Dissertatio mathematica de linea. Qvam permissu & approbatione venerandæ facultatis philosophicæ in regia academia Upsaliensi præside ... Johanne Bilberg ... publico examini submittit ... Sveno Joh: Bierchenius w gothus. In auditorio Gustaviano maj. ad d.

1. N. /. _tö

SD1SSERTATIO MATHEM ATICA

•de

LINEA

Qvam Permif?u& _{Approbatieöevenerandae} Facultatis _{Philofophicas} m

Regia Åcademia Upfalietifl

PRiESIDE

AMVLlSSlMO _Vl\6

DN.

_JOHANNE

Machet: ProfcfT: Ordinario

pubheo examini iubmittil ;

Stipendiatim

SVENO _{JOH*. B1ERCHENIUS}

W Gothus.

la _{Åndjtono Guftaviaco Ma), ad d.io. Marti)} -Anno MDCLXXXIII.

■

UPSALIÄ

. s

■

m

DISSERTATIONEM DELINEA

luvenu _{Prafiantijfimi,}

D, S VE NO N IS BIERCHEWIU

arw ^anu» uwiiu» _{|;uyu(>.« «>&•} «„SpiSt

Principiura curfus3 ft&diiqne repagula

fignat

Linea: fic vit» ftat nota ccrta pias.

Fincmcurricoli rnox ukltna liaea monftral:

Sicftatukrebus linea nots roodum*

Attibi _{procedit,. SVENO, febtilius} afta*

Et fliaic islaudes linca _{longa taas9}

ffl (pIATATfl BlEPXENin

T?* ***X*PtdV' antXit <** $*yi**

Air, *p* lijuewnf t*iy if* ™

öoroeßic«_quondamgratrquefamiiiati*

Satis mf mar, gratuiai&cSicaufa fcr.

JO H COL

UMBUS.

lvv«»v _tft.

/

Litterarum_{bmarum mormquc elegantia}

_folitißimo

Dno svfnomi bierchenio

De

LINEA _MATHEMATICA

Eruditc _{difputaturo»}

Rcftum

Etvarios

qvi

certo

curvo

deduccre

dftcerrcrelumineproropto»

_{pollice flexas}

Nonvaluit _{,{imul&qvadris muta re rotund«ra>}

Legefenex Samius Mufarum limine motum

Valvisadfixa _{voluitj (icnamqae licebic} Arte humerismeliore alas _{aptare veloces,}

Q^cis &

öatdaleas

multo foperemus, _iifcjue

Longe tutius Icariis fcandatur adaftra y

Iamqve taas hoc BiERCHENX ffonuiffe _peraares Dixiffcm monitum _{fapientis, nibeoe noffera}

A tencris multum _{geniohos placuifle iabores;} Nunc age confcendens Mufarum pulpita, doftas

Pcftore _{prome tuo, raras& pinge figuras.}

Sic _{gracili acclamabat}

avena T.

SKENo _jOiitfitrfå

P R M F A TI O.

Dlfcipltnamn

praßantia, nec

Matbem-,

nonfemma

qvanta

ßt

utilitas & necej-

mbilltäi

dtque

fitatnoftfolumaäalw_{petfetlé peräifcindas}artes,

'verumetiam ad_{Reip: aiminißratiomm prcmsvendam,} nimmt,ql'ivcllcviter_e<vsinjbexit, obfcurum ejje_poteft. Ele¬

ganterßmcoßendü Proclus neminiad metaphyficam aditum

pann, _nißpermatbematicas _{difcipltnas: Kam fi} arebus fenßbtlibus, _{qvasphjßcus conßderat}s adresabommm.tt•

rta fenßbilt feer_{etas, qvaslontemplatur MetupbiVires ad*}

fmqtte noßri intelleSlus attollereabfqueullomediotentemusf

ssofmetipfos excacabimus,non fecusaceicontingit, qxn e

car-ceretenebricoß_inlucemfolis_{clavtfsimam emittitur.}

Antequam

itaque arebus Tbyficis, qihtmateriafenßbilt/mtobnoxlt,

adresmetapb: _quijmt ab_{cadem maxime}avuljie

>intelleSlus

afcendat,

nccejje

eßeum priusaffveßertrebusminus abßra-Bis, _gua/fj£ matbematicis conßaerantar, facilius iUas

