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dissertatio mathematica;

de

TRIGONOMETRIA

ε

SECTIONIBUS CONICIS.

<vüam

VENIA AMPL. FACULT. PHIL. UPS. .

publice ventiland am sistunt

SIMONANDR. CRONSTRAND

rmi. mao.

ET

SFENO FREDRICUS LIDMAN,

ostrogothr,

hf åudit» öustv maj d. xxi apr. mdccciv*

ab m. s.

P. I

upsa liMt typis εdμ α ν ν i α ν i S,

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(3)

DF.

TRIGONOMETRIA Ε SECTIONIBUS

CONICIS.

■. «a&v

5. x*

StatlausneminiSc ufura.non perfpedla CirculiCanceliorum veroin&Gonometricaanguilias impadens,analyfi Sc

unam inilidsfe viam non fatis habet eaccultior Geometria. Fa- dbs improbi iaboris <5c multimodi periculis, liberius evagari Sc in majus excrefeere in dies Sc debet Sc araat. Nos vero, ίι ηοα omniao nullius cenfebitur juvenilis nifus, operam non plane per-

didisfe pufabimur, ituri, quantum penes nos, oftenfum, quaami-

citia in eandem anaiyim Sedlionutn unaquseque Coaicarum mire

conipiret, ^

Si f-Fig. I.J denotet a majosem EÜipfeos AG Sc trans-

verfum Hvperboiae , b minorem ejusdcm Ellipfeos Sc

fecundum ejusdem Hyperbclae femiaxem, y Sc χ orthogona¬

les coordinatas Pp, Qg, CP, CQ; ex elementis liquet,

fumds fuperioribus pro EMipii Sc inferioribus pro Hyper-

b* -

_

boia ilgnis, aquationem yz ==—o—-.2 Q i«* 4- ) utriiis-

que curva exhibere proprietates. Poiito a b , allata afquatio,

in y2 ijssHh a2 Up x2 transmutata, Circulum Sc Hypcrboiam se-

quilateram erit defignatura. Si vero y Sc χ eodcm , ac in Cir-

eulo, modo anguli refpondentis ACq Sinus Sc Coilnus Elliptici

Sc Hypetbolici nuncupentur, erit, fubüituds his denominationi-

b2

h«s, Sin2 = —. (-ha2 Zp Co/.1*)* ubi y Sc x, prout Circu¬

lum, Ellipfin vel HyperBoIam feparadm rcfpexerisj Sinus Sc

Α Co-

(4)

Cofinus Circ-ulares, Ellipticos & Hyperbolicos fignificabuat Quapropter, ii reda AD in A tangat, Sinus <5c Cofinus ex data

tangente cujusvis anguli vfaciii negotio eruentur, Frodit vidcii°

+ a* b2 ΊΊΖ b2 Cofv2\

tet Sm u* ( ——α2 /): Co/υ2 : ; Tango2 : a2 t

ab b Tangv

Cofv ξξ ~ " " & Sinυ zza . Pariii Vb7 ± Tango2 Vb2"4- Tangυ2

* modo, fi Jungentes vocentur redae^qu# centrum C & termina*»

toria arcuumAq, Ap punda q, ρ conneduntj habebitur Jung υ ΞΞ b,\/a2 -f- Tangua

-J/Sin u2 ·+* Cof v% =

S/b* ± TWgir

allat# funt, formulse iolumniodo obtinent, fi

ponatur α ζξζ radio & anguli υ e vertice Α fumantur^

il vero b ξξ: radio fingatur, anguli w de G numerentur,

& ad CG5 ut iemiaxem tränsverfutn, Hyperbola GV de-

a Tang w

defcribaturj erit Sin w ^ ' ΓΖΤΙ7ΙΓΖ ' r Co/" w

|/r.2 + Tang w2·

ab a. Y b2 -j- Tang wz

£ jfowg to = . Ut ex

j/aa Hh Ta»g· ffl2 γa2 + Tangio2

bis mnotefcat ratio inter angulosu & ?o, quanao o CofWy>

b. υ ab

jonatur " ~ """o ande evadet

V^2 i Tango2, Ϋ'α* Hb Tang W9

ab

Tangυ rr '' · 1 · Sic etiam, fiCo/"u ί^: w*

Yaz n* H~Tang

€ef

(5)

