DD /!'PROBLEMATA GEOMETRICA

CIRCA

ELLIPSOIDEMCOM PRES SAM,

Qu«,venia ampliss. facult. phil. upsal.

PRASIDEVIRO CELERERR1M

Mag. fredericomallet,Math. Inf. Professor. Reg. et Ord.'

Acad. Reg. Scient. Stockh. et Reg. Societ.Scient. Ups. Membro.

PRO GRADUPUBLICO SUBJ1C1T EXAMINI

Stif, Wrgd»

JACOBUS VENERBOM,Vestbogothus.

In Auditorio Gust. Die I. Maji mdcclxxjx»h. p. m. s.

UPSALIiE,apud JOHAN. EDMAN, direct. et reg. acad. typogr»

i* #fi/ S % I- o * J t Ü ti fl • * -

. ... d:x •

*

W.'.'*'., " ■_ „•,;■■

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I

Quemadmodum figura Solida,quam Semicircu-lus rotatione fua eirca diametrum generat,dicitur Sphaera; ita Sphasrois appellatur

quando figura plana, in fe rediens & a femicirculodifcedens, convolutione fua fpatium determinat abomni parte convexum, licet a Sphxra abludens,Quia vero natura figur® volventis indolem Sphaeroi*dis determinat, illiusque varietas fere infinita repe*ritur, diverfa orientur Sph®roidum genera. Interh®c Geometris maxime familiäre eft Ellipfoidicum,quod ab Ellipfeos rotatione circa axem minoremvel majorem gigni poteft, quodque in fe duplex eft,Ellipfois videlicec Oblonga dicitur, quando axis majorEllipfeos generantis e(t axis rotationis, CompreJJa au-tem vel Oblata vocatur, in qua volvente Ellipfi axisminor manet immota. E®dem figur® folo Sphxroi-dum nomine apud Mathematicos fiepius indicantur,non folum quia fimili ratione cum Sphsra gene-rantur, fed pr®terea quoque maximum inter circu-lurn & Elfipfin inveniatür connubium & quafi mu-tua qu®dam dependentia. Fingitur etiam Sph®roisComprefla nata ex compreffione Sph®r® ad polos,ita ut aequator magis elevetur, & Ellipfois Oblonga

A 2 ex«

cfWfe \ A ( &&J 4 v,

extenfione axeos gigni fubddenre ipfo aequatore, Depnori five Fpbaicide ccmprefia noftra in praefenticomrnentatio aget, quoniam Tellurem, quam inco-limus, hac inprimis bgura repraefenrari inter omnesconvenit, indtque omnia problemata circa Ellipfoi-dem obiacam ulum in Geographia maximum habi-tura evi&om eft. Poterunt vero disquificiones Geo»metricae circa Sphaeroidem oblongain ex iisdem prin-cipiis inftitui, & fimilia Theoremata ex eodem fun-damento circa utramque coliigi.

§. II.Uc propofiti ratio eo magis elucefcat, jam in li¬

mine obfervandum ducimus, Problemata omnia cir-»ca Sphaeroidem oblatam ita debere refolvi, ut proquovis Ellipfoidis punfto, ex data inclinatione Nor¬malis ad planum sequatoris quantitates determinan-dae inveniantur« Hsec inclinatio apud Geographosdicitur Latitudo loci, quam, utpote facillimam de-terminatu, principium calculi femper adoptare cu-piunt» Scientia itaque Geometrica circa calculumdirigendum ita verfabitur, ut breviori opera perfi-ciatur,& (i quid abfolute nequeat obtineri, celerioriflpproximatione eruatur. Inter Methodos Mathemati-cas huc facientes eam fumfimus dilucidandam, qu®breviore compage traditur in Matbematica TellurtsDefcriptiorie (Sveth. Verlds Befkrifning i Del. IV.Cap. $'. $• 2i. 28.) a CELEBERR. PRiESIDE an. 1772edita, cujusque uberiorem illuflrationem Tironibuseo magis necefTariam ducimus, quod, calculorum

in-

å% ) y (interdum fa£ta fupprefllon^, omnia ejusdem artificiap a legentibus vix hauriri expcr--—

^ tus fin?, Sit PAE quadrans El-lipfeos ? cujus Semiaxis minoreft PC & major CZs, & a pun«8:0 A (it du&a AB normalis

£ ad Ellipfin , erit AHE latitndoC B D loci/4. Ponantur CE=tf,CP=£»

a* — b2 = c2, ABE = z, du&aque AD ad CE nor-b2

mali, fit CD = DA =^, unds BD = ~ JamJ2

vero erit j s — * :: Sin z : CoJz> & per naturamEllipfeos ä2 : £2 : ; a2 — .v2: ;y2, adeoque a* : b* t l

b4, x* Sin a2 b* x2 Sin z2a2 — x2 : — , five /?4 — a2 x2 == .

