D. ΰ.

SPECIMEN ACADEMICUM

DE

PROgfECTIONE ORTHOGRAPHICAORBITARUM ELLIP TICARUM,

> "

QU OD

ΓΕΝΙΑ AMPL. ORD. PHIL. UPS.

PUB LIC Μ PROPONUNT CENSURS

OLAVUS F,Ρ Η I L. MA G.

ET

DANIEL ECH. HOLMQUIST,S T r P. REG.

HELSIN Glo

IN AUDIT. GUST. MAJ. D. XI JUNII MDCCXC.

UPSALI/E, TYPIS DIRECT. JOHANN. EDMAN.

sacrie regle majestatis

μλχιμ-JE FinMXΕ C C LS SI ΜSVIOGOTHICMARCHIEPISCOPO

ET

REGIM jéCADEMIM UPSALIENSIS

PROCANCELLARIO ,

NEC NON

REGII ORDINJS VE STELLA POLARICOMMENDATORI,

GENEROSO ac REVERENDISSIMOZDOMeffflO 3)0<SMQ3iI

MMCENATI SUMMO

DICAT AUCTOR.

GENERAL MAJORSKAN,HÖGVÅLBORNA FRIHERRINNAN

FRU "ULM.WJL ΜΣ,ΜΟπΟΜΛ

Född $>Ö3¥3)§;MIN NÅDIGA FRU MORMODER!

f^enna Afhandling vågar jag min Nådiga Mormoder ödmjukafttilågna. Den ömhet, fom min Nådiga Mormoder altid tåcktsvifa for mina Syfkons och mitt val, och de vålgerningar, fom afmin Nådiga Mormoder jag i fynnerhet fått åtnjuta, fordra et oån-deligen högtideligare prof af min vördnadsfulla erkånfla. Måtte dockdetta obetydeliga tjena tii et bevis derpå, at jag åtminftone icke glömtmina heligafte plikter!

Blifve min Nådiga Mormoders dagar många och fålla! Fa-derlöfa barnabarn fkola aldrig uphora at vålfigna Forfynen for en fådyrbar fkånk.

Framhårdar i djupafte vördnad

MINNÅDIGA FRU MORMODERS

lydigfte Dotter-SonD. E. HOLMQUIST*

DE

PROJECTIONE ORTHOGRAPH1CAORBITARUM ELLIPTICARUM.

Theoria Orbitarum Ellipticarum, quas Planette ornnesdemetiuntur, licet usque a tempore Kepleri tamaceperit incremenra , ut vix quidquam de Ulis co-

gitari posiit, quod non antea rnulti magnique nominisviri dixeruntj nemo tarnen, quod quidem nos audieri«mus, illius figurse, quae per orthographicam harum or*bitarum in fua invicem plana projedionern oritur, natu»ram & fpeeiem determinare, & in primis, quo ad omniafua.Elementa, debito calculo fubdere docuit. Verum qui¬dem eft, quod in plerisque problematum Aftronomicorum,quae legibus hujusmodi projedionis nituntur, fufficiat tan-tummodo cognofcere planorum inclinationem & anorna-liam lineae Nodorum , quarum ope cujuscunque radii ve-doris projectio absque ullo negotio invenirur; at veroeas quis negaverit oriri posfe quaeftiones, ubi, nifi exadecognofcatur indoles figurte, per projedionern in pianoortae, minus bene procesferis? In primis Refolutio illaProblematis De Appnventiis Statiombus Planetarum, quaeAdis Reg. Acad. Scient. Holm, pro anno 1786. pag. 193.inferta legitur, animum nobis injecit vires circa hancmateriem tentandi. Id vero praecipue praeftitum volui-mus, ut magnitudo & poiitio axium in Elliptica proje-dione, locus foci & quae cetera, ex datis in Ellipfi pro-jicienda Elemenns per calculum eruantur. Qualiacunqqeergo iint, haec conamina noitra mitiori Ledorum cen-furae fubjicimus.

A

2 De Protections Orthographien

§. T.Si Ellipßs, α piano utcunque fecla, in hoc planum ortho·

gvaphice projeff<1 fuerit; srit quoque figurα in piano inåe or-ta Ellipßs, cujus Centrum eß proje&io centri Ellipfeos dat<2;

axes projezti erunt diametri conjugatce in hoc Ellipfi.Sir AEBD (Fig. 1). Ellipfis, cujus axes funt AB öl DB, accentrum C. Per pun£lum quodvis F axis majoris planumfecet Ellipiin , exfiilente LK linea interfeclionis. Si jamex punilis omnibus Ellipfeos AEBD in hoc planum de-mirtanrur lineae normales, uc AG, BH, DO, EP öl CM;dico figuram GΡHO esfe Ellipiin, cujus centrum Μ öl dia¬metri conjugatae erunt GH öl OP.