3

S/Tangw2 .(bz b2) +a7 b2

fiet Tang υ Ξη , «Sc utroque ca-

Tang w

r-, °2

fu, in Circulo Tang υ = atque in Ellipfi Tang υ m Tangw

ab

**ve unius anguli Tangens alterius Cotangens, Pofitfr

Sinν Sinw

y Cofυ m Cofw, Jungυ m Jungwt prodibit

ab Tangw

Tang υ zzr ' Tang υ n=

\fα2 b2yjjf Tangw7. a7 -b2

ab Tång w

\fTangw7J-a2"jf.b2, Tang t/=n _

\Λϊ2b2 ip Tangw2.a7 -b2

Primo itaque «5c ultimo cafu una eademque manet angulorum ratio, in Circulo vero & Hyperboia »quilatera per iingulos cafus

habebitur υ ns w. Porro fi Sin υ mCofu; reparietur Tang ν

ma &, quanao Sinw m Cofw ; erit Tang w b.

Ex prascedentibus porro patet, Π fuperiora adhibeantui figna, nec Sinus nec Coiinus nec Jungentes, ilve a five b radio

aequalis ponatur, fore imaginarios; hoc vero in Hyperbola eveni-

re, &, quando \a m radio atque Tang > b, &,quando b m radio atque Tang > a. Semiaxes itaque Hyperbola? limitem conflituuut, quem tångens anguli excedere nequit. Pofito radio

m:a & Tang = o ; fiet Sinm o, Cofst Jung a; pofito vero b radio <3c Tang rrr o, proveniet Sin m o, CofmJungm b.

Sic, ίι « =3 radio, Tang m b vel b nr radio, Tang m α;

crit Sin Hyp = Cof Hyp = Jung Hyp n= oo :fed, ii azzzb

<5c Tang m o; habebitur SinCircnr SinHyp. ceq.ηζ ο,CofGirc,

nr Co/ Hyp. o??, nr α, //. aa, n= Or m c, Λ 3

quem

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quem Jungentij circularis valorem pro quovi* anguio & arcu conftaxitem deprehendimus.

Hjec, quae attulimus, coroliaria ipfam folummodo figuraro

intuenti iunt aperta. Tangens enim femfanguli, qui ex Hyper-

'bolae afyrnpfotis conficitur, uno axe pro radio furnto, alteri »<- quatur.

§. II,

Füsxunt praecedenti §:o ad angulos relati Sinus, Coiinus

& Jilngentes. In Circulo aufem, ab conftantem inter hos reia-

tionem, utrum angul©, arcüi, an fedori adfcribantur, perinde

«Λ; pro Ellipii vero & Hyperboia haud eadem regula obtinet;

fed, ut trigonometricse harum fundiones, circularibuj, quantuin licet, analogas cvadant, ad refpondentes illas referre fedorespras-

ndCof

Äat. Sit ideo η : b :: ι :η; eritque ip fluxio

2 ι ψ Cof*

Eliiptici vel Hyperbolici fedloris, qui axem inter ipfumque ar-

cum eomprehenditur. Duclis perCof-p \/ipi .y-hιT-Cof*

Ρ ndCof

numeratore & denominatore, provenict /Zjp ss

2Vdz1Hh^0/2

η|/ZJZ ι.Cofd Cof

~ nd Cofip ""

Ρ 2 ztz 1 F- Cof2 %

/ —- — - _ " Log. Nep. (Cof,

"

>(Cof-p-i/ipi.i/-pi>+-Cof2) 2\/Z|Z ι

^V^P1· Vzh ι + C°f2)? ^nde? ß Neperianus mimeri Κ

to-

(7)

;arithmu* unitati· »qualis ponatur; deprehendetur,

fiuiito fecftoris inlegrali zs φ& loco Cofψ ^1·VzL·1 -F

ipfius valore fubflituto, ISf ' r^°J ^ "^F 1 · Φ&

-φ]/ + 1_ηαο/φ-γΖΕ1 I Sinφ ^ τ .