poßitcomprebendere. (hocirca diyinus Plato matlem di• Jciplinastrigereammum dJdivmrumrcrum

contempU-tiomm exaCuere mCrJisdciemaffirmat, _{sfd bas utdita:es} o-mnes,ma>imaettamaccedtt lucunditas ac VoUptas$ qvi

tujufque awnus

his

artibus excolcnduexrnenJuquepnfun» ditur, in _{q\>ibus non}

_/olum

_ingenfti

_{ädolefentes;}

_{fed etiam} nobilesVtrt> Prindpes, Reges, Imperatoresadhoneßißimam

maximeque

liberalem

animt

deleSlationcm,

quam fumma cum

uti-mittäte cönjutiStam

_{ejje viJcbantßkmultäqi}

_{Ver[artfolebanf*} jirchimede_{SyracujaruMaibeatLü, tantumbuk arudediits} futffememoriaproduntutcibi potufqueoblivifteritur, corpo* rifque Curamprorfasnfolgeret%tumörn

rapereturfabinde in*

vitus dd_{ungvendum corpus &ad}

_balneum,

_in

_foio

_figur

_at jirithmciicas exdrabdt? dumquemigtretur,dtgito

tmeas

du*

cebaty tama mmirumdulcedineartistion modocaptusj [ed

ttim omnino_{inflamnatus, Horum}

_ßudiömm

_jucmditatt

non minus_egoab in _{ernteafafe r & qvam} primum

fcirem

qyidJfetlineamducere,afleclus fum>eaquejemper

di!exi&

in

pretiohabuimdximo.Quapropterin

hoc

vita

meaficantea&a

currkuto_{yVelutipcnfumameipfo qvotidte exegi,}

_fladiumque

percunendum, Volente Deo, mTarnafio Mu/arum mihhuet

propofuijdcircb cumjamineo

ejfem

utjlecmen

academicum

cd _{remt materiamuttkmfimulacjucundamdiu inqvirenti$} tbtulit _{fejebacprafensquam}

_expltcandam

_afjumere

_diu

du-bitavi _{obfummam eksdifficultatem& ingeniimestenuaatr,}

fid aborimprobusomniaVtncit.Ideoquefvafueorumqvorum

interfuity virescxpeririQFexerierc

pLcuit. ltihac itaque

hncje Tbeoriavariorumfintentias_acopinioties proferre &

di-rimere animnsnoncfly_cumbocneque_[cripturamodus_nequevirct

rcci, tant:_(edmßramfaltem_fententiam

Excellenttum

Ma*

thematicorumratimbus_{innixam%qvantd}

_fieri

_{pvterit breVUdtt} Atqueperflicuttateexpromere„

Jdfititaque

Dm

cvpfitf

C AP. 1

Arpumnti _{bujus ^ovcpit-nhoyKu txltltns}

Thes* I'

Ufpicatioremut nobis _par.damus difputandi viam opcrae _prccium effc _{pjtamus,abinitiodc vora*} bulo Lineae nonnulla _prselbirc, utdeindc _{inpcnitiorem ipfiusret}

explicationem, in=offenfo _pcdc dcfcendere_{qveamus#Lmeam itä«}

queåLino dici vulc_{Jfidorusl. 19. c» 18. cum qvo fa¬}

cit_{VoffiusinEtymologico}

aliique.Becmannus

qvo-quede origine lingvae lat, in vocabulo liter* perfpi-cuéoftendit illas ccrtis linets _contineri»

atque

pri-i»umlineaturasdi&as,poftroodum

vjitcras; qvod ipfum etiam vox _graeca confirmat _{y?*w*Tawaptfryw}

-y/*wwdifta. _{Aitquc porroex Scalig: de C. ll# linea} effe roinitnam _{corporis dimenfionem: qvia}

