li SIMON

DISSERTATIO MATHEMATICA,

de

TRIGONOMETRIA

ε

SECTIONIBUS CONICIS.

s======é==i==g^a ^,

qjjam

VENIA AMPL. FACULT. PHIL. UPS.

PUBLICE VENTILAKDAM SISTUNT

SIMON ANDR. C,

phil. mag.

ET

CARO LUS BEHM,

OSTROGOTHI,

in audit. gust. maj. d. xxi apr. mdccciv.

η. ρ. m. s.

P. II.

ups a lije, typis edmannianis.

GROSSHANDLAREN

Hü GADLE HERR

MICHAEL _PETER GERLE

«SH

BÖGJDLA frun

SARA CHRISTINA _GERLE

F OD D

. ■ ;·.·'·# "·7-

WESTBERG

TILLEGNAS ©ETTA A'F SANN VÖRSNA®,

' - <' · - t.r ■■ ■-,·"··'·",·.' -

'

'· * ·"*■» ^.·. , "I ■'

CARL B EHM.

£

HERR

PEHR' KIM

SAMT

FRU

BRITA HELENA BEHM

född 50X)K23.5X:RAls 3D,

MINE ÄLSKADE

Emottagendetta* fclfotn etfvagt bevie min

vek den oinfkränkta med vårdnad, jag fårblifvtr

EDER

Lyciigße Son

€^ARX. SEHMi

DR

TRIGONOMETRIA Ε SECTIONIBUS

CONICIS.

. . . 38» ..ττπτ·

5· v.

..E^-elata hucusque ad axes Hyperboia, fupereO: ut etiatn ad afymptotos transferatur. Quam ob caufam ad centrum A (Fig„

3) et axem AB deiinietur Hyperboia XBZ, cujus afymptoti

AS, As rcFlum primo conftituant angulum. Recfla Nn Hyper-

bolam m puncflo quodam Ρ tangente et demisils de Ρ in unum afymptotum alteri afymptoto parallells PM,Pm; PM\ (zzzAm)

Sinus, Pm φμΑΜ)Coiipps, PN Tangens, PnCotangens, AN

S&cans et An Cofecans Afymptoticus refpondentis anguli ΡAM

audiunt b) Evadit exinde, ficurt in Circulo, Sin ΡAM ^~ Cof

(go9 ^— ΡAM), CofΡAM—Sin (> ΡAM)et ex naturaHy-

perbolae Tang ΡAM—Cot ΡAM. ^— AP, Sc€ ΡAM —; 2 Cof

ΡAM et Cofec. ΡAM= 2 Sin ΡAM.

Quando angufuts PAM< femiredo asquipollet, ex ipfa figu»

ra facile conftat, eise Circularem Skmm,Coiinum &c. afympto-

tico sequalem illoque exinde cafu, quofit ΡAM ^> 45°vel < 45°,

eruendam esfe ituer fundiones circulares et afymptoticas ratio«

Hem, ut ex Ulis datis hae eliciantur. Du<fh*s itaque de Hyptr«

bol«vertice 5ad afympfofösAS, As pairallelis BGtBF pofitisqiie

AB—R et ΡAM~z; obtinebitur AM. PMzzzBG.BF, Cof

i?*

Ar.z.Sin. As.z=z£R*, CofAs ζ Eil vero R %

C Tang

b) Vide Disf. De Elemeutis Trigonometrie? Byperbolicee, fub Prsefidio

Cei, Maltet editani,

j?s Mgm ^.

