PRIMO AC SIMPLl·*

O IS Ρ U T AT I O Ν a Μ G

METRICARUU PRIM ■ ,"t

Dt*

/: Λ,

PRIMO AC SIMPLl·*

CISSIMO GEOMETRIjE

PRINCIP Ip,

Iii

Quain-»

Divina auxilianté

gratis

SUBPRISIDIO

M; Martini Erici Ge-

5 Γ R I

jLU Μ A T Η

I Ν F

I Ν

Rcg^^^adercja Upfaiienfi

''reiTorispublici,

Publich tuericdnabitur

PETRUS A. SCHOMERUS

MATHE M. STUD.

Bahebttur Diß>utatioinAuditorioMajori die_27. OÜobJhorismatutinü^

opsali

ms

Pxcuäebat Escuillus Μα τ τηι

\Λννο Μ. D C. Χ Χ ΠΙ.

Mi

Ηι

Η

Γ/7\ C ST\^/τ»7» ss* ·νΛ.->*t ,-· τ*>, p*.^?is<**/Ti i ;

\£LtisTRi

^{g εχε} R o s o ε τ

MAGNIFIC O DOMINO

dn:johanni qhttc/

L.B. DΕ

£)iit>frejf/Dn. D Ε ©rénpco

et ©tr^mprwm, ε qjj ιτi aurato μ a- ximeftrenuo,inclytiregniSveciasSenator*

aropliilitro Camera Regi» Adnoini-

ftraiori & Academi» Upfalientis

Cancellario Magnificen- tiiTimoj&c.

Domino

fuo gratiofe S.

τώ μελετών

7Ρ>λισ·σ;^^ων βασιληων

Αζζχγ $Yj _7τχξ εμν

uW^j&ßqf/,aliX.>}V

Κ' ctvT'jV not ψϊ μ,ετίαμ,6<τώπ1& τ%

χαζ/εν&ς

&άλ7Γ$ς &αλφ6ηη οιρφιβάλγς λίτν^, SubjeftiiTimus-*

cliens

letrmΑ. Schomerw*

DlSΡU TAT I O G SO ΜST RICA

PRIMA*

DtL9

Ρ Ν C T O.

Rcfftondentc*»

PETRQ A. SCHOMERO

UPS ALIENS I·

Thesis I.

I alla intotoEucliMeo

lumine materiapsculiariini*

tiocognitionedignay &Ven¬

tilatiomrebieinfiituendaap- prime neceffaria judketur#

iüaprofefib,quade Puncto fufcipiturtpraaids ejjepotefi, quippe^quacognita&prehe

perfiecla,

'

terGcometrasdemagnitudine^conttnuaagitarifa*

péfolent,accuratérefolvttt&ådfmgularemrerum Geometricarum cognitionem viam pr&monfirat commodißmam. Quart ^nos amplißmum Geo*

metrißcampumdifquifitiombiu quibufdam aggref

Juri, harte ^rem dpunfto ordirsopsrßpretjum^ar*>

bitramur.

2. auteminpr&fenti curruulo fintΡbo^-

ΐ jffafar* ac Duct ^certo divagtmur9\hane

de punclo

A theo*

thcortamad hospotißimum 4. qu&ßionesrepoca-

bimu*i(filtrumprimaerit: Atifit in^{reruru na-}

iüfä pundtum? 2. Quid fit? 3. Quaenam f- jusmiütatcs ? ^4. Quamara infigniora inre-

järn natura ριτη$3 ? qusbm ^cum fuisrequifitis

f$sdjunBisdextrepoßmodüm explicatis,plcraft

quidde punctodifferuntur,manifeßiora evadsnt.

ι»Anfit Circa cju&fiionem an fit Pun&um, dua

fuoäfi? ßixi extrema pbilofophorum fententu^{y una} (im- pliciternegerns* alteraftmpltctur äfftrmans Pun-

ßumψrempofitivuntjquampoßerioremcttarrLj

nos tanquam maximé & nccejjarib veram inhoc dißurfuacceptabimuéi

4* x^Antequam igttur PnnBum tn rcrum

natura effe cerita demonßrattonibm oftendamtu,

•prsfuppommuicum exEuclidis Pleroentis Geo-

merr ci$>tumexArißote/einCat. Quant.item

4,Phyi. t.25. 49. & 91. Hem1.1. de An. c. 4.

t. 70« trtnamdtmenftomm, extraquamPun&um

reale non eß> revera poβtivameffe,potißmun.z_>

cum de cadem vera (f realesaffeBtonesdemon~

βrentur , quemadmodumpaßim videre eß apud Jfcud,^& Ram. tnfuü Geometriciselementü,

f. Hoc pofit0, fm modoexißere&maximé neceffartumeffeinverumnaturaPunBum, contra rigorofosquofdam^> exißsmqntes

ipfuw,

tatcm»»tura(U£,nihiltjje, quodinrerumnatura exißeret, (ed efje velpuramnegatimemContinus-

taasine»hnea?quaabipfoveltnipfo terminatur,

vel negattonem divißonut inparitbw hneaconti-

mani, htjce rattonthm Geometrtcis eptneereco*

nabimur» ^» j

6.Si

61 Sr Corpusfinitum tetminatur fuperficie, EuclicLz.dcf.I.ii.cl. Ram. 1.22.Geoai.e.2.

&Superficies linets, Eue!. ^6. dcfi. zh Ram. 1.1.

