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CL. GEORG. IGNAT. p& METZBURG.,

IN UNIVERSITATE VIENNENSI AA. LL. ET

PHILLOSOPHUE DOCTORIS, MATHESEOS PROFESSO-

RIS CAS$, REGII PUBLICI, ET ORDINARIEI

INSTITUTIONES

MATIIDLNLATICI

IN USUM

TIRONUM CONSCRIPTJE.

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INDEX CAPITUM

TRIGONOMETRUE

pag

Car. I. JVotiones generales Trigonometria.

i

CAP.

H. De quantitate Sinuum , Cofinuum

€c. determinanda ad conftruendas T'a-

EUZaS Slnuum....- *?*9 9999 9»9 999?Í9.092o99€

6

CAP

IIl De principüs caleuli trigonome-

trictin refolutione triangulorum ; cum

explicatione 'Tabularum Logarithmi-

Carunz'.. e€e€9*$ 95252929?*9? 99069*9999» $9429 $5$

17

CAP.

IV. Refolutio pradiica triangulorum., ,

3

o

GEOMETRLE PRACTIC/E

CAP

l. AVotiones generales dtmenfionum,...

46

CAP.

44

ÀI. .De dimenftone bafeos. .. .. .. .. ss

CArD.

IIL . De anguls, et inftrumentis ad eos

48

accipiendos neceffarüs.. .. .... ..L

CAP.

IV. De correcitone angulorum relate ad

centrum ftationis , et ad horizontem, .. 60

CAP.

V. Deconftruciioneretis per triangula,

et calculis trianguloerum obférvatorum. '12

CAP.

d

INDEX CAPITUM.

CAP.

VI. De calculis refpectu linee meridia-

A

d

A

n& , 8 perpendicularis , & conftructio-

;

Ü

|

ne mappa per puhcia fixa. .... .. ..

94.

v::nc]e. Y

|!

Pä

CAP.

VII. Refolutio problematum, quae in

Eä

praxi . Euthymetrica, & quidem in

-u

z

dimenftone horizontali occurrere pof:

—— Á

Jänt....æ,.,, €*6€9 99 909 *?9* . 0909€279 9$9€* 9-

95

Car.

VIII. Refolutzo problematum, qu& ad

]

altimetriam , feu determinandam alti-

b

3

tudinem objeciorum pertinent... .. . .. 10

"m

Car. IX. Zpplicationes principiorum ad par-

4M

tiales dimenfiones, ut agrorum , fluyi-

-

orum, f'ylvariit Qc.i i. eeo o n EE

CAP- X' -De LibäZlationEOO e**9 06( *** -29- IZO

Tabulæ Logarl't/zmorHNZ,. ...1. eoe€9oee€et* $64 149

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OTIONES GENERALES TRIGONO-

METRLE

DERINITTIO I

AX

te ( — —(21

Ie]iej — — (2]

[ vigonometria plaua. €ft fcientia refoluendi triangu-

Em-—— s lum re&ilineum in plano defcriptum, hoc

&wa eft: determinandi quantitatem laterum, & an-

*s

gulorum.

COROLLARIUM

Cwm etiam triangula in fuperficie Sphzre defcribi pof-

fint, quorum proinde latera arcus circulorum funt, fequitur, al-

teram etiam trigonometrie Ífpeciem dari,

que. de refolutione

triangulorum fphzricorum agit, & frigonometria fpherica dicitur,

€ujus elementa ad aítronomiam pertinent,

AMetzb. IMaih. P. 1II.

A

Scuo-

—

Y2

CAPU

l.

CHOLION I

3. Ut huec'innotefcant, ex fex tri

uli partibus, tribus

fcilicet lateribus

& tribus angulis, quibus omne triangulum

conftare diximus, tres cognitme effe debent, que dari feu data

dicimus, & inter has tres datas, unma faltem pars, latus tri-

anguli eife debet

Cum enim triangula duo, longitudine late-

rum diverfa, szquales tamen angulos habere poílnt, fequitur ad

determinandam quantitatem totius, longitudinem unius falt

lateris darl debere, unde reliqua innotefcant, quemadmodum de

vqualitate triangulor

in geometria differuimus: Dato ieitur

uno latere cum duobus angulis innotefcit per regulas Trigono-

metri

tertius & reliqua duo latera: datis duobus lateribus cum

angulo innotefcit tertium & duo reliqui anguli: datis tribus la-

teribus innotefcunt tres aneguli,

SCHOLION

A.

4. Ut autem dato latere angulus innotéfcat, neceile eft

2

23

ut aliqua proportio determinata inter angulum & latus detur,

quam dari inter angulos & latera oppofita in geometria $. 121I.

infinuavimus, ubi triangu!lum quodvis c1rcu10 infcribi poffe de-

immonftravimus ,

quo efficitur

ut anguli exiftant in peripheria,

& latera fiant chorde eorum arcuum , quorum dimidia anguios

his lateribus oppofitos menfurant; & cum dimidia fint ut inte-

D.

| L.

$ra, etiam inter chordas dimidias & angulos dimidios-proportio

l

fubfiftet; jam cum omnis angulus ad centrum, fit duplus anguli

ad peripheriam eidem arcui inüftentis, erit dimidius angulus ad

Hinc fi ex centro C

|

centrum aqualis angulo ad peripheriam.

]

CFig. Y.) demittatur ad latus oppofitum perpendicularis CE,

|

ea angulum, chordam, & arcum biffecabit ( $. 78. geom.) proin

|

angulus a zqualis fiet angulo 4, & perpendicularis Bm erit di-

midium latus anguli oppofiti ad perlphcrlam & dimidia chorda

!1

arcus BED «etiam determinatam proportionem ad angulum .4

habebit.

SCHOTLTON m

5. Cum igitur ex vertice cujusvis anguli tanquam centro

arcus inter crura interceptus defcribi, & ex extremitate uniu$

lateris ad aliud perpendicularis demitti queat, fequitur hac ra-

tione omnibus angulis, & arcu^us correfpondentes lineas rectàs

proportionales inveniri & determinari poffe, qui pro diverfitate

"du majores qmdem exitant, proportionales tamen femper

ma-

ä

35

NOTIONES GENER, TRIGONOM.

2

maneant, quemadmodum de gradibus circuli mutato licet radio

tribus

in geometria diximus, uti v. g. chorda arcus 60? femper equa-

"ulunm

lis manet radio,

Id igitur bene notandum eft,folum hic aei

| data

de ratione & proportione laterum, quam ad angulos oppofitos

; tj

habent, nullatenus autem de eorum magnitudine abfoluta, qu*

late-

exinde'determinari non poteft, nifü unius faltem lateris magni-

ür M

tudo in triangulo detur,

Hinc jam fequentes definitiones & pro-

dtem

prietates fluunt :

n de

g

DEPFINITIO II

20n0-

i as

Curm

6. Defcripto (F/g. 2.) ex vertice C anguli « ar-

; h-

cu.AB, erit ex B , ubi latus unum occurrit arcui, de-

miffa perpendicularis BD /mus re&us anguli a, & fi A4B

ad F producatur, donec .A4F. fit squalis quadranti,

ef,

du&a FC perpendiculari erit ? angulus complementi

anguli & ad rectum , & BG erit finus hujus anguli 7,

j

I2l

que cum femper fit xqualis linee CD, funt enim, ob

de

perpendiculares GC & BD, parallelze inter parallelas,

d

etiam CD pro finu complementi haberi poterit: eft

jlos

KE

igitur CD finus complementi feu cofimus anguli a & CG

rtio

cofimus anguli P.

a

li

sad

DEFINITIO

IlÍ

Qv

("E,

7. Ere&a in .4 perpendicularis, donec occurrat

01f

alteri lateri produ&o in E, feu AE eft taugens anguli

[

4, & ere&a ex F perpendicularis, donec occurrat ei-

'di

dem lateri produ&to,

feu FH, erit tangens anguli

b, tangens igitur complementi feu cofamgens anguli a.

D.ERGINITIO.IV.

AU

8. Latus CB produ&um, donec occurrat tangenti

i

in E, feu CE eft fecans anguli complementi /, proin cofe-

49

*4nf anguli a.

6

eu

A 2

D &-

CAJIUT .T

L m

By, ?

DEFTNMTTIOT:T7V

peret

qu PA

9. Pars, qua in altero latere, feu in radio CA vper

s dim

finum D abfcinditur, feu pars intercepta inter initium

tintis,

arcus Z4, & ejus finum BD, id eft AD, eft fimus verfus

anguli 4, obtinetur proinde ablato cofinu a radio; cofi-

L

sur verfus eft FG , finus verfus fcilicet anguli comple-

cum fin

menti L.

tendun

COROLLARIUM

l.

eit, d

guli 1

10: Ex datis jam definitionibus fequitur, etiam .47 (Fr'g.

tadio í

2-) eife finum anguli a, cum fit perpendicularis ex pun&o, ubi

den (

unum latus occurrit arcui, ad aliud | latus démiffa, que ex de-

les fin

finitione lI. eft finus.

los t«

|

COROLLARIUM II

finus

!

