Specimen analyticum, XVII. Problemata geometrica, algebraice soluta, continens, ... Præside ... Erico Burman, ... Publice exhibitum a stipendiario regio, Johanne Rudman, Gestricio, in aud. Gust. maj. d. IX. Novembr. anno MDCCXXVIII. Horis ante meridiem co

D.

_D.

Specimen

_Analyticum,

XVIL

PROBLEMATA

Geometrica,

_Algebraice

_foluta,

Continens,

&

Cum

_confenfu

_Ampliff,

_Facult.

Philef.

in

_Reg♦

Upfal.

PR£SIDE

CELEBER

Ri MO

_V1R0,

Mag.

_{Ε R}

_I

_C

_O

BURMAN,

Aftron. Prof

_Reg.

_{& Ord. R.S.L.S.}

publice

exhibitum

a

Stipendiario

_Regio,

JOHANNE

_RUDMAN,

GESTRICIO,

In Aud. Guft.

_{Maj.d.IX.Novembr;}

Λ ΝNO MDC CXXVIII,

Horis ante _{meridiem confvetis.}

SummeRever.

_atq;

Celeber

rim

_o

Vi RO,

D

N.

d

Λ

N I

ε

L i

»3U-9l»<53ie/

S S.

Theol.

Doft.

&

Profeß. Prim.

Archi-_Prdpofito

_{& Paßori}

_{llpfal. longe}

meritiffirno, nec non

Ven.

Confift·

EceleEAdfeßori

Graviilimo.

Adm. Rever.

Pneclarijjumque VIR

O,

Un.

MAG.

Ε

R 1

C O

ajsDisesi/

S.

Theol.

Leflori

in

Reg.

Gymn#

GevaE nec non Pajlori in

Hille

vsgi-lantiiIimo,/ip««i«/ö

meo

propenfiiTImc,

Mxcevatibus iß Patronü

Hafce.lucubrationes.

ob.

_fumwa.

_in.

_me.

Academicas.

_collaia.

benefi

in.

tmendi,

_certijfimam.

_cum.

_calidiflhms*

tis. vener

abimdus.

offer

o.

Summe

Reverend.

PraeclariiT-cultor

devotiffmui}

)

VIR

O

_Nob;li([imo

_{^Amphlßmoque,}

I

DN.

SA

Μ

U

Ε

L I

KLINGENSTIERNA,

Α,

In

Regia

Academia

Upfalienfi Mä¬

re

thematum Profejjori

defignato.

)

VIR

Ο

_Clarijßmo,

)

Dn.

ANDRERE GAß«.

DUHR

_E,

ru Direäori Socieratis

Mathernatico-;i-

Oeconomicae lllcuneniis dextcrrimo,

o,

prudentiilimo,

vis weis Maximü<% Oftiwis₅

ii

_fiu

_ac.

fubmijji\

ariimu

documentum,

$

«ί.

ér.Jfiem.

_{favorit ulterius.}

ob-is*

_pro.

_peremu

_jviguioriim,

_fehcitate,

vo-d.

Nobiüiiim.

_AmpliiT.

_&

ίί· Nominum

*/>

obfervantisßmus,

färgaren

οφ 0uö<fnuben iGefle,

2trebovnc

_{ί>φ ^onfierfarnc CD?aftaren/'}

<2<M

AUGUSTIN RUDMAN,

€Ücm _{£ogidrabe$dre gaber.}

Dg i;ar jag Idngefelm fbrftdbf/ £ur' obcffrifitg mpefet gobt/

0om 23awen _{jidbfe njuta}

Utaf gördlbrar bag från bag;

£n>araf oif tfyeraO pjerte^ag

jag npgfamt*funnet fluta.

Sm oeffa barnen tfu bete

0ig fperemot/ jag «tal fan fe:

€9?en _{l;urn tpe ffu i;inna/}

- Sit fdbant

tpém forftpüa vatt/ Sfjeriil fan _{jag päintet fatt}

$?ig ndgon ufiodg finna. 2p jag/ fa mal fom anbre fler'/

3 min formdga tnfet fer /

0om _{ridgovlebé fan fmava} 0not _{tbeti gobbei/ fom jag (jas}

Sif (£ v/ min fromm' οφ fyulbe gafjr/

0d _{riflig fdif forfara.}

Slllenafi fan min milje |W

Verebb/ at _{(£l)r tiifmnba gd}

C9?cb toorbnab/ facE οφ

_fjeber,

£>φ trenne dr tl;et/ fom nu jag

ä£etpga tdnfec tljenna bag/ 0amt tvalmeni _{()dr for} Sbec

50?et> _{tijejja blaben offrar opp.}

Sag iefmer t iljet fafla f)opp/

$D?m fare _{gaber Idrer}

$be famma taga gunjligt an.

<£[)r gtftoe 0ud/ fom alt ge fan/

♦ptoab gobf 3 fjelf begdrer!

Ä _{£ogtdrabe$·}

£pbiflffe ©··/

J.

R.

ooooooooooooeoco

D.

D.

PROBLEMA I.

A

_tis,pr

_o

_Rbomboidejjm

area

Qequali.

dal#

cujus-dam

reåds

_quadrato

_j

at-(jue

/umma

bajeos

&

perpemliculi; b

a-fin

ac

perpendiculum

feorfim invenire,

&c.

_{Fig. I.}

SOL UT10.

Sit data area zz Q ΤΗzz aaj

Tum¬

ma bafeos &

perpendiculi

_AB

~/>;ba-iis AE ~ χ : erit

perpendiculum

CHtz

b χ_j &

_proinde,

per^r.

/.

£#z7.

jFQvatio. *a~ bx·— xx

Add..v.v

xx _fy aa zz bx

$ubtr»^4

bx—. SubtrJ.r ►V AT ►VA' "bx ZZ— ΛΛ bb\4 b b>4 A A-.V

ι

·4Φΐ> )

Ο

(

χχ bxfybbiq.— bb\4— Λα

Extr.rad,;

χ b\zzz Fbb\4 _"""Λα

Add.^

Λ·;Ξ /^£/4 —· <*^) >Ε

b

2.

Constructio.

_{Superi zzb\z}

_de*

fcribatur femicirculus AHG $ in eo ap·

plicetur

GH

—

FH

zz a,

eritque

AHzz

Vbb\4 — aa)} cui

ii

addatur

HE — b\z\ erit AE Z2, χ

quaefita bafis.