^ . Ex his ltaque datis

η

ajquationibus pro Eliipfi formule Sinφ~η'(^ ^ )

2γ- Σ

KT^V""1 f λΤ"Φ]/"I

(i),Cofφζξζ _____ (2) &pro Hyp-erboiaSinφ™

ζ

*·{Ν(Ρ~ N

(i), Cofφ=iN® i (4) ckducun-

2

f I

*ur. Quoniam ed Ν—,ηΦΥ-j-1 ^nm ^ Cofφ + yCp ι.

Sin φ*)™j erit Sin tn φzs

(nCofφ-4- 1/+ iSinφ)"- (η Cofφ-yqz13ΐηφ)" & _

«-1 . 2W V+1

(ηΟο/φ+Ϋψ ι&ηφΐ+^αο/φ-Ϋψι&ηφ)7' ^

m

di,

(8)

ftt— ι in-i,m-2

dibit, Γι pro m9 m. , tn. —— &c. ponantur A, B,C

t 2 2 · 3

C

&c., &7»mφ A Cof(pm~1 Sinφ-q- a . Cof(pm-l Sin φ3 -f-

jg >

Cofq>™-5 Sin Φ* Zf. &c., .(5) & Co/m φ = Cofφ™Zf.

B D

~ Co/φ*-* Sm φ* -f- ~ Cofφ*1- 4 Sm (p4· -f-öcc..„ . (6). Por-

ro, quoniam 2™ Co/φ·«1 -t- W"^ ; erit,

fumto μ = φ v"-j- 1» fusama iérierum

JV » Ρ +.ΑΝη'~τ'μΝ~μ·+.ΒΝ'ηι~3·μΝ~'ικ·+·&.:c.( __

Ν~ΜμΑ- ΑΝ~'η~1·μΝμ ■+■ΒΝ~™~2μΝ 2μ -+- &c.)

2m-\~lCofφη9 vcl imCofφηςζ: Cofmφ ·+■ A Cof^m-2) φ -f·

i? Co/(»»-4) φ -f- &c.

Idcirco, ίϊ duplicentur termini, qui fedoris pofitivi cofi«

num continent, ut evitentur, qui fedoris uegativi coflnum in- cludunt; habebitur, prout m pari vel impari numero aequetur,

a"1*1 Cofφ™ zzz Cofmφ ·+. A Cof (m 2) φ Η- &c... .

m- (jjW+ 1)

#».—; : (7); & 2m-1 Cofφ"1 z= Cof m φ'~\τ

2 . a .3....£#j

Cof(m-a)φ ·+· &C. (8). Sic etianv priori cafu proreuict (ob V'-F* = i ilftt zzz4g·, fcd V-)f 1 ssZp 1 fi m zz. 2g

(t]

(9)

7

aaåm». ^

(31) a Sm

4~ (Cfo/**«<£) -A Cofm-ζφΆ°Β Cofm-4. φ

&c...

»Z - I.fR-2r.". (|W-p Γ)

) (9)

r ,

<& pofleriori (*r~)

2.2.3 ·· · · -yψ τ·

τη- l m m-ι Κ τ»

2 .Sm(p Ξ=» Sinin.φ-Α Sin{m-2) φ-jrΒ Sin(m-4)Φ-&c.r

(10), quoniam ySf 1*=yqp ^fi ~= 4gJf.1 &γ:fi"*—Zf yzpι fi m= tg. -f· i.

§. III.