Geom.^c. _5.& 1.5. 6. etfam linea finita clau¬

deturPelnegatione,velfetpsd,vtlalquotnaxtfim-

tinaturam dtftinClam alt ipfahalsnte. _Nonnega- tione, ahequin quslibet linea circulans formali¬

tereffetmterminata,quippecujusparsomnislike-

ra ab ißanegatione, aBu continuatur-, ^non(etpsa;

quia quemadmodum ^res terminatatermtnumdβ

natura difiinclum requirtt, utdemonfiratArid.

6. Phyf.c.r,& Euclid.l.i.el. def.$.6.i3.& 1 ir.

el. dcf. 2: Ram. l.i.Geom.e.5.1.2.0.3« 1.5 c.3.

1.22.c.2. italinesconvenietterminärt non(etpsa:

Ergo altquo alio inexifienti, quodabeafitdtftm-

8um: illud aliudejjenonpotefiquamPunBum^s,

Eucl. l.i. cl.dcf-3+ Ram.1+2^Geota.^c.3.Datur ergo ΡunBum.

7. Deindequoniamontneterminatum unica dtmenftone fuperat fuum ternsinum, Ram^,1.r+

Gcom.c. 5.fic _Corptu Superficiemprofundiiate, Superficies lineam Utitudine; Ergo etiam linear fuumterminumlongitudineiontumfuperabtt, qui

cumneclongttudoipfafit,utin priorithefioftenfum

efl, multominus låtitudopelcraßuies: eritquid

Individuum.IlludpunBinominevenit; Daturer¬

go Punftum.

8. Ex Continui definitionecfiendeturtdem.

Eil enimContsnuumcufus partescopulan-

tur communi fcertxiino. Ariit. 1.6. phyf ^{c. r.}

1.1.Eucltf.dcf.l.i. el.Ram. 1,i.Geom. e.4.

'jam ^verq terminus in hnca bac _egseßma-dbC9

A 2 v.g.B.

*b.g.Bäutt*itquidcofi"

ttnuumtautnom stcon~

titiuum, efut partes fi»

malcopulabuntufy paté»

bit_fyfic aittui m infint*

itftri-j * Sinon,autertt

quantitiS dsfer^{et4 au t}

quid indivtfibile:

in

continuanullsu datur aBu numerus:reiinqui'ur

ergo Cijod Bfitquidindivifibde quaititaticon ι~

nutedBCadjacens,quodpunBum appeäamus. Da-

turergopunBum,

p* InfuperFunBumredeeffi Gcowtricedt·*

monfirabimut ex redtcontaBulintere&Jd &^ctr·

culi· EtenimficircuJostangit re£tam, inpun¬

ctotangct.CoroLprop.16.I.3. eL

io. Tangatctrculus CFD, cujtu ^centrum Ft rcBamAB inC» Dicotüumipram tcngeretn pun- ß> ^ BoC,adebut Cre-veraftt

ρunΒum. Ducaturemm

ex centro circuH Ξ,re- ßa EC, per 1.poft. qua erit ad contmgenten.z_»

perpendicularü, ^per 18.

prcp. 3. el. fcuclid. vel

c.f. e.rs- ].i5.Ge*Ram.

&intangente dB fuma-

turquodvis fignumύ.g+

A%

coniungaturqfrtßa

port. ffuoniam

igiturintriangulo

AECduoangult

minoresfint duobusreßia, per 17« p*r.el.

Euch

velι.c. 0. 1.6#Gcom.Kapa.^- * & angulutadc eii

eft r?Bm per το. def.τ.elem.Faclid. Brit &nύ

v, guljuΕ /iL,reBoa*mor.^uareper 19, ρ. 1.1 el.

JmjcI. vei ii. c. K6. Gcorn. Ram,majoreritre-

B*/1B quamEC> Badem eftratio deomnibas aitjs

knmduBteex Ε adquahbet punBainter A&C.

Lfihtarectrcuiu* CFD^, tanket lineam ABfolumin

4L punctos qmcontaBm^tumfitrealis: realeerit A

punBum & pofitivum; quod erat 0ßendtndums*

η. Hute immohUifundamento adeo tnmxus eH Ari/l.irfic.8 Mechan punBum·, nonfolum-»

popttvum, verum etiamcaufameffe%curl·figuris,

waxtwemobilesfint rotundaⁱⁿpiano,afferere non dubuarit· Cum enimcontaBusmuh^afuipartefa¬

ttatbareret &fimuleffeea, quafefefic contingunt:

quo ergo major erit hie contaBus,obdtutius,qué

minor eo cittui har^erefaciet. Si igiturmn effet

•punctum,m quorotundumplanumtangeret,fané iüudcontingeretinpiano,acproindl multdfuipar-

tebarer_et, Cfficmajoremfacilttatemnonhabere^t globus quam tetraédr&n»oBaedron, dodccaedron velicofaédron, necbdberetfigur^amperfcBe(pba-

rieam,(edplanum, quod utrumq, tumzxperientia

&ejusnatura, tum

24,de£u.eL£uci4rr/»/^»4^

i ζ. Batetergoexbisdemonßratüpunetare-

verapofittva effe}qualicet in continuofint,nscex~

trat[ludjubfiflantsxifiimandumtarnenoneil tlla%

utpartesintegr&nteslineamcomno-mre,cum indi*

vifibiliaqoantumvis multa& juncia,non fa~

csant aliquid dividuura ingenere quantita**

tiscondnuje, utdocetAniUfé.phyi.c.i.