Intervallo

-dein AE fiat arcus £0, fecans re&am

AB — b in O:

_eritque

_BO

_zz

_b

_{~ xy}

perpendiculum

quamtum

5

&c.

Theorema etRegulaArithmetica.

A dlmidiee fumma

quadrato

fubtraftk

areaj refidui radiei Ipfum fumm<e

dimtdl-um addaturj & habetur bafis: qua (i

ab

ead m _{fumma dematur,}

_{reliqvum dabh}

perpendiculum

)

&C.

SCHOLION,

Eadem conditione _{parallelogrammum}

reftan-gulum, nec non _{triangulum, quadrato dato}

aequa-le, conftruipoteft. '

PROBLEMA II.

Data

_{portione minori redde media}

&

extrema

ratione

_{fed<e> ipfam}

_re¬ darn invenire , ex qua

tota

in

por

tio*

nem

_majorem

_{reäangulum fit}

_quadrato

•m)

o

(^·

}

SOLUT 10.

Sit

_{Qnadratum datum}

_□

_ABzzaa,

pars

qutciitae

minor AC

—

b;

qujefita_,

ipfa

BE

zz Ar;

adeoque

pars

major

rr .v' b: crit per

conditionem

Pro-blematis, ducendo *

in

*—· iEQyATio. χχ —

b

χ zz aa b_bl4

_bb

₁₄ xx—* bx *ϊ*bbl4~ bb/4*baa

Extr.R·

x~>blzzz Vbbl4>fcAA

Add%b\z

X7Z Vbb{4>fa aa))%4

b\Z,

Constructio. Ponantur ABzz a>

& AG zz b\z ad invicem

normaliter

;

dudta:que

BG

zz

Vbb\4

^

aa

)

addatur

EG ~ _bjZ : erit EBzzχ. Porro

ab

hac

EB fi dematur EM zz AC* habetur

fe-gmentum

majus MB;

&c.

Theorema et Reg. Arithm,

J>>ua-drato _portionis

_minoris

_dimidia

_addatur

quadratum

datum;

e

fumma

extrahatur

radix ; huic itidem ipfum

portionis

ejus-dem dimidium addatur : _fumma

hac

da-bit _qua

_β

_tum.

PROBLEMA III.

Data ,

pro

Triangulo

reäangulo

,1

4

)

o

(

§φ-praeter

b

a

fin,

portions

majori

perpen·

diculi media

&

extrema

rationefeäi;

invenire

_portion

_em

_minorem,

_&

_con·

fequenter

reliqua.

Fig. III.

S 0 L HT10,

Sit A ,

perpendieuli

AC

portio

major

AD zz a;

minore

λ·.·

erit

tota

ACzz χ_^a i Sc

quia

_{xfya : 4^ a : x,}

per /7.

VI.

yEQvATIO, xx^axzz aa

aa\4 aa\4

xx α χ4^aal4 Ξ3 jaafj

χ& alz z: Vyaa 4

Subtr.4*

xzzVsaa,4) —. a/z.

Ccnstructiö.

_Jungantur

_ad

angu-lum rectum ADzz a,Sc EAzz alzij

erit-que£/)~ V^aa\4:

unde

demta£G:z:

alz]

habetur DC_{(zz GD)zz} χnec non,huic

fiaddatur^/^totura

_{perpendiculum,&c}

Theorema etReg. Arithm. _J^ua'

drato data portionis aådatur ejtudem dl' midi£ quadratum,· & a radice _fumm<z

ipfa -porrio dimidia

fubtrahatuy

;

reßduum

dabtt

_{fortionem minorem quafitam,}

) ° C §d2&- 5

PROBLEMA

IV.

Datis-,

pro

Triangulo redangulo

,

ejus

area , atqtte

bafeos

&

perpendicu

-li

_chfferentia;

_{perpendiculum}

_bajin

tnvemre.

Fig.

_I

V.

S O LUT 10.

Sit _Δ EFB > area = Q AB zz _aa,

differentia bal. & _{perp. ACzz}

_b$

_per¬

pendiculum

BFzz χ: erit

baiis

EB zz

x^b·, Sc, vi 41. L

AQv^tio. xx\z^bxji zzaa Mult.t

xx>i*bx zz 2 aa b b14. bb\4

xx_»{< bx bb,4_{zz 2aa bb'14}

x»l· b/zzz Vzaa^

_bblq)

_Subtr.bjz

χzz Vzaa _{Hl·bb\4)— blz,}

Constructio»

_Jungatur

_re&ae

_AB

zz a ad

_angulum

re&um alia

AD,

_ipfi

ae-qualis

: Sc erit DB zz Vzaa. Tum in_»

D

_erigatur

_{ad BD normalis DGzz}

b\z,

ut fiat GB zz VLaa

_^bbij.

_{Inde vero}

demta GHzz _{GZ>,eritBF{zz HB)zz χ:}

nec non, eidem GB additå EG zz _[DG,

_(i.e.

perp.

additå

EH

femidiffer. duplå)

<5

·«)

o

Theorema Et Reg.ARITHM. Area < dupU addatur dimidia

diffcrentia

quadra-tum: ab _aggregati

radice quadrata

fubtrahendo

femidifferentiam,

habebis

per-pendiculum

,* nee non c

andern

eidem

) v

ei

•perpendtculo integram

differentiam^adden*

doj bafin.

PROBLEMA

V.

Datisy

pro

formando

Parallelogram-wo,

differentiis

inter

latus

_utrumque

&

diagonalem, invenire

latera

figillatim*

Fig.

V.

SOLUTIO. i

Sit

_diagonalis

_FN

_&

_lateris

_majoris

AF differentia AB zz a ;

_diagonalis

FN

_laterisque

_minoris

_AN

differentia

ACzz b;

latus majus

AF_{zz x.}

Erit

dia¬

gonalis

~

x^aj

ideoque latus minus

ANzz x^a— b. &, per _47. L ÅLQv> zxx zax·—■

zbx

aa ^

zab

bb

zz xxHbζ axHbaa Subtr. xx, 24x

&

λλ xx—1 zbx —· zab bbzz o Add. zab xx „

•Φ3 ) o

_{( *£».}

₇

xxι—ι 2bx\%ι bbzz _2ab

Extr.Rad»

x *"■* b zz Vzab _Add.£ λ·ζ5 V

_lab')

_{+· b}

Constructio. Inter ABzz _{a, 8iAC}

zz _b,

quarratur media

_{proportionale}

,/?£— _Vab

i

_fumtaque

_{AD zz AE, erit} ££>=: /^24^.