Tan°o-Ζ :: ~ —: SinAs^% etideo SinAs.ζ s=y{ R. Tang%,

2 Sin As ζ

R2

Eodem modo piodibit' CofAs ζ ξξ ——-et TangAsζ

2V%R· Tangζ

R.yf ^R. Tangζ. Si itaque poiiatur Rzz: iooooooo,re- Si»£

perietur:

Log S"m As ζ 4.8494850 4- ^Tog Tang.ζ Log CoJ As ζ = 14.S494850 — £Log Tang,ζ

Log TangAszzzz 14.8494850 4- § Log Tfmg,2—Log Sin^ζ ( Sin Asζ _ ^ ^

10.0000000

; As3 "4

Lop " ^—Lo£. Tang,ζ<

yCof As ζ) 6 ö

ΓTangAs -T\

LogI ^:—: 1~ 10.0000000—Log Sin ζ Sin Asζ )

Hinc porro,it Circuli peripheriaΒEBB rrorfumatur; dedueetur

Sin As~Tizzi00,CofAs \ττ~ο,Sin As\^7t s=o.CefAs_\7t=—

SinAs I 7t = — «>, Cof As | of Sin ArTnsrv,CofAs^7t

ΞΞ 00. Patet itaque, differentiam inier circularem et afymptoti-

cam trigonom.etri.am ineo praeeipue coniiftere, quod, pofito err- xuli radio et hyperbolac axe==i, ümites Sinuum et Cöiifiuum afymptoticorum fiant —00 & 00 , cirfcularium vero —1 <Sc 1, Sinusque idcirco et Coiinus afyinptoficos majori femper, quam

refpondentes circulares et crefcere et decrefeere ratione. Ex ai-

latis _prona omnino fed falfa conclufio flueref, per methodum ^a- fymptoticam, quando quteftio de angulis rninufcüiis exoritur,

multo accuratiorem quam per cireularem haberi trianguli cujus-

dam refoiutionem. c^. Ut vero asqualitas inter minusfeuli »ngu- Ii c) Hac opinione perperara abreptus deprehenditur pr. cit. Disf. Αθς

&or? unde nos in hane materiem ipfi incidisnus..

—_·

*5 —

Ii cofinum ef radium perfectam inejusmödi trianguioröm analy.fi<

circuli _opem, fic etiam xqualitas tangentem inter et angu au- nufculi cofinum hyperbolse fubiidium repeliif. Id tamen ρ aevul¬

gär! trigonomefria commodi habere afymptoticam, ut, quando

ex finibus et cofinibus anguli refpondentes quam accuratisfime

quatremur folito exactius quxfita edat, ^nemo, infpedia, quam ad- exemplum fapputavimus, Logarithmorum Sinuutn et Cofinuum Tabula, inficias ibit.

Ut eruatur ratio inter Tngonometriam afymptoticam, ii afymptotorum angulus ^{redtus, et} hyperbolicam, fi uterque hy- perboiac axium ponatur sequalis; defcribatur (Fig. 4.) hyperbok ZBXj cujus fit femiaxis BA (10000000) et afymptoti AS, Ar.

Hyporboiam in vertice B tangat Bp, de puncto quodam Ρ du-

catur tangenti normalis Pp jundtisqne Ρ A ^, ρA ponatur ΡAM

εξ#, pAM~ß ^et fedtor ΡAB ^™ λ. Erit exinde

A = Log Cof Af. f*45°— a) ^—9.849485°

LogSin Hyp.Λ. = Log Sin Aß.β^— Log CofAfß-f-ιό.οοοοοοο

LogCofHyp.λ.=Log TangAfß—·LogCofAf/3-4-10.0000000.

Causa, _qua dedueuntur inventac formulas haud diffieuiter enodabitur, il ad AS _ponatur angulus SAtt angulo ρAM sc*

qualis et de pundlis Β,η, Ρ demittahtur verticalcs BF, mn et PL. Ex natura fcilicet ipflus hyperbolse dabitur fedtor ΡAB

έςζBFLPmLog Nep,AL——LogNep. AF\cum vero eademper-

manefinter fedtoreshyperbolicosratio,five perNeperianos, five per

Briggianoslogarithmosexprimantuiqfaciiitatisgratiafy/temaBriggta-

num. adhibeatur, unde proveniet ΡAB—Log CofAfPAL—Log.

AF. Propter fimiiitudmem porro triangulorum BAp et wAn

mn.AB An. AB

erit PM(- BpJ=3 ~—fjfff et AM (x=lAp)Π atque,

ob ΡΜ; By;:AM:ABetPM(Bp): vc;; ρA (AM); vA(A3),

C ζ m.

vt -3 Br, five Sinβt: λ. Cognito igitur Uno angulorum

κ st /3 alter nullo negotio obtinebitur.