t}.* Namfipuncto,utpartesintegranteslines-

componerent,fausLontraäimonfirata Euclidts fc*

A3 querem'.

qwrentur al·furda. ^/. lntru triangulum duBas reftu parallelas baß eademeffemajores. 2. Dia-

mctrum quadratt äqualem effe Uteri ejufdem,ac

proindé βmmetram, id quodnosGeometrieebifie

rationihui ofiende^mus*

i4. Sittriangulum lm%eXeg(de reliquistri^- angulorumffeciebw idem eftoJudicium)A B C,^cu¬

jus bufis AC^y componatur ex

χ 4.tantumpunftis, Dico inträ

£

I \

triangulum ABCy β bafis

"7 fünftes compofita effet»plures

i/ aj

^ A ι — χι Jem ejp majoreSm yer punftum j""""""" ~'\ß

du-

j. —CAiur dgy parallelabaft ^AC,

c— Ά

per 31.p. 1. cl, Euclid. qua

quandoquidem dividuaeB,duobu» faltemconfiabit indiwßilibiü. Jnfrabanc ducanturalia 4.ρaral¬

ias FG> Η /, KLy ΜΝt_{per prop.} eandem,qua quo averitceB funtremotiores>

eb

evadmt,· Tuncenim F G, conftabit tributρunftisi ΗIver oquatuorjK Lquinfy& Μ Νfeχ. fffuare

cumbaßsACexbypotbeßconftet^tantum 4. &re-

vera exdemonßratisKL f.&MN6:^eruntba ßn- guU"pertict Bpropioresy majores

baß

tf. SitdeindequadrattABCDdiameterAC·

Dicoβ iineacomponereturexpunftis,fore^utdia¬

meterAC fatsquaiis Uteri AD. A quovispunftet

latens _DJ,ducaturperpendicuUresper11.prop.

ί,ι, cl. adquodvis punftumlaterisBC ^eregionen paßtum, quaneceffarto^peromniäpunftadiametrr tranf-

tranfibunt, cum tota quadratiarea

tftis linets

pleatur♦ <§uarediameter

ACoppofitaanguloma^

ximo^Dt & exdefimtione qua- ρ q drattreßotintriangulo A D C,

aqualis ertt Uteri AD

oppofito

quippé di-

enidto reßi, contraea, quade- monftratafunt abEuclidelib.

i. elcm. prop.i9*&Rara.1.6.

Geom.e» iz. acprotndéfymmetra,cujustameη

contrarium demonßrae Euclidesprop. 117. lib.

10.clcm. & Ram. 1.12.Geom. c.$< e^. Namfi

diameter/tCfit 4. lattu tero<lAD,y eritqua-

äratumdiagonij ιό.dupium ad

quadratum lateris

per47-prop.l.i.

elem. fcuclj&

i

Geom.Ram. &itaquadratum lateris

erit

(f

idem erit _g.nempéexyfic%par

ejjet impar* quod

abfurdum

16. UtitatybacceinGeometvia

abfurda evi-

tentur,fatendum eil,^non

expunßis, fed

lineam coalefcere, & inter duo quavis

aßignata

punßa, Imeam quamtis exiguam

ventreyjuxtaillud vulgoGeometrarumreceptum:

Continuum in infinitumeftdividuum.

77· Hincfequtturpunßacuminter (t,

titrtLj

d linea, cui infunt,realiterdtfttnßa

effe.

fi

unumputtßum ne%lineam,^ne%

altudpunßunt-»

componitt & *hnea,

(f abalio punßo »in

loco

diftatyutmodo

oßenfum eft

baud tnconveniens

ritcolligerty inter

bacreale intercedere dijermtn■«.

ig. Et

baßenuspunßumejffe monftratimWr j^Ouia

:jamverbquidfituUrb

nobis inquirendum

flat.

J 4 *ß.€*r-

tρ. Circumferunturenimvariaαvarps au- thorihut Funßide/criptiones, qualicet verbutdt- »

fcrepent, reipsdtarnenquämproximtcotäemunt*

Jlij emm Pund:um affirmativeperunitatcm

iitiirn in continuo habcntem definiunt, ut

PytbagorasSc Αriß. alpperfloxum lin

Cuianus:quidam^perterminum communtm

I

Jinesr, ut Hiero. in üb. deßnit. nonnüüi _per

iignum in magnitud ine Individuum.« ^s ut ^,, Ram.I.i.Georme.é-PetrusRyf.part.i. Quas-

ßiOQ.Georn-c.2.