E

II. Sequitur ex eadem definitione finum anguli obtufi e

( Fig.4.) effe etiam perpendicularem BD; eundem omnino, qui

]

eft acuti anguli b, qui eft prioris o^tufi complementum ad duos

ctefte)

redos; igitur ut habeatur finus;anguli obtufi v.g. I20?, fubtra-

perfit

hatur 120 a I80, refiduum 6O eít angulus complementi, cujus

jot ef

finus fimul eft finus anguli 120?.

bus e

COROLLARIUM Il

lm h

ls

"aheb

I2. Sinum vero anguli re&i C, feu a -- b ( Fiz. 2.) effe

ipfum radium; cum enim una tantum perpendicularis ex punc&o

ad latus duci poífit, radius autem ex conftru&ione fit perpen-

dicularis, & finus etiam ex definitione fit perpendicularis ex eo-

efï, [

dem pun&o, fequitur CF & radium & finum eífe , & tunc finüs

quoque veríus radius eífe debet.

ft r&

Ieitur patet in omni triangu-

fnita:

lo re&angulo, fi hypothenuía habeatur pro radio, cathetos effe

(Bp

finus & cofinus refpedivorum angulorum, ut Fig, 17 oftendit,

fi vero cathetus aíffumatur pro radio, alteram cathetum fore

h pai

$enp.

tangentem anguli oppofiti, hypothenufam fecantem, ut in Fig. 18

at

cxhiþetur.

$ I

COROLLARIUM Iv.

in n

Vllel

13. Sequitur quoque finum quemvis effe dimidlam chordam

p

d

arcus dupli; fi enim (FEig. 5.) arcus BE producatur usque in A,

donec .4E fit wqualis arcui BE, «erit BE.4 arcus duplus, finus

Bn,

NOTIONES GENER. TRIGONOM.

5

o m R

Bn, produ&us ufque in 4 fit chorda arcus BEA, fed cum CE $£t

perpendicular's & arcus biffe&us ex conftrutione, chorda quo-

que BA in n biffecatur, ($. 78. geom.) igitur Bm finus aneuli a

per

eft dimidia chorda arcus dupli angulum totum a -- b menfu-

ti"r'"m

tantis,

erfur

COROLLARIUM V.

j

ple

I4, Uti cr;efcunt arcus, ita quoque finus crefcere patet,

cum finus fint dimidie chordze, chorde autem ut arcus, quos fub-

tendunt, crefcant; hinc (inus anguli o? squalis quoque zero

eít, dein finus confequenter ut arcus creícunt; finum vero an-

guli re&i finum máximum effe patet quoque, cum fit vqualis

Rg.

radio feu dimidie diametro, qu» chordarum maxima eft. a qua

, Ubl

dein finus rurfus per angulos obtufos decrefcunt, cum fint squa-

Q«

(d

les finubus angulerum acutorum complementorum ad duos anegu-

los re&os , donec in angulo obtufo maximo penitus evanefcant ;

finus anguli re&i fus totus vocatur.

COROLLARIUM VI.

Wi a

qui

Is. Crefcentibus finubus decrefcunt cofinus, & viciffim de-

du05

crefcentibus finubus crefcunt cofinus, & cum finus verfus fem-

jtfae

per fit pars re(idua radii ablato cofinu, ($. 9,) patet, quo cofinus ma-

yjus

jor eft, eo (ünum verfum minorem effe, hinc crefcentibus finu-

bus etiam ínus verfi crefcunt; & cum ífinus totus cofinum nul-

lum habeat, finum quoque verfum maximum feu equalem radia

habebit.

elle

[el

M

OROLLARIUMM VItL

H

Qo

el-

16. Tangens cum finu crefcit, ad quem etiam parallela

£0*

eft, cum utraque fit perpendicularis ad C4, & dum aneulus

nus

fit re&us (Fig. 6.) & üinus totus equalis radio, tangens fit in»

finita; latus enim CB eft ipfe finus, proin tangens AE ad latus

u-

ei

CB parallela eft, hinc cum eo nunquam concurrere, proin nul-

lit,

la pars abfcindi poteft. ^ Aucto angulo a ut obtufus evadat , tan-

0fe

gens DF ex altera parte determinatur, & fit wqualis tangenti

anguli complementi ad duos re&os, quemadmodum de finubus

I$

$. It. diximus.

Idem de fecante intellizendum eft, qu* etiam

in angulo re&o infinita fit; cum enim latus ad tangentem pa-

rallelum maneat,

nunquam cum tangente concurrere,

aut pars

m

quepiam pro fecante abfícindi poterit.

4,

us

A

3

Scuo-

CAPUT. L

SCHOLIO!N

E

17. Ut igitur he linez pro calculis omnium anegulorum &

jlaterüm triangulorum

adhiberi queant,

determinata quzedam

fi

menfura unius adhibenda eft, juxta quam line relique exami-

Ti

nari & determinari queant, quod fequenti capite trademus.

les

Ne autem longiores he voces finus, cofinus &c. moleftiam

3n demonftrationibus & calculis adferant, finum fcribemus /im.

,rel f. cofinum Cof, tangentem T. cotangentem Cof. fecantem

fec. cofecantem Cofec. finum verfum Jin. verf, cofünum verfum

lat

Cof. verf.

ad

fur

m

h

CATUT II

DE QUANTITATE SINUUM, COSI-

NUUM, TANGENTIUM &c. DETERMINAN-

)1

DA AD CONSTRUENDAS TABULAS SI-

NUUM.

THEOREMA L

jj

r9. In circulis quibuscunque zequalis eft ratio

radiorum ad finus, fangentes, fecantes, & ad co-

finus, cotangentes &c. &c.

1:;

DEMONSIR—ATIUO

Du&is radiis ca, eb, & CA, CB (Fiz. 7.) in utro-

que circulo ita, ut squalem arcum intercipiant, de-

mitantur finus d & BD, erit triangulum cbd fimile

E ss OM

triangulo CBD. — Nam asgilus c z C ex hypothefi:

D —dquia re&i, ergo B z b, tertius tertio, ergo

iriangula funt fimilia, proin latera proportionalia , er-

go eft CB : BD z o :bd vel CB : CD z :«l &c.

Sed CB & cbfunt radii, BD, CD , & id, cd funt finus

-

Q

k

DE QUANT. SIN, COSIN. TANG. &c.

& cofinus, itidem in triangulis 4CE, ace eft AC : AE

n&

z a6:a4, vel AC : CE — ac : ce &c.

Sed AC & ac

lm

l

funt radii, 4E & ae tangentes, CE & ce fecantes, ergo

radii funt utfinus, ut cofinus, ut tangentes, ut fecan-

am

tes. Q. E. D.

fu.

COROLLARIUM,

em

iftt

I9. H ergo lines refpe&ivam tantum magnitudinem re«

late ad radium habent, ficut magnitudo graduum circuli retate

ad radium coníderatur.

Ieitur (icut octava v. g. pars circuli

funt 450, feu circulus fit magnus feu parvus, ita etiam fíinus an-

D

£uli 45? aut altérius quantitatis femper idem manet, feu circu-

ius magnus feu parvus fit, ad quem refertur,

quia .femper

proportionaliter ad radium affumitur,

THEOREMA II

[.

20. Latera trianguli funt ut finus angulorum

"

A

oppofitorum.

DEMONS.TRATIO.

Infcripto triangulo circuto. latera funt chorde ar-

cuum duplorum angulos oppofitos menfurantium, ( $. 13.)

ergo dimidia latera funt finus arcuum dimidiorum an-

gulos oppofitos menfurantium , cum tota latera fint

chorde, ($. 4.) fed dimidia funt ut tota, ergo latera

funt ut finus angulorum oppofitorum. Q. E. D.

COROLLARIUM I.

2I. Maeg!s geometrice id ita demonf(trari poteft.

In ttian-

gulo ABC (Fig. 8.) ducatur centro C, radio BC, arcus BlI,erit

BD ünus anguli C5 prolonegato latere 4B, centro 44 , radio A4E,

— lateri BC, ducatur arcus EG', erit EF finus anguli 4. Jam

in triangulis fimilibus ABD, AEF et AB: BD — AE: EF

vel, quia .4E — BC ex conftrutione, A4B: BD — BC: EF.

hoc eft latus 4B ad üinum anguli oppofiti C, ut latus BC ad fi-

num anguli oppofiti 4, Q, E. D,

A 4

CORO:

t

—À

p

APUT II

COROLLARIUM I

22. Affumta ergo quantitate quadam pro radio, determi-

Santur in eadem quantitate (inus, & ad hos proportionaliter la-

tera inveniri poterupt, cum fint ut finus, quod inferius decla-

jl

*abimus

l

(J

THEOREMA

III

25. Sinus anguli 30 2qualis eft dimidio ra-

dio circuli.

DEMONSTRATIO

K

&

1

Radius eít chorda arcus 60?, (r154. geom.) fed

dimidia chorda arcus dupli eft finus arcus feu anguli

dimidii, ergo dimidia chorda feu radius eft finus angue

li dimidii, id eft 50*. Q. E. D.

