturDiffatatio in JuditorioMajori

dieρ. Februarijhorismatutinis»

UB

S

A LI Ms

κ) rf

¥/

Excudebat Efchillus Matthias

ANHO M. DC. XXXIII.

(2)

REVERENDOifö

AdmoduminDominoChrijio v ,

D*-

JOHANNI ΒΟΊ1

VIDI

Norcopenii Gotho. S.S. Thea

ι

logi#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

V°nerando,

Clarißimo

>ac

Tr&ßantißimuM

SenatuiibidemConßßorijEccleßaßkkunacum

ffk

Gymnaßarchis ittic folertijßmis, pra-

ceptoribu*, tutoribusacpromoto=·

fp;

ribut·,absfy uttatemporis6x~

ceptionecolendis.

Kee nen ^

VHI8. um

Honorandis, Do&iilimis, ac

Humaniffimi^

Dn.Pr#poiitis&Paitoribusin ^ärtecfittb$pe

Öoberwibboocontraftibusvigilantilli- WUt

mis, fautoribusacbenefa&oribus pri

#tatém amore pro-

fequendis. hv.

Bxerritium hocPhiloMathematicuWyinfuirffrh

corda-tionem■>gratamentis remunerationetn^H ,obfervmti£dpteßificationem, dedicatquiaVfyai

lutty datqumiemdebuit fa Defeniöxibus Deditus

Defenib^

(3)

:

·

Hfl Μ1Ü1I Ü

,ec: DISPTTJTIOGEOMETRICJ

\m|

3di·.

TERTIA

Dcj>

no

L I Ν Ε Α.

Refponientο

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-

(4)

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 ? riteexpoßtüßrefolutis,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,ut

docet

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

(5)

'.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*

Difcretanon

eß,

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 abfurdeconcipereρ

J?

tousexierminofuperficiei alieujuspartimimi

'

1 ρ ^

(6)

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 linü

nontangeret, 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'· ζι

Sit 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 fr

liqua extenfio planas, fßuoniam ergo

ABpL

extenßoy Cylindru*ΑΒβplanumCD,qua

fe

modo tangunt, adeequabuntur ineo>ιΛ quo

fe

ta&\n

i piano enim corpori non

poteB nißplanum

^^ dftv

At

te

&

m V(

Vi ac

qu

no

gunt

(7)

im

is

g||

mά

da ipa fett

,MH\

lintr 'am

I

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 -

(8)

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 fleri

de·

jecliopotefl.Majorergo

inharentia

majus eflimpe-

dimentums. Patet ergoutrumq^.

14. Exhifce demonflrattsmanifeflum efjepil¬

tarnasLinearareverddariinrerumnaturaex-

?+ Quid

*ftw> jam

vero

quid 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 latitu

dinisexpersiflve abfq;latitudino.

16. InhacEuclideadeftnitione duonobiscow

fiderandaproponuntur, j. q

uid

fit Linea

l dein-

^quidnonfio?

17. Quid fit LineaexplicatEuclidesperW cabulum μηκ®». fljuoniam enim Linea natura

naturampunclilonge

excedit,

itaut

mintl·sfimpkx Ψ

fitpunBojquodomnisprorßtsdimenßonis

eft

expem

i.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

(9)

m

iorei de npe-

ϊ8. Perperdm igiturfacitRamus,

qui lib.

3.

Schol.Math. addef.i.&z.EucLdefmitionem hane

linea Euclideamtam

anxiereprehendit,

quafipure negativa effet,quanegatiod

definitionibus

,te

fe

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

principium

ab eis

qua

funl

tk Φ™)**** aliend

conflat effentid

>

ξβ hor

um

negatio-

(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

tantum

longam definitl.

2.

G.

e. 2.

lex 1uafi tantum longum afßrmaret

ξβ

longutru

I

abiq;

latitudinenegaret,non

animadvertens hac

di-

eife

Synonyma^. Tantum enim

longum & lon-

ltj, gumabfq;latitudine

idem denotqre,

quis nonvi-

htutf

Exclufivam

ergo

particulamRdmi tantum>

,'ip.

qualatitudinemdlineaexcludit,

non

magis 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

<8

Γ

ein

(10)

Jt

.fitenuiffimumfilumarancαaut bombycis,lineail¬

la,quam

Anettes

tertiocoloreduxitinProtogenis ab- igntts.tabella,prioresduas aß ξβProtegenecertatim duclasβcans>nullumrelinquensampl/usßubtilitati

locum_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

quod'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 cum

impojji-

hdefit * ut

indivifibile

βnonq%antttm> quale

eB j.

pun~

(11)

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-

(12)

wientesi tranßrenl

per unum idemop punBum minoris■>

nonin idemma¬

jorüdeßnerent, ut funtADB) ξβAD

C) tuncdu# reBa DB, DChaberent idem communefe-

gmenlum 9 quod

contra

td

9

quod 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&dcentro

ejusdem

circuli ad eandemperi- ι jfåeriam dueuntur contra 15,d. 1.i.elc Eucl.; cum

j

duo latera ÄD> DCtrianguliADCreliqno ACßnt

|

majoraper20. ρ.1. 1. el.