Si fuerit axis major AB ad angulos reclos lineae in-terfe£lionis KL, res per fe patet. Sin minus, producaruraxis minor ED, donec occurrat LK produflae in punftoN, Sc jungatur NP, quae etiam per Ο & Μ tranfibit. Perpunfhim quodvis R in axe majori ducatur ordinata ST,quae occurrat lineae LN in Q; demitre normales SX, RF,TU, öl junge Q^, X, linea per U öl F transeunte. Quo-niam igitur lineae NE, QS parallelae iint, ut öl lineae EP,SX, erunt plana NEP, QSX, adeoque etiam linea? NP öl,OK parallelae. Praeterea ob AG, CM, RF öl BH paralle¬lls , erit AC: AR = GM: GF, itemque CB: Rß = ΜΗ: FH\\quare componendo "fit AC. CB : AR . RB = GM . MH:GF. FH; fed per naruram Ellipfeos eil AC, CB : AR . RBzezDCq: TRq; ergo etiam GM. MH: GF. FH= DCq: TRq.Quia vero NDO öl QTU{unt trianuula fimilia, erit ND-.NQ= QT: QU; fed ND : NO = DC : OM, öl QT : (QU —TRTuF; ergo fit, DC: OM= TR: UF, & Z2C/ : 7%==OMq : Hihc tandem evadit GM . MH :GF. FHzzzOMq : £/£^. Quum ergo G7/ & ö/5 fe invicem bifariamfecent in A/j & praeterea OP öl UX iint parallelae atquebife£fcae a linea GH, earumque quadrata iint inter fe ut

re£tan~

Orbitartim Ellipticarum. 3

re&angula ex fegmentis ipfius GIi\ fequitur, quod %u-ra GΡHO iic Ellipiis, cujus centrum A/, & diametri con-jugarae Gli Sc OP.

Scbol. Quomodocunque linea interfe&ionis planorumpcfita iir, vel inträ vei extra ElÜpiin 3 htec omnia eödemrariocinandi modo demonftrari posfunt. Quum vero com-muniter ufu ,veniat, ut linea Nodorum transeat per fo-cum, Sole communi Planetarum centro gravitätis exfi-ftente, asfumamus in pofterum, punctum 'F es fe iocuniEllipfeos AEBD.

§. n.Dicatur jam femiaxis major ACz=zc> minor CDczzd^

excenrricitas PC — e, angulus inciinorionis planorum — J^Sc anomalia lineae Nodorum, feu ang. CFK^a. Ex cen¬tro C demittarur in LN perpendicularis CJ, & junge JM%quse etiam cum LN re£tos angulos faciet; quare fit ang.CJMzzzJ. In triangulis igirur re&angulis CFJ Sc CJMeft i : fin a; : e r CJ = eβη a, Sc 1 : fin J \ · CJ: CM ==efina'fmj; unde in triangulo CFM, ob CF : CM =R \(in CFM^ fit fin CFM == fin afin K; adeoque cofCPM.=Ϋι —finn*(inj7·. Eodem modo,quia, ob angulum FCNrefturn, fit fin CNF ~ cof CFN = coj a, cfolligitur esfefin CNM— cofa fin J ^ Sc cof CNM = ^ 1 — cof a* fin J*.Hinc paret ratio determinandi femidiametros conjugatasGM Sc MO. Eft enim R : cof CFM = AC: GM\ ScR: cofCNM = CD: MO; quare GMzzic ^ 1 —fin a* fin J2 ,

Sc MOzrzd^ι — cof'a* fin J2; vel, fi brevitatis causfaponitur ang. CFM=zB, Sc ang. CNMzzzb, erit GM =zc cofB Sc MO d cof b.

Ut

4 De Proje&iov? (hihographieüUr inveniarur angulos FMO vel ÖMB^ obfervandum eil>

quod fit FM = e cnj B ; MN == C.V cof b ~ e tang a cof b\MJ = CJ cofJ = e fin a cof J; FJ = e cof as &

CJ25>N=~= e fin λ fang α. Eil vero in triangulo FMJrFJFM \ FJ-=zR\Rn FMJ, &FM:MJ~R\cof FMJ ; un-

cof a fin a cof Jde fin FMJ =—■ <5c cof FMJ =·——-™ "· Sie7 J cof B cof Bquoque in triangulo MJN eil MN t NJ = R : fin JMΝ,

βη α& MV: MJ~Rx cofJMN,- adeoque fin JMN— -j,.

fö/* a cof J _ .

& re/ 7Λ/ΛΓ = —-—■-7 Sed ex Trigonomerricisro/ b

novimus, quod iit fin {FMJ 4- JMN)~ fin FMJ. coj JMN4- cof FMJ fin JMN; & quoque cof (FMJ 4- ^M/V)= cof FMJ cof JMN — fin FMJ fin JMN. ^ Si ergoin his expresfionibus fupra determinati valöres lubilituan-tur, evadit fin FMO =r —■■ I^~7T" & cof FM0ro/ B cof bfin a cof a fin J2 cofJ

adeoqu'"V™0* jm.cof.fi» 3*'Coroll. i. Si Radius ve&or quicunqtie in Eliipfi

dicmir r, <3c angulus, quem facit ille cum iinea Nodo¬rum, vocatur qt erit pfoje&io hujus radii ve£lorisv ^1 — fin q* fin J2*.