_{Quare,ii fiat}

_CFzz

£Z>;eique

addatur AC; erit = λ·. _Huic

ad-datur _BAj

_&

_{erit BF zz χ-i-a} 5 nec non, hinc demum fubdu&a _AC9 an ZZ χ Ή a—1

Theorema et Reg. _Arithm. Ε

Ju-plo-

faclo

_{differentiarum}

_{diagonalis αό u·}

troquelalere extrahatur radix _quaårata

,

& huic addatur

_dijfercntia

_major,

qu<eesb

diagonalis

a latere minori : ex

aggregato

provcniet latus_{majus, A lateris vero ma*}

joris &

_differenti£

_{minoris fumma β}

diffe-rentia _{major fubducacur}

j

reliqvum latus

erit minus.

PROBLEMA VI.

Datis,

_{proTriangulo reftangulo,}

_ejus

area,

_dupla

_{quadrati data}

_A

_{B atque}

differenti

a

bajeos

_&

_{perpendiculi;}

ba-fin

_{&perpendiculum}

_invemre.

_{Fig. VI.}

SOLU-g -m ) o

(

Η*·

SOLU TIO.

Sit data reda ABzz a

(ideoque

_area

dupla

zz

zO^B

zz

zaa)

i

difFerentia_,

CDzz _b\

baiis

_{AB zz χ:}

erit

perpen-diculum BCzz,χ b, &, per 4/./.

ÅLQv. xx\i

+-bx\z

zz ΖΑΛ+

Mult.

2,

Λ·,ν ■+— bx ZZ 4aa

bb,4 bb,4

xx4—bx 4— _{bb\4ZZqaä 4—}

_bblq.

Rad.

χ-l·- b/2zz Vj-aa+»

bb\4)

Subtr.

b\i

χ ZZ _{Vqaa -h- bblq)}

_bjz.

I Constructio. Inter FO ZZ a,

&

OP zz _4*,quaeratur

_media

_{proportiona¬} le OB zz ^4-aa $

cui dein

normaliter

jungatur OE

zz

CDjz

zz

b

2:

eritquo

BE zz Γ4αα ±-bb\4).

Huic

vero

Π

de·

rnatur vcl addatur EHzz b2 ·, erit

ibi

{/IBzz ) Blizz xi

heic

(BC

zz)RDzzx

+-b:

quianimirum

_{V^aa +-}

bbi4)+~b'lt

ZZ _V/j-ax -i—b b 4) —

bjz^b.

Theorema et Reg. Arithm.

druplo relia

data

quadr.no adda'ur quA'

dratum

_{femidifferenti<eaggrep}

_att

r.4-dix ryiinuattir _augeaturve

_{fcmidifferentix}

ipja: [te habebitur, ibi

baßs\ heic

4M ) o

(

§&·

9 PROBLEMA

VII.

Datis , pro

Triangulo

rettangulo,

media

_{proportionali}

_inter

_{cathetum mi¬}

norem _,

_atque

_hypotenufe

év catheti

majoris

_fummam;

_differentia

_utriusque

catheti; év

_perimetro:

_invenire

_cathe*

tos

_{figillatim, φ*}

_c.

_Fig,

_VII.

sol ut to.

Sit, _pro

_Δ

_DGH,

_{media illa}

_pro¬

portional^

ea ss a i

differentia utri¬

usque

catheti

HO zs

b}

perimeter

~

Ga s c} cathetus minor GH x:

eritque fumma hypotenufae &

catheti

majoris

DG * DHzz c ~ χ}

cathetus

autem

_major

_DGzz

_x^b.

_Quare

_per

17. vi. AlQVATIO* 44 S CX—< ΑΤΛΓ 1^4 .v*— 44 f* Λ"Λ"—■ CXZZ *— 44 ff/4 ff/4 xx τ* *_{^/4 =: fr/4ph 44}

_Extr.

_R.

Λ-— _{f/aZ5 /^ff/4} ρ-*44)

Add,f/a

Λ·_J5f/2Hh /^ff/4—H

44),

Constructio.

_Super

_(7^/*^

_ss

B Ä

ίο ) o (

c/z

defcribatur

femicirculus

GFB;

in

>

quo

applicerar

GF(pz

AE)~z

a \

& erit

Fß zz s/cci4 —· aa),

Hac

autem

fubdu-da å GB, remanet G Ηzz χ: cui

addi-tå HO zz b, habeturredae GO

aequalis

GD zz x>& b}

&c.

Theorema et Reg. Arithm. _aqua*

drato dimidia _perimetri

_fubtrahatur

_medk

proporttona

Iis

quadratum', radix

itidern

quA'

d-ata _refidui

ab

_ipfo

perimetri

dimidio:

& habetur cathetus minor; cui_β

addatur

amborum

_diger

_entia»

_fumma

_dabit

_ma>

jor em }

&

c.

PROBLEMA

VIII.

*

Datis ,

pro

Triangulo

obtusangulo\

area, &

differentia

_fegmentorum

ba·

Jeos,

perpendiculo

media

ér

extrema

ratione

_feiles;

_{ipfa mvenire}

_fegmentaι

nec non

_{perpendiculum.}

_Fig.

_VIII.

SOLU TI o.

^

Sit Δ:ί

4bc

area —

(□/)£=:)

aa\

differentia dida HCzz

_{b; &}

_portio

_ba·

feos minor AD zz _{χι erit}

_major

_DC

zz _{xJ&bj baACzz zx^bj}

_perpendi¬

culum£Z) zz aatx b\z)\ &,

_quiai*q^'

#_{»i< bzzx Λ-b; χ}

j per 16.

VL

•φι

)

°

(

«* jeqvat« 2χχ>$<bxZZ χχhp

zbxbb

·—. XX_{5 — bx} XX ZZ bx ►£bb Subtr* bx XX— bx ZZ />£/4: bb/4 xx~ _{bx^bb\4~ <;bb\4}

_Extr.R.