Si jam in Trigonometria AfyRipfotica Sinus et Coflnus ad

itdores Hyperbel#:, de reda BF (de vertice B in afymptortnn Tiormali) /umtos, perinde ac in Trigonometria Hyperboiica ad

fcdores de céntro liumeratos referantuc.; ope aquationis KtzLog NepCofAf(45*«— a)~~'Log Nep^FcomperieturΛ {=: Log Nep, Co/ AfΛ

vg|^ retento valore ipiius Ν et poüto AB =3 i,

AF

Ν'"Α V CofAfA> 2\Γ*λ^{tn V}2· SinAfλ^% unde Νηλ^~_{V*·Cof Afη λc:(V:,} CefAfκf.

N~n^Er|/i.SinAs λErf\/2· η

jyA±* e1/2. Co/4/" Γλ ±zj, ΛΤ " ±*AsVf? §nAf}rß),

Pro fedoribus itaque quibusvis λ et κ fequentes ^ex Iiis prodi-

Ibunt afymptoticas formulas:

Sm (X-4-%) =3 V2· Sin λ.'Sinμ,

Sin (h—n) =3\Λ· &'» λ. Co/κ

Cof(Χ°%-n)=3 V2·Co/A. Cofκ, Co/(A— κ){=:/^2.Co/A,Srn*

i

Fang As t^zQSinζλ ΗΗCof2λ)

5ί/ιλ-+»Sinκ=32 5t» f A-+»κ. (Tang\ λ—kJ

eS?»Α—Smκ ^2Sm§λ-f-κ. (TS?» §Α^—κ—Cq/|Α^—κ/

CofÅ-bCofK-^zCofjÅ-i-K, (Tanglλ ^— κ)

Cofλ—Cof^κ- ^2. Co/fA-fr»κ.(Co/§ Α—κ—5m J λ—κ)

Cof2 λζΐFz CofΧ9·,SinzX^ Sin X%

, Cof3Α,s=2 CofΧ*, SinjA t:25m A"5&c.

Si ulterius, qui inter afymptotos includitur, anguius SAspzk ponatur/h. e* <, =3, > 90° atque, ut pro Hyperbola *qni-

låters^

latera, {Flg.5) Sinus dicantur redas PM, quse de pundo quodam Hyperbolae Ρ in unum afymtotum alterLparalleJas ducuntur & ii

AM refpondentis anguli ΡAM Cofinus afymptotici nominenturi

oportet, ad inveniendos Sinus & Cofinus, de puncfb's Ρ, r in

AS normales redae Pm, rv demitti, Po cum eodem AS paralle¬

le duci jungiqug Ao. Sumto enim angulo oArm v\ & ΡAM

zz ζ^; vi confirudionis obtinebitut Pmzz ro zzz Tang tj 0c ex Tangη

analogis Sin h : 1 :: Tang^η:PM, Sin Αβζζζ—^—ζ··" eil vero,

pofito BG. BFzzρ, CofAf.ζ. Sin Αβζζζρ & ideo CofAfizz

ηSin h ^_

. Cognito angulo κ» cognofcetur quoque reipondens

Tang.»f

Cof h. Tang, η

qnoniam Sin h: Cofh :: Tangy]:Mm⁼ _^— &

Ρ Sin h1 HbCofk. Tang*]*-

Sin h. Tang g—* f* ^' £*$ ζ ^~

Sin h. Tang η*

VSu,k'-±Cofk. TaVg^· InVent0irc ^ofreperiemrqooque

Seeζ("pSinh2jhCofh. Tang15®) ^■

APzl ^, quae reda AP refolutioni

Sinh Tang^τη

trianguforum", triangulo APΜ fimilium, eodem modo, ac Tang

AfPAM refolutioni triangulorum redilineorum infervif.