Coroll. 2. Quia eft FJ—eeofa & = tßnacofj;eric (? ro/ « : r fin a cof J — i tång KFM / adeoquefang KFM = rof^ ta»g «. Eodem modo , ob NJ =:s fin a tang ay & NJ\ JM — R : /TW, erit ta»?KNM

Orbitarum Bllipticarum. 5

KNM = fö/ 5^ Cotang a. Et in genere eft Tangens an*guli, quem facit proje&io cujuscunque radii ve<ftoris cumlinea nodorum, = coj J tang q.

Coro//. 3» Vicisiim, ii angulus, quem facit proje£tiotang Q

cum linea nodorum, dicitur Q^, erit tang qrzz coq-j *CoroIL 4 Si vero anomalia cujuscunque radii ve&oris

in ElJipii AEBD vocatur /, & angulus, qui ipii refpon-det in piano fubje&o , dicitur S , erit tang S =

cof 7 tana s•—~ — ——-—-— Sit FF ejusmodi ra*cof B3 Ar ßn a cof α μη y2 tang sdius ve&or, cujus proje&io eft FZy tunc eft ang. CFF=s öl ΜFZ S. Fone igirur FFK =: q, eritque tang q =

— tang a — tang stang a — s znz 1— ; unde tang K FZj =0

ι Hh tang a tang scof J fang Λ — cofJ tang s

cof y tang q = — . Sed efti Hr~ tang a tang s

tang KFM — tang KFZtang $z=ztang (KFM— KFZ) =2 — ——;ά b v / ! + tang KFM tang KFZ*quare fi heic valor tangentis KFZ fubftituitur, & obtang KFM = cof y tang α , prodibit tang S =3

cofy tang f (ϊ + tang a*)Ponatur deinde

I + cofJ2 tang a2 A- fin y2 tang a tang iin hac fun&ione (ec. a2 loco 1 -f- tanga2y mulripliceturtam numerator quam denominator per cof a2 y atque, obcof a fec, /? = (, & coj a tang α = ßn a > erit tang 6 res

. fed eftcof a1 -f- fin a2 cof J2 -|- fin a coj a fin y2 tångs*cof a2 Ar fin a2 cof y2 τ=ζ i — fin a2 fm y2 ~ cof B2 f

A % qud-

6 De Proje&ione Orthographiea,

cofJ tang square tandem tang Sz=.

cof B2 Hb fin a coj a fin ff% tang sSi fuerit s > λ, vel q = s — , erit tang q =tang s —tang a— : atque tang S = tang {KPΜ4- KFZ) =ι 4- tang α tang stang KPΜ+ tang KFZ

TT—:— ; unde, fa&a fubftitutione, idemϊ — tang KFM tang KFZprodibic valor tangenris S.

Coroll. 5. Hinc fequitur etiam, quod, vice verfa, fitcofΒ 2 taηg S

tanü s co/jf — fin a coj a fin J2 tang S§. III.

Cognitis ilc in Elliptica proje£lione duabus diame-tris conjugatis, una cum angulo, quem comprehendunt,ad axiurn magnitudinem Sc poiitionem determinandamprogredi liceat. Reproefentent AB & DE (Fig. 2.) has dia¬metros; iitque AC r= c cofB = m, CD z=z d coj b = η;ang DCB — p\ Sc praeterea femiaxis major zzz χ} minorz=y atque excentricitas ~v. Ex natura Ellipfeos cogni-tum eft, quod, ii per vertices diametrorum conjugatarumdöcantur lineae tangentes Ellipfin, ftt parallelogrammumex ipfis contentum sequale re£langulo ex axibus. Seu,quod idem eft, parallelogrammum ex CB Sc CD fub an¬gulo DCB aequatur re£tangulo ex femiaxibus. Ideo ficmn fm p zzzjcy, vei etiam mm (in p ezzixy. Praeterea quo-que innotefcit, quadrata diametrorum iimul fumta aequa-lia esfe quadratis axium fimul furntis; quare m2 4- n2 =2λγ2 4-y2. Si ergo ad hane aequationem addatur prior, fietχ2 4- 2xy 4- y2 = m2 4- 2mn βη ρ 4- n2 , vel χ 4- y^m2 4- 2*»»,fin ρ 4- η2. Sie etiam , ihbtrahendo priorem a

poile-

Orbitarum Ellipticarum. y

pofteriori, fit x2 — 2xy by2 = m2 — 2mnfinp -fr- η2 , &x~- y-=z Vm2 — 2 mn (inp -bn*. Hin c in ve η i tur

χ =z-I. ^Vi2 -h2mnfinp -bn2 b ~ ^m2 — 2^0 βηρ -+.

J =ir + 2mnfinp -i- n2 — m2 ~ 2 mnfin ρ -b a2, &v =r χ -+-y . χ—y =2 ^(/ä2 -4 »*)a — 4m2n2 finp2.