COROLLARIUM

92 4. Determmato ergo radio v, g. Iooo partium, habebxt

finus anguli 30* partes 500, & üinus 90?

erit ad finum 30

ut

Q1OO00 :

500.

Si latus angulo re&o oppofitum in triangulo re-

&angulo habuerit 100 hexapedas, ut inveniatur latus angulo

20* oppofitum, erit ex 'Theor, II. finus totus feu IOOO ad latus

oppofitum feu IOOIW, ut finus 30? feu 500 , ad 150"**,

THEOREMA

IV

25. 'Tangens anguli 45? eft aequalis radio.

DEMONST

A 4 1

(Fig. g.) Cum angulus 4 fit 45*, & in C fit re-

&us, erit in E etiam angulus 45?, proin triangulum

^

ACE 1fofceles,

cam

i duos angulos habeat xquales, er-

go latus AC zz lateri CÉ, fed .4C eft radius & CE eft

cahgens, ergo tangens zqualis radio.eft. Q. E. D

THE-

h

—

— M M—

RE ICNOS

"

—

9

DE QUANT. SIN. COSIN. TANG. &c

THEOREM4A V

tettmy

26. Dato uno ex his quatuor

finu, cofinu,

ter ]

decl

finu verfo, vel cofinu verfo, determinata femel quan-

titate radii

pro quovis arcu reliqua reperiuntur

( Fig. 2.)

DEMONSTRATIO

) [à

Triangulum CBD eft triangulum reCangulum, in

quo quadratum hypothenufe zquale quadratis catheto-

rum, ngtur dato finu BD, ett CD* zz Cb* — BD

feu cof. —

(R* — fm.*)

) fd

2*. dato cofim, ett BD* z CB* — CD*, feu./fin

mparemer

Lsl

Dgul

agl

V^ (* — af*?)

*. dato cofinu eft D44 — CA4 — CD, feu /m

verf. z R — cof. vel dato finu verfo et CD — CA

yebrt

4*. Dato finu BD, quia BD — GC, erit FG z CF

0

ut

— CG, feu cof. verf. & R — fin. vel dato cofinu verfo

) f€

erit CG — CF — FG, feu fnus — R —cof.. Q. E. D

eulo

latus

COROLLARIU fHMM.

hablto

27 Hinc inuento finu 40*, habetur finus 50

enim finu de 4O habetur ejus cofinus, qui eft finus anguh *O,

idem de aliis intelligendum eft.

4 H EUORERBEBMÁ.A —Vi

29. Dsto ex his tribus

finu

COS

& tang

duobus invenitur tertíium cum fecante, cofecante

te

& cotangente

um

DEMONSTRATIO.

er

ell

(Fir. 2.) Sunt enim triangula CBD, »AE, item

tria nguld CBG & CFH 6milis, & cum fit CBG femper

equale cum CBD ob latera GB — CD, & GC z2 BD

A $5

&

i2

E

——

m s

IO

CA PUT-.'I.

& CB commune utrique, in omni cafu erit etiam CAE

fimile FHC, hinc erit (radium fupponimus effe unitatem,

hinc in multiplicatione & divifione omittitur) CD :

fjl

DB z U08: ÁME, hoc et £. w um R

T, ergo

fin

i

ImO

Va

ucueeym

—— al o u m s. x T. vel g. m

.

d

, b

. Rurfus et CG: CF— GB: FH, & quia CG — BD

ek

erit BD: CF — GB:FH, hoc eft /in, R — cof. : cat,

fit

—

ergo 4?. cot, —

vel 59. cof, z cot. X fin. vel 6^,

T

"

d

€0

:

K

Jm. m c-t?, Item eft AE: CA22 CF : FH, fed CFzz

g

CA, ergo AE : CA z CA: FH, hoc et: T: & —

in

—

Tan-

R:ct. ergo . T — Zt., & 9?. eot, —

T

fit

dem eft BD : AE —— CB : CE, & BG : FH - CB:CH,

t

hoc eft /m. : T'z—R :/fec. ; & cof. : cot, z R cofec, ergo

(

9. féc.

d

& IO*, cofec, z

;-.—

7»

Q. E. D.

Jfn.

J

COROLLARIUM.

29. Habetur ergo ex 2*. & 6'. i. — cof. X T —

COÉ.

ïg

ex 3* & 5*,cof. z — —jX tot.

2

fmn. -

L

i

ex D n TER —

—

H

cof.

cof

V ODL. Em i

cof.

|

ex 4^. & 8

ä:

THE-

n d

E

DE QUANT. SIN. COSIN. TANG. &c

II

AE

THEOREMZA VII

!tnm,

J)

40. Inventis üis lineis, quaze ad quemvis an-

gulum pertinent ,

determinantur etiam ea, qua

fgO

ad ejus dimidium vel duplum pertinent.

DEMONSTRATIO.

BD

Detur finus BD anguli BCA, ( Fig, ro.) ducta

chorda B4, & radio ad eam perpendlculan CE, angulus

at

C & chorda AB biffecabitur in. F, & erit BF, ex con-

ftru&ione & definitione, finus' anguli dlmldll feu £C.

9.

in triangulo re&angulo CBD datur finus BD, proin

datur cofinus CD, ($. 25.) qui a radio CA4 fubtra&us

l

p

relinquit finum verfum feu partem D A, igitur in trian-

gulo BD datur cathetus BD, & cathetus DA, proin

memi

invenitur' hypothenufa B.A

íft enim BD* -- DA* z

Atl

B/^; extra&ta radix dat B4, & divifa per 2 dat que

fitum finum £* £4 — BF. Cofinus CF ex theorema-

H

te V. erit z

(R* — AF»),

"o

Si detur angulus ECA (Fg II.) producatur ar-

cus, donec EB fiat — AE, erit BCA angulus duplus

anguh dati, & BD erit flnus anguli duph, produ&a

chorda 4 usque in B, erit triangulum BDA fimile

triangulo CEA; eft enim in F et D rectus, a ad utrum-

0j

que pertinet, ergo P —— c, ergo erit CA: CF zz AE

df

BD, hoc eft hypothenufa ad cathetum majorem in utro-

que triangulo, fed CA eft radius, CF

ft cof, anguli

dati proin notus,

BA eft duplum de FA4, proin duplus

finus datus, ergo etiam quartus termmus BD finus anguli

dupli innotefcit.

Q. E. D.

COROLLARIUM

2I. Hinc dato finu ane2uli 30

1nnotefc1t finus angnh »PODO-

&

hoc noto habetur etiam finus anguli 7? 30 ; 3? 45 &c,

ng

(TAPÀ

tandem poft duodecimam diuifionem finus anguli 52" 44" 3

n"n70g9

45

*

THEO-

— m i

Y2

CAPUT

IT.

.THEOREMA VIIT.

42. Datis finubus. duorum arcuum habetur etiam

finus fummze utriufque: id eft ( Fig. 12.) dato finu

(ne

BD anguli 4 & finu KL anguli b inuenitur finus,K M

M0

lum (

totius anguli C feu 4 -4r- D.

OLz

DEMONSTRATIO.

O;W P

etpo

Ex L demittatur LP perpendicularis proin paral.

K0]

lela ad BD, ducatur perpendicularis LO, erit OM —— LP

QN

parallele inter parallelas ; invento OM -4- OK habebi-

tur KM. ]Jam ob CBD, CLP fimilia habetur CB:

0b

fim

CLzzBD: LP, hoceft R: cof anguli dati », ut /fm.

(

anguli dati 4, ad LP, cui xquale OM fubftitui poteft,

pte

eft ergo OM z cof. b X fin. a$. radium enim unitati

ep

equalem ponimus, hinc omittitur; porro triangulum

Klk

KOL « OQL ob perpendicularem OL ex angulo

upe

re&o ad hypothenufam demiffam ( $. 1371.geom.) Item

OQL w QCM ob verticales Q, &rec&os ad 0 & M;

& COM « BCD ob parallelas OM & BD, igitur

KOL wv CBD, ergo erit CB: CD— KL: K0 hypo-

EW

thenufà ad cathetum majorem in utroque, hoceft: R;

cof. ang, a z fin. ang. b:. KO ; ergo KO Cof. a x fíin.

etit

AB

b, igitur KO -- OM, feu /ft, a - b — fin, a X cof.

HI

b -- fin. b x cof. 2. Q. E. D.

£H

p

ten

THEOREMA

IX

n

[

45. Datis finubus duorum arcuum habetur etiam

fex

finus differentize utriusque, id eft in Eig. 1. dato

angulo 4 toto, & ejus finu KL,, dato b & ejus

finu BD, datur etiam angulus & & finus differen-

3

tie K M.

DEMON-

.

x—.