Relaus ad Hyperbolae fedores Sinibus & Cofinibusj ex

' ·

U s:; CofAsK

aequatione λ iz Log (moct^ ρ Sinh) Hyp, ~~ffT prodibunt

t=— CofΑίλ, Η t=— SinAfΛ &e., ubi Η baiin it»

Yp Yp ^ :

guificat lyüematis logarithmici, cujus modulus eilρ Sink» Si_ita-

*8

itaque ad Hyperbolam aequilaferam applicantur sequationes in fu-*

perioriSbus emtas, vel, quod idem eil, il Sin hz: rad. 0c Cofh Z3

ofumantur; erit

Log SinAfζζζζLog. Tang >j

Log CofAfζζ=ζ19.6989700 ^—Log Tang'ή

LogTangΑ/ζ:=9.6989700 ^—Log SinAfζ-\-LogSee ξ.

§. VI.

Vellet inflituti tenor, ut trigonometrix jam hyperbolicse

In triangulorum fphasricorum refoiutione iifuin & adplicationem

etiam exponeremus, fed, cum Cel. Lambert ej id dudum ^pra>

fiitit, <3c ignobilis & irrita esiet ulterior opera, ϊη transcuriii

vero:, ut & afymptotic2; Trigonometrie, ab Illo nee tadas',

nec cognitse, δί hyperboiieas una compareat in problerßatibus

iphaericis utiiitas unum δί in hujus <3c iiiius indicium edemus

exemplum, mox declaraturi, quo pado in ioivendis problemati«·

bus, trigonometrie plane propriis, utraque adhibeatur.

Si hunc in ilnem fignificet (Fig. 6.) HZh meridianuni,

ZCF primum verticalem, Hh horizontein, ARr ileliae cujusdam

circulum horarium δί Ν poium* il arcus iemidiurnus AR ~τ,

arcus AS = t, ßelle de polo diilantia zz: b, fupra horizon¬

tein altitudo = h <3c poli de zenith diilantia dicanturj eril

Sin hz$ 2 Sin b. Sinc. Sin | r+ L Sin ξ t--·—* t, ex~qu aequätiο πe inveniatur Log Sin h, ad conflantem Log2 Sin b. Sinc jfolum-,

modo addendus eil Log Sin *^r-J- t.■Sinjτ-1. Si vero ilél»

ja, de _qua ferino movetur, infra horizontem haud defeendat vej,

fi circulus ARr horizontem nullibi feeet; imaeinaria evadet re¬

da SP, quae, ob dudlurn diametrum SD δί ideo ilmiles trian-

SR.Sr ^-

gulos SPR δί SrD3 fit =! —"zzgr_AL atque, ob Sin ~ τ_\ -i- t"

Sin ä) Cfr, Hißoire de VAcademie Royale des 'Sciences de Berlin? mnk 2768.,

SinI τ— t:: Sr: SR^> Sin h imaginsrie exprimetur. Hoc ita»

que cafu non obtinst pro inveniendo Sin h ailata förmula. Ha»

- Sink

betur autem generalis formula -7 7—ξξ: Cot. b.Cot c -J-Coft

Sin boSinc

Sin h

δι, poilto CQ~* i, -ττη■ — ps ~ bC--4- CF. ^unde³ ob

Sin b. Smc

bC^Cof.t, emerget CFrr Cot. b. Cotc. Sit ideo OCa^gS, qCa = qCF~ 45°, ad femiaxes QC, gC& g'C delineentur hy»

perboias aequil^terae FQT, vqt Sc vqt Sc ponattir arcui as refpcm»

dens area hyperboliea CMQsss ce atque CGQpzz-ψ. Ex iηde

exfurget Sm/i^2 Sm b. Sinc, Sin hyp.fψ Η- ce. Cof hyp. \ψ ^— a?·;

atque, il eidem arcui as refpondentes aresc afylnptoticas qmbn

^ ce' Sc qdMgss.ψ' fumantur, Sin h zzz 2 Sin b. Sinc. Cofaf.

2 ("ψ*-f- Tang,af~ (ψν^— ßy)*,qu3equaeflti Sm /i expresEo«

nes pro quavia generaiiter flella adhiberi posfunt,

TRIANGULORUM PLANORUM RECTANGULORUM PER 'TU. AS. RE*

SOLUTIO.