Si paullo attentius coniideramus has formulas,protinus apparet, quod iit »2 = n2 finp2 -b n2 cofρ2,ob βη ρ2 -ρ cof ρ2 = ι. ΕΟ: i g it ur χ -b y zzz* m2 -b 2mn fin ρ -b 722 Jm p% -b n2 cof p2 r=.

Ϋn2 cofρ2 -μ (w 4" nfinp)2 — η cofρ ι (n nfin / Vv η cofρ J

Εådern ratione diet poteftf—-yxzncofp J/^i 4 Γη n fiu!)sf\ ncofρ )

vt -4- η βηp ni — ηβη ρPonatur ergo ζ=ζίαη?φ,δί — — — tann Μ:öη cofρ ώΎ η cofρ ώ '

atque fiet χ -b y ζ=. η cofρ ^ι -b tang φ2 — ncofρ See. φ ι

nec non χ — y = η cofρ ^ ι -b tätig ω2 — η cof ρ fec. ω.Quare χ ζ=ζ \η cofρ (Jec. φ -4- fec. χ)\ y ·=ζ \ncofp (fec. φ— fec. ω), & ν2 = η2 cof ρ2 fec. φ fec.ω , vel ν =

η cof ρ Ϋfec. φ fec. ω.. Cognitis iraque tang φ Sc fang α>,quarum valöres admodum fimplices fünf, ut in pofterumuberius exponetur, horurri angulorum Secantes in tabulistrigonometricis qtiiert posfunt, id quod labori juvabit,calculum abbreviando.

Ut determinetur pofitlo axis majoris , fit CG zz x»&

g De ProfeSItone Orthographien& ponatur ang BCG^z. Per B, verricem diametri, dii-catur tångens axi occurrens in T, Si ex B dernittatur i ηaxem normalis BQ. Tunc erit ang. CBTzxzp, δί ang. C TB— DClzxzp—- z. Igitur quiun m quovis triangulo iinusangulorum fint ut latera oppofita, erit in triangulo CBT·,

m fin ρfin p ~z\finp\\ m : ; Ted per indolem

χΊ

Ellipfeos eil CQj CG = CG: CT i quare CTz=z—j & id-x2 m fin p „ ^ x2 fin{p — z)

circo eriam — = 7— : ? ieu CQjr. rCQ fin (ρ — &) m fin p

ln triangulo autem re£tangulo CBffi efl R : coj ζ zzzx2 fin (p—z)

CB : CO, vel 1 : cof ζ m \ j u η α e ritJ tn fin ρm2 fin ρ cof ζ z=z x2 fin {p —- z). Sed eil fin (p —- z) =finρ cofζ — cofρ fin ζ quare m2 finp cofζ = χ2 fin ρ cofζ

Χ 2 - ffl ^— Λ'2 fo//> fin z\ feu —— fin p cofζ = cofρ fin ζ.Si ergo utrimque dividatur per cofp cofz^ fiet tång ζ ~^2 — /Λ2

~7T~ "'»sρ ")§. IV.

Ope hujus calculi facillima panditur via Geometriceconflruendi axes, quam, quia calculo noflro inferviet,ilatim adponere lubet. Ducatur fcilicet per centrum C li-nea ΜΝ ad diametrum AB normalis ; fint CM ~ CN — CBEf jungantuv MD ac DN. öltevtus pvoducatuv ND ad Ρ >donec fiat DP = DM. Si jam, bifeäa ΝΡ in ex centro

C per

*) Lecpns Elementairee de Mathematiques par dela. Caille l'AbbéMarie, §. 683 & 684.

Orviiavum EllipticuYum, 3

■C per U diicia fnerit liven CÖG zz NU vel U:Perit CG fe-miaxis major; Cf fi ab ND aufevtur DR zz DM> erit NR$quaIis axi minori 'IJjL

Qnoniam enim MCB fir angulus reßus, erit ang,MCZ)zzcompL ang. Z)CZ? ad 900, adeoque cof MCD zzΓιη ρ, & /20 MCD—cefρ. Sic eriarn eft ACZ? zz fin ρ.Ex pundo Z) in MV demirte normalem Z)S, ac fiet Z)S— 6 Ζ) (in MCD — η cofρ, & CS— CD cof MCD — n (in ρ ;unde , ob CM zz CN — m, evadit iVS zz 7/2 -l· 0 (in />„ acMS — m — η fin ρ, PJinc ulterius .fit AZJzz ^ ΝSz + &/)2— ^H- 277/72 Snp ~l· 722 / & Λ/Ζ) zz P'MS2 Hl· Z)S2 zz

—2mnfinp-\-nz; quare A7/5 z= AZ) + =^*«2 + 2mnßnp -+- «2 -l· ^m2 — 27220finp-i-n* —2X (§. 3)

•r— axi majori; & ATZ?zz AZ)— DM zz 23* zz axi minori.Porro, fi ducantur MP ScMRy erit MP parallela axi ma¬jori & MR minori. Demonftrandurn enim eft, esie angDCMR zz HCM. in trianguio CND eil ND : CD zz (in NCD :ün CND, unde öb AZ) zz 4- 2mn(inp4- »a } CZ) zz