ER

DE QUANT. SIN. COSIN. TANG. &c

t3

DEMONSTRATID0

tram

Du&a perpendiculari LP, & prolongsta KM in O

ül ii

erit

7

donec linee LO parellele ad JP7M occurrat

"

MO z LP ob parallelas inter parallelas, & triangu-

lum CBD «» CLP ob parallelas CP & BD, igitur CB

CL -L—BEBD L/J&qula LP —0M,erit CB: CL — BD

OM, hoc eft: K ad cof. auguli a —— ffn, anguli b ad O.M.

ul

ergo OM z vcof

X fm..b; rurfus eff triangulum

KOL w KOM^ ob parallelas LO & QM; KOM v QPL

.L

ob verticiles ad O , & M & P re&os, & OPL v PLC

Debi

(B:

ob PL perpendicularem ex angulo re&o ad hypothenu-

fam; & PLC v CBD,

ergo, KOL v CBD,

ergo eft

t fi,

CB: CD z KL: KO, hypothenufa ad cathetum ma-

el

jorem in utroque, hoceft: R ad cof; b — /fin. a ad KO,

titati

ergo KO z cof. b X fín a, fed KM z K0 — 4M0, ergo

jlum

KM zz cof. b X fin. a — cof. a X fm. b

, que formula à

go

fuperiori folo figuo differt

(em

M;

COROLLARIUM

1.

LUE

4. Utrumque in Fig. I4, exhibetur

EF eft finus 6, BD

7p0-

finus a, EG /in. a-4- b feu finus fumms. Produ&o EF ufque in H

IH

erit ob chordam biffetam EF

EH, &proin EB—BH, erso

d7

AB — EB id et 4B — BH —— AH, ergo .A4H arcus differentie, &

if

HI ünus differentie

Du&is parallelis FK, HL erit ob EF:

FH-EK: KL etian EK — KL; Jam ob triangula fimilia ut

prius CHG, CFP, CBD, MEF, FEK erit CB: BD —CF: Fp

item CB:

C'D——-EF ER. ergo FP feu il zquale KG, & EK

innotefcunt ,

fed FP feu KG -- EK —— EG í[inui fummæ &

FP — EK ideft FP — KL feu FP — FN z NP — HI finui dif-

ml

ferentie

lato

COROLLARIUM II

awä

5 Ex his jam duobus theorematlbus invento finu anguh

n-

X & Is* habeb1tur finus 45

25*&'c iiém éx go" & 2*

39 habebitur 92?

30' &c,

THEO-

2

14

CAPUT-H

THEOR E MA X

!(

36. Sifinus K/M arcus K minoris 40 gra-

Qp

dibus addatur radici tripli quadrati KI finus diffe-

It

rentize arcus ejufdem K.4 a 40^, habetur finus EN

arcus F/4, qui tantundem excedit 50 gradus, quan-

fum arcus K A ab iis deficit ( Pig. 15. ) id eft EN

& 60

—VylK -- KA.

( -

QUE aC

DEMONSTRATIO

Sit arcus BA z 40^, BK transferatur in BF, dus

catur FK , qua in J per CB erit biffe&a, ducatur finus

FN; & OK parallela ad CA

erit SIF — SGQ. ob

fintum

Akwy

re&um in J & G & verticales ad $, igitur tertius 7 z

tertio a, qui ob parallelas QOG, .CV aqualis eft an«

U r

gulo c, qui cum fit 30*, etiam «, &7 erunt 30?, ergo in

dee

yus &

triangulo re&angulo FGK, fi KF affumatur pro radio,

proced

erit GK finus anguli 7 (&. 6.) feu finus 30*, qui finus

fni

cum fit equalis dimidio radio ($. 23.) erit GK —— 1 FK

$000

GK — IK vel FI

angyl

igitur

Eít autem in triangulo

Yf

re&angulo FGK, FK^— GK*zz FG*

feu quia FK

Tent

biffe&um eft in J, poterit 21 illi fubftitui, igitur loco

6^

FK* poni 4JK*

& loco GK illi equale IK , ergo erit

AIK* — IK* - FPG* hoc eft:/541K^ — FG^, ergo

ad f

V5le —

FG, & cum GN fit equale K/M, erit to-

Tx

Tünt |

tum FN z [/31K2-1—KM Q. E. D.

üimi

Qu i

THEOREMA XI

er

Ü

41. Si ad finum FT arcus FH (Fiz. 15.), minoris

k f

6o*, addatur finus F] arcus differentize a 60*, habebitur

Qnd

N n

finas KO arcus HK, qui tantum excedit o^ quan-

T

tum HF ab illis deficit, id et KO FT -- FI.

Tuy

fum

DEMON-

DE QUANT. SIN. COSIN. TANG. &c

15

DEMONSTRATIO

KO enim confiat ex GO & KG

fsd GO

FT

g

ob parallelifmnum , & KG z— FI ex priori demonftratio-

]iffe

ne, ergo KOzzFT-r- FI

Q. E. D

EN

COROLLARISU !DMM.

]u

E

38. Ita (inus42^ -- fin. I8^ — fin. 78, quia 60 — 42 — 18,

& 60 -- 18 — 78

Item fin, 53 -- (in. 7

fin. 67, quia

60-—— 5327 & 60-]- 7 — 67 &c, hac methodo alli a 6O vf-

que ad 9Q inveniuntur

, ür

CHOLION L

finus

) ob

39. His jam principiis innititur conftru&io tabularum, qua

w

finuum & tangentium vocantur: & fub hoc nomine vulgatz funt,

sl

L

Aifummitur in iis radius circuli, ad quem angulus datus refer-

ane

tur IOOO0OOOOO00O0 , & proportionales ad hanc quantitatem chore

de earumque dmudw. feu finus quzruntur, quibus inventis cofi-

(

)

nus & tangentes cum reliquis lineis determinantur

lta autem

dio,

proceiferunt ii, qui tabulas compofuerunt

Ex Theor. IIÀY. cum

h-

finus 30

fit zqualis dimidio

radio ,

habetur finus 30" —

$000000000;

cum angulus complementi fit.GO , erit coíinug

,

anguli 30

fimul finus aneuli 60

qui invenitur ex Theor.

)

"ulo

V. fubtra&o quadrato finus 30? a quadrato radii, & ex diffe-

FK

rentia extraca radice, qux erit zqualis 8660254, habetur finus

6? equalis 8660254,

CO

erit

Habitis his (inubus ulterius juxta TheorzVIII, procefferunt

ad finus dimidii & dupli, inveneruntque finum I5

7 30 &e,

f go

Ex his per Theor, IX, invento finu fummz & diífferentie erue-

0«

*

runt finum 45

?

75 5 37 39 &c. qu$ iterum ex Theor, VIHT

dimidia & dupla dabant, uti 22

305 18 45 &c,

quae rurfus

cum inventis comblnata fummas & differentias dabant,

ti 67

E

: 56? 15': 82? 30' &c,

Per finum anguli 7 3O' habetur 2 l

52' 30'' &c.

j

& üc per blü"eötlones ad finum valde exiguum aliquot tantum fe-

LE

cundorum 52"

54 34

'" devenitur, qui ab arcu ipfo fenübili-

r

ter non differt, proinde illi proportionalis eft, hinc fieri pote-

n

Ylt proportio

ut hic arcus ad fuum. fnum

lta arcus

I

ad

habetu

fuum finum, habito 1/ habetur duplun I xX 2

2 ;

fumma 2 -- 1 — 3' &c

Ta

A

na

dp st

16 CAPUT II. DE QUANT. SIN. &c

u

Tandem pro teliquis refiduis determinandis, qui nec per

fummas & differentias n

per duplum & dimidium, nec ratione

cofinus habebantur,

recurr

unt ad 'Theo:

X. & XI

& finus

omnes a 3O ufque ad 60 , & a 60

ufque ad 90 methodo i^i-

dem prafcripta invenerunt, & tabulas finuum confecerunt, qui-

bus competentes tangentes & fecantes ope folius regulz aurez ad-

DE

diderunt; fecantibus hodie non utimur, cum problemata omnia

trigonometrie fine fecantibus folui poffint.

SCHOLION II

40. At habitis his finubus cum refolutio triangulorum per

regulam proportionum fieri debeat

multiplicatio igitur medio-

rum & divifio fa&i per primum facienda fit, fequitur, ad loga-

rithmos iterum recurrendum eífe, quorum ope multiplicatio me-

Lo-

E

diorum in additionem , & divifio in fubtra&ionem mutatur,

garithmi hi ex fabulis logarithmorum, de quibus in algebra egi-

lii

mus, conftru&i funt, magno fane labore, atque vel foli, vel

ipfs tangentibus & finubus annexi in tabulis reperiuntur.

Me-

thodus autem , qua confe&i funt, diverfa ab illa non eft, quamr

jbidem expofuimus,

ita finus 30^ qui eft z— 5 , adjecis novem

zeris habebit logarithmum . 6989700 przefxa chara&eriítica 9.

potti

qualis in*tabulis numerorum naturalium numeri 5, reperitur,

Oppo

Aliorum finuum correfpondentes logarithmi per regulam propor-

tionum queruntur, quam ibidem in Prob, III. tradidimus, 1igl-

Ye] 1

£ur diutius his immorandum non effe cenfeo.