<24 Co/a2 Cb/ Z2T*- TT« A a2 Cofz2b* Sin z 2Hinc deducitur = ar2; led po-

Cofz*fito r— Radio, eft Cofz2~r2— Sinz2, ergo a4 =a2r2—a2 Sin z2 -\-b2 Sinz2 a2 y2—c2Sin z2

Cof z2

2Z =

a2 Cofv2

•X2=,Cofz-

x2 8c fa-

/7 ^V^ ■»/79/ff8:0 £ Sinz =2 a Sin v habebitur a4 = —

Cofz2x Cof v a Cofz

r , ~—- tf1, a = — five x = —\ adeo-Cojz2 Cofz Cojv 5l2 x Sinz a b2 Sin z Cofz

que ob y = ——,— = — erit y =3a2 Cofz a2 Cofz Cojv

b2 Sinz——. Aflumatur z = 59*, 51', 3o"=Lacitudini Up-

A 3 fa-

» ) 6 (faliae proxime obfervatse, fitque a: b: ; 200: 199,

c ^399 „ /^399unde -== ■, Sin v = = ,5/« v =a 200 ä 200

»S/w (40, 57', 173'). Idem angulus u evanefcet inipfo a&quatore; Sed pofito 2 =^0% erit proxime^=5°, 43' hujas vero ufus in fequentibusabunde conftabic.

§. III,Ponatur arcus EA = .r, & pofito 2; = *, erit

etiam j== o 5 Ted pofito 2; = 9°° eft j = EAP : ertauccm ds2 = dx2 4- dy2. Aft— rf# :Sin z: Cojz,

dy Sinz , Sinz2,ideoque — dx = - - & 4- ———•^

Cy» Co/a2/ a2 \ 4M r2 _

, r/vzsjy® fi4- ■ ) = , five . Si-y X ^ Cof a2 ) Cojz2' Cojz

f Cofz2\ fSinz2-\-Co[z2\militer efbds> =dx\i+/ % ydx

= d*2 f \ & d/ = — -—, quia crefcente ja2 y •$>/; adecrefcit *, fumta abfciflarum a centro origine,quod etiam facili negotio eraitur, fubftituendo in x-

rdy dy . dxquatione ds = — pro —ejus valorem- —«X proportione nuper alkta inventum.

§. IV.a Co/z f _ fQnandoquidem x= vel *CoJv—aCofz

una

«ÖS3 ) 7 ( iuna crefccntibus x, Cofv, & Ctf/2;,erit— Co/vx Sinvdv a Sin zdz aCofzSinvdv

= adeoque —dx~—?rr~z—r r ^ rCefv2a Sin zdz a f Cojz Sin vdv Sin zdz \ af Cof v r \ CoJv2 CoJ v J r

/cSinzCofzdv Sin zdz \( —— —7— J obcSinz—aSin v, Hinc\ a Cofv4 Cof v /

rdx adz cdv Cofz - , . ^

ds=— —— = — —-i . Sed fite Co/z =5V« & Cofv CoJv*

»An<rr r cdzCofz advCoJvadv Cofv, five —— = dv, & dz = —777—>aCofv cCofz, ad% c2 Cofz* dz az Cofv2dz —-c2CoJz2 dz

ergo dsz= - = iCof'v a Cofv'1 a Co/v%

dz== ——- (a2 Cofv2 — c2 Cofz2). Verum Cofv2 =

o

fa—o/ff9*;undeds~ (a*r2—c2 Sinz2—c*Cofz2}aCofv3

dz b2 r* dz l1 v2 dv■I m 1 m \ (L^ Jf ^ ■ /*^ ZZZ I ZZ ■ ■'

aCofv3 a Cofv3 c CoJ z Cofv2 "$. V.