20* Verumlicetha PunBidefinitionssinge-

nere optiméfalv&ripoßint,inßecietarnenearum nuiia srit, qua effentiam punEli Geometmcifuffi-

cienter mbis monßrare videtur, cum non tdrru»

fecundum particulartm feientiam Geometriam^* *

quamuniverfalemFunElumdcfinlmtj&cnaturam *

ejus explanent«,

2i. Nos proindé FunElum Geometrieccon-

templantes, ejtuqs[implicitatem, quaommbusijs praeft, quafub cognitionem ipß cadunt,intuen·

tes, dpartiumnegatione,quaeffentiamejuspro-

xtméconfequitur r«wEucl. 1,i.el.def.i.Punftu;

deferibimm,

l·μεξος xbh:

2 2. Uta.bacdefimtio plamüsacfaciliüsper-

j

eipiatur,fciendumeftQuantitätecontinuamtres haberepartes>unamfecundum longitudinemyreli- \ quiisfecundum latitudine&craßitiemfeuprofun-

diiatem, itatarnenutnonomnüquantitasbafccj»

omnes habeat,fedquadamunteamfecundumIon- gitudinemtantum, quadam duasfecundumlongi-

tudinem if latitudtnem, quadam denid przter

j

bafee

h»fce dtm altitudinem feuprofunditAtem,^quem»

» admod'um dsmonfirat Clavius in^comrn. iupc?

SphairamjohdsSacro Bofcopag.r4.

2f. Hocis^aoflenfofactlecögnofceturyquidfit

PunBumpoidehcetid,quod in talimagnitodi-

nefine omni_parteexiflit, ka utnequt Ion«

! _{gum, neque} latum,nequeproiundumello cogketur·' quod licetmagnitudoipfamnfst, til

. tamen, utpoßmodümofiendemm^?initium&c(un>*

damentum omnis magnitudmit & infpeck tins£>

qua ubi Imea incipit Pe(etUmeffedefinitytbipun- Boftatimterminatur, utdoset Eucl. Ui.cl. d.^3.

Ram.L2.Georn.e-3.

24. Punéti bujut Ge&metrhéexempiumins

rebusnaturalibusverumreperirinußumpotefi. St

» quamPünonullaadebexiguafintyut β inteäigantur

* dipidiiparteseorumomnemfinfumpoftea effugiet*

ctijufinodi efi vtfiigtum tenuißimiftil·, lemter itt-»

ceraimpreßi, extremstasi(km reBd line^quam...»

Apeliesfertur poft Protbegenem duxiffeinJabel-

layextremumcaudaScorpknbsffic.tamencumhac

omnia inmfinitu fecaripoßmt%non>»umpunBum reprafentare, (ed qualemcung, ejushmm,udmc/n

nobtsmculcarerexiftim&rt debent.

I 2g. HincttajjAppAfetyPunBumfenfuhatid-

quaquam perceftibiU

effe,

fit

t Tantumca_, quacorporafuruinmfinitu>u äivtitbi- It», fedmente tantumappr&hcrfwiletpAfdtioncm

nuttamadmtttem, ßveinlongum}fvein latum-*,

cumutrodp deßituAtur,

26. ffuoda.punBum defimtu-rpernegatio»

I ^-nemfidnecdefinitionenstpfamaliqudρ tccatswo

A f »ecu-

accufatj neeexiftentiamtjuscrerumnaturλtoHit.

Eil enim funclum cmnium quantitatum conti- , nuλτumprim um acfimpUcifitmum prtnctpiunLs,

quo(ubUtadefetet magnitudoynanfecusac

fublatd

unitat^eipfenumerus,Cuf.1.3.de^ffiente cap+9.

principiorum verb saeßnatura, utnonnifinega¬

tions explscenfurtutdocet Platoin Parmenide,

&^notat Proclusaddcf. i.l.i.el. Eucl. Deinde

multainrerumnaturacxiftunt, quorum cumdtf- ferenda pofitiva noslateani, negationeeadefini-

re, (f häc rattone ihrum cjjentiam indagareo*

portet.

27. Solum ita% PuncluminGeometriapar-

titioniseftexpers> utunitasquidem,(edinArith-

mctica-) πvuvfeuinßans-, fed inPhyfica: Sicfybac defnitioputifii,qua

apudprimumphtlefbpbum ob

forte

cam materUm ejus_fyprtncipiafufftcieniertradi-

tafit>&infuoproprio(oro tlluftrata.