Probt. I. Datis cruribus, qucevitur angutorum cuieruttr.

Reg. Logarithmusuniuscrurisde alterius Logarithmofubtraöus dabit

Sin

Log ^—— anguli qusjflti. k i-jOJ

Probt2» Datis hypotemfa C? crure, qncsriturangulusattirutei\

Reg. De Log. hypotenufe fubducetur Log. cruris dati & erlt

• Ttmg· 1.

renduum ξτ Zog- ^—:—_Sin anguli, eidem cruri oppoiiti»

Probt, _3. Angiitis & crure datis, quceritw ^eras reliqmm Reg. De Logarithmorum Sinus anguli, cruri reliquo eppoiiti,

& cruris dati furnma fubcraelo Coilnus ejusdena anguli Lo»

Rarithnioj obtinebitur Log. cruris cjuasfith

. · FrobL

probt. 4. Datis angutis hypotenufa, quceritur criii utmmlihet.

Reg. Ad Log. hypotenufa; addito Log. Cofinus anguii, quasfito

cruri contermini, & de fumma fubtrado Log. Tangeciti*

ejusdem anguli, dabitur quaefid cruris Logarithmus.

Probt. _5. Datis hypotenufa & crure, quceritur crus reliquum.

Reg. Dato per Probl. α angulorum alterutro atque de Logarith¬

morum hypotenufa; & Cofinus inventi anguli fumma fub-

trado Log. Tatigentis ejusdem anguli; dabitur Log. quasfiti

cruris.

Probl. 6. Datis angulis & crure, quceritur hypotenufa.

Reg. De fumma Logarithmorum dati cruris & tangetitis anguli,

idem latus adjacentis, ducatur Log. Cofinus ejusdem anguii;

eritque refiduum = Log. hypotenf«.

Probl. _7. Datis cruribus, quaritur hypotenufa.

Reg. Invento per Probl, 1 angulorum alterutro; de fumma Lo¬

garithmorum Tangentis anguii dati & cruris, angulo con¬

termini, fubducetur Log. Cof. ejusdem anguii ut invenia-

tur Log. Hypotenufa;.

TRIÄNGULORUM PLANORUM OBLIQUANGULORUM PER TEL. AS' RESOLUTIO.

Probl. I. Datis duobus lateribus & anguTo, alteri datonm op·

poßto, quceritur angutus, lateri reliquo cppoßtus.

Reg. De fumma Logarithmorum Sinus anguii dati & lateris, et«

dem angulo contermini, fubtrado Log. lateris reliqui; da¬

bitur Log. Sinus anguii quasfiti.

Probt. 2. Datis duobus lateribus & angulo ab iudem iftchfo, qucsritur angutus alteruter retiquorum.

Reg. Erutis primum per Probl. 4. Tr. Red. Ref. Logarithmo redte, qua; inter cognitum angulum & de ignoto angulo

in oppofitum latus demisfam normalem interjacet, <& per Probl. ₃ ejusd. Ref. Logarithmo normalis-; dabitur, ti de Log.

Log. normalis fubtrahatur Log. difFerentise iηter oppofitmn

Sin" ^. .

Sc interjacens Intus, Log_v ~_Co/ anguli quaditi.

Probt. _3. Datis tateribvis SA baß, quwruntur anguli,

Reg.-Majori iateruni. ut radio, beicribatur circttitfs bafin fecaixs.

De Logarithmis fuitima; Sc difFerentiss laterum duca'.ur Log.

baiis, ut inveniatur Log. dilFerennas inter baks iegmers-

ta, Sc Logarithmo fummaéj quse ex h;.c difFerentia Sc drmi-

dia parte, qua ex bafi Fupereft, conficitür de Log. la-

t r Tang

teris majoris iubtracto, dauiiur Log - - '^· unius acuti an«

Lcf

guli; Logarithmo süteort·dimidi« partis, ex bafi fuperkitis,

de Log. nimoris latcris iubduclo, tmerget Log ---Tang

' Löf

alterius acuti anguli. Dati fic anguli,, de 18op fubtradti»

angulü'm obtufum redcenf.