Sit radius curvaturas iny/= /?, erit, ob r\dz:idz ds Rdz b2 r2 da

Kjöfx, — = -, leu dj = ==—— unde#=r R r aCofv*

b2 r3

7c—, qua expreflio, cognitis atque datisa& j,pro quavis latitudine ex angulo v facile computatur,

£a-

6 ) s C $1 b2 r* . (iRCo[v%

Eadem vero aequatio R = —oat b ——,* aCo[v3 br>five cognito radio curvaturae pro latitudine quacjaro& data ratione axium, dabitur magnitudo axeos mi¬noris, & reliquae Ellipfeos generantis dimenf\one%Ex. gr. Sic a: b : : 200 : 199, 2»= f9°> J1 > 3°"> pn-deiT= 4°, J7',- 17I '; Sic etiam 598, 9*27> milHar.Svec. eric £ == 595,1994 miil. Sv. & 0 = JS>8,1902mil). Sv.

$. VI.b2 v* åz

Ut aequationem ii" = ff^'Qrff a(* *ormarn reda-camas fimpliciorem, ponatur a Tang y = b Tangz.Ell igitur, ob r2 =5'/» y2 + Co/>2, a2 r2 Ce/s2 =

, a Sm ya2 ähj2 Co/a2 + o2 Co/>2 Co/a2, & ob —=

\ a, feu (PSiny* Cojz1 —b1 Sin a1 Cofy2, habe-Cojz

"tur ä2 r* Co/s1 = i2 57fl s2 Co/3/2 4- ö2 Coyv2 Cofz*a2 r2 £2 5/tf s2 4- Ä2 Cojz2

atquc —— = > quomam veroCo/ y2 Co/ *2Co/u2 =a2r* — a2 *>2 = a2 r2 — c2 5V« z2

c= a2 r2 — <T Sin s2 -4- b* Sin s2 = az Cof z2 +a2 Cofvz a2 Cofz2 -}- b2 Sinz2

b2 5/0 s* , erit — = — =Cofz2 CoJ z

m2 r2 ar2 åy a Cofv2 dy _ , ,

FTi'&F7T== ~F7T-1 Sed quiaCofy2 Cojy2 CoJz%a

) 9 ( ^_ . ar2 dy br2 dz . abr* dz

a Tang y ene ==-—-—, ergo etiamCojy2 Co/s2 * Cofz2

a2 Cofv2 dy poi TT* • »= —--,five br2 dz= a Co[v2 dy. Hinc erit ds=

Cofz2 ' j jb2 r2 dz ab Co/v2 dy bdy _ , _ _ rCofz

= —± d — —L} & ob Cofv = ——,a Co/ v3 /2 Cofv2 Coj v Cofy

bdy Cofy ady Sin yds = ———= —— . Uc vero in sequatione

r Cof z r t>m zady Siny"rSinz exPnrnatur &n z Per y> coniide-

randum eric h Sin z Cofy = a Siny Cofz adeo-que b2Sin z* Cofy2 = a2r2 Siny2 — a2Siny0- Sinz1,five a2r2 Siny2 — Sinz2 (a* Siny2 -f- b2 Cofy2 )= Sinz3 (a2r2 — a2 Cofy2 + b2 Cof y2) = Siw 22„ _ . , ady Siny(a2 r2 — c2 Cofy2 ); erit ergo ds = X

r

l/a2r2— c2 Cofy2 dy \d a2 r2—c2 Cofy2t = — , quas ror-

ar bin y r2 1mula, quoniam in figura telluris, c ed ad ä in ra-tione admodum parva, in feriem maxime conver-gentem reducitur* Notandunv vero infuper ds perdifFerentialia arcuum z 8z v expreffum, quantitacesprodere magis complicatas, vel minus convergentes*ut cuivis tentanti apertum eric.

S. VIL

Quoniam jam habemus ds =~ Fa*r*—C* Coh*

f| ) 10 ( $aJ-j fl — r3 Cojv* a3 .

== ^ • fic — = «2, enc ay == —r u2 v2 f2 **

CoJy__^ atque, extra&ione radicis quadraricx per«nz r2

^ t / Cofy2 Cofy* Cofy6ra£ta, dJ" — ™ 2?;2^2 *88*r+" i6;;6r6

$C°fy8 lCoJyl° i\Cofylzi2g«8r8 256»I0r10 1024 nx2 r12

.—22_!l0~ &c. V ponendo /4, /?, C, />, &c.2048 fl1**-14 /

pro termini? Seriei inventae, eric /4 =ifin2 r2#= —ACojy2, 4»2r2 C= B Cofy*, 6n2r2 D = 3 C Cofy2,S ii2r2 E—f D Cofy2,10 n2r2 F—y ECofy2 &c. adeo-

✓ A Cofy2 1 B Cofy2 3 C Cofy2que i-f r v^1 2n2rz 2,2.n2r2 2.3.n2r2

5 D Cofy2 7 ECofy2 3, FCofy2 \,4 h -t- 8zc. } quae fe-

2./j..n2r2 2.y.n'1rt 2.6.n2r2 'ries quantum Übet protrahi poreft, quia ex hislex progreffionis terminorum fatis perljpicue intel-iigicur.