fit é li*

no: Isfcilicet

fubtiHffimusSuenuifiimusfuDiculusqui

fcrme

_{übtutumfugit.}

II. _{Refteitaque &ntruramfitt Lineaidero eft}

qvod

flamen,

_{foniculus,perpendiculunijacidcirco}

roathe-roatice _{confiderata, lorgitudo qvzvis}

fcu extenfio

fine_{latitudine* linea dicitur,qvodin gloffis}

reéfreex-prirnitur _{yfi«wv>™p<*,ntycv. Removeareus proinde å} coniidcrationenoftra lineana _{pifcatoriamdeqva}

au-A 2

Itorcs _{fsrijj lineara roargariurum}

deqfaICti;

linc-am _qva fabif

utanturad

materiaro

fignandamutcft

apud

Cicrroncrr.Excluduntur

et

i

a ro

fila

tcDuiffima,

atpotc Äranex 8c

botnbycis,

ut

8c

iinc*

penicilio

labtililfimo duöx, qvalis illaqtram

Apellesinfignu

illcpidor tertiocoloreduxit in Protogenis

ablcntis

labella,priores duaså

fe

&

Protogsnc

certatim

duftas

Iccans,nallum relinqvcnsfubtilitati locum, Accc*

ptioocs alias

ubiex

fiois,terminus,

oradi-citurficut _{apadHoratjmors}

ultima

iinca

rcruni,apucS

Euripidem » extrema

lincaamirs

Ter: in Eunuch: 8cidgenus alia. Sic qvoquc

fen-tieadam de lincafivcinTheatro, fivcin Curriculo* fivcinludo, fivein

agro,fivc

in

Mufica, fivc

in mana,

quaead propriam

fignificationcm

facile

revccantur* utcrudite docctBetmannus in iua ad lingvam lat:

manuduétione.

HL Lineaminrerum natura dari&åfc&cseteris magnitudinis

fpccicbus

diftin&am,

mclioribuspro*

bare non poflumus rationtbus quam

beatiGedrinii

in diftertatione _argnmenti

_ejufdem,

_{ubi roonftrat}

cor_pora exißerc _eorumque

multa

fuperßeie

uifihili

clducfi,

fauntibus &

iuipßm

Euchde

a

dej.i

Ham.

z,

l.iiÄ

cl. geom.

qvopofico

,

fefvitur

lineam

Velfeipfa uel

ne-gattot.e

ad

Mo dquo

inbar

tv

ti,

cujus

natura

ab

ea

tß

dijtmfia, termtuari#

Uttum

a.

boc eß

tmi

fitri

a

fcipfa,

quia

term-m*kokeß_{/ars trrminaia mp^nttvdtnU; aliud n.}

eß

terminom altua urnianm l Urmtnu*n> umttitcryaSo _infe**

rior_{eß ic mmnio:ftdtec ntgdiontfetta/terms lOntitmatiomi}

frvvathx , emu, emm magnitkdo /itfinita a&u> jirtß.I. $.

c. 6*pk

tnfinitapoflulatm

_{lamumdijciplimcauja,dt idem}

itCdloc i./. 10,F uhoah^mtnbitrentt_{obca dißmSlo} ,•qVoJ

lihCm mminamm. Porro Imeje exiftenriaex mcn(ur4

continaa_{monftratur,cojus partesadterminum}

com-R.uncm _{coaptantur, cuiabutraque diftin&o etiam}

fuperficiei partes conne&untur, qvi lineae nomine venit-Sit _{E.g. in fig. iuperficies abed. continua,}

cujuspartium af 8c de,terminus communis ef

Gfue-ritcontinuus , cius partes alio infinite _augerentor;

fin_{minns,effetqvantitas difcrera,qvod contradtÄ o»}

nem _{involvit; critiraqueextremitase/utriulqoepar*}

tis_{af 8c cdqvoad latitudinemindivifibileqvidjqvod}

lineamcfleintelligimus E. linea datar. Adhoc de-monftrabiturlinearealis cxreali contadu _Cylndn,

qvod licet ab aliis deroonftretur , placuit tn. uti

de-monftratione Clarifl. Geftrioii. Sitin _{fig. z. Cylindrm}

ab_{tandensfupcrficicmplanam}

cd}dico (jhnJrtmplanumnö poffetango calibiquaminlma, cum CtreuIm in Cyhndro

ptrfcde

iQtundoanceptusßt

inum juperfide, cruhtpur,ila contaclus _{plantcd &periphere<e lüius ttreuhm unartclaex} dc'noußratts Eutl dts, fdtaqueCjbndrusin