2g. Hane Pun&inaturamfimplicemaccurate contemplans Lucretiusftpfumtumexb^acpartium negatione,tumreliquisejus,quasnosinhacqua»

ßione tetigimus, proprietatibus^ utPoetaPbyfcus eleganter bifceverfibus defcripßt:

Tum porrbquoniamexrremumcujuf4uecacumen Corporis eft aliquid^,noftriquodcernerefenfus,

Jam_nequeunt,id nirairumfinepartibusextat>

Et minima conftatnatura, nec fuit unquam Per fefecretum, ne^uepofthac eflevalebitt

3,0-a.r- 3 β* *ta

fi*

flum Teidimut*

san? pftA rb tuet minimafit^natura ζ£ entttatis^> infignes<

t amin

tarnenin fe habet utilitates, quaexiguitaternejus

fingulari quodam

encomio compenfdnt,

quadsmfunt intpfout

eft

tmmobile,& d

titate,in_quaeft,infeparabile, quadamPerintpfo,

uteft mobile, &d quantitate,per mentü[altern^

abfiraBionemfitparabtle.

jo. Utilitates priorisgeneru

potißtmum

funt^t quAruntprimaeft, quod

lincam, cujus eft

terrninus definiat, &ab alia quavisexclu-

dat, & quod munus eft

jufticiae,unicuique

ficircularisfue-

ritlinea,poteftatefuaipfdmdeterminat, prineipij finisfa vicefimulfungitur, ftcutpalaⁱⁿ

annulo; fi

autemfinitanecinfe reflex^{α, α}flu

tpfam

ejus prtneipium eft & finis,juxta3.

def. 1.1. el*

Eucl.ubi dicit lioearaclaudiieuterounariu-

trinquepundis*

β /. SecundapunBiutHitaseftt

quod fit

quoomncsli-

neae partes interfevindas

contineat, &ex

multis unamcontinuamlineam faciat* De~

monftrabimus: Sintdus lineaA B,BC·Dicopun-

Bum B effe communem [\ -6 ς

copulam^, qua lineaifla b

copulanturinterfe,^ittu*

ut tfla lineafiantunali-

/ \

nea continua♦ Si eninL» f

\

punBum B utrtusfa li-

Μ 4}

nea AB, BCnon effet

communis copula, nonfecundumfetotunsyfedβ-

cundüm aliamatfaaliamfuiρartemlineasAB>BC

• copularet9 &ficpunBumBeffet

divifibile,quod eft

contra

contra ejus nåturam. Quare B eft communisco*

fula: quoderatoßendendutn.

$2. HmcfecjuiturPun$umt om oisquaa- titacis continuae primum ac iinopliciffi-

mum eile prineipium, Cum enimpunBum, u$

jam oftenfum efi, uni&t ^omnem Untam,Itnea o-

mnensfuperfictem,(uperßeiesommcorpus·>*ηαηι~

feßitm eil punBum omne corpus untre, Quod

enitfl unit unientia, maxime unict _{uQita_..}

jf. Tertia TunBiutilttaseil,conftituers

circa fe ipatium ubi ubivis fit in piano,x~

quabileå 4.angulisplanisre£Hs; quodficde»

tnonftrabimus:SitpunBumΕtnpiano. Dicoipfum confiituere efreafeJpatium aquabiled ^4. angulis

reßts. DuBisentmreßU

ΑΒ, CD feie(ecantibus

in Ε dato puncto,· etunt duoangelt CEA, Α Ε D, duobusrtBU,äquales^per

13.prop, l.i. cl. Euclid.

vel per 1. e. el. 8. 1.

Geomet. Ram. hodem modoeruntduoanguli D ΕΒ, ΒΕ CduobusreBis äqualesi quibuf ad priores duos duobus reBisa»

quales,additityßent^tx1,axj.c}* omnts 4.an-

gult C Εät AED,DEjB,BEC tn fiatio cir-

ca ΕpunBum, äquales 4+ reBüi quod erat de- tnonßranduttLjii

$4. Quarta utilitas eil, quod iit com¬

munis fe&io omnium linearurn fefeiecan- tium ; velquod tdtm eß> id, quod pluitscoro- munitefiioeas feeat: quoajicoßendemw* StnC

d&4

duslines. J B, C D fefe fecantes inÉpunBo* Di¬

te B effecowmunem illarumfeBionem, feuunum piano,punBum, quod duos^t(lasiine&s communis terfccat. VunBoenim B exreBa A B, (umantur

utrtncfjporttomsÄqualesG F, FHt

quibutyCodern

A

1 α

<? ^{_ . Ε} ,.Η HS

K

D

funBoy non <varisto circinoexCD³

(umantur

quates Fl, FKper 3. prop. 1. cl. & fit Fpun-

3um mediumitneaGΗ, Ε y>erοpunBummedium

linea IK. SU dico F & Ε effe^unum idem^pun¬

Bum. Si enim fmt diverfapunBa, punBum Ε,

aut erit extralinear» AB» ausinipfa. Si extra,

tnia^r» non fecat, cum m comurratquidem^>quod

eß ^contra bypothtftn. Si vero intpfa,auterit

ad

pé rtesA,Velad B:&cumduo

punBa

im«

medtata in eddem Imea» ut oßenfum eßtntheii

36, femper tnter iüa

cadet Ums,

fi addatur

adm tatam G F, velad H F,faciet dimidia^ma- jcrem, &(teGΗ,nonerttperl K bifartamfeßay quod ftem eß contrahypotheßn. Εβtgitur F&£

idempunBum, quod plurescommumter

lineasfe*

tat;quoderatoCundendunu*

55.guin.

qχ* Quinta utilitas eß, in eodem plano-

recipereinfc piures lincas, nonin directum jacentcs, fcdüproducancurfcfeiccantesr%

adcujufvis anguiiftve rcäilinei, utBAC,ftioe

curvilinei«* Η D G,

HDEtFDBjfivemixti- lineiIDG>IDE con-

fhtutionem, ut inde

ornmum rerum certtL»

habeatur menfuratio.

