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*
h» ab m. s.
P. I
upsa liMt typis εdμ α ν ν i α ν i S,
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-
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~Tang0®
€ef
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
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
2y± 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-
;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
2»
di,
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]
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(φ—;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 &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-$>-φ)-—^(φ-·ψ)
ια^·+ψ)4-|αί^ψ}
£-C.(ip~\^)-i-|-C(φφ-φ)
is.ji φ+φ,αΐφ-Φ
2Ο.^φ-ήτφ, S.iφ—φ
2S.£φΑ-φ.C. ψφ-φ
2.€,£φ+>φ.£ i φ-Φ
2θ4Φ+·Ψ·0·τ<Ρ—<l· 2 C· 5φ+ψ.Ο. §<?—ψ -~·&ί<ρ+ψ>. i<p-4>&c.|φ+ψ.5.|φ ψ&c
Ce^+'Co/ψ—
α<?/φ—θο/Φ=
Sin ζφ ss:
j3iS".φ.C.φ.
&».3 φ = iS,<p.C.(p*-~S. φ1
Siu 4 Φ = 4&φ.£φ3-~7&φ5.£φ
T &c.
ϋο/2 φ = C4p*--—&φ*
Co/3 φ =
^>-~αφ.&φ*
Co/4φ = c.φ*-~^α.ψ* s.ψ*-h φ4·^
2 8ΐηφ* == »*.(i—C.e(p)
4 Sm φ3 == n*.{iS^-~S 3φ)
5 Smφ4 = w4-.(C.4^-4C.2(p+3)&c
3 Co/φ* ΖΞ C.3φ ·+" I 4 Cofφ}Λ z=z Ο.ιφ-Ι-ιΟ.φ
s Co/φ4 = ^4φ-4-4^.2φ-+-3 &C.
2Sinφ. Cofφ
3£φΧφ?+Π£φ*
4§.φ0.φ**4—,2&φ''Όφ
Ι &c.
ο.φ*+·^.φ*
c.f>*+$S4*a<p
Ο.φϊ+^-Φ^.φ*-η4ϊ
S φ4.&ο.
«a(C.2φ—ι") η2.(Ξ.$φ—35.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/3φ = 4Co/Φ* — 3 Co/Φ .
CV40 zr gCo/φ4 — 8Co/φ3 4- ι <&c. h. e. fi m fo*-
miila retro fubfUtuatur β,
Cofrφ zz 2θο/φ.0ο/{ν- ι)φ-α*Cof (r- 2)φ.
Sit
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
m» 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: 2ß & 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
3©
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
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»
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.