Eucl.

W7. e.1.6.G,R,

Batet ergolineamex

hujutmodipunBis

non

eße

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^uale

punBumy^tQmm quorunda indivifibilium quidems

fed

tarnenquantorum* cujmmodifunt

quaaamadeo

- minimalinea,quaatomafunt^tß nuüam

divißonem

admittunt) exquibus> (licetnon exqunBis)lineas

totales ac dividuas>tanquampartibuscomponi? ac proir.dequamlibetlineam in

mßnUumfecari nonpoß

p>

(13)

fe,fedtandem

continud quadam divifione ad illas li~

neosindividuasdeveniendumejftj,.

28. Piimoqutadantur linea

commenfurabileSy

Ergo aliqua erit communis

menfura omniu

>quam

neceffe eB ejfeatomam-%

Namfi illa div

i

du

a

efiet,

pofet

femperfecari tåfubfecari in infinitum

per10.

ρ.1.1,el.Eucl. quarecumpartes

hujusmodi fint

to-

ticommenfurabiles,fequitur aliamtåaltamininfi-<

nitumexifieremenfuram, qua omneshapartes,ac

proinde

tota

linea commenfurentur,quodferi nequit,

cumunafit

longitudo

omnium

commenfurabilium li·

nearumcommunismenfuraindividua,exquaomnes linea

commenfurabiles,quapradilla

communimen*

furafint

äquales

,componuntur,

20. Secundo;Sijam e

B

communis

menfur

a

atomaomniumlinearum, ejusfiguraplana, reliqua-

rumfigurarumplanarum,

quafitper linets Ulis

com-

menfurabilibus confiituuntur>communis

menfura

e-

riti cumlineaillafit

individua,

erit

tåfiltra

pa-

riterindividuas.

jo. Tertiö; Si communis menfuraomniumli-

nearumfiatuereturdividua,

effet nulla amplius linea

rationalisnetirrationalis,quodeflcontra

demonfira«

tionesüb.ίο. el.Eucl. Namfi communis

menfura

ejfetdividua,

tolleretur

ea

de

rerumnatura,

tå fc li¬

nearis

etiamfymmetria,tå

propter

hane rationalitas,

quaexßmmetria

oritur

; quare

nedp data

>

ad

quam

catera reläta rationales velirrationalesdicuntur,

rationalis eritι E. irrationalesnullaerunt;E. nec illairrationalis erit, quam vocant

Apotomen

ex

Binomio,five

ex

duob. nom'm ibus, de

qua

Euclides

Prop. 74,1,10. £1.

tåfequentibus fufepcrtraBat,

jr.Sed

(14)

J/. Sed huictripliciArgumentamied refionfio-

neoccttrremui·, negando omnes lineas commenfwA-

biles unicdquadam

eadernq^

determinatd menfuvA tmenfurari oporterefedomneslmeas->quAadinvicem

funt 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-

(15)

<\: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,

quadrat 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

(16)

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, Ε per 47. p.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,qutatunt

diagonalisföret 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

divide

lintam,

qua

(17)

quiipfamttrminat,

AriiL

1.u.Μet c.z.Ram.L

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,utdemenßratum, 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

(18)

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«

totofuoflaxu

ne*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

(19)

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, quantadtßantiaquainterejus terminos inter9 qua£

tarnfola mettatur, ut ait Proclus. Siclinea Α Ct, reffa eßtqutainterfitumprineipiumA &fi-

nemB itaaqualiterinterfacet, ut iüudyquod in¬

tereosterminosintervallum 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.

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

(20)

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 ut

di-

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

fecarc

jubet:

talis iüafuper qua r. prop. 1. i.t\. Trianqulum a-

gquilaterumcenflitui

poflulst.

Νam

ft

Linea

hi

proponerenturinfnits,non

pojfunt

neg, bifartam

fecari,nctfr

fuper

jjs Triangulum ullum t

mftitui,,

tum bifeflio (f trtangult squilateri conßitutto, finitistantiemcompetant.

f§. Nontcrminatafeuinfinitadicitur,eu·

jusarta nenponiturqusntttsa, fedtants aGeo-

metrafumitur, qusntum pro(uo ufuopushabet, refiduonegleflo. Talis

efiiäa,ad

quam

å

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 tält

punflo

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

(21)

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·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*

(22)

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$

&

(23)

tå DO Singuhautemduo latera

quomodo

cunq; allupta tertio eile

majorayoßendit

EucL

p.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 fets

arcuifuo , inßar nervi fubtenditur*

qualiseßBD^'

in fegmento

circuliBADmaiore,vel BCD minoriveletiam

AB infemcirculo ACB.

S7. Tangens eftilla

reBa,

qua cum

circu-

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»

inträcadat, dicetur EG reBa iecans. Et

itafe habentLinea reBa. Sedanteqnam

ad

circ*~

larem

explicandam deveniamuiy

LineareBa im-e~

ginemprotinus

intuebimur*

fl.%«*"

References

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