χ_ b\z zz V\bbl4 Add,bl2 xZZ Vjbb/j) _$/2.

constructio.

_Jungantur

_ad

angu-lumreftym £/>=: _(//C=:

)b,8c

_ILzzbi2;

eritque

IP zz

Pjbbfq.. Cui

dein fiadda¬

tur AIzz _b\z\

_erit

AD,

fafta

acqualis

reilas AP,zz x. Turn verofumatur DR zz AD,

_{eique addatur}

_CHzz

_b

_,·

_&

ha-bebis/)C;&c. Denique

properpendi-culoj fiat ad lubitum

_angulus

_CDF,

_&

aa/xyf*bjz) in hanc refolvatur analo¬

giam,

utDGzzx*blzad

DF(zz

DE)zzα,

ita DE zz a ad DOzz BD.

theorema et reg. arithm. £jua-drato _{differcnti# datde addatur}

_ejmdem

di-midi# _quadratum;

atque e

fumma

extra-ff<e radici _{ipfa itidem äimidia:}

aggrega-tum dabit _{portionem minorem 5 ut &,} addira _infuper

_{diferentia■,}

_majorem.

_Si

autem area _{per baftn}

_dimidiam

divida-tur _i

in quoto

prodit perpendiculum.

pro-12

_4fr*

_{) o}

(S&-PROBLEMA

IX.

Datk^

pro

_{Triangulo reåanguh)}

differentiis

mter

bypotenufam

ércathe·

ioj, uraλ

_{trianguli; bypotenu¬}

fam

invenire.

_Fig.

_IX.

so LUT10.

Sit,pro

Δ

Λβ£7,«rfC·— i?C22 (αζζϊ )λ\ Ac~AB22 _{(Λ/Cii )£,area22 (QDN 22)} <r;

_hypotenufa

Λ6' 22 * :

erit cathetus

minor BCzz χ —· a_;

_major

_{^2?22 χ—}

ex dimidio horum fafto _{area; &}

_fic

^BQy. xx/2—·bx/z—· ax/2►£·λ£/222 Mult._2.4 xx— bx~* axab7Z zcc Subtr.^i■ xx— bx—ax~ Zcc~1 λ£

0^/2ι^/ί/2)

Q^/2ψΛ/2) xx— bx— ax _{^Ob/2 λ/2)22} + ^

Extr.R·

x~b/2— al222 ^2ίί·_^

_Q£

_2ψΛ/2)—

_Η

.Add. bjzψ 4ji Χ22 _{b<2 4- λ/2}

_ψ

_{^2(fi ►{κ□}

_bl

_ζ ι^ιλ/2)—·

bj)

Constructio. In D

_erigatur

_nor¬

malisDG22 _(ZW22

_V;

_{critque duiU}

~Vzcc

_{DeinadyiiV22 Gi^ponatur}

nor¬ malis NF~

_{blz^alz (22}

_AZ/z

ψ

MC/z)\

eritqoe

-m) o

(.%&>

₁₅

eritque

AF zzVzcc +- Qb\z 4-alz).

Por-ro inter NH— AZ& HI ~ MC quae-ratur media

_{proportionale}

_HK~

_Fba5

&

_fuper

_AF

_{defcribatur femicirculus}

AOF, in _quo

_applicetur

_{JO s} e-ritque AO ~ Fzcc Obl* *-a/z) ·— cui fi addatur0C:=: ZZ/2 λ-MCtζ ~ biz

ή-alz5 erit(AO +-0C~ )ACzzχ. Abhac

denique,per

conditionem problematis,

dematur _ZZ;

_&

_{remanebit CZ JC;}

fimiliterque

fabtrafU

CM,

reftabitZA/

:= Z2?.

Theorema f.t Reg. Arithm.

drato ex _{fumm/i åimidiarum}

differentia-n/772 addatur are<t

_duplum

_{j ex}

aggregato

fattum

_{differentiarum}

_fubtrahatnr 5 λ

?·£-fiduo extrahaeur radix

_quadrata

_{j £5* hac}

dtfferentiarum

femi_geaugeatur:

aggrega-tum dabct

_hypotenufam

_{5 a qua}

_{β differen¬}

tia _{feorfimfubducantur,}

uterque

prodibit

cathetus4

PROBLEMA

X.

Datis,

//i

_Triangulo

_obtusangulo,

777^/0

_{proportionall}

_inter

_latus

maxi-w/77/7;

_medium,

differentia

lå¬

teris _{minimi & maxim}

_i,

una cum

_perl·

metro \

_{laterafingula}

_{invenire. Fig.X.}

tfojtf-14

)

°

(

SOLUTIO.

Sit i _pro Δ

media

illa

_propor*

tionalis BDzz _4;

_perimeter

_ss

_(OCzz

_)£j

differentia ΑΗτζ c_j

latus

_maximum^

AC — χ₇ & medium _{7. Eric per}

/7, F/. vel 20, VII.

.dSQv. *7:=: 44D iv·_y 2*4-7—<<

*ΖΞ _44/7 —7 +-f'

2*ΖΞ £+-f—₇

Div,*

*ir: _{b/ζ 4*}ciζ—« y!z AMiyz^ b/'z -5— f/2—« y/2

Mult.)*.*

1 _{zaa ZZ} by -i-·cy —1 yy 4—_{yy —1 2ΑΛ »} yyZS by+· cy—« 244 Μ by _cy yy~by-* cy zZ —<244 □£/2 +-_f/2)

_q^/j

_4-

_f/2)

77—· by^cy 4-C}£/2 4-f/2)s

_Q£/2+-

_i/0

— 24^ Extr.R· 7 2 — r/2^

Vjjb/z

c/z) —< 2aa) f 4- biz-f-ί/ί

I

7~ b;ζ 4- i/2 +7

VQJbjz

Λ-c₁₂₎

24/«)

CON'

) o ( *f * _Constructio.

jungantur

ad

angu-lum reftum BD s a Sc DE ~ a5

du-daque

BE ~

deicribacur

ferni-circulus

_fuper

_D/*=:

_£/2

_{ή f/2 5}

_in

_quo

applicetur

Λ(ϊΞ2 Z?E, ut

fiat

DG~ ^

Q£/2 4-_f/2) —. 2^4)*

Intervallo dein

DG dueaturarcus_GFy fecans DAln fjatquc

eritAL[zzAF)zzy> Porro

quia

λγ;τ aay, fiat ad Ibbitum

_angulus

_KAC,

&infe-ratur , ut AL~ y : AI(~ BD)~ a ,ita AK— 4.· yiC. Ab hoc

_denique

fide-matur _AHi

_refiduurn

_HC

_dabit

_Lc.