_extenfione

platta

tangent,

mejjéiiumfe

habtra_exlenjmcm

_{pLuam, cjyeA}

ibfurdum

eß;

cjviapUmmnon

fließ

mß

plam afaqym,

CyhnJruspiättutn

hoc

modotangtris

ßctriunftohätur

Berfas

d

tu abw_qväfit

contaSitét^afcifditin

_g

höfg htiftta

defctfd

t

iftäi

n9(yimdri

_nulla

itiäryenientecoficflaßathneymm

_in

_hnea

tdtkuilnUexpcrtefiät conictBusjnqyampxuciorzsabaxein•

ßßtmt

perpendic

ular

esmottim morantes.

hinc

Ii

qvet

Cor

Cyündrus

facüias

movcatur

in

piano,

qvam

aliud

•folidum _reguläre

_qvod

_figurata

_planorum

_adeptU

_cft.

Cap. 11.

CompUüem Lima Definitionem ^ Divifimem

Thcs. h

f _ine®Definitionem_{nobiseshibetipfe Euclides} 2. 1.1. ubi diciteam e(Te: Longitudincm htitudinis

expertem. Nonnulli lineam fljxum punfti dixeront,

alij punfti

evoiutionem,

ut

poftea

Euclides

efficic

Circulum fluxufeumotaiinearcurvae, 8c _Sphatrara fluxu _femieirculi, 8c conum Cylindruroque motu

irianguli8cparallelogrami. Primam

fpeciem

magni-tudinis definithic Euclid: quam dicit effe qvantira-temabfquelatitudine,

äqva

removetur etiam pro¬

fundus five fbliditas, _{qvaeeft diffareutia corporis:}

ficutlatitudofuperficiei. Lineam hancmathem:con« ciperc poffumusexratione

luminam

8c

umbrarum,

jnter corpus

luminofum

& corpus

oppofitum

feuob-nmhratum, eftlinca naturalis lenfilis, lantudincm

ffrdccn

w?g*nationUaffumendiL ift _{itaqö€ afts*}

gnitudius I Ju*gton etric* intelligentia_mcnris^ma«

gnitidotn ipla, t nea, fopetficies _&c. revera cft i»

/enfiliår _{fhyficccorpore: Q}

vare magnitud ines ii~

cet mente _{feparcntur, revera ramen funt dias}

ipüß magnundines, _qvibuscorporaphyfica_{eorporafunt»}

id eft _{longa, lata,(olidaatq}

ie ocrnino locum

occa-pantia,qvaregeometrice 8c _{phyficejodicavic <driß:}

cumdauit_{magnituåh^reVera}

fbyfr ejje: fed ahfraElte

icateru_{pbyjictsq^alitatibusageometrismenteConfideruri.} I L _Mathematici ut nobu» inculcenr veramlineae

intelligentian, imaginanturpurdlurn _{defcriptam é}

Jocoinlocum moveru Cum m_{punctum fit prorfqs}

Individuum _{»relinqyeturextfiomotu irnagirario}

ve-ftigiumqvoddam lorgumomms erpers latitudmts,

nt fi_{punftum a infi;Euere}

intelligaturveftigiuroeffe» étumlinca_{vocabitur, cum intervalluminter duo}

punéta a8cb_comprthenfum, ut apparet exfig.3*fit

longitudoqvadam

carensomnilxutudinc, proprerem

qvodpun&um4 omni privatumditnenfione_eam

efft-ccrcnulla ratione

potuerir* Neque exifiitnandpns

cftÜneamcx _{copofitione pun&oramconflatameflfe,}

fqvidem trnpofibilccß utivdiyfthik(*f non qffavium rem fVavtdm_{icmponat,mquitQarJJ.}

_Gcßtin:

_{1.1. ad.}

_def

_2»

eiEpfJ. _jedex

fluxu ipjorim counmo,

quoccjjamt

ItneS

non-noll! _{contendutit, fed}exlineiiperpetuis 8eio

infini-turndivifibilibos,qvod_{ex motu} duoram Circalorum

fefeinvicem fecanrium oftcndi _{poteft, fir} in _fig. 4.