Probt, 4 Angiitis SA tat^ere datis, qucer.itur altmitrum reliquoram.

Reg. De fumma Logarithmorüm lateris dati & Sinus anguli, eidem

läteri conteimtßi, fubtr, dlo Log. SiiL anguli oppoiiti, da-

bitur Log. lateris qujeiiti.

Probt. 5. Datis duobus lateribus SA angutö imerjaceftii, quvritur

latus retiquum.

Reg. Inventis per Probl. 2 angulis, ex praecedenti Probleixiafe in«

innoteFcet quatiltum latus.

Eandem viam fequendo, ex datis Sinibus Cofinibus Sc Jun- gentibus Hyperboiicis eorundein quoque Probienlatum efFedhii

dabitur analyiis.

§. vir.

Evocata jarn ex Ellipil & Hyperbola diverfa pro ilngulis

Trigonometrie fpecie; facile infclligitur, in figuris dimetiendis

fuum quoque habere ufum Parabolam. Defcripto fcilicet circu«

D iov

lo, cujus radius ilt quarta parametri pars Sc centrum vertex Pa-

rabolse: ope circuiaris Sinus Sc Tangeiitis «Stper Parabolse,ad exte- riorem axein, sequationem 4a Sin. Par.φzz CofPar. φ2 dabi-

4Tangφ2

tur Sin Par φ=3 - -> Cof. Par. ^<p~4 Tangφ Sc fung.

4 Tang φ2 ^{. -}

Par. φr ==- — , unde, poiito α = ιοοοοοοο, Sinφ

Log. Sin. Par.φ ^ 2Log. Tangφ—*^ 10.6020600 Log.Cof. Par.φ ^ Log, Ταη§φ-4- ö.6020600

Log.^fung Parφ ^ 2 0.6020600 — -/bog. «Sm φ.

Quarum quidem formularum in obvios cafus adplicationem ^cum

ex didiis evidenter perfpicit quivisj ab hac jam deilflamus ope¬

ra, quam, cum, praeter ceteram, haud poenitendam quoque, ad

dodlrinam Seddionum Conicsrum larglus ditandam^, prüftet utili-

tatem, non omnibus numeris nos lullsfe iperabimus. '

SECTORES & LOGARITHMI SINUÜM, COSINÜÜM &c. ASYMPTOTICORUM.

Gradus

j

Secior.

0 Infinit.

I 0.S79°3S3 0.7284581

3 0,6403021 4 0.5776782 5 c.5290241 6 0.489*899

7 0.4554281

8 0.4260988

9 0.4001438

10 0.3768406

11 0^3556739

12 0.3362628

*3 0.3183180

14 0.3016145 15 0.2855738 16 0.2712518

*7 O.25733°5

*8 0.2441120

*9 0.2315141 20 0.2194671

21 0.2079113

22 0.1967967

=3 0.1860746

24 0.1757085

25 o.i6f6638

26 0,1559°9*

27 0,1464171 28 0.1371628 29 0.1281240 30 0.1192803 3* 0.1106132 32 0.1021054 33 0.0937413 34 0.0855063

35 0x773866 36 0.0693695 37 0.0614428 38 0.0535953;

39 0.0458154 40 0x380933

41 0.0304185 42 0.0227813 43 0x151721 44 0.C076814

log, Sin.

e.ocoooo

8.9704457 9.1210269 9.2091829 9.27!8068 9.3204609 9.3602951 9.3940569 9.4233862 9.4493312 9.4726444 9.4958111 9.5132222 9.5311670 9.5478705 9.5635112 9-57S2332 9.5921545 9.6053730 9.6179709 9.63OOI79 9-64*5737 9.6526883 9.6634IO4 9.6737765 9.6838212 P.6935759 9.7C3C679

9.7123222 9.72I36IO 9.7302047

9-738871.8

9-7473796 9-7557437 9.7639787 9.7720984

9-780**55 9.7880422 9-7958899 9.8036696

9-8*13917

9.8190665 9.8267037 9-8343*29

9.8419036

log. Cef.

infinit.