Theoremaet Reg. Arithm. _Qua* dra ttim meidia _{proportionale}

_dttpfum

_a

quadrato

dimidia

fummapertmetri & dif-ferentia fubtrabatur* atque

vefidui

radixr

β ab

ipfa

perimetri

&

diß'erentia fumma di~

midia_{[ubducatur flatus dabit medium.Per} hocautem_{fiquadratum media}

_{proportionale}

data_{dividatur^habcbis etjam maximum\nec}

non ab hoc demta_{differentia, minimum.}

PROBLEMA

XL

Datis , pro

Jriangulo

obtusangulo,

media

_{proportionali}

_inter

_{ejus altitu}

-dinem

&

_fegmentum

_{majus bafeos;}

differentia

jegmentorum

; atque

[e*

■

s

gmenti

major

is

&

perpendiculi

aggre-gat§

ι£

_4Φ4»

₎ o (

gato;

invemre fegmenta

_feorjtm,

nec

tion

perpendiculum.

_{Fig. XI.}

soLUTίο.

Sit media

_{proportionalis}

_AC

_{=: a\}

differentia GE 73 _{b; aggregatum}

_Λΰ

*·;

fegmentum

minus

_per¬

pendiculum

BK73 yi

erit majus

EK 72

x b\

&,

_{per /7.}

_VI.

■dBQV". xy+- by 73 aa Div. λ·-t-1 y73 aa\x +-b)

Proindeque

AA\x4- b) 4—χ4—b C Mult.Λ"

Ή

ta +- xx ibχ 4— bb 32 cx bc «—· AA bb~* cs xx 4—zbx <— cx 73 bc aa ~ b b Qb —· c/2) bb 1 if +■ XX 4-ibx—·fAΓ*- _f/2) _£3 ff/^—Al ar 4-b-« Ci2 73 _{Vcc/4 —« 44)} ·—· b _4-" Φ

I

χ 32 ~ b-t-f/2 4—_Fff/4 —i aa).

Constructio.

_Super

_AFI33

_(ADH

32) cjz defcribatur

femicirculus; &

'n

) O

(

17 jΓ/7ri _Vcciq.—« ^

Deinfidt/f/ζί

_.F/i/j

eritque Λ/ =3 r 2 Fcc/q -*

af).

Ab

hac vero fi dematur Kl _(zz

_GE)

_{zz bi} erit λK ~ χ_j

_ideoque

_χ

_b

_{zz KG}

(zz AK) HB GEzz KE. Porro, quia_y

zz_{aa/xtrfr ad lubitum}

_angulusCi£;

& _{inferatur,#/ AI(zzKE) zzχ b ad AC} zza·) itaAGzz aadAOzz {BKzz)y.

Theorema et Reg. Arithm. A qua-drato dimidia_{[umma data dematur} me-dia _{proportionalst}

_quadratum

_{j a refiduo} radix _quadrata

_extrabatur

_{} radici}

adda-tur idem _{fumma dimidium ,·}

_&

_{ab boc}

ag-gregato fuhtrahatur

dijferentia

:

quodre-flat, dabitfegmentum minus;

# buk

eadem

addita

_{dijferentia majus.}

_Quadrat

_umde«

nique

media

proportionalis dividatur

per

fegmentum majus :

&

quotus

dabit

per»

pendicutum.

PROBLEMA

_XII.

DatiSypro

Triangulo

reflangulo>medi&

proportionali

inter fummam hypotem·

(<e

&

_perpendiculi

ex

angulo refto

in

hypotenufam

demijfl,

atque

fiimmam

cathetorum;

_dijferentia

_perpendiculi

_<&*

hypQtenufe; dijferentia

etiarn

catheto*

18

4M

) o (

|φ·

rum; ma cum

_permetro:

_invemrt

.

hypotenufam

,

cathetos

Jeorim,

&c.

Fi g.

XII.

SOL UT10. Sit _{Δ AMC·, media illa}

proportio¬

nale EX=: difFerentia

perpendicu-Ii &

_hypotenufae

_{EUAzz bdifFerentia}

cathetorumETzz_c;

_perimeter

_{(= VX)} zz _d$

_hypotenufa

_{AC zz x\ erit}

per»

pendiculum

zz x~ bi fumma perpen-diculi &

_hypotenufae

_{AC f FMzz zx—ί>·,}

fumma cathetorum AM ψ MCzz_d~x; & quia zx~ b: α zz α: d~χ, ex

qua-drato medii &c« vi /7. Vh

JEQv. >aaZZ zdx~« _**.*·—> ^2ΛΓΛ: —^ zdx αα ζχχ—« zdx_ζζ~bdfybx —b_{χ — dA} Ζχχ zdx~ _bxzr,—αα— bd Div.i ~ */*—· bx/izz~αα ζ — Μ* Q///2

_£/-?)

_dd/4ψ ΑΤΑ— _{S bx/z *} o d/z

b/4)

ZZ ddiqψ

_{££//1aa/zExtr.}

_R« | Α<//2~

%

s-ψ ~ <M/2. ~

bd:4)

_% d\z*bli * _{Ä dl2 *Ji}

blf

Hh

_Vddl4

_^

_bblzff

_aa/2 ~

bdrf)

CON'

) ° C ip

i Constructio. InterAGzz _(vxuzz ₎

d\iy & GL zz _(btt?zzz ) b/<ζ quaeratur

media

_{proportionalis}

_GHzz

_Fbd(4

_i

fu-per GD

{zz

EX)zz λ

defcribatur

femi-circulus GID5 ecentroO

erigatur

nor¬

malis Ol:

_du&isque

_Gl

_&

_ID,

_erit

_GL

zzVaa\i. Dein huic fiat

_arqualis

_GN:

eritque

duéta

HNzzV-«

aaji — bdj4\

Tum ad AG zz _{(PX/zzz) dlz ponatur} normalis GP zz _{\EWi4zz )bj;}

eritque

APzz _{Vdd/4 ifr bb i6).}

_Ulterius

_fuper

AP, tanquam

diametro

,

defcribatur

femicirculusASP; in quo

_{applicetur a^}

ZZ HN; Ut fiat

_PQjZ

_Vdd\4

_4.