Uwe faßum eil, ut a

Cjutbufdam angulus (it

dißrnMagißerMathefeos. Linde illudGeometri-

cum: Fonmm anguii maximaexparteeflo

åpuncto.

$6. Sexta tandem uttittaseß,quodfit in-

ftrumentum _quo refta & psripheria fefo

tangentesita coar&antur, utintcripfasrc- aiia locum haberenequeat, quin circu-

lumiecct.Demonßratto:SitpunßumJcoarßans peripberia per-

feßam ABCy cujuscentrums D, Gf reßams

Ε Ffefe _fangen'

tes in J.- üico

punßdm Λnun

. ptrmit&rt put 'ttUre&ainteripfastranfeat, quinfeiet ptyipbi^·

triam* Tranfeatemmalia reih_quaftzΛIL, &du-

i■ A; quaadconttngeniemperoendt-

ffuldm_mt,per*8.proo*j,ehEuch vd cap.v

elera. 15.1.15.Geomet. Ram.fat angultu ADI aqualisangulo FAH, per 23. p.l.!«eL£ucl.JjPuo-

niamigituranguli FAH> A D IexconfruHione*

juntÄquales»additocommunianguloDAΗ»erunt

duoanguliAD/, DA HȀqualestotianguloreHo

DAF,per 2, pron.iåcoqp duobus reHis minores.

jQuare pern. pron.l.i.el. Eucl.veli.c.c. 12.

1.5.Gcom.Ram, coibuntAH, DlreHa inalt'

quo puncto, utinI. Quomamigiturin triangulo

ADIttresanguliA D/, DA/, DIA Äquales fmt

duobut reHtéper^32«p.l* 1. eL Eucl. ve!^9.e. 1.6.

Geom.Ram. &duoanguli DAUADloßenftfunt Äquales reHo DAF: ertt rtliquus AID reHus%

atej.^adeomajor,quamDAI»acutus i JguarereHa

DAmajoreßquamDI,per 19·prop.l.ι, el.Eucl.

veli2»c.1.6. Geom.Ram. Ί^οη igiturDiad circumferenttamperveniet,^acproinäépunHum._»

I,intracirculumABCexißet>atcpadeo reHaA//>

circulumfecabit. Coar&at igiturpundu A ^pe- ripheriam ABC, &tangentcm EF inA ka,

11t nulla inter ipias alia reda pertranfeao, quin circulumFecet: quoderatoßendendum.

Αγ. bafuntprincipalsspriorisgeneris punBi utilitates, quaeitnfuntquatemueflex/0 tmmobtle&aquantitAte tnfeparabile: pofiettorts

generisfequuntur»quatnpunHofunt»uteßper/e_»

tnobtle,&aquantitatepermtelietiu*operattomm

feparabiit^.

38* Haautemutilitatesemotulocali, quem

punHoimaginattonoßra attributt,potißtmumdi-

manant, qm eilvelunm» velmultiplex^eoaenua

tempore(actus. ^{- -}

3P%Uno

i λ 1

C

qρ, UnomotulocalimoDeturPunBum,ium jfeorßm extßens, cujufdam animalculi infiar,vel

retio9 "bei circulari velmixtodeniqimotuferrttn-

telligitur, itauttn motursciotermmum acqusre-

re velnonacqmrereDaltat,

40. ReBo motumoveri&terminumacqui*

rereputatur, quando tnter dtßantia tranfeunda

terminos ne% tn banc neg^ inillampartsmdefle-

Bit)[cdaquabtleminceffkmtentt,utipfamdißan-

tiam veftigio rtltclo adaquet, Hujusficmotiutili-

taseft, quod fuo fluxurectamlineacDnobis^

cfformet, é cujusrattoneformalt illaemanatpro-

prietas;qudd fit omnium inträeoidcm cer-

rninosbreviflirna, utbancdefimt Arcbimed.

Incon£ i. ad1. poft. deSphar. Utβcogitetur

punBum ^Αfluere ^ex Α ^υ

inΒβtautnonfulfultet, [ed direBo itinerepro- gredia^tur,effuieturfå¬

nevefttgium^ACB,quod

linea reöl^aappellabitur,

quacomparataadADB,

A.

AFB, inträA (fBterminos,omniumbre-

vtßimaerit, utconßderantipatet.

41. tiinc apparet å quovis pundto ad quodvis pundtum ducircbtamlineampofle.