Hos losarithmos

finum

in ordinem redegit Vlacquus & ordinarie fub nomine tabule

tiote

Viaqui indigitantur,

Gardinerus Londini accuratiores impreffit,

I"fbl

adje&is lovanthmxs fecundorum. omnium pro 72 minutis primis,

qux Avignone re*ampreffæ funt additis logarithmis pro omnibus

gus

fecundis ad quartum gradum ufque, quz ingentem utilitatem ma-

tur;

joribus in calculis adferunt.

In ufu quotidiano & menfurationi-

d ef

bus ordinariis ez fuffciunt, quse logarithmos fingulorum minus

torum continent, pro fecundis enim íi opus fuerit, per propor-

[

tiones facille determinantur

Il ge

SCHOLIFON

IH

müc,

AT.

Ufum tabularum methodumque inveniendi loganthmos

intermedios inferius dabimus,

L

EM LUE RI EAC ISFGFRnz SMIEU AAE

T UE m n R m TA

QUs

C A-

l,

À

AE

O

MM

EZORSK

17

: per

um getset

ISGONGESM

tione

finus

jhi«

CA.PUT II

qui-

: ade

mitia

DE PRINCIPIS CALCULI TRIGONO-

METRIC[ IN RESOLUTIONE TRIAN-

GULORUM.

n pet

edio

THEOREMATL

ln

42. Si inter duo ex tribus datis fit oppofitio,

) me-

Lo-

id eft: fi latus & angulus datu

, Et«

liqua hoc ipfo innotefcent.

$ fibi opponantqr, re-

,d

Me

quami

DEMONSTR'ATIO.

orem

a 9

Inter finus angulorum & latera oþpofita datur pro;

ritut,

portio $. ao. eft enim finus anguli uniu$ ad latus fibi

oppofitum, ut finus alterius ad latus huic oppofitum 2

opor-

L

vel latus ad finum anguli oppofiti, ut latus aliud ad

hmos

finum anguli huic lateri oppofiti, fed datis in propor-

jbua

tione tribus terminis quartus invenitur, (veg. auv. im

refít,

rinisy

elgebra) ergo ubi angulus & latus óppofitum datur, an-

gibus

gulus quoque vel]latus dato tertio oppofitum inveni-

) m

tioni»

da eft,

tur; que operatio pro toto triangulo bis inítituen-

mn.

In refolutione pra&ica attendi debet ad illud, quod

.ægofl

in geom. $. I29. de conftru&ione triangulorum dixi-

mus, & quod inferius $. 6r. repetemus..

of

THEOREMS.SA i.

43. Datis duobus lateribus in triangulo rectan-

ulo invenitur tertium,

Ld

Metzb, Maih, P. I1I.

g

DE-

18

CAPUT T

tad

DEMONSTRATIO.

e

In friangulo re&angulo quadratum hypothenufe eft

r

gequale quadratis cathetorum fimul fumptis, ergo datis

n

duobus lateribus dantur eorum quadrata, quorum fum-

eati

ma vel differentia dat quadratum lateris tertii, ex que

extra&a radix Vdat ipfum latus. Q. E. D.

:

^«

"A

THEOR

]

m

l*(ï A. III-

faf

44. ln frianqulo rectangulo finus eft ad cofi-

num ut catlietus ad cathetum.

DEMONSTRATIO.

,

Eftenim in triangulo abc reCtangulo (Fig. Y7.) be

finus & «c cofinus anguli a, vel «e finus & bc cofinus

diio

ieitg

anguli ? ex definit. II. $. 6. pro diverfitate centri, quod

velin a vel » fumitur; igitur eft finus ad cofinum ut

Cath

ade

cgthetus ad cathetum. Q. E. D.

fpef

t

THEOREMA IV

48. In triangulo rectangulo radius eff ad hy-

pothenufam, utfinus unius eX acutis ad cathetum

ïïï()q

oppofitam ( Fig, 17.) vel cofinus ad cathetum ad-

pred

jácentem.

patte

(eque

DEMONSTRATIO.

Tteru

&ultm

tertia

Si enim ex acuto angulo « defcribatur arcus radio

Veniel

ab, erit ab fimul hypothenufa & radius, /c autem ex de-

Yanat

finitione finu$ anguli « & fimul cathetus; igitur eft ra-

Tüy ;

ht

dius ad hbypothenufam, ut finus ad oppofitam cathetum;

squalium enim eadem eft xatio.

]dem erit fi cengfo b

radio

3

d-

a

DE PRINC. CALC. TRIGQN. IN RES..&c. 19

tadio 4» defcribatur arcus, erit t;c- finus agguli » & fimul

eathetus, j« autem hypothenufa & radius, igitur eadein

T

qua prius proportio.

Q; E. D.

Data igitur hypothe:

latis

Afl«

nufa in pedibus aut alia meníura, invenitur in eadem

que

cathetus angulo dato oppofita,

THÉOREMA V.

46. In omni triangulo rectangulo eft radius ad

eofi

tangentem, ut cathetus ad cathetum (Fig. 18.).

DEMONSTRATIO,

Si enim catheto ae tanquarb radio defcribatur arcus

sd, erit ex conftru&tione Pc tangens anguli «, . & fi ra-

v)þl

dio /c deícribatur arcus ce, erit ac tangens anguli 2, etl

finus

igitur ac. fimul radius & cathetus, /c fimul tangens &

;uod

f üt

cathetus, ergo eít radius ad tangentem. ut cathetus

ad cathetum , & cum altera cathetus fit cotangens re-

Ípe&u anguli complementi, etiam erit , ad cotangen-

N

tem ut cathetus ad cathetum. Q. E. D.

COROLLARIUM

hy.

47, Patet.jam ex his theorematibus, cum angulue rectué

2u

alioquin femper detur, & notus fit, fi duo a!ja dentur, per

| i

predi&as proportiones triangulum re&angulum folui, & omnes

partes determinari ; proportiotes hz, ut facilius inveniantur ,

fequenti tabula in ordinem reda&s funt.

Columna prima nu-

merum proportionum exhibet, altera data 1lla, qusc proeter an-

&ulum re&um nóta funt, velduo latera, *el aneulum & latus,

tertia illud , quod queritut, quartà proportionem, qu» ad in-

fidio

veniendum illud, quod quieritur,

adhibenda eft, quinta theo-

remata , "unde regzüla proportionum defumitut, & fexta nume-

: de

[

rum figure in fummitate eifermate, quos ad eam propqrtionem

pertipnet,

tullly

(tü !

Ll

Gaíus,

liO

u-

* fner n

u

20

CAPU T.HI

yitli

L

*

C

O..'

lu-

tb

2![(2/

Ca-

.L)d.ïao

Q *

ue

»

fus,

rene

4 X

da,

(*/ï P»/fj Bþ Cl

BC

Bc — g/(CA

- AB^)

Th.

ïþ

w

AR WHCocemR

Th,

*

^

AB, AC|. B

Tang.. B

AC

AB

M

ezdtdams.C

MRU

n

z

e

|

z

AU

Quor

——

AC

AC

( BC* — AB*)

h

1

dem

AB

Bc| B

BC: Ab

—— R

Cof, B

Th

Hl

*

P5

iis |

BC; AB

L

Sin,

u

Tn; Il5

l

z u

|

*

L.I N

ÁÁ

Q1f

AB

AB zv (BC^—.AC^)

Th, T

[

AC, BC| B

BC: ACz R: Sin. B

1 n II]—»

*

ZZZ

*

BC: AC-— R: Cof.C

Th III

l

MO CRO

AC

AC

R: Jang.

B — AB

FhO V.

*

ä

?

AB, B

BC

Qu B: ( m AB

BC

Th

Til.

z 4

*

t9 O2

|

r t ] r r

|

J

R

Cot, C.zz3 Ab

4€

rTh

V

L

4

*

t

WEB, C

"

BC

Th III

Z

*

PA t2

:

rE RE iA

BC

Sin C: R. AB

k

1gltm

AB

f)

Cot

Bo C

-B

T.

VO

*

—

14 4c

B

-

4

I5

Bbc

Sip;B: R— At

BC

Fh, IIT

z u

[* :

(

B du

16

AB

R: Tang

C...—

4C

AB

Yh

XT'

*

BA

i. £

BC

COl C- R L54€

BC

Th

Wl.

e u

E

| N

literj

R

AB

Ln;

Iu

L]

I8

BC B

€of, B —— BC

fünt :

T9

AC

R

$Sin

B

BC

AC

Ih

l,

z

*

Ro k

"

Fh

W ;

20

AB

Sin

Fa .BC,

AB

i.