_bb\t6

—.

I

aa\i

bdl4).

Denique ab

AG

zz

d\i

4»

GR zz _{b\4 (zz AR)}

_dematur

_CRzz

PJgj

atque

erit AC

zz x.

Porro autem _pro

_Cathetis,

_fit

hypo-tenufa ACzz _f;

_differentia

_cathetorum»

ut

_fupra

_{9 c;}

_cathetus

_major

_zz\y*

erit minor s _y _ c.

Quare

_per47.1.

jEQvatio. lyy~ icy^cczz

f

Div.

z*

yy cy^cclzzz

f/z Subtr.rf/a

yy—« cyzz

_f/i

τ- cc!z cc\4 _cc\4

yy~, iy^ccl4ZZ

fiz<~ccl4

Extr.R.

20 ₎ o (

y— c!ζzz

_Vffz—

_Add.r/i

yzz

_Vff\z~

cc

_4)^4

d*.

&

y~czz _Fff/z — ccl4)~* c]i·

Constructio.

_Super

_^C:=

_/

defcri-barur iemicirculus ATC_{5 é centroΖ}

crigatur

_{perpendicularis}

TZ:

du&isque

TC & _{AT, erit} TC zz

_Ff/z.

_Super

_hac

itidem fiat iemicirculus

_TJgC;

_{in quo}

applicetur

TK zz (TlF/zzz )c,i: eritque

KC zz Vjf\} _{— cc\4 i nec non ab hac}

iubdufta KM zz c/z , MCzz >·> &C.

TheqremaetReg.ARITHM. <ffuå-Mrato dimidiθmedia_{proportionalesaddatur}

quarta pars

produtti

experimetro in

dif'

jerentiam

perpendicuii

&

bypothenufa;

dg-gregatum fu^trabatur a fumma

quadra-torutn exdimidia per_{imetro atc,ueejutdtt»}

differentia parte _{quartai refidui quoque}

radixa _{fumma dimidia perimetri 6f}

quAt¬

ta _partis

_ätfferentia

_:

qnod fupereß,

äA-bit

_Bypotenufam,

Adimtdio dtin

_{quadrato hypotenufa}

_ful··

ducatur

_quadratum

_{dimidia cathetorunu}

differential nee von ar■βetui radie

efemi'

diffrentia ipfa: fi. babebtscathetummino¬

rem_{j eadem v erofi addatur,}

majorem»

) ° ( Mß- **

PROBLEMA XIII

Dal_{is, pro}

_{Triangulo aquicruro}

_,

ßra,

d/fferentia bajeos

$r perpendi

_·

cuhex

angulo

alterutro

_ad

baftn

_in

_crtis

oppο

β

tum

demrjji,

una cum

perimetro;

invenire crus &

bajm.

Fig.XIII.

SOL UT 10.

Sit AABC\ area zz (□ AIzz)aaj

dif-ferentia Hata AHzz b-,

_perimeter

AB

crus utrumlibet r= >>5

bafis ACzz χ \ erit

perpendiculum

CG

zz χ— b_j &, _{per 41.}

L

ΛΕΟν. xy!i~

by!2

ZZ aa

Mult.i

xy■—·

by

ZZ 2ΛΛ

Div.

y

Λ*— bzz zaaty Add.b

χ zz 2aa/yHb b.

_Quare

2y 4— 2AA!y 4—b zz e Muit.y 2yy 4-zaa 4-byzz cy Div. I

yy +- aa +- by12 zz cy/2

Subtr.

aa yy 4-by/2 zz cy 2~ aa _Subtr.rjV* yy 4— by/2—- cy/2zz·—aa Qb\4 —« c\4) CJb.f— c4) yy 4- by!2 —· cy/2 4- □

%

—· c14) zz

}

Q «7^ ~ -λλ

Extr. R.

C 5 y

4-ϋ _{-Φ§ ) o (}

_l·®*

y +- bj4 — _{c/4 ZZ} V^b/4 ~ _c/4) —« aa) >

-η b

4+-c14

y ZZ— £/4*-cl4 _+-VOb/4—> ₄₄₎

Constructio.

_{Super aflumta}

_CE

zz _{b/4— d4 ,}

tanquam

diametro,

de' fcribatur femicirculus_CDE;

_in

_quo

ap-piicetur

CD{zz dl)zz a:

eritqueduåa

EDzz VC2bl4—1 _{ci4)~ aa).} Huicdein

aequali

BE

eadem

CE

_{addatur: eritquc}

BCzz _γ.

_Cujus

_{duplum fi}

_a

_perimetro

fubducatur;

ipfa

remanet

bafis,

Theorema etReg. Arithm. _Jquar·

ta _{parte perimetri futtrabatur}

quart tu

differenti* parsj ut & a refidui quadrato \

area _{; radici}

_åenlque

_pofterioris

_refidui

quarta pars perimetri quarta

dfierentie

parte multtata addatur. _/lggregatum

_dat

crus unum : nee _{dijfieulterinde}

reliquapel

cemditionem _{problematik babentur.}

PROBLEMA

XIV.

Dat

_is,

_pro

_formando

_{Triangnlove·}

åangulo,

media

proportionalt

inter

caihetos, perimetro,

_atqne

_differentk

eatheti

_major

_is

_&

_hypotenufa:

_invenl·

re cathetos

_&

_hybotenufam

_jeorfiffl.

Fig. XIV.

'

SOLU-) ° ( 2?

ΰΠΟ,

Sit Δ ABC, media

_{proportionaüs}

data _{BDzz a;}

_perimeter

_{^ f- BC}

+-CAzz _{difFerentia didla^A/'zr}

r;ca-thetus

_major

_{*j minor BC~ y.}

Ergo

vi

/7. /Ύ.

iEQvATro. xyzzaa _Divy

xzz aa/y Sed novå denom.

_perim.

2x +- _{y -f- czz b Subtr. y, c}

2XZZ b~ _{y~-< c Div.2}

χZZ biz~myiz— c/z

Quare

vi Ax.