» cof ρGc fm NCD zz ro/>, fit fin CND zz ■ ■ ,y Jtl ° V\m°- imnJmp-\-nz)

m H- 72 fin ρSc cofCND zz Deinde ex puncto

^702 Hl· 2mnfin ρ Α-η2)R demittatur ad NM perpendicularis RXj tunc, quia eftR: cof CND zz NR [NX, erit AX zz NR cof CND, adeoqueAfXsz: MV— NR cofCND; lic etiam eft XXzz NRfmCND.Sed in trianguio MXR eft MX\ RXzz R: tång RMX, un-

NR Γιη CNDde mg RMX =___ Subftituantur jam inhac fiinftione valöres ipforum IVÄ, A//Vj ßn CND &

B «/ C7V7?

ιer, De Proje&iotte Orthographien

zofCND fupra determinati, atque prodibit tang RMX =nycofp

jzwzw2 -}~ 2z/zw (inp -f- n2) — my ·— ny ßn py-7 - - ——; :— ny cefpVm2 Hr 2-mn finp -\-n2 —y rrx) =: ———Sed quia

mx —~ ny (in ρxyeft mn ßnpzzz,xy, erit η =;—-—ΐ quare. inferendo hune

m (inpy2 cofpv&lorem r fit fang RMX - . Si ulterius ία

(in2 —3z2) /?zz /?aequatione zw2 + n2 = λγ2 -f)/2 ponirur hsec eadem expres-

x2y2fio ipfius «j fit m2' ·+■ =z x2 *4- y2\ adeoaue y2m2 ßn ρ2 ^ J ^

ix2 ·— zw2) m2 ßnp2 (χ2— m2)m2 ßnp2χ2—m2 ßnp2 x2—vi2 ßnpm2x2 cofp2- y2 (x2 — m2)ßnp·

; ergo erit.x2r—m2 ßnp2 D ' zw3—x2 cofp2r y* cofp (χ2 - z»2)Hinc. randem fit — =. £7^ /> r(zw2 —j2") ßnp x2 * ^ 5

{X2—ZW2)adeoque etiam fang- RMX — : fang ρ» Quum ve~

ro haec fun&io itidem exponar valorem tangentis %, urad finem §. praecedentis oftenfum eft , erit fwz?£ a r=·

RMX, & idcirco ang. RMX~BCG rz HCM; undeeft 3/Ä parallelä axi minori //7, & ob angulum infemicirculo reftum, eft quoque Λ//5 parallela axi majoriCG. Et quoniam ΜΡ Sc CU iint parallela?, erit NC: CM= M7: UP, & NU =2 UPy ob NC=z CM, Ergo fi permedium lineae NP ex centro C ducaiur CGzzz NU ζζ^,^ΝΡχ,esit CG femiaxis major, q. e. d.

CorolL

Orhitarum EUipticnrimi. 11

CoyoU. i. Hinc quoque paret, quod media propor-rionalis inter ND Sc MD iit aequalis excentricitati v.Nam fupra oftenfum eil, esfe ND = χ y y Sc MD ταχ—y\ quare, ob υ2 cxi^x-by) (χ-—yf, eil v2=ND.MD^Sc ND:v=zv: MD.

Coroll. 2. Qu um fit DS 2= η cofρ, SN=m -bn ßnpSc MS 2= m — η ßnp; atque praererea DS : SN ~ ι ·tangNDS, & DS: = ι : tang MDSt erit tang NDS z=w -f- w ßn ρ m «— η ßnp

& tang MDS= Sed eil quoque,η cofρ ώ η cofρ

m π- Vjßnput in §. 3. Tuppofuimus , tang φ — ————Sc tang &2Z2

vi — η ßn ρ* ——unde fequitur, quod fit ang. NDS-^ιφ . Sc

η cof ρ Dang. MD $2=1 κ.

Coroll. 3. Ex allatis Üuit etiam a 1 i a methodus inve®niendi angulum %. Scilicer oftendimus, esie tang z =

ny cofρtang RMX222 — ^ - .

m v {m2 -4- 2tnn ßnp -4- n2) —■ y(m 4- η (inρ)Sed eil ^ vP- -f 2 mn Tin ρ -{- η2 — η cofp fec φ (§. 3); fubilb·tue igitur hunc valorcm , divide per η cofρ, ac pone tang φ

m 4- η fm p y y cofφloco —-—, erit tangzM — = —-—.