§. VIILAd perficiendam fingulorum integrationen! ter-

minorum, in ferie jam inventa, plsniorem nobisfaciunt viam theoremata, a Celeberr. Prxfide in A£tisAcadem. Scient\ Stockh. exbibita anno 1758- Eft vi-

flyCofy™ Siny Cojyw—i tn—t f-yCoJym—%delicet 1 — = - 1* / ,J fm viy w—1 m J rm~<2

ad-

H I ) »I ( -CÜT

adeoque la-lS°lt - *&3<Vi +"

* r r <* zn2>* 2.zn2 r2

ay fady Cojy* aSiny Cofy2 ^ o. a ✓ dy Cofy2' ./ C»4yr Q A. «4 fi 4 J2.2. n2r 1 8 8.4. 8.4 •/ &4r

+ -—-3-V, /7.4.2. «4r ^7«ÄV/ry Co/y5 3. a Siny Cofy % ny *adyCojy5

8.4.ff4**4, S^.4 i»4'"2 84.2.ntr* J i6n6r7a Siny Cojy* y a Sin y Cofy5 ^ 5 q.aSinyCoJy16.6.n6r6 16. 6.4.»6r* 16.6.4. 2n6r%

5.3.^16. 6.4. 2»6r

z

& fic porro. Undc fiet

<57 (i

iSiny Cofyn2r%

_L*L_ + — — + &c \128.12806 256.256z?8 * J

_ »Siny C"Jy% / i + $. f-7- ?B4r* \ S.4 I6.6.4»* I28.8.6.4»* 4

256.16.6.4»* 4 ^C*)aSi»yC„ry< f i t y. 7. + 7-9-7 + &(. \

»«r« V 16.6 128.8.6»'* 256.I0.8.6«4 — /iibi/ntilla eget corre&ione, qufa dato j==o, eftquo-que <?, ob £ 7<mg 2 = 4 Tangy^ cvanefcentibus

B 2 in

/

ü ) J2 (in hoc" cafu omnibus feriei noftrae terminis. U'umvero ejusdem, non opus eil:, ut mulcis doce3mns:ponamus enim s = EAP arcui, eric quoque y asqua-lis qaadranti Peripherie ad radium r, quem dicamus= #, fimulque Cofy = o: in hoc igitur cafu omnescermini evanelcunt, qui potentias Qofy continent, &fiet s = (l 3 — &c.)

r V 2.2 «2 16.16ö6 /

AfTumatur jam r : 2q:: 113 : 3^5 & a : b :: 200:199,ut antea, atque == 598 > 1902 Will. Svec. erit — =

cujus poteftates feriem inventamcelerrime con*40060

aq 355rergentem efficiunt. Eft namque —= —^=939,63 j

mill. Svec. = 2,3438 mill. Svec. & - =4722 r 8-8 rn*

0,00437 m. S; hinc quia terminus quartus non nifi perdecimalia o£tavi gradus & inferiora poteft exprimi,idem cum reliquis tuto negligitur, eritque adeo pro-xime j = 937, 287 mill. Sv. Ex allacis vero patet,in aliis ipftus 2 valoribus asque felicem obtineri adarcum AE valoris approximationen), eandemque, obcelerrimam terminorum convergentiam , aliorumGeometrarum methodis minime poftponendam efle.

IX.