_i,/.

aalyzzb/2~y!i~c!z

_Mult.y,

ζ 2/ιαζΖ _by —yy cy Add yy,cy yy cy 4- zaazz by Subtr. zaa,by

yy +- cy—- byzz —· ιαα Qf/2 —

^/2)

_{Od2 ~}

_£■'*)

>y4-ry— by+-Qc/2—

b/2)

-Qc/z- £/2 — 2aa _{Extr. R.} 7 +-f/2_ b/2 ZZ VQC/2 —,

£/2)

_{ρ-1 2,*4}) — _{C/2 -3—}£/2

yZZ ejt

+-b/$

+7 ^Qf/2—« b/i)~* 2,1a)

Constructio.

_Adangulum

_rectum

ponantur C77 & C7, ambae

_aequales

re-dacSZ) = ₄₅

_eritque

_dudta

_IHzz

fzaa·

Deinde

_ioper

_{BS zz c/2 -» £/2} defcri-batur

24

4*5

) O

(

batur femicirculus BLFE ; in quo ap-

_j

plicetur

BF

zz

IH;

eritque

FE

zz FQ

'

cn — ^/2)— -2^). Radio

demque

_££

(i fiac arcus redarn BE fecans in C; ha¬

betur £C = 7. Porro , quia χzz a,ny) fiat ad lubitum

_angulus

_KBD$

_&

infe-feratur, ut BC—γ

ad

££( s ££>)z= ita BD s λ ad BK. Hoc autem radio BK fiat arcus KA

_perpendiculo

_BA

oc-currens in A:

_eritque

_AB

_cathetus

major;

&

c,

TheoremaetReg.Arithm. _Duplum medi£ _{proportionale quadratum a qua·} drato dim i di£_{perimetri jemidifferentia}

_di'

,

minut£ fubtrabatur_{i ut & reftdut}

_radix

ab _{ipfo perimetri dimidio eadem femid/ffe·} rentia mulciato : _{fic catbetum b abcbisml·}

norem, Detnde 9 _per aquationem primarft)

tertia _{proportionale, ad catbetum mino·}

rem , & mediam proportionalem datam,

dabit catbetum _{majorem; nec non huit} addita _{differentia bypotenufam.}

PROBLEMA XV.

Datls,

in

_Triangulo

_{reftangulojif*}

ferentia quadratorum

laterum, &

a*

rea;

_{ipfa lat}

_era

_invenire.

_Fig.

_XV·

SOLU'

·&■§ ) β ( ij

SOL UT I Ο.

Sit/\ABCy differentia

quadratorum

» area =2 _{(Q52) =: )rr$}

latus

_majus

_{AB zz χ: erit minus5C=3}

ut & — zccix.· Unde jEQV. XXS 4CCCC/XX Mlllt.XX

XXXX—<24Λ·ΑΓ _4CCCC aaaa _{4 aaaalj.}

xxxx~-■_{ααχχ^ααααίψ^ζ aaaa}

4%*^tccc

R.

x*·—« _{aatzzz Vaaaa\4}

^4cccc~)

Add.aal

_t xxzzaa/z Vaaaa 4►£4cccc) Extr.R»

x ~ _{Vaa\i\$\ Vaaaa!4 dH4cccc~))}

Constructio. Adumta redta BGzza pro unitate, ut hat quoque

eadem

zz

aa; & faciendo 9 utBGzz aa ad BFz; zc, ita BD zz c ad _BH$

habebitur

_zcc.

Dein

_jnngantur

_{ad angulos re&os}

_BL

(=5 BG/z) ~ aa/z,

&

BR

(0

BH)zzzal

eritque

du&a

LRzz Vaaaa/4 q* 4cccc):

cui fafta

_aequali

_LK,

_{eademque reftae}

BL addita,

prodit5A'~

aa\z^Vaaaa ₄

*4cccc)j nec non, inter hanc &

uni-tatem IB—BG fi _quaeratur

_media

_pro*

portionabs

, ABζ: χ. Porro , quia_*

iiiinui s zsc/x, _pro

_hoc

26

4**!)

Ο

(

m-tur, Ut BÖ

(dB)

zz χ

ad

BP

(ζζ &F)

S2f, ita

BJgzz

c

ad

BC.

Theorema et Reg, Arithm.

ddda-tur

_quadruplum

_{bi quadratum}

_{area qua«} dratoexfemiffe data

quadratorum aifteren·

tiafé

e fumma

extraüa

rdit c

i

i

t

ide

mipfum dtjferetttia.

ej

't*

de

m

dimidium

· agg regatt

hujus radix dan»t perp

näiculum

?

five

lams _majusi per quod

ft duplum

art a

diviadtur, halebis etjtm minus,

PROBLEMA XVT.

Dath,

pro

Triangulo

reåanguloy

media

_proportvmah

_in:er

_bypotenufam

φ1

are(g

triangulär

i*

radie

em;

chjfe-rentia

_quadratorum

_areg

_ipfius

_&

by-fotenu/g

;

atque

differentia bafeos

&

perpendiculi

: invenire

bypotenufam>

perpencticulum &c.

Fig.

XVI.

SOLUTIO.

Sit ΔdBC , media

proportional/s

diDzz a₅

differentia

quadratorum

₍₌₃

D dR) zzbb; differentia bafeos &

per¬

pendiculi

HKzz c5

hypotenufa^Ä^:^;

perpendiculum

BCzz y:

erit

areac ra¬

dix s _dAix\

_&

_confequenter

_ipfa

_area

s=iaaaaixx)

bafi$

4Czz~ y<~* c:

&

·φ£ ) O

(

ty ΛΐΟν. χχ ~ aaaalxxzz

bb

Mult-

xx xxxx—» aaaazz bbxx Add.saax xxxx ZZbbxxfyaaaa Stibtr<££x¥ xxxx bbxx zz aaaa blib14 bbbb\4 xxxx <-« bbxx >E bbbb/4ZZ aaaa^

bbbbl4

Extr. Rad,

xx —« bb/zzz Vaaaafy

hbbbi4)

hdd-bb/i

xxzZ _bb\z »Jt Vaaaa >{< bbbbi'4) -

Extr.R·

XZZ _VbbjzfyVaaaa

fybbbb\4))