*

2I

BC. C

AC

Coí; C « DC

AC

Th,

liE

z

CEELI

Qp

fem

el

SCHOLIO?S

uule

Winu

48. n formula 4 eft (BC* -—-—ABQ), fed conftat ex al-

plus

gebta factum BC* —- AB^ oriri, fi BCf* .4B ducatur in BC—— AB

3eitur fi loco BC -- AB (ubflituatur ejus logarithmus,

& 1loco

Ug ;

BC — ÁB itidem ejus logarithmus, erit fumma horum lozgarith-

äï)gn

morum equalisfa&o ipfarum qvantitatum ; per logarithmos enim

$huMiplicatio in additionem mutatur,

Si jam hec fumma divi-

oæq

c

datur per 2, qu divifio pro extra&ione radicis adhibetur, ha-

bebitut losarithmus radicis equalis lateri AC

Idem de fore

Qe

mula 7 notandum,

Regula hec fic enuntiatur

feimnifumma logae

38

n

vithmno-

"

benr- v d

—

ia

DE PRINC. CALC. TRIGON. IN RES. &c. 21

!

vithmerum fumme &

& differentie ex hypothenufa G caflz to una, eft

d

m

C

a'gualzs logaritluno catheti altevius,

THEOREMÁA . VI

| LX

A nE

49. In omni triangulo rectilineo eft fumma

peu u dELDE MM

—

l

CO2 x

duorum laterum Z4C -- 44 B ad differentiam. eorum-

N

dem AC — AB , ut tangens femifummse angulorum

N

N

üis lateribus oppofitorum b-t- c ad tangentem íemis

MI

differentize eoruridem angulorum. (Eö. 16.)

A.2

N2

DEMONSTRATIO

N

(UN O

Latere /AB minimo defcribatur mrculus & produ-

AB z AEs;

eatur /4C ufque ad peripheriam, erit. DA z

J 1

ï

igitur DC z AD -- AC fummz laterum , & EC zz AC

H

N,4

D differentie laterum; conjungatur B.E, & ex D per

N.2

B dueatur indefinita DBF, demum

N d.

exv

C ducatur ad

i

N À!

BE yarallela FC; jam »--c eft fumina angulorum datis

['N. 2

- ——

lateribus oppofitorum, «qui utpote interni ümul fumti

NI

funt equales externo o, ergo etiam o e(t squalis fum-

,h 21

—i

ma angulorum , qui cum fit a

ngulus ad centrum, eft

N,1|

duplus anguli ad peripheriam ?, igitur ; ef —— fo-—

[4

N.2|

femifumm:» angulorum »-f-c, & ob paralleias RE & FC

eft ; zquale « -- e, ergo etiam e -- e z femifumms. Jam

confítat ex algebra $. o9'7. ex duabus quantitatibus ter-

minum majorem eífe zqualem femifumma terminorum

ex al*

plus femidifferentia, minorem vero zsqualem femifum-

,AB,

m minus femidifferentia terminorum , icgitur cum c fi

& 1000

angulus datus minor utpote minori lateri oppofitus, erit

Oaaï fh

0f emr'l

is'zqualis femifumms minus femidifferentia , fed angu-

ius c habetur; fi a femifumma ec--c fubtrahatur e, 1gt-

j dv

uï, ha

tur e debet effe femidifferentia, eít igitur « -i- €

Inis

if

ü*nma & e femidiiferentia.

Praterea patet ex CODs

j*

Qie

t

33

22

CAPUT HL

ftru&ione, cum DBE fit in femicirculo , in B effe

re&um ( $. 93. geom.), proin etiam in F ob paralle-

lifmum re&us effe debet , igitur DF haberi poteft pro

!

tangente anguli « -- e, & BF pro tangente anguli e, ( $.

I2.) ergo eft DF tangens femilumme, & BF tangens

It

femidifferentiz angulorum.

jam in triangulis D&£E &

t

DFO, ob parallelas BE & FC, ett DC: EC — DF:

BF, fed DC eft fümma laterum , EC differentia late-

j

rum, DF tangens femifumms angulorum, bF tangens

Hs

F

femidifferentie , igitur eft fünma laterum ad differemiam

u

laterum , ut taugeus femifumma ad tangentem [emudi]jerenie

€

angulerum, Q. k D.

t

COROLLARIUM.

^

l

li

£O. $i igitur fit refolvendum triangulum obliquangulum, ln

«uo duo latera cum angulo intercepto dantur,

hoc ipfo etiam

fumma reliquorum duorum nota erit; fubtra&o enim angulo dato

i

Jum

a 180^,relinquitur fumma duorum aliorum ($.I17. geom.), qua

divifa per 9 dat femifummam, queratur hujus tangens ex tábulis,

& fiat per Theor, precedens: fumma laterum datorumr ad diffe-

fentiam eorundem, ut tangens femifumma angulorum prius in-

vente, ad tangentem femidifferentiz,

que femidifferentia ad-

dita femifummze priori dat angulum majori lateri oppofitum,

E

fubtra&a dat angulum minori lateri oppofitum , quibus habitig

datur oppoíitio, & invenitur latus tertium per Tlleor. L, $. A2«

ii

$«

iv

THEOREMRA VII.

Vet

$1. Siad latus maximum-vex- angulo oppofito

demittatur perpendicularis, erit latus maximum ad

fummam reliquorum ut differentia eorundem ad dif-

ferentiam fegmentorum per demilïam perpendicula-

rem facdtorum. (Fzg. 19.) |

DEMON-

DE PRINC. CALC. TRIGON. IN RES. &e. 23

effs

DEMONSTRATIO.

alle«

Latere minimo 44B tanquam radio fiaf: circulus ,

pro

(j

toducatur BCin G, erit GC fumma laterum BC-- AB ;

& CE differentia laterum &C — B ob BE zz BA,

;ens

Jtem eft FC differentia fegmentorum 5 nam per BD ]1a-

L

)F:

tus AC dividitur in 4D & DC, fed 4D — DF ob per-

ate

pendicularem ad chordam ex centro demiffam ( $. 78.

geom.) ergo DC — 4D zs DC — DF — FPC, ergo

jc

ii

"FC differentia f«gmentorum ; afqui' conflat-ex geome-

(ittt

(EA

tria g ISo. totam (ecantem CA effe 2d partem alterius

extra circulum CE, ut altera tota fedáns CG ad par-

tem prioris CF, hoc eft: C4: CEza CG: CF, fed CA

eft latus maximum, CE differentia laterum, CG fumma

laterum , CF autem differentia fegmentorum, igitur eft

latus maximum od diffeveutiam latevum , ut. fumma laterum. ad

m

:tiam

differeutiam égmentorum; & alternando, latus maximum ad

dito

qua

Jummam laterum &c. — Q. E D,

julisy

COROLLARIU M.

qife

g i

j àd*

-

52. Datis igitur tribus lateribus, ut inveniantur anguli,

fiat proportio predi&a ad inveniendum FC, quod FC fubtratum

itufty

abitif

a latere maximo AC relinquit AF, cujus dimidium e& .4D, &

idem dimidium additum ad FC dat DC, igitur in triangulig

2

zecangulis 4D2B & BDC nota funt duo latera cum angulo recdto,

jnveniuntur ergo per Theor, V, anguli 44 & €, quibus notis ha«

betur tertius B.

Sequens tabula omnes hos cafus exhibet,

fito.

gäd

üf

la.

B 4

€aíius,

(

j*

1

24

CAPUT m

DW

rat

Ca- |Data, ; Quz-

fem

,

ren-

'fus.

da.

u

aihi

rith

AB

BO iSim, C: AB z Sim, 4:,BC

Ob op- [N, I

&

Sin. C: AB z Simn. B:.AC

pofition,|vel 9

fun.

AC |B. enim dunbps notis innotefcit,| Th, I.,

gyan

Til

AB

C |BC: Sm. A: AB: Sin.C,

Ob op-

N. I

BC

| &

Sim A: BC z Sin, Bs AC

pofition,jvel 2

l

AC

noto C & À nrotus fit B.

Th. L

LO

AC

*

|

dB - 4Cc: AB — ACz Ti

eud

AB

&

(C-- B): T£ (C— B)

Theof. N. I

B & CX Sum. -- 7 diff. — C.

' VI.

999

Tiet

$ $Sum.— £ diff.— B.

habito C & B eft oppofitio,

T

lum

AC

A^lA4C: BCM- AB z BC — AB:x

AB

Theor,

NO 3

BC

B |4C—x-—AD& 4C—AD-—DC

V,

i

E

j

den,

AB: AD z R:Cof. A

tuor

Bg; L.)-C':.-..- 3: Cof, C

coli

dex

scHöLloN.

col

$3. His jam principlis, qusx omnes cafus refolutionis tri-

Di«

te 1

angulorum complecuntur , tote practica refolufio innititur.,

fpiclendum fcilicet eft, ad quam «ex predi&is preportionibus

2,1

m

cafus, de quo fermo eft, reduci poffit, vt juxta regulas ibi-

dem traditas ignota invenlantur.

Utimur autem in his propore

yd

tionibus tabulis logarithmorum, que in fine annexe reperiunturs

|

Vr

logarithmi laterum in tabula numerorum naturalium queruntur,

ünuum vero & tangentium in tabulis finuum.

f

Tier

EXPLICATIO TABULARUAM

pri

LOGARITHMICAR U M.

fe

Ti

Ttoi

.54. Ordo fequens in his tabulis eft: prima

pagina logarithmos numerorum. naturalium. ab

ft;.