Fiam cumlineafitfiuxm quidamimaginarws,atg,

adeo linea rtBafluxus reffaprogredient, [tut β punBum quodpiam ad altuddsrecié progrsdiin- telligaturf du&a fåneßt4puntload punBumretia

linea. Ut inhoc diagrammateapuntloΛadpun- B^um Bduftaefl nBax abeodem^du adD(fEt

imb

'■j■

imo innumcrA alte ab eo*

dem ductpoffunt, tå quod ^*

i* fua petitions poßulat**·

fcuclides* _^—χ>

42. ReBo motu moveri &tcrminumnons

acquireredicttur Punélum, /V* moveri ima- gwatury utnulltbtaffurgateUiiusynulltbi fubfideat bumtltus, (edaquabiltterincedat,/i /'«infinitunz_>

fnoPeretur, Hujusßc moti utilitas efl, quodli-

nearninfinitam nobisdelineet, (»0» /'«- fint^ta reveraftt, efttmpcßibile y(edreß>eBa defcnbentüicui conccilum eßre&ara lineam in continuum reda producere per z. poft*

1.1,el. aututentés, qui alterumejusterminum^

noncurats cujufcun^βtlongitudinis) de quapoß-

Modum tantum accipit Geometra, quantum pro

fuo u(u oput. habet, refiduo negleBo, ut hoc modo

α,π&^ν declaratArift.I, 7. Phyf.c. 7»

Hujuu proprioa* eft, quodadeam,å datoqao- vis pundo, quod extra eameil:, perpendi-

cularis bnea duci poflit, quemadmodumdocet

Eucl I i. el. prop«i2.

4$* Circulari^motuintelltgitur ferri Pun-

6lumj dumcircaaliudpun- 61umfixum^movetur,donec adcumlocumrede^at,dquo Caperatmoveri»itaquidem

ut eandern dtßantiam abeo

femperteneat· ütβcogite-

tur tunBum Α moveri a-

qmdtfianter

Oquie-

Dquiefeens, dorne adtundem redeat Ucum, dquo

\

dimoveri c&pit,circularimotufirri dteetur. Hu* f

juspunßiβcmotiutilitasey?,quod peripher!a ra circülinobisdelineet,cujusproprietatesmaxi"

me funt admirabiles, quemadmodum id ofiendtt

Arift,inproaemio Mechanicon.

44. Adixto denitfo motufluere inteüsgttur

Punctum,quando inmotu ineeßumaquabilemnon^ tenet,fedtitubat, modoinhanc modomillam^par*

tem defleßens. Bujut ßc^moti utiltt&seß, quod quamplurirna miftarum linearuta genera^

nobis depiogac, ^quarum quadam funtumfor-

mes,utbyperbole,parabole,e[iipfis,combiüycittoi*,

beltx;qu&damvetodifformes,quarumcopiofm eß

numerus, quas cumfuisinßgmortbm proprtetati-

busaliasουνθεω enumerabimus,

\ _4f. Multiplici motu movetur Punctum vel*

inpiano, Pelin folido+ Inpiano,quandoduobuswo*

tibus fecundum reßam, velmaliqua ratione,vel

innuüamovetur.

46. In ratione aliqua dupücimotumovetur Tunßum, fttmagtneturperβmoveri fecundums

latusunumparaflelogrammtrtclanguli,fecundum

alterumperoperaccidens, ita quidem, utißt ^mo¬

tas rationem quandarn laterum inter fe jervtnt.

Hujusßc motiutiiitaseß, quöd retta ngul1dia->

metrumnobisdeicribac, utdocet Arirt.cap.i»

Mech. &noshdcrationeoßendtmuf. ittrettan- gulum ABC D cornpr&henfu*»fub reßisAB, AD,

quafint int^erfeinratione,quamdualationesipfius

Ababent.MoPeatur Aduplicimotu,alteraquidem

tendenstnΒ, alteroverb admotumIiηes. A B fer

ratur '

raturinD, fervata in-

terim Uterum rattone:

ΨΚ, !

lta% ponatur ex motu ab A versus B, perve·

mjje in Εt exmotuau*

Ρ

tur cumlineaAByfaBa

tem_,quoin rationefer~ Ε C

£

ip(a

perveniffein

tur. Erit igiturparallelogrammum A EG f,pa-

railelogrammoABC Dproportionale) & fmileper

i.def.6-eI.EucI, & circa eanderndiametrum per

conf._prop.24.1.6 el.Euclid. _JgiturpunBum-*

K-Αfi duabus lationibus feratur, laterumpropor- tione fervata, lineam reBarn producet,Videlicet diametrum A GC. Εfl enim omnis diameterre"

Banguli reBa:quoderatoßendenduml%

47. Hute confentit & tllud}quodaProclo

ex Gcmino aeeeptumftc expofitum efi·: Si qua-

drangulum duosque rootus,quia?quali^ce-

leritatefiant, alterumquidern perlongitu-

dinem, akerum _perlacitudinem intellexe- ris, dimetiens produceturreda exiftensli¬

nea? 1.2. com.indef. redas liaeas. Proprietät Diametriei?,quodparallelogrammum omne bifariam fecet, utdemonßrai Eucl.prop. 34.

> L1.elen1.6c Rarn.2,c.e-6.Lio.Geom.