Circulus abc.qviefcens,/^ _{c. v.} egrediens, _{dc a} priore manente difcedehs, iam in hoc_motavidere licet

nul-lam _{partemneque Circuli qviefcentis, nequé}

_lineae

Circulo content«

remanercqqftiondividituraCir-euloegrediente, propterea

nihil

qvidquam indivi-dnam _{relinqvitur velin lincareéia}

_{velin Circulorum}

Circumferentia.

III. Pundanonfunt _{parteslinear, utunirates mi¬}

men:fed tcrminiquibus

terminaturfiveutrinq;

clau-ditur,non qvod omnis linea terminoshabeat;

Peri-pherian.cum necinitium

oeefinem haberecenfer^r,

videtur _{pundis nontcrminari,}

_{atcumdefcribitur,}

una_{parsline* a}pundo

aufpicaturaltera

in pundum

terminatur. TanElum _{itaque eßtermmus Imc* modoaSlu}

ntin lineareStä, modopolentia_titm Teriph.

perfecla

inquit

fym.l,ii. el.geom. $.defAilineam puridis tanquam

partibas

integrantibus conftarecoocederetur

#qvod

minimetamen concedendumeft, _{fcquererur, linea}

redarn Circulum _{tangere pofieinpluiibus pundis}

quamono,quod eft contra

Demonftrata Euclidisfit

vgjn

Fig.

Circulus

abc. redarn

dg

in purido e

tångensi fitconiadus in unico pundo,

ad

pundß e line« _dg*

_{excitoperpendicularemexcentioi}

_ex

_il«

F

p.el, Eue!* c*eodem ccntroducoftaliam lh% ad _pen« éhimqvodlibetlinear _{eg.erittriangulum cih cujus}

duoanguliA*eduobusrcétisminores per 17 p.1» 1 El, lud.Cum _{angulusade fit redtus per 10. def.l.i el.}

Eue!» E._{angulusiie.rc&o minorideoque major etic}

re&a _{*7;.qvamieper# fpfpj.i.el. Eucl. cangit itaqae}

Circulus reftam inuno_{puodo e quoJerat}

demon-ftrandum. _{Peripheriamtangere periph: in unico}

pun-fto, demonftratuf ex Theoremaieiz _{p. xm. l.w.}

el. Eucl: _qvod

_{adfcriberenoapigebir.}

_Tangat

Ctrcu-Aueab, in_{Fig. Circulum d}

ah% Intusß_{ßeripoteftin}

pluribuspunBuqu imuno,ntina. &b.ponaturCitcult c ab. centrume.Circuit v.d h_{å.fimatur Cent.fper 1 p. lib.}

5» qyiftentradrvtrfaerunt, per_6p.ltb. 3. ducla E.reBa per

centra e(.protraBa_{cadettncontaSluma}

per 11 p.1.3. Si

praetera ponaturpunSlum contaBus in bconneBatur& iüud

cum_{utroquecentrocÖ} freBis be _{b f lam'Vero}

trianguli e

fbduo latera_{rekqvofuntmajora qvomodocunque f*mptayperé} 10p. i.ficommunisreBaci auferatuvjemanebttf b _major

quamfa p.yaxiomi t .ideoquepunBum bcadet extra

Grcu-lum_{interiorem, ubi nuVus fitcontaBus, fititaquc contaBus} in unico_{punBo. 1. Tangat Qmulu*abcCirculumai o}

rx-terius in_{pluribus punBuquam uno, fifieripotefi,utittdem in}

aGf bydutla_EreBa

g e per Centra g etueeßario perc

tf-taBumtrattfibitper

np.$.fietiaminbfieretcontaBusJum

duBa b_g,b

_{c,aqvaltsfieren$}

_{lineisga e a}id}

_cfleagpcr

ic

iéf: t\.qvodper _{ip. r.}

eflabfurdum%

Circulus

(irculum

extra

non_{tangitinpluribus punftis}

qvauno,qvod

_erat

demon(Iradu.