?; cojp vi fec φ-y tang φ m —y fin φPonatur ulterius in hac expresiione valorem ipiius y 2=jrft cofρ (Jec φ—fec &>), atque, ob cofφ fec φ 22i\x Sc ßn φ fecφ

β η cofρ (ι — coj φ fec ω)=tang φ5 fiet tang z2= ——-—- 7 ;

m^-^ncofp tang φ -b ±n cofρ βη φfec ωfed eil \n cofρ tang φ ßn ρ ; fi ergo hoc fubili-

B 2 tua-

sä De Projcfiione Orthographienm—n fin ρ

tuatur, divido fiat per ^ncofp, Sc loco — ~~ perna-η cofρ

i — fec ω cof φtur tan? ω, evaferit tangz — · —■ —: multiplica;

tang ω 4- (in φ fec ωtandem Sc numeratorem Sc denominatorern per cofcoy ac«que, ob fec ω cof ω zzz i Se cof ω tang ω zzz· fin ω, pro-

* cof ω — cofφdibit tang & = — —Eft vero cofωcofφ =s0 fin ω 4- fin g)

Φ "Hl· CO Φ ύύ Φ —J- CO φ — CO2 (in ■ .fin- ; & βη φ-{· (in ω =2 ftn .cof *)>

22 22

φ—-ω φ —counde fang % ~ tang—-— 5 adeöque ang, 2» = -——.

§. v..Supereft, ui in omnibus illis expresftonibus, qune

§. 3. inventas funt, ii fubftituantur valöres, quos 2. pree-ber, quo tandem proje&ionis Elliptica? axes, excentrici-tas Sc. focus, quoad magnitud i η em Sc poiitionem, rire;determinentur ex datis Elementis. Eft vero, id quod;prius monendum, m — c cof B = c ^ 1 — (in a2 fin J2 ;

cofjη ξξ deofb ~ d V\—cofa2 fin ff2\ finpzzz-—j \ 6c:

cof B. cof bfin a cofa (in J2

cofρ =. —fofB cofb ° igitUr per $ 3' fltjfVml·· -f· 2tnnfin ρ 4- n2' -4- •iE'm2— 2mn (inρ η2; yzcz

IVm2 4- imn finp 4- η2 — m2— 2mnfin ρ 4* η2 , atqueρ2 sr 4- «2)2— /\m2n2 finp2, reperietur , iniertis va·loribus fupra di£lis,

φ) Eümri Intr» in Ana!, Infinir* Tom, !.§· 13.1·

Orbitarum Ellipticavum» 13

-^cofBU icdcojjf+d1 cojb* + ^c*cofB~--icdcerj+d-co[b*— 4 l'cHofT1 TöfP-— i vdcofjid'cojb*·

Sc v = V\c2 cofB2 d2 cofb2)2—^c2d2coj J2Si< vero loco cofB Sc cofb ponantur γι—fina2 [inj2

Sc Ϋι—^coja2finjf2^ erit mzA-n2!z^c2 A-d2-—finj2(cifina24- d2 coja2 ); fed, ob d2z^c2 —-e2 3. eft c2fina2-\-d2cofa2nr r2 — e2 coja2 i unde fiet ?n2 4- «2 ~ c2 Hl· <^2 —

(c* ~ e2 coj a2) = c2 cofj2 d2 4~ e2 cofa2 fin J_2.Ideoque etiam exponi poteft

—^(c cofj+d)2 + e2 cofa2Jrnj2 4- -I^Cccofj-d,2 +e2 coßa2/mjf*[c cofff\å)2 ^e2 cofa2finff·2 —C0JJ ~ dfi\ cz coja2 finf·2

(x2—^m2)läJiterius, quia tangzz=: —tangp^Sc x2z=:^m2OC

,4- ^\m2 -h n2)2 ■— 4m2»2 finρ2 , eritfin p {n2 — vi2 -j- i/((/»ä4-ö3)2— 4m2 n2 finρ2)) ^

^ cofp{m2-4-n2-\-P'Q{m2-{-n2)2·—4m2 n2finp2f)'ergo in hac quoque expresfione valöres ipforum /», 0,finρ 8c cofp ponantur, ut &, brevitatis causfa, v2 loca/(c2 cöfiB2 4- d2 cofb2)2 *— 4c2 d2 cofj2 , riet tång %

cofjf (d2 cofb2 — c2 cojB2 4~ v*)fin a cofafin J2 (c2 coj B2 -+·a2 cofb2 Hh v2)

Si denique angulis φ 8c ω> ut antea monftratuni eft5lvi 4- 0finp

in calculo uti velis, erit, ob ί^φ~ ■ JiC0j~—ί 8c η cofpB % ZZLdfitl·

14 JDt ProjeSiione Orthogvnphicadfin a cofa finj* d cof3

TtfB ' atque "β"ϊ = -^fB ' Unz φ =ccojB2 -\-dcof3 c cofB* ■— dcofjTT τ „ ' Sic quoaue tanpvzzz .dfin a cofafin J* 6 dfin a cof a fin J*

i — ro/2J fin 2 aVeletiam,<ob fin32 — ,finacofa— , &

2 2

fin2 (α-{*3)Ίτ· finita—J)βη ia cof2j= · & praeterea* ob

-2

I-ff- cof 2 BcofB1 z= — *); fiet fin a cof a fin J2 —

2 Rn 2a — fin 2{a +3) — (in 2 (λ —J)— : ^ adeoque tang φ ξξ

o

r (4 -f. 4ro/2#): </-KS ro/J2 fin 2a — fin 2 {a-k-J)—fin 2 [a — J) ' ^ t<lf7^ ω ~~