I ufjue YoO continet 5a, ut numerus ipfe qua-

j

ratur

DE PRINC. CALC. TRIGON. IN RES. &c. 25

ratur in prima columna, G in fequente flatim

fecunda. logarithmus illius: in tertia 8 quinta

iterum numerus, in quarta vero & fexta loga-

d

rithmi correfpondentes ; characterifticam - loga-

N1

rithmis non prafiximus, eo quod ex algebra no-

vel 2

tum fit, " charadleriflicam unitate minorem effe

quam numerus notarum , cujus logarithmus quo-

MI

ritur, hinc facile quiuis eam ipfe prafiget , ab

Yel 2

X ad 1O ftilicet charadleriflica prafigitur o, &

n je€-

IO vero ufqgue ad 1co prafgitur 1 Gc.

cunda pagina logarithmi a 1cO incipiunt & ad

NI

999 feu icco continuantur , qui fimul pro nu-

merl$ a 1000 ad 10000 férviunt; i trium nota-

rum efl numerus in prima cujusvis pagina co-

lumna quaritur, 63 immediate ilh ad dextram

Ó 3

in fecunda adfceriptus eft. logarithmus correfpon-

ens, cui prafigenda efl charadleriflica 2. ft qua-

4uor notarum numerus detur, prime tres in prima

columna querantur, dein in eadem ferie verfus

dextram tamdiu procedendum eft , donec ad eam

columnam perveniatur, cul fuperius in fummuta-

; ttl-

te ultima nota data refpondet vel o, vel 1, vel

Di

2, vel3. Gc.; quatuor veltres nota in hac co-

nibuf

; ii«

lumna occurrentes adjungende fünt üs duabus

vel tribus , qua in fecunda columna , cui fuperius

jpote

'tüt

xerus o adfcriptus eft, puncío a reliquis quatuor

H

feparantur , & omnes-fex fimul logarithmum nu-

meri dati effidiunt, ita pro 1915, quaeriüur im

prima columna 19x , dein procedendo ufque ad

Jextam columnam a dextris, ubi fuperius 5 legi-

mus, inveniuntur 2169, qua adjundis duabus

notis 28, ante pundium in fecunda columna po-

"u

db

Jitis, dant 382169; logarithmus igitur qu& LUS

pnia noia

t

5 has

- €fü4.282169. lta Iggs.gdantfubfupre

26

CAPUT II

lur

Lj fza.& quatuor 3363, que adjediüs prz'mis' 29

nu

dant logarithmum 3.293363 ; & fic de reliquis.

dii

In uitima columna. differentiz minoris a. proxt

D)

me majori logarithmo annotantur. d unice no-

(

tandum, fi in his columnis e quatuor notis prima

Jit minor quam ea, quo e quatuor ultimis prima

(ua

7i

(

poft punctum in columna fecunda flat, tunc pra-

cedens feu prima ante punctum nota unitate au-

ger l

jri

genda efl. lta pro 1906 reperiuntur note qua-

tuor O124, quibus non 9] ex fecunda columna,

a

fed 28 preponendum efli pro 1909 reperitur

$9A

C806,

0i

quibus rurfus non 27 fed 28 propont

3

debet, . Partem jäqueng tabella exhibet.

Jl

2

Nnm-' o II i'2 !

[

ENESUBENERER:

190 |27.8754|8982|9210

0123 |0351 '0578 |0806|228

|

943919667 |9895

jae

I9I |28.1033|1216|1488

1715|1942'2169

2395 0622 2848 3075/227

€0

"m

192

330113527|375313979

42085 4431

4656 4832 5107 5332 |225

193

8557,5782,600716232

6456|6681

6905|7130|7354. 7579 224

j

übu

194

7802|5025]3249|8473

8696 |8920

|

9143/9366/9580

9812|223

222

m

195 |29.6035/0257 |O480/0702 |0925 |1147

1369|159111813

2034

196

2256|2478

4246

221

l'ein

bombd

26991 2920 13141 3363

3583 3804i4025

gua

Vai.

Si quinque notis correfpondentem Logarith-

mum habere velis, juxta regulas in algebra tradi-

g

as per proportionem queratur, inferendo, ut dif-

pam

jferentia tabularis numierorum YO, aut 100 , au£

Jf

1ccOo &c.. ad differentiam logarithmorum, que

IT

inter majorem, & minorem reperitur, ita diffe-

41

rentia inter meum numerum & proxime minorem,

ni

0

ad differentiam , minori logarithmo addendam;

Tiun

atque ita hac tabula pro invenienats oninium nu-

n

merorum logarithmis aut logarithmorum numeris

Tuni

n

Ia pro numero 29 logarithmus eft in

ferviet.

Tth;

pagina prima 1. 4623985 .pro numero 346 —— 2.

lwg

5390765 pro numero 3460 idem erit. prajfixa. fo-

£

-um

DE PRINC. CALC. TRIGON.IN RES. &c. 27

TT

lum characferiflica 3, fcilicet 4. $39016; pro

Iquit

numero 3463, progredere a numero 346, verjus

TOXI*

dextram ad quartam columnam ufque,. cui in

e n«

fümnmutate 3 adfciptus e ,/ & invenitur $39452

rima

cui charaderiflica .5 præflgatur; numeri 5849

um

quaratur 5845 & in eadem ferie verfus. penulti-

mam columnam , que numerum 9 in fummitate

pra-

? Qs

gerit, progrediendo habetur logarithmus c82, cui

91[0.'

prefguur 67 , id efl 467082, cum charaderifti-

V,

ca 3; pro numero 18365 queratur 7836; qui erit

Titr

894094, differentia a proxime majore in ultima

cotumna annotata eft Ss,ergo erit proportio 10 :

pont

55255: X; erit x 27 , qui additi ad logarith-

mum inventum dant verum 4.894121. Gc.

,[Lg

Tabule finuum & tangentium brevitatis cau-

6000

22f

fa in minorem formam contrade funt, partum

eo quod ordinariis geometricis operationtbus

1

224

abunde fufficiat , angulos fingulorum minutorum

2

habere, cum in inflrumentis goniometricis mino-

A.

(

2

rem quamdam divifionem difficulter quis. difin-

guat, partim quod ii, qui accuratioribus obfer-

zrztll

vationibus fefe applicant , majorem canonem lo-

rüdi

garithmorum aut ad manum. habeant, aut com-

parare fibi certo debeant. .Noftre tabule ita in-

tdi

aut

Jirude fünt: Jfiniffra pagina logarithmos Jt-

nuum, dextra tangentium exhibet ab 1

minue

(lU(B

jf

zo incipiendo ufque ad. finum totum feu go*. Pri-

mos quinque gradus cum fuis minutis folium pri-

rett

iam;

mun & alterum habet, gradus fuperius adferipti

fünt, minuta in prima finifluma columna qua-

p f

wers

runtur, ita fí 2 graduum 44 minutórum loga-

rithimum habere velis , 44 minuta in. prima co-

n

z

lumna qugre, dein tamdiu verfus dextram pro-

a$, ubi

3

gredere, donec ad eam columnam veni

J.UP&!

il

58

C APU IT R

füperius . gradus adfcripti funt, & erit logarith.

qu

pl

mus quaefitus 8.618405.

4 fexto gradu inci-

piendo ad finem ufque gradus & minuta decadica

[

[ 4

1O, 20, 30. 40, 5O. in finifiima prima columna

e

Jflant,eorumque logarithmi in immediate adjacente

fécunda columna , ubi fuperius zerus videtur, an-

logi

C0li

notantur, ec vero, que inter decades continentur

novem unitates in fummitate paging quaruntur,

qui

ubi fuis columnis adfcripte funt,1,0,3:9€. lgt

zur gradus & minutorum decas in prima: columna

dià

quaritur, tum tamdiu verfus dextram. proceden-

gan

[

dum eft, donec ad eam columnam perveniatur,

cul fuperius unitates decadi annexa. refpondent.

nj

Gradus fextus cum fuis münutis in fequenti fche-

Inr

mate exhibetur.

;

il

n

Logai*ithmi Sinuum,

Tun

PG.M.] [o)

|[:de p5 p4 H5 16 p pe- 19

Ti

65

0|01.9235|043

1632|2825|4016|5203

638617567

8744

9918

10/03.1059|225

6:3421|4532:5741,6895

d

$048/9197

|

0342

1

|

1485

que

20/04.2625/3762|4895

!6026

71548279|9400,0519| 1635

2745

üm

80/05.3859/4966,6071

7172|8271|9367 0460 1551 2639

|

3723

40|06.48€6,5885|6962

|

8036|9107

i

0176 124212305|3366 4424

et

071911759

X5

5o]c7-5450/6533|7583|8631 |9676

|

|

?797» 3832 4964

90

l6t

lta 6* 44' logarithmum fi cupias, quere 6* 40

tn prima columna, dein in eadem ferie horizon-

tali verfus dextram progredere, donec ad co-

p«

lumnam pervenias, cui zn. fumitate 4 adfcriptum

T'tni

eft, ubi he quatuor note leguntur 9107, qua pri-

bt

rmis duabus notis columna fecunde o6. adjungan-

tuj

tur, G habetur logarithmus finus 6* 44 —9.069107.