IV, Definitionens_{lineaeexcipit divifio,qva lineat}

eftvel re<5tavel curva.rÄ_{eflqua ex<eqyaltfmfmSlain¬}

tet_{jacet 4 def} L _{%. cl*}

Eucl.yd

_ut rl\am:

vuli

₅

d.,

Ii i.efl

quainträfuosterminos

aqualiterinterjacet^uwa

con¬

tra._{atqvaiiter inter}fuos_terminos in&erjacet

linea^va^

dononhic humilius illic elatius fublblcat2 _ledseqya—

Iis& uniformis efl:_{fpatio inträeoldem comprehenfö»}

utlinea^i _{Fig. jificqvi reélaro iterfarit,}

_vulgo

di«-eiturtantum itineris _{conficerc, qyaiihironeceffé efb:} qai facitobliqvum , plusqvamoporteat. A

roh

iro

ed

i

dicitur bteviffima inträeofdem terminos; Cujus me¬

diaextremis_{officiunt> utin folis felipfi, fi Imea reri-gt}

jfole _{perlunam adoeuium noftrurnduceretur^tTncd!«}

ipfäluna noftris luminibus

öffrceret, folifque

afpe^V

äum nobis _{ädittieretqvod exopticis deluirrptomv} m _{inqvibus doeetur vifurn redtisradiis}

_{fieri. linear}

rationeufus&accidentiurn varig indait nomina, di— eitorvel finitaieuterminatai, infinite feu nontefm!*

hata:illa efl quam _{bfläriarn fecare}

jubet

_{Eucl. p. iQ.'}

i# fuper_qvaTriangulüm äfcqyilåreratn conftitui _po—

ftulatp. id.iihaeceft

_{adquamåpun&b,}

quod ine&i

2*onefl: perpendicularem _reélam

dcducerelubetp

_it:

iicaeteraefpéeies line^reétsin Figt7, cognotcuntur.

"w n3

V

Diameter dt finus re<fhi» _{Jlngult fe fi«u» rcåtis co*ij»lc*icati} t* Semidianict: db hnusverfm t c b _{fg finus /crft>* comp: V}

ee, »remHadius I h _{Tangens &arcus g q Tangen» conipJ:}

y

9_{g&Gn»tot^difi9eh fecans}

tk it fubtenfa feu choida >

c* _{fg fagittaidqj rcfpcöu}

chordse»

V* Greularu_{lineaeßqy<edißat}

_aquähter

_{a meäto}

um-prebcn/ifyatij. ifytn.def.8.1.2» efficitur h*cmotu pun*

dinon vacillantis,fedin orbem_{uniformi motu}

atque

diftantia,å _{certoaliquopundo^ad qvodomnesreftas}

lineaedudae interte_{kmtseqvaleSjficutexponitEucbd*}

i).d. i» ex hisliqvet duas fantum lineas fimplfces redarn _{&circularem, omnes aliasjqvsreunque funt} mixtasappellari#qvodex illis corrponantur,

juxtaq,-triplicetrjilineam trestantuno eflferrotus, duos quide

fimplices redarn8c Circolarem tertium a. mixtum

cx duobus hilce _{compofitum. Linearum mixtarum}

plurima funt genera: quaedam uniformes _qvaedam

difibrmes, uniformium aliae funtin _{pianoaliat in}

fo-lido,ijn _{piano firnt hyperbcle,ellipfis, ovalis/Helix}

conchcidis etc. in {olidofeufuperficie _{curva funt} al-terius_{generis lineae helicae,utpote}

quae circa Cylin-drum _{convclvuntur,} Diffotmium cum fit infinitus

nurrerus eas recenfere non_opuseft.

VI» Exoc caforehuju_sdi_vifi_{cnisqu aßionon i}

a-jucundaoritur, L'trum ^Ftuwotiiefvo filfr'm natura.

deqva n agnaolim cofitroverfia iuit inter

Pbilofo-phos.utramqueparum comradidioirishujus;

predi-dit Arift. 1.4.c,a.de coelo. _{difputatn~ Peiifhetiam reSta}

ctrculumretlibneö

prtoremeffendtnra. t.qvid

perfe«

Bitr _efltcu

_nihil

_ei

_{pofjtt adJt:}

_poßit

_autem

_%e£la.