0.7285243 0.5779431 0.489787*

0,4271632 0.378509*

0.3586749 0.3049131 0.2755338 c.2496288 0.2263256 0.2051589 O.I857478 0.1678030 0.1510995 0.1354588 0.1207368 0.1068155 0-0935970 0,0809591 0.0689521 0.0573963 0.04628*7

0.0355596 0.025193?

0.0151488

0.0053941 9.5959021

9.9866478 9.5776090 9.9687653 9.9600982 9.9515904

9.9432263 9-95499*3 9.9268716

5·?»88545 9.9109278 9.5030801 9.8953004 9-8875783 9-S799°35 5.8722663 9.8646571 9.8570664

5/n

log. Tang.

infinit.

10.7285904 10.5732077 10.4503827 10.4282223 10.3801649 10.3410605 10.308^624 10.27583c9 10.2549988 10.2325742 10-213212 3 1.0.3953433 10.17507' o 10.1641553 10.1505150

*0.1378551 10,1262^-92 10.1153506 10.1(53250 10.0955662 ι IC.0S72445 10.0791129 30.0715324 JC.0644632 10.0578729 10.0517339 10.0460201 10x407129 10.0357898 10.0312347 JC.C270325 10x231699

Jex156349 JO.CI64170 IO.CI35071 IC.CJ08968

ic.cc85792 10.00C5479 10.0047978 10.0032242 10x021226 10x011928 10x005256 10x00132.3

log.

Cef ini. Negat.

2.2419215

^.'543o838 5-7*939?8 2".8446437 2.94.155*8 1X216202 1.0891438

Ύ.1478024 Ϊ.1557124 Y.2463188 1.2^ 36 522 ϊ.327474,4

X.3633640 7.35677IO y.4280524 T.4574564 7-485339°

T.5117760

"1.5365718 1.5610658 y.5841/74 1,6064066 1.6278518 y.6485 830

j.6686724 i.688181g T.4071658 T.7256744

j.7437'20 ΐ.7614394 T.77^7736 7.7957892 T.8125174 T-828P874 1.8^,52268 y.8612610 1.877* *44

1.8928098 1.5083692,

1.9238134 1.9351630

1.9544374

7.5656558 T.5848372

Infinit.

I-7580785 1,4569162

i.2806042 1.1553564 1.0580482 0-9783798 0.9108562 0.850*976 0.8002876 0.75368*2

0.7113478 0.6725256 0.6366369

c.5032290 0.5719476 0.5425036 0.5146610

0.4*882240

0.4630282

°-4389242 0.4158226 9-3935934 0.3721492 0.3514370 0.3313276 0,311818a c.2928342 0.2743256 c.256248,0 0.2385606

c.2212264 0,2042108 c.1874826 0.1710126

0.1^47732 0.1307390 0,1228856 0.1071902 O.C9I6308 0x761866 0.0608370 0x455626

0.0303442

C.C15 1628

infinit.

*-758*447 1.4571808 1,281*998 i-*564155

3.0597140 0,9^07654 0.9141055 0.8564447 0.8056676 0.7603298

<3.7194012 0.6821211 0.6479120 c.6163248 Ο·587°°38 0.5596619

°-534c^47 0.510,0176 O.48735St 0.4659483 0.4456708 0.4264246 0.408*220 0.3906867

O,374°5*7 O.358*5S°

o-342953a c.3283907 0.3144288

0.3010300 0.2881607 0.2757903

0.2638912

0x524383 0.2414087 0.2307813 c.2205370 0.2106580

0.201282*

O.I9I9325 0.1830571 O.1744891 0.1662167 0.1582287

ro<iιςο

o.ocooooo

o.ooooCtfi 0x002646 0.0005956 0.0010591

0.0016558 0x023856 0.0032493 0.0042471 0x053800 0,0066486 0.0080534 0.0095956 0.0112760 0.0130958

o 0150562 0.0171583 0.0194037 0.0217936 0.0243299 0.0270141

0.0298482 0.0328342

0.0359728 0x392697 0.0427241 0.0463398 0x501190 0x540651 0x5818°8 0x624694

! 0.0660343 0.0715795 0.0764086 0.0314257 0.0866355

0.0920423

0.0976514 0.1034678

0.1094974

°.**57459

0.1222201

0.1289265 o>i35£7=5 0.1430659

0.1505150 90

89 88 87' 86 85 84 83 82 81 80 79 78 77 76 7?