Ex formulis fupra inventis parallelifmue inter Trigoneme*

friam EUipticam & Hyperbolieam, vel, il »= 1 ponatur, inter Cjrcularem & Hyperbolieam} utroque Hyperbolie axe scquali, fa«

ciie ci'uitur. Reperitur fcilicet pro fectoribus quibusvis

φ & ψ

Sin ((β-

Sin—;ψ)=

Cof(<p+^) = Οο/(φ-·φ) =

Άηφ.αοβφ ~

Οο/φ.Ξΐηφ ϊϋ

8ΐηφ.Sin\p ~

Cofφ.Cofφ ~

StfiφΆ~Cofψ

&'»φ-Smφrs

in Eilrpii in Hyperbola Ξΐηφ.€ο/\ls>-t~Sm\p.Cofφ

Sinφ. Cofφ—Sinxp.Cofφ

α.φΌ,φ-^.φ

$ΐηφ.Cof\p~i~Sin\p. Co/Φ

Sinφ.Cofφ- Sm-ψ.Co/*φ

βφύψ+£$φ.&φ>

c.φ.αφ+·— s.φ. s.ψ α,φ.σφ-^s. ψ. s.φ

Ι£(φ+φ)+ί$.(φ-φ)'^.(φ+φ)+^.(φ-φ)

±ξ.(φ+φ)- iS.Qp-ψ).££ί^+ψ}-|£(0-ψ)

—C(φ-φ)-.—α(φ-$-φ)—ο.(φ-$>-φ)-—^(φ-·ψ)

ια^·+ψ)4-|αί^ψ}

£-C.(ip~\^)-i-|-Cφ-φ)

is.ji φ+φ,αΐφ-Φ

2Ο.^φ-ήτφ, S.iφ—φ

2S.£φΑ-φ.C. ψφ-φ

2.€,£φ+>φ.£ i φ-Φ

(10)

2θ4Φ+·Ψ·0·τ<Ρ—<l· 2 C· 5φ+ψ.Ο. §<?—ψ -~·&ί<ρ+ψ>. i<p-4>&c.|φ+ψ.5.|φ ψ&c

Ce^+'Co/ψ—

α<?/φ—θο/Φ=

Sin ζφ ss:

j3iS".φ.C.φ.

&».3 φ = iS,<p.C.(p*-~S. φ1

Siu 4 Φ = 4&φ.£φ3-~7&φ5.£φ

T &c.

ϋο/2 φ = C4p*--—&φ*

Co/3 φ =

^>-~αφ.&φ*

Co/4φ = c.φ*-~^α.ψ* s.ψ*-h φ4·^

2 8ΐηφ* == »*.(i—C.e(p)

4 Sm φ3 == n*.{iS^-~S 3φ)

5 Smφ4 = w4-.(C.4^-4C.2(p+3)&c

3 Co/φ* ΖΞ C.3φ ·+" I 4 Cofφ}Λ z=z Ο.ιφ-Ι-ιΟ.φ

s Co/φ4 = ^4φ-4-4^.2φ-+-3 &C.

2Sinφ. Cofφ

3£φΧφ?+Π£φ*

4§.φ0.φ**4—,2&φ''Όφ

Ι &c.

ο.φ*+·^.φ*

c.f>*+$S4*a<p

Ο.φϊ+^-Φ^.φ*-η4ϊ

S φ4.&ο.

«a(C.2φι") η2.(Ξ.$φ35.φ)

»4'.(C.4^-4C.Sf^-Hs)^0

C.2φ-4~ ι

0.3φ-ί- 3C. φ

α4φ+4^2Φ*+·3 *c.

Si formula (6) in Co/m φ = (ι + δ -4-D-J- F

m-i

Äc.) Cofφ —-f 3 2 D-f- 3 F 4^"+" &c·) Φ "4"

m~ 4 si r

{D-hiF-\r 6H+ &c) Co/φ _(3-4-4#-+. <3cc.).€ο/φ

. (ίο) permutetur; in Cirevalo^ Ellipii <5c Hyperbola cvadct

Co/3φ ZZ 2Co/Σ)2 I

Co/ = 4Co/Φ* 3 Co/Φ .