0/f

compendii caufa ultime tantum note variabiles

uj

in his columnis intermediis exprimuntur. | Id

g&lz

ipitur bene advertendum efl, fi in his columnis

e quatuor notis prima fit minor quam ea, qua e

g ude -

DE PRINC. CALC. TRIGON. IN RES. &c. 56

qrith.

quatuor ultimis prima in /äcundd columna flat,

: iCl

tunc pr&cedens feu pruma ante punctum untate

"udica

augenda efl , ut prius de logarithmis numerorum

|umnd

diximus. lta in 6* 45 fLa 40 ad eam columnam

TCente

pervenias, cul 5 adfcriptum efl, deprehendes

m

logarithmum 0176, quibus non cÓ6 ex fecunda

'entur

columna fed o adjungendum

TA

guz's.

gz, & fic de reli-

, Igi-

Pro fécundis minutis proportio, ut prius

u

dictum efl, infituatur ; accipiatur differentia lo-

cedan,

garithmorum & inferatur 6o" dant hanc differen-

iadut,

£iam, mea fecunda quam dabunt? quae addita mi-

vdett.

nori logarithmo dat. verum.

i$2 füb ipfa opera-

i fole

£ione charactertffica major a minore fübtrahenda

Jit, addantur fuperiori charadleriflice 10, & fiat

—À

fübtraciio.

$z fumma duarum | characteriflica-

—

rum major it quam 10, quoties fieri poteft, xo

|?

rejiciatur, refiduum erit characleriflica vera. . Sit

n

quarendus logarithmis 6* 4s' 40"5 cum 6* 4s' lo-

88

n

garithmus Jit: g. o10116. G 6* 46' fit 9.071242.

1ar2a

u

erit differentia 1066 ; igitur infer 60: 1066 — 40:

X ; erit x z I']'], quod additum logarithmo minort

9.070176. dat l

ogarithmum verum 9 010353. e

fic de alüs.

T '40i

yaur.

$2 cofinus querendus eft, fiat angulus com-

plementi, fubiracio angulo dato a oc*, & diffe-

zd (0*

rentic finus in tabulis queratur ; fi obtufus detur,

g'ptum

1 pïi-

fübtrahatur a 18o^, finus differentie eft finus ob-

tufi que/fiti, & huius differentie cofinus eft quoque

pnpane

Qr07-

cofmus obtufi.. Ita Cofin. angult 73* eft finus an-

guli i. Sinus anguli 12* eft equalis finui an-

abilt

]

gult 44^, & cofinus anguli 156" eft equalis finui

'umnz's

anguli 46*,. qui eft cofinus anguli 44^.

quæ d

C A-

u

30

C—APUT IV.

PUT V.

y!

RESOLUTIO PRACTICA

hg

TRIANGULORUM.

T

PROBLEMA

I.

zs. Datis in triangulo JABC rectangulo ad A4,

bifi 4B — 2295 , & angulo ad bafin B22 32* 49

L.or

invenire & hypothenufam BC , & perpendicularem

re(i

&i

AC. . ( Figura 20.)

Hl

gui

RESOLUTIO.

Subtrahatur angulus B a 9o* feu a 89* 60 erit

angulus C z 57^ I2/, & habetur oppofitio, igi-

tur erit fin. C: A4B z fin. Tot.; hypothenufam; vel

p

fin-. C: AB zz fin. B : AC.

piati

Exemp. Logar. AB — 3. 516535

addatur ad fin, Tot, — IO. OQOOOO

—S

13. 516535

pty

fubtrahatur fin, C z 9.924572

———H921

)

üt ]

3.591963 e

ad

2? Logar. AB 2 3. 516535

rit h*y*pothem»lfa -- 3908

fin, B 9. 733765

ih 1

13. 950300

fubtra&o án, C —— 9. 924572

adt

?

— ÀÓ

9.325728 erit erzgo latus AC z 9

n7.

PROBLEMA II.

$6. Datisin triangulo rectangulo 44BC catheto

ix

BA-35, & AC 52, invenire hypothenuíam

i.ü'xq

& angulos. (Figura 21.)

RESO:-

RESOLUTIO PRACTICA TRIANGUL. :

RESOLUTIO.

Logarithmus utriUSque catheti multiplicetur per 2»,

ut habeaturlogarithmus quadrati, quadr.ta addantur, &

logarithmus fumms dividatur per 2, habebitur logarith-

mus hypothenufs»,

Exemp. Log, 33 — 1,518514 X 2 z3 3. 0370928 refpond, 1c89

Log. 5222 1, 716003 X 2 —2 3. 4320C6 refpond. 2764

d Á,

3793

t

Loz. füumm2 3793 — g; 578982: 2 -:.1. 789491I

refpond, fub characeriftica 3. 6158 , ergo hypothenufa xqualis

jlarem

61,58, feu 6z, 6

Habita hypothenufa habetur oppofitio, unde per Prob, L am

£uli invegniuntur,

COROLLARIUM riz

j, etit

n jgi

$7, Sidetur hypothenufa & una cathetus fiat refolutio uf

]j yel

prius hoc folo difcrimine, ut fummse quadratorum. !oco accie

platur differentia eorundem , ejus radix erit latus qu'efitum,

COROLLARIUM

T.

58. Si immediate angulus querendus fit, adhibeatur una ek

o

proportionibus fuperius expofitis, v

oe

—

ut latus 4B

95 M

——

I. 716003

ad fin Tot,

Rad,—: IO

o

A

11.716003

ita latus .4C ,

AB z 1. 518514

——

ad tangeïntem B

IO. 1974892—57* 36".

Af

PROBLEMA IIIX,

$9. Datis in triangulo obliquangulo angulo

stheto

B—92*, 20' A2 6o^o0 &latere AC — 56589.

nilafi

invenire reliqua.. (Fig. 22.)

RESO-

$0

52

CAPUT IV.

RESOLUTTIO.

lat

Ex notis duobus 82* 20' & 60* oo' habetur tertius

ap

^'7? 4O, & hinc ob regulas oppofitionis fin. B; 4C z fin..4:

PL

BC, & in. B : AC fin. C: AB

fubtrahatur fin. B ab 4C, AC z 3. 565244.

11u

fen

B zz9,996100

——

Wt !

tium

4. 567144

tüt ,

eri£ differentia communis 3. 567144, ad quam additus

gler

finus 4 & C dat latera oppofita BC & AB.

3. 567144

3. 567144

o

9. 93753t

9. 786088

3. 504675 —3196. 3.353232 222355.

,

PROBLEMA IV.

gu

60. Datis in triangulo obtufangulo AB — 4. 526,

feif,

AC z 6254 & angulo C — 29* 58' invenire re-

liqaa. ( Fiz. 23.)

RESOLUTIO,

Cum fit oppofitio, erit 4B: fin. C & A4C: fin.

üen

fub

B ; addatur AC & finus C; AC z 4. 5946

4j

— 9.0985323

ten

13. 493299

àdq

fubtrahatur 4B —— 3.626087

a 1

9.857212 cui logs.

zithmo refpondet 46? 2', igitur B inventus eft 46? 2/,

habetur ergo tertius 44 complementum ad 18o*, eít

ergo z IO4* , O.

FE

ad |

SCHOiIONa

Jfe

61. Reflectere fefe hic opottet ad ea, que in geometría

$. I29. de defcriptione triangulorum datis duobus lateribus &

üf

angulo uni ex his oppofto diximus , alterum angulum dato alteri

OR

lateti

AR

RESOLUTIO PRACTICA TRIANGUL. 35

lateri oppofitum vel obtufum velacutum fieri poffe, ut in

Fig.9

B*

3.

apparet, detur enim latus 4C —-6234, .A4B — 4326 angulus

fertiug

38 ,

invariato A4B & angulo C latus AB

vel

1n

C — 29

z nl

B vel b cadere

& angulus B vel acutus ut B

vel obtufus

ut b fieri poteft

in quo eafu non inventus angulus 46 2', fed

& tertius ;4

ilius complementum r33

acciplendum effet

fieret — 16 "^ 4 , cui latus oppofitum Cb refponderet 2596 hexap, ;

ut ieitur hujus

generis problemata

rtite folvantur

necelífa«

ut v1dea«

fjum eft prius fieutam trianguli quam proxime dati,

für;

an oppofitus anoulus quaerendus obtufus vel acutus accie

N

plendus fit.

PROBL

MA V

62. Datis in triangulo ABC ( Fig. 22.) latere

BA-—24188, & BC—*;224I cum intercepto an-

gulo B 25 82* 50, invenire angulos 44 & C &latus

46

tertium AC.

Ie [c»

RESOLUITIO

Summa laterum datorum — 5429, differentia eorun-

dem —' 947, fumma angulmum incognitorum 97* 3

( fin,

fubtrahendo fcilicet 82? 30 a 180* ergo femifumma —

A? 45 , fiat jam ana]ogn cafus tertii.

Summa 1a-

terum ÁAB.-- BC

4239 z

.

e

L]

9

3.7234720

ad differentiam laterum .4P — BC

947 2 2.976349

ita tangens femifummas —

48* 45 — 10.057012

fogte

13.033361

3. 734720

;60 24

94

$

ei

ad tangentem femidifferenti» z I1? 15

9. 29864I