4g. Innufla rationemoveturPunBum,ftab nnoparallelogrammireBangult anguloy eodem du- plici motufecundüm eadem Litera, fedininßanti indivifibtlt moveri imaginetur. Htijwftcmotiu-

f tilitasefly quod viciikni peripheriam circuli

nobis dcicribat, ^ut doeecAriic.cap.£- Me-

r, t ,:

£2 chani-

chanicon, idquodnos aliter hdc rationeofien-

demta: SitreBangulum A BCO. Cogitetur enim

I

A mövertininfiantiversus D,per(insamA O, &fimul

A

werjusB perlineamA B,ita tarnen, utlineaAD,ipfiAB

^

femper (tt perpendicuUris.

Sitautemcum ApunBuwLs fuomotu perlineamAB per*

B

veneritin Ε, idempunilum

A fimiliter fuomotuperItneamA Dlatum,perve-

niet tn G: erit ergo ex duplici bosmotu·,quo.per

ADquafidiffunditur &difceditforas; 0JperAB

versus centrumB, (efienimBcentrumqttadran-

tis BAFC)retrabitur, nekvageturlongiys, quam

åqualitas dtβantia undifi d centraferbandaper- mittit%nonquidtmin Ε necin G7fed in F, eritfi ^>

punBumFinctrcumferentia circuli.Quare pun- dum A duplici hocmotu fincratione late- ; rum , peripheriam defcnbet: £)αοά erat o-

|

ßtndendum. fjjuanta verofitutra%latio,men- | furatur Itneis reclis, quarumalteraeBfmusre- I

dusΕF, altera finus verfus EA*

4p» In folido duplici motu movetur Puts»

Buw,quandocircafohdum^unomotu perfefécun-

dumreciaminfuperficie fohdt, ^{alteroadmctunt-» f} ejufdem recla,Aqualtcelerttate delatst,circaaxem

ipfim foltdi defert^ur. Hujus(te moti utilitas eil, quod^variaslineas mixtas&uniformes,_pro

rationeiolid* nobisdepingat. Siemmhocmo¬

tu moveaturreBd mfuperficte &circaaxemcy-

hndri) ^ortturbeltxcyltndrtaca;fireBdfecumdüm

longi*

Ungitudlnem com, (f circum ejus axem movere

φ intelltgatur,fiet beiix^contca: βinfuperficiefpha- χ

r£_i & circaejusaxem, ßiralisßh&ricagignetury cujußnodi eil iäa,_quamfoiabortuin ocrafumten- dens nobis deßribif,de quihm omnibiufuoloco.

$o. Ettantumde Puniii utilitattbus. Ante-, quam verb ad ultimum qu&ßtüm perDentamus,

ι quarerehtcnonimmeritoÜbet,cummaginatione concipiatur itaperlocum moveripoffe Pupilumsy

anei ficaliquis locus attribuendus} Ad^quam quaßionem refpondendum arbitramur, PunBurkj»

quatenm reßdet in corpore, locum^nonhabere^per fetfedrationecorporis cui ineβ:quatencu vero per

imaginationem,fecundum ßmuttudinemquandam,

* qua cumcorpu(culoconfertur, per(emovetur,ni¬

hilobftare> quin per tandem analogiam

lociuipß

' *

attribuatur, tta ut inteiltgatur^t(jeinpunctocor"

porisambientis,hoc incorporelacato·.

jt. Tandemquaßunt in/tgniora in quantita- ^4. Qu«-

tepunBa Didendum eil. Conßderanturautcmilla naminh~

velabfoluté velrelate. Abfolutein fuotantum

(üb- naturL/

je&o, tfquidem velinimtioyDelinmedio, velin-* gun<ä*?

finesundeetiam triplex oritur Punftum: Incho-

ans, quod eil 'mitium, utA,Terminans/i?#

fi-

B ^ niens, ^uod

eß finis

C, Continuans, quod

inquovislococonceptum (fpriortm (fpofterio-

rem partem refpicit,ut

B. Ex bis incboans (f

terminans utrobifj uni¬

kumeßV conttmmtiaaufemppffunt

ejfe

B * ■ j.j.Re-

$2. Pelate conßderaturΡunBum,quatenus addiverfarefertur(ubjeBa, Eftq,Df/PunftutrL» ^»

contaåus, vel Pun&um interfediionis, W Centrum,veldfPiqi Polus.

jj. Puo&tim contadus proprieeß, i^iL·, |

quo linea ctrcularis circularem cangit ,vel i iupcrficiesglobofa globoiara» "beiquandotc-

j

da lineaaapcripheriam circuli vel iuperfi- \

ciem giobi ita pofita eft, ut cum diametro :

angulumred^umconilituao» Tale tBtnboc

|

fchemaée punBum-^

/. - ««ς- - {,

£ qUo circulares ;

linea HC, IC,

KCfe_tanguntjitem

rebla FG & diBi circuliHC, IC, K C,

HocpunBum effeu- nicum, demonßra·

vimut in thefi 10.

Alias PundumcontadusdiciturAn quo ρ

linea

lineam quam- cunq; quocun- que loco noiu

plane interfe-

JjT11"1 _—j-j *r-l — p· *

ied

tum cootingic*

Tale ettpunBum B & F in dtagrammate boc appefito.

S4· PundumInterfedioniseft,in _quo duse linea? velreda?, vel reda &curva, vel

deή"o; curv..a?·te^mntuofecanr.Sic_quoniadu£

■

: ' ^- VM