Re&ificationem arcuum Eiiipticorum, quam hicexplicavimus, Geometris non folum ex eo capite

com-

«s8# ^ t $ ( e8&

commendandam exiftimo, quod celeriori conver-gentia peragacur in cafu minoris Excentricitatis, fedquia pr&terea quoque calculus terminorum in ferie§. Vlbmse adeo facile peragatur, ut totam clafiem Problemacum, qux ad -arcus Eilipticos reduci poflunt,ad debitam fimplicitatem in eadem redu&arn eile cre-dere liceat. Ut vero in Ellipfoide comprefla manea-mus, obfervandum hic erit, omnia plana, Ellipfoi-dem lecantia, ad ipfam fuperficiem Ellipfes forma-re, omnesque fe&iones inter fe fimiles fieri, qua-rum plana aequaliter ad planum aequatoris inclinan-tur; hinc, fi arcus, inter duo punäa in dato pianointercepd, defiderentur, iidem per modo allata faci¬le determinabuntur, modo fitus axium inveftigatusfuerit, ut angulus z, omnesque ab hoc dependen-tes,poffit inveniri. Varia quidem circa hanc mate-riam oriuntur problemata, in primis fi ad Magne-tismi Telluris Theoriam illuftrandam, aut attra&io-nuni diretfiones in planis per axem magneticum in-veftigandas applicatio inftituatur, fed quae a praefen-ti argumento aliena jam brevitati ftudentes omit-tim us.

$. X.

Si aequationem ds = ——— refumamus,&, as*aCoJv3

lumto p pro peripheria circuli cujus radius ef\ r9confideremus eflfe r: p :: x : — adeoque diffe-

r r

ren-

41 ) i+ (r-1. . nCo/z

rentiale fuperficiei Ellipfojdis; erit, ob #==——,t

Lo/vPfl'L = & fiquidem ris Ca/s = adv Cofv

r - CoJ v4. . , . ab*vpdv . .

habebicur — = — ♦ Hmus vero nuantitatis>* c CoJ v1

Integricioni lequens infervic theorema, quod ex (u*

/dvroi Sin vrm — iTTT—• — T™-Co/ V» f»—X CoJ vm — i

—2 j dvvm—2. ^ jab* rpåv ab* p Sin vm—t / Co/u«*—a* ^ cCofv3 2cCoJv

/ab*pdv ' ab*p£p . .———

f & Integrale ipfius -—lpeciaum m-icrCnfv & zcrCofv

L/ab*pdv ab*p T r-r-SinvJL/~7T"—9zcrLofv 2Cr Loj vdeflgnat Logarithmum Hyperbolicum quanti-

tatis, cui praefigitur: quoniam igitur evanefcente arI" Sinv jr

cu % evanefcit v, atqueCoj v

.

. „ t ab* rpdv ab*perit Integrale completum lpfias /-—r— = —-

/ CLo/V* 2C

(Sin v i f r+Sinv\- -4— E,———}. Sed Cofv i r + Sin vwrxCofv* r Co/v J J

Tang (4f + Zv) & Cofv: Sin v :: r: 7>wg v; ergoab*p flanav T Ttmr,Uf°-+-±v)idetn Integrale = — ( tt-1- + JL2cr \ Cofv v

jam vero pofito z = po*, erit v = s*, 43', J5V,''

m> ) i y (^S® J * J K <?$>

CV J}Y CV ^(§♦ II), & Sinv=z —, Cofv = —, Tangv=z s

a a b

~CoJv~ =: "T""' er^° faperßcies Eilipfoi-<z2/> rf£2/> I" tf + f a^p / b2 T

dis = — + —- b-r~ = ~1 (i+ -L-~)2 r 2cr u 2v \ ac b /

c- ^ J a + cbive ut 2 ad i -+■ 1^-7—, ita iuperficies Sphse-ras fuper diametrum secuatoris ad ipfam fuperficiemTelluris» In (upputanda vero formuL pracedenteproceflum ita inflitui oportet:quia 4* = 1,005025 &bc

, . "+ . r

y = o, 1003768,erit —f — M054018 cujus Loga¬rithmus Tabularis eft = 0,0435266 &in 2,30253509mulriplicatus dac Logarithmum Naturalem bve Hy-perbolicum numeri 1,105401g =0,1002237; hic ve-

b2ro numerus duttus in — = 0,9912649 dat produ-

QC

&um = 0,9934826; ergo fnperficies Ellipioidis erica^p— C1 -9934826) = 4481987 Mil!. Sv. quadraticis.Eadem methodo area? zonarum Ellipfoidis, utcun-que fumtarum, fbpputantur, quibus vcro ipfo cal-culo determinandis jam non immorabimur. Super-funt prseterea varia in Ellipfoide Problemata geo¬graphica, qua? ex allatis facile refolvuntur; ne vero

ultra Geometria? porrceria nimium diva*gemur, eorum principia nobis

fufficiat explicafte.O O 0