_i

_{qvia cir*}

euluseß(smphkr cum

ßt

uttius

krmim

%

Rt&dintumv»

jrmboru ttrminorü. Acut

peripher«

_{nihil addi poteil,} arpcriph:(5t:fic

Rc&aenihiladdipoteftutreda fit.

Neque

perfe&iorcft

hocargarrento

peripheria

qua rcfta:neq5tacn

perfe&ius

ftatim

natura prius

ed:N5

pcrfcdiuscft aoicnalfcminc,neqac

tarnen natura priuscdö. Quare

fi

argumentum

prim

um verum

fit

nihilcamenqvidquamconcluciir^neqxic iecundum quidquam

robuitius; quia

fimplicitas

noo

arguitur

nameroterminorura: fedporias_rerom

terminata-rumjfiqvidcmanico

termino

comprehenditur

muiv»

das, nequetarnen

cft

fimplicior arbore,

Leone

^

homine autqvabbetalia mundi _parte; qvin

rotun-dum _{jplum Pcolomaeo polygonia,irrov.}

_Ariftotdi

ipfi cfyyevM totusangulus>8 c.3.

lib»

de coelo«

dicitur,

tanquam figurarotunda

fit,

nonfolum

multangufa

fed_{totangula. ttaquerotundum licet}

_unico

_termino

cornpre-htnfummagismultangulum

tft

magifpe

multilatcrum

quam

uTiumomhinoreSldmeumirquit idem

Arid.

HuncnodS

folverevidcturPlutarchusin _{qiaeftionibusi-latonicis}

unopraseipue_argamento* RiFla

yait,eßmater

pcii-pherix, motuquüextreniimrecläpunSlif circa

fixum

reliqrS

pui.FlumdeI rbit periph: demuetotusjutmotu

defi

tibtum (b^ftorechiineum tummundoLucuhm Sic plana

luper-ficies defcribitcorpus mota fui fecundom_redbarn,ii

reftiiinea (i?

_prifma,

_{circaredarn, fi trungola}

parti-Iclograrra, fenicircuUriiconum,

Cylindram,Spb*-ram. Videruruaqocreda _{*i agnitudoptrens Sc}

ge-nitrit_{obliqva?, ideoque natura prior; Etiam Arift:} inmechanicisobfervati"rorundoiwtfTWifcffccÖTéri

åtconcavi_{qvodProclusad4 d.i} jcpetit&diffirriilim-dincrr» nullamaiteffein _{reda,quactamen(itin}

pcri-pheriaoteonvexi ScconcavuItern

ti

peripheria

fit,ef-fc_{redarn,licetnon juxra generationein, verum}

jux-ta _{rtfpedemad centrum,ur>de concludit redarnpe¬}

ripheria fimpliciorcm* Ex hac redi priore&anti-qoiore natura,meritonarum proverbiom ab Arifto-telcipfo celebratnm 5c. 1

die

anima T« *v9-«*«u rosvjv

t!#roKét/unvAovytvurKofAtv u-pat^ stpQwo kopmv, rt5 KafxvvÅav

*Tti*»T*bTt (ftcfto&ipftim _®obltqvum

cognofcimu*; Regula n.cßjudexutriufque

>obüq\u

v ntqjui nfq-nilt_judex

eß: ficredtmagnitodocrit judex tedx &_obliqv«: fic_{Angnlusredus redi&obliqvi judex}

eft^icTrian-gulum Redangulumjudex obliqvangulorum. Hinc redum _{provero Sc jufto,obliqvum contra pro falfo} Sc

_injufto

_{aeeipitur. Ideoqae icdum in partiendt}

linea _{prxpooaturobliqvo8c ficdeineepsin}

qvoefi-qccgencre redum

obliquo

priashabeatur.