Constructio. _{Si b pro}

unitateaf-fumatur, ut fit ARztxzmzz

bb}

fiatque>

Ut AR zz b adANzz ay ita AD zz a

ad

AOzz aa\ erit,

_fuper

_AE

zz

bb

zy

erc-dta

_{perpendiculari}

_AF

_zz

_AOy

_{clu&a_j}

FE zz: _{Vbhbb\4 fy aaaa);}

cique

aequali

fa&a & redac AE addita tota

A^

zz _xx\\mmo

imtvAgJzAZzz

(ARzztb

quaefita

media

proportionalis AV

valör

ipfius

χ,

Deinde vero haud

difficulter

areas

radicem habebis

_(inferendo,

_ut

_AF

zz *ad AM zz _{a·) ttzASzz a}

ad)

AT\

&

confequenter

pro

Perpendiculo

>

fi

fa

ATzzf:

erit

vi

41,1.

2S _{4M )} o

(

JEQvat. yy af— cyjz = f JMuIt.ι

-yy ~ cy == zff

c_{c',4 cc<4}

φγ~ ry CCI4 —

_zff^cc

₄₎

_Extr-R.

^— r/ass

_Vzff

_{ff -f) Add.f/i} > ~

Vlff

cci4)^ _cjz

ConstructiO.

Ad

_{angulum re&um}

ponantur AT z: fik

_{ipii sequalis}

eritque

dudta

TG=z

_VzffTum

_vero

fi fiat GLzz TGj & IL z. ciz ad hane

normalis: _habetur IG m

_{Viff-J^eci4)\}

cui addita, vel

_demca,

_{Klzz IL dabit,} ibi KG

pro BC zz y , hic GH pro

AC

ZZ, y —> c.

ThEOREMAEtReG.

aru·™. Bi CjUtl-dratitm data media _{proportionalia}

_addi·

tu*

_{quadratorum femidiff'erefiti<e}

quddrA'

to} & e _{(umma extraffa radici ttidem.»}

quadratorum femidifferentia

? dgg rega

ti

bujus radix dabit _{hypotenufam, Porto,} β duoio area addatur

_{femidifferentia}

_b

_A'

feoi & peroendiculi

_quadratum

_{: funtma}

radixt

ipfd

femidijferentia

auftd,

dabit

perpendieulunt}

nec non eddem

dminu-tA_,

_bafin,

PRO-) Ο _z9

J

PROBLEMA XVIL

Όät

_is,

_pro

_{Tfiangulo retlangulo}

,

quadrato hy

potenta

, #ra?; /»·

venire htera, _{Fi g}

_{Χ V"}

_11.

SOLU TI O.

Sit A ;

quadratum hypotenufae

=: _{(Q ) 44 j area— (Q}

Jo

_=)bb;

latus majus /?c=s *·. eric minusAB~

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urjitate _{pojnatur; erit eadem =:}

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applicetur

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Pereximie atque

Eruditisfime

DOMiNE,

JOHANNES

RUDMAN,

Amice

_integerrime.

Μ

Ens

fiupet ipfa fuas

ignea vires,

contemplans

Miraturque

novum,

jam

fibi

nota,

decus;

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excelj<e

dum

plurima

munera

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_facit

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_{aliis innotuifje}

_[imuL

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precor,

fauflo,

quo

niteris,

Opti-me, gr

efin,

Profiterafintque

Tibi pr^mia,

lau-J-S.

Pereximio Domino Auctori,

QUemadmodum

grcsius atquc

omnia,

perfeftio iägacitati

quorum

ortus

dcbencur

,

pro-humani _{ingenii, ab cxiguis initiis, ad}

_illud

quo nune vigent, faftigiutn pervenifte, neminijCui

quid iapicntiae, non patet: fic eriam Mathemata, licet ab _{ipfis incunabulis, ob evidentiam &}

certi-tudinem, fummaui femper habucrint dignitatem;

non tarnen nifi _{poil tot fecula ad prxientem emi·} nentias _{gradumatiicendere} potuaunt. Antiquiflimis

fcilicet, &c _{pauciilimis illis inventis, feniun tam}

multa_{iuperflruxerunt iequentiumtemporum eruditi,}

ut jam veluti ad apicem pejventum effe videatur. Praecipua vero incrementa hx ceperunt diicipliri®, poilinventam Anah finSpeciqiam,Methodum i!lam

Froblemäta iolvendi, panter atque Tfieoremata

in-veniendi,admirandättt. Hxc cum', ioiis iniiar, tan-,

tam adfiidit orbi Mathematico lucem, urjam ilmv

mo _{compendio, & qnafi iponte, icié oileraritill.v}

qux veteres per _{longas ambages quaefitum} iverum; eaque ex occuito quotidie prodeant cie quious

oltm nemo _{cogitavit quidem, hrto, per liauc} tam facilis _{patet ad abitriiläs veritatcs aditus, ut} hodie _juvenes ea in medium proferre poilint, qux

olim, _praeter acatiilimum ingenium, vjnie_quoque

ikexercitatillimurn _{requirebant judiuum.} Tua

dif-fertatio _{praeiens egregium hujus rei iuppeditat}

ex-emplum; in qua nova, vel prius non äudita,

ριο-bkmata Mathématica 6c iolvis, & _{proprio Marie}

quemvis iolverc doces. Hoc, cum _ipccimen fit indolis _{induitrixque Tue. prorius eximix , ac}

proinde opus laude eo _dignius, quofpesomnibus ccrtior eil, Te, qtii juvc-nis hxcinvenilti, olim

ad-huc majora, ik neicio quid_non prxiliturum _:ore,

dum matuiior acceiTeiit xtas; nollra vetuit

ami-eitia, utinter alias _{adplaudentium voces mea}

deil-deraretur. _{Qu ia tarnen prxeonem laudurn Tuarum} agere mihi animus non eil; quippe quas a iummis Viris celebravi convenit, & celebratum irr _lcio:

pio-fe&ustantum iilos a-ieo _{iingulares,&quales in hoc} iludiorum genere prxltatniilimo vix prius noitraz

viderunt Camoenx, Tibi exanimo iincero_gratulor,

ifinnilquefaufta,fdicia Scprolpera quxvis adprecor. feßinunte calamot