r (4 + 4cof 2B): d—% cof31 fin 2a —fin 2 (λ +J)—fin2{a — J)' quiE exPresfl0nes.licet prioribus prolixiores ^ideantur, ufui ramen magisinferviunt faciliusque computantur, ob evitararp multipli-cationem. Et inventis hoc modo angulis φ&ω, ceteraomnia, quae inde pendent, & in §. 3. ac §. 4. Coro//. 3.aliata funt, absque ullo negotio eruuntur. Eft enirn

dßnacofa fin3zxzz&cofp (fecφ +fecao)2= — (fec φ Hl· fec ω)

dfin a cofa ßnf1y—\n cofρ [fec φ —fec ω) =- —~ QfecQ —fec cd)

2 COJ D

Orlitnrum EUipticavum.

dfinacofafinj2 —-wzz n cofp VJec φ fec ω =. — YfecCp fec ω Sc

φ —- ωang. ζ =s —-—ο

§. VI.Repraefentet jam KFL lineam Nodorum, fecantem

diametrum AB in pun£to F, quod eriam focus eft in El-lipii data. Tuac eft FC — ecofB. Fiat CP aequalis ex-centricitati inventae v, vel ^ND . MD ', adeoque eft ^fo¬cus in hac Ellipii. Jungatur porro FP, Se producaturFL, donec occurrat axi GO in pun&o Z. Ergo in trian-gulo FCP, ubi cognofcuntur FC, CP Se ang. FCP 2= z,erit FP~ Ϋe2 cojBz v2 ~—iev cofBcofz, Sc tang CFP

v (in ζ1 Ulrerius eft ang. FZCzz: KFC — FCZ

e cofB'— v cof %z=. KFC — quare,. cbfin FZC \finFC7j~ FC: FZ, erii

e coj Bfin %FZ= runde, cognita FZ, Ii pundum Ζ

nn (KFC ~ a)jungatur cum centro Ellipfeos datae, dabitur illa diame¬ter, cujus projeöio' efficiet axem majorem in quaefita.Sed aliter quoque reperiri poteft pofttio bujus diametri,.Jungatur enim FG\ Si in triangulo CFG, ubi cognofcun¬tur FCyCGzzz Sc ang. FCG'= compiemenro anguli -a.

χ fin ζad i8o% erit tang CFG ~—— quare fi hic&

e cofB ·*- x cof ζvalör fubftituatur pro tangS in formula, §. 2. Coroll. j.allata,

x cofB2 fin ζerit tang s~-ecojβcofj^ x cojJcof%—xfina cofafmy*finz~Eft vero heic ang, s anomalia. illius radii ve£loris in El-'

lipii,

Be PvojeSiione Orthographien

lipfi data, cujus projeetio eft FG, vel a cujus exrremit-a-te demisfa normalis verticem axis majoris in projedtiö*ne tänger.

Fatet quoque .ratio comparandi radios vedtores mambabus Eliipfibus. Ponatur enim radius vedtor quicun-que in Ellipii projicienda — v, & ejus anomalia = s -}tunc eil q-=z.a — s Ί vel s—a. Sit FFprojedtio ejusmodiradii vedloris, quae ergo fit ·= r Vi—-finq* fm ψ7- (§. 2.CorolL 1); ut öl tangKFFe=. cofjtavgq. Quseftio eft,ut pro pundto F inveniatur radius vedtor VF & ang,CVF, in Ellipii projedta. Quoniam däntur anguli VFC& CFK, dabitur etiam VFK vel VFL. Sed innotefcitquoque KFV\ unde cognofcetur ang, VFF VPK—-KFF. In triangulo igitur VFF, ubi dantur FV\ FF Scangulus interceptus VFF> erit VF zzz' i/( VFZ -4- FF2 -h2 VF. FF. cofFFF)'j ubi fignum + adhibendum, di obtu-ius, iignum vero —, ii acutus fuerit äng. VKF. Inve-niri etiam poteil ang, CFF= S per Coro//. 4, §.2, quiad ang. VFC additus, vel ab illo fubdudlus, prour pun¬dtum Fpoiitum fit, dabit angulum FFF. Ulterius, ii di«

y*citur ang. FVCzzzu, erit VFz^z — ·— : unde χ .VF

χ — v cofuχ .VF— F χ F

VF .cofur^y2 & cofu= — -—= .1 J Jv. VF v v.VF

Adeoque, datis radio vedtore ejusque anomalia in Ellipiiprojicienda, ut & pofitione & magnitudine lineae FVtjungentis Focos, determinari poteil ille radius vedtor f

qui ipli refpondet in figura projedta.§. VII.