74 73 72 71 70

69

68

67

C6 65 64 63 62 61 60 59 58 57 56 55 54 53 52

■5*

50 49 48

47

46 45

ΡKBorr/«·«// '.)<·.

tOGARITHMI SINUUM & COSINUUM ASYMPTOTICORUM PRO ANGULIS MINUSCUL1S.

26 27 28 29 30

Log. Sin.

7.1922724 7.3427874 74308330 7.4933024 7.5417.574 7-58f3480 7.6148214 7.6438174 7.6693937 7.6922724 7.7129688 7.7318630 7.749244t I

7.7653564 7.7803180 ι 7·7943324 i 7.8074469

7-8199087 7.83*6492 7-8427874 7-8533821 7.8634837 7-873*363 7-8823780 7.8912424 7-899759*

7.9079543 7.9158514 7.92347*4 7.9308330

Log. Cof.

12.5066976 12.3561826

12.268*370

12.2056676

12. I572126

I2fII7622O

I2\C24I486

12.0551526 12.0295763 12.0066976 II.9860012

Jr.9671070

IX.9497259

TI.9336336 II.9I86520 II.9C46376

ΪΙ.89Ι473Ι

it.879061.3 11.8^73107 11.8561826

ιί.8455879

11.8354863

ir.8258337 11.8165920 11.807727 6 11,7992109

**•79*0157 11.7831186 11.7754986 11.7681370

Ζθ£. Tang. £

r>

Log. Sin,

12.5065975 7.9379533 12.3561825 31 7.9448474

12 2681369 32 7.95*5294 12.2056675 33 7.9580119 12.1-572126 34 7.9643064 12.1176219 ³⁵ 7.9704237 12.0841485 ³⁶ 7.9763733

12.0551525 37 7.9821642 12.0295763 38 7.9878047 12.0066975 39 7.9933024 11.9860012 ⁴⁰ 7.9986643 11.9671069 4* 8.OO3897I 11.9497259 42 8.OO9OO66

**-9336335 43 80139987 11.9186519 44 8.OI88787

11 9046376 ⁴⁵ 8.O236513 11.891473* 46 ^{8 02831*3} 11.8790613 47 8032893°

11.8673207 48 8.0373704 11.8561825 49 8.04*7574 U-8455879 ^{50 8.0460575} 11-8354862 5* 8.0502741 11.8258336 52 8.0544103 11.8*65919 53 80584693 11.8077275 54 8.0624538

11.7992109 55 8-0663664

11.7910157 56 8.0702098 11.7831185 57 8-0739864

**-775498f 58 80776984 11.768*369 ^' 59 1 8.08*3480

60 Γ

Zo£. Co/.

II.7610167 II.7541226 II.7474406 II.74O9581 II.7346636 ΙΙ.7185463

II.7I25967

II.7IÖE058 II.7II1653 II.7OO3057 II.6959729 II.6899634 11.68497*5 H.6800915 11-0753*87 ^. II.6706487 II.666077O II.6615996 II.6572126 II.6529I25 II.6486959 II.6445597 11.6405007 II.6365162 II.6326036 II.6287602 H'6249836 ji.6212716

11.6176220

11.7056676

Leg. Tang.

II.7610167 11.7541226 II.74744C6 II.740958I

TI.7346636 IT.7285463 II.7225967 II.7168057 II.7II1652 II.7OO3056 II.6959729 II.6899633 II.684971I II.68OO913 II.675 3186 11.6706486 11.6660769 11.6615995 11.6572*15 11.6529125 11.6486959 11.6445596 11.6405007 11.6365162 11.6326035 11.6287601 11.6249835' 11.6212735 11.6176219 11.7056675

sä