CV40 zr gCo/φ4 8Co/φ3 4- ι <&c. h. e. fi m fo*-

miila retro fubfUtuatur β,

Cof zz 2θο/φ.0ο/{ν- ι)φ-α*Cof (r- 2)φ.

Sit

(11)

9

Sit jam ζί+ί;V+ 1 radix pöteflatii re bin^mio* qiiodam

. ' ~r

Z7+ /^1/ip ij len U-4- VS/ip i == w-j-vt· ip ι & U

1r z*~7~~ """

|/zpi znu—v|/ip i , unde Uz ( V V-F 1 )*

(^yj/ipi)1, U :rz fuminas impårium & Vyif.i zz fura-

r

parium in zi +i; |/ ZC ι terminoruin habebitur. Si vero 112 f/|/~f—1 a* ponatur & circulus, cujus radius zr a„' Eilipiis, cujus major axis zz: ia Sc minor zz 26, flve Hyperbo- la, cujus axis transverfus z: & fecundus zz 2& defcdbaturj erit, fomto fuperiori in j/Zp1 iigno, u sequalis Circulari vel

'

a

Eiliptico Coilnui, cujus reipondens Sinus, in - du&us,per t/ ex*

t - -- ' #„,% SS --

^ 1 —a

ponitur, quoniam in hoc cafu fit Cof* Z_J/■—1 zz: α- J;

b2

& eodem modo, ii inferius in j/Zp 1 adhibetur ilgnum, « Hy-

a

perbolicus evadet Cofinus & v == - Sm. ob Co/** —5 ßÄ

Sin9 zz ea. Quoniam ifaque CofrCp omnes ordine impares,

»- ~r

λ a

& - SmΓ*φ pares omnes in Cofφ -4- ~ V^+i Sinφ (5, β)

a

termijaos centinent, erit U zz CoJrφ & V= - Sin r φϊ

* ,a

B

(12)

quapropter* Γι binomium, cajus radix ^useritur, fub eadem vene-

rit forma ac U ± V V"1'■> cum Cof Ε1νφ±-V-ιSin.ΕΙ.τφ eomparari poterit, fi vero binomium fpeciem binomii U + V

a

reifert; ope Cof Hyρ Sin Hyp vφ radix determinabitirr.

Evidsns eft, hac methodo binomii radices tum tan tum in- dagari posfe, cum \/U* {V V4Z 1) perfechm aditiiferit

evolutionem. Sin minus; per/ ita muItiplioabiturU/3

ut produftum perfe&a quidem, fed minima nat,

z~Vy^i ,

quae per nu-

merum v exprimifur, diguitas, Provenit exinde (U-V/-4-1) sz(u^vqnae radix, per allatam métfoodum erufa,

Χ γ —*—■

& per fr in reciprocum dudla3 fubminiftrabit \/U-h

For*-o, liquidem J VCofνφ 4- VCofr φ' ι 4~

§f/Cq/V<p 's/Cofrφ*-—ι Cafφ, crit, Γι Cofvφ datse

v ~ . ""

quantitati R ponatur aequalis, Cofφ = § VR -fr- %/R% l

γ - - '

/' ψ

Hh ι VR V-^2 i· Eft vero (10) i?=s 2 Cofφ -*~

T% ^ Cofφ =·$-r 2 Cofφ ="åiC* (ob

(13)
(14)
(15)

Γ-χ r~3

t— a 9 5-f« 20+3-^4" &c. &c), unde radix hujus

r ~~~ r ~ "~~ ~

sequationis ξξτ I γR-*r YRX—- i -f- § \/i?— \fRz —- I, «J.

Νθν itaque acquilateras folum, verum etiam cujusvis Hy« ,

perbcrlae Fedorum Coimusasquationum refolutioni adaptari qneunt,

pariter ac, ii i? > i, tam Circularium, quam Eilipticorüsn fe¬

dorum Coilnus eidem fini iafervient.

$. IV.