Quod reliquum eft, Geometricam conftrudtionem ad-forre liccat, quae exponit magnitudinem ac poikionem

axium

Orbitarum Ellipticarnm. 27

axium proje&ionis Ellipticae, data Ellipft, quae projici de¬bet. Sit ADBΕ {Fig. j.) hsec Ellipiis 3 AB öl ED ejusaxes, F focus & FN linea nodorum, cum axi majori an·gulum faciens NFCz=ia% Öl axi minori produ£lo occur-rens in N. Super AB defcribatur circulus AGB; fiat HCG5= an gulo inclinationis planorum = J. Ex Η in CG de-mitte ^normalem HLy fume CK=:CF vel excentricitati,Sc fiat KM normalis fimiliter. Deinde ex pun&o Ν inlinea nodorum abfcinde NO =: KM, per 0 ducatur OPparallela ipil CN, occurrens HL in P. Si jam junganturEP öc PD, dico, esfe EP Ar PD = axi majori Ellipfeosproje&ae, Sc EP — PDzz: axi minori; nec non mediamproportionalem inter EP öl PD == excentricitati.

Dicatur, ut antea, ferniaxis major AC~c, minorCD zs d^ & excentricitas CFz=e. Quia tunc eft CH =AC~c, öl ang. HCG™ J, erit CLzrzccofJ, adeoqueELz=zc coJJ-}-</, & DL ccof J—d, (vel,fi pun&umL inter C öl D ceciderit, DL ~ d — c cofJ). Porro, obCK=zCFzze, öl Β :fin J = CK: KM, fit KM — e (in J;ergo etiam eft NO™KMz=ze fin J. Sed, demisfa ex 0in NC perpendiculari OQ, eft R ifinN^x NO: OQ; quareOQj=ze fin J fin Nz=ze coj a finj; nam in triangulo re£lan~gulo FCN eft fin Ν=: cof ΝFC zu cof a. Hinc, ob OP par¬allelem ipli NC, eft etiam PLzz: e cofa (inj. Igitur fitEP = SEL*-*-PL* = r{Ccoijjf </)* -+ &PD= ν~ϋΓ*-t- TL*= S(i coff-^y*ande EP -+- PD — ^(ccofj + j)* + t* fm j*■+- r{ocojJ — d)? +e* cofa* fin J* = ix (§. 5.) = aximajori; öl EP—PD — 231 = axi minori. Eft quoque EP=■*■-■+-y, öl PDczzx—7, adeoque EP. PDzzzx Ά-y, x<—yξ= u2 , öl EP : ν = ν : PD.

Ulterius ducatur ex centro C in lineam nodorum nor¬malis CJ, fiat, ut Rad. ad cofj, ita CJ ad JS', junge SF,

G. quse

t g De VrojeSiione Orthographien Orbitavum Ellipticarum.

quae prolongetur, donec in T occurrat normali ex pun£toA in lineam nodorum demisfae. Producatur itidem FS adalteras partes, ut fiat SF =:%{EP ·+■ PD) = ±ER = x.Sume SX aequalem tertiae proportionali ad SF & ST; ex Ferigatur perpendicularis FF-> & per X ducatur XF3 paral¬lele lineae V3>, (ii jungererur), ipii FFoccurrens in Ζ'". Situnc jun<£la SF producatur ad lineam nodorum in Ζ, eritFZ pofirio axis majoris in Ellipfi projeéla.

Nam, quia eil CJ\ JSzzR: coj jj3 repraefentabir pun-£tum S centrum, proje&ionis; (conf. §. 2. & fig. 1). Et, obSjjzzefin a cnfj^ & FJ— e coja, erit FS = ^FjJ2 -f SJ2

tcoj a* -4- fin a2 cojjl·7·. Similiter, ob AU: UTzz CJ\ JSzzRicofJ, coliigitur esfe UT = AFfm ® cojjf, & FU =r.

AFcof a, adeoqpe TFz=z ^FÖ2 4- öl2 = AFF[cojal -j-7?»λ2 cofj-y, unde —fin a2 fin jj2 zzzm. Exponit i»gitur T^proje&ionem ferniaxis majoris ^6'. Eodem modorepraefentabir ΛΦ, fi jun£la fuei ir, pofirionem, quam haberproje£lio axis minen is, ideoque erir ang. NSF~ ρ. P*rro,quum fit SF~x, & SF: ST zz ST : SX3 erit SX ~

ζ»2 χ2 — m2—, adeoqueFXzzx—~ =· ·,. Sed, ob .Υ/Ίρϋparallelam, eil ang. FXFzzpy & propterea in triangulo

χ^ ι ■. fjj^' X2ffi2"XFF eft i: tangpzz \FF\ quare FFzz tangpoX. X'

Igitpr , quoniam in. triangulos SFF fit; SF : FF =rx2 —m2

R.\tång FSFy erit tang FSF = -——— tang ρ = iang %

(§. 3). Unde coliigitur , quod fit FZ pofitio axis majorisin Eliiptica proje£lione.

Plura quidem funt, qu® hac »h re ve! addi posfunt, vel politius eli-roeri; quum vero inträ fat enguftos limites diverfo modo Hmus coar-$ati, heic fubfiftere cogimur.