CaP. III. ' i

.'Linea _{affeüumes exbrimcns.}

Thes» I. •

fjh

fiedscwicsuniasmagnitudinKfunt

terminari

Sc

fecari, veIre&itudo& Curvitas. Affcdioaes planum raagnirudinum communes qvatvor

potan->

tur,fymmema8cratio qtiae

Arithraeticar, Adfcriptio

8c _{eongruentiaquaegeometri® compecunt. Duarfi}

vjinearuminterle funt _{Perpendieulum& Parallell,}

fmus. Eft Parallelifmus communis tam redaruuj qnamohiiqvaramlinearum,

Euch d.

34

l.

fomtimeMpirallelMeßein

eodetn

piano,

& ex utraqye

parte in infinitum

ß producantur,

in

neutram

ftbim'4-tuo incidere <fed _.ubique éflant tqyähter, iHz

inßgnem nobisprjebencufum inOptica,

Mechanica,

Pi&ura Archite&ara8cc. necminori nobis ufui eft

lineaPerpendicularis, cuiusopc exploramus num ftrufturae fublimes exade re&aead _{planum horizon*}

tis _{exiftant, nee non planitiera, num fit piano} ho-rizontis _parallela,

_qvod

_exfilo

_in

_bafin

_fincidentej judicatufacileerit, dicitur libra vei libella fvet:;

SBftftipaß. Perpe n

d

icu

I

□ m

f/2 Itnea altm

in

üflens,

qutt facit angulos,qWfuntdeinceps, dqvales d. 10.1 ié elt Euch

eft hacc_{femperdeterminata ac}

_definita

_interomncs

ilineas,

_fifcique

_femper

_{aequalis,unicaeft}

8c

Angularis,

Jdeoqaenonpatitur

alias

plures

ab

eodem

termino

Headern parteereåfas, qvomihosilü acclihareni*

nequCvaecjvaliterinterjacercntiConducit nobis Pcr-pendiculärisin omni menfuranda magnitudiiie: ita

utrantahabcatur_qvjevis longitudo, Iatitudo& _pro-*

fbnditas, _{cjvaatatft perpcndiculifis ab} uno larät*» ad latus _{öppofittjfnvelprctra^um, fi itaopusfuenrf,}

ére<5ta ut_{patetex Ffg*} 8'longitudbliujus

figurs

eft

Ferpcndieularisabvc\fg.excitataåiatere fr ad

op-pofiturtiop. latitudo eft perpertdicularis ou;it\

duftaä_{ktcre/p. in latus protra6tu^ r,hzetiam lineas:}

perpendiculares aravales luntper 34b. 1. i.» el Eucl: IL _{Hujus linear perpeodicuiaris ope, ftudiofus} Geometrie , multaspropofitiones (olvere Åreamqi

cujufcunquefiguraefacilfnegotioinveftigarepoteftj,»

fic,Fxempli loco,Triangulum cujusarea mihi mqvi-* renda _fu> eius _{itaque latusin perpendieularem ab} » oppofito angulöderoiffam maltiplico, produdd

di--raidium dabie mihiarcam ut latusinfigurap. _hc fit:

Toped. Perpendiculårisrfi, yoped^produétusesinis

erir3500.cujusdimidiurn 17joped.dataream unius TriangulLTres proinde magnitudinis fpeciesiineat

fuperficiem &corpus, extrina dimenfione

longita-dine,latitudine& profunditateonri conftat- Nam fi >

ad _{qvodvis pun&um\inabqvo corpore, iufceptuirs}

tres(oium linear ad feinvicem perpendicularesduci

nnoexiftuntplan«, rcUqtat in divcrfo, manifcftum

«fttrcstantum cffc dimenfiones, _ideoqnetres

ir.a-gnitudinis

fpecies

cx naturarci inanaquaquere cor-porca

diftinftas,

Hifcc

pro temporis 5t inftituti ratione, lineam geometricam adombravi, non ut

roloifed utpotui,_cartera

_qv^iic

_letiter

_{taftavel pla¬} neomiffafant, demonftrationirelinqvantur. _DßO

Triani,qvi omnia conßituit infiiadimenfioiie, na«

mero nimiram pondere5c menfara, fit honor,_glo¬