Sit (Fig. a) fedor Ellipticus five angulus PCA de-

fcribatur fuper majorem Eliipfeos axein, ut diametrum, circulus,·.

iangat App in A, protrahatur PM donec circuli eircmiiferen«

tiarn ίη Ρ offendat, ad minorem Eliipfeos axem normalis duca«

tur PF, ad punda ρp' producantur redas PC, P'C & ponatur P'CAz^. ψ. Hinc exfurget analogia Tangψ: Tangφι: ι: ηΒ

cujn« ope uiio angnlorum φ & ψ dato,, alter faciliim-edetemiinabi-

tur; fit enim Tang φ rr: η Tang ψ. ^Sic etiam dabitur

Sin El. φ zzz n Sin Grc ψ Cof El. φ Sin Grε (go°ψ)' jfungELφ See φ.Cof. Grcψ.

Ex Iiis itaque adparet, quomodo., ex trigonometricis Circuli fusi»

dionibus, Eliipfeos etiam invefiigari posiint.

Si ex. gr. pro quodam planeta fupputata foret Elliptiese or- bitac trigonometria, anomaüae de pundo A fumerentur 0c fol in

uno focorum S iubfifieret; ex data quadam circuli excentrici

anomaüa ΡCA, planetas de centro orbitse & normalis de majori

orbitas axe diflantra innotefeet. Per refölutionem trianguli PCS,

quo cognitas fnnt redae PS, CS & angulus PCS, invenientur.

etiam planetas radius vedor PS& vera anomalia PSA.

Qüemadmodum "ex circulari Eiiiptiea Trigonometrie, ita.

etiam ex Circuli redificatione Ellipieos redificatio pendet. Quse-

B 2. ra-

a). Quomodo hsec formula re ipfaadhibeatur, commouftrat Disfer- tatio/fub Prsefidio Cei Prof. Landebbegk edita: De ufu nmliiplke*

iionis rationum in aquationum refoiutions»

(16)

XI

ratur fcilicet Ellipfeos arcus BP, eritque, pofito g == i η-

dCof Elφ. yι -e*Cof Elφ1

Üiixto arcus BP ss ~~. Snmto ve*

Vi—■ Cof Elφ*

roZsr9©*φ, eritSmsrsCo/*Ä/(p,undeBPzzfdz]/i-e2 Sinz*

td Sinζ fe* Sin z2 d Sinz

= / —= &c....^rconfi, Γιve

J Loj ζ J 2 Cofζ

,1O

r- t' 3«4 45 e *575 « 99225g1- -

< ■—&c. C ζ

t 4 04 2304 147456 :474560ο S

3«4 45 *6 1575g8 99225g10 7

■1 ■-{· ""f™ "fr" ■"I-<5tc. >Smζ»Coj%

i 4 64 3304 147456 14745600 j

rg4 15g6 525 g8 33075g10 7 _

—I—ί—-# -f- —-f- \-öic.>Sinz'.Cefz

L 32 1152 7372B 7372800 ->

c 3g6 IOC g8 6615 g10 7

^.3L —.4 j- g-&c,?Sinz*, Cofζ 288 1843a 1843200 ■>

-15g8 945g10 -5

*f- 3 4* —4-<SccSSinz7.Cofζ

, C3072 307200 J

io5g1^'

j, C t-&c.7SmiS9oCq/^4-&c

138400 5

«ui integrali con/lantem addere haud opus eft; cvanefcerite enii»

z, evanefcit Sin z. Quan'to minor itaque e/i Ellipfeos excentri¬

skas, tanto accui'atior lJlius redificatio per fimiiitudinem Trigoa

»omctriae Circularis δί Ellipticas enodetur.

Sic ß globo terre/lri applicatur allata feries, habebitur η

sr 0.995 > ^ 0,009975, g4 ΣΖ 0.000099500625, €6 SS

0000000992518734375 &c., quapropter, pofito ζ go° & sr

rrperipheri;» circuli, cujus radius eil; erit quadransmeridiani Ei»

liptici s3f ^.0.9974574205768470 &c. & tot» terra per polos

cireumfereatia =4#.0.997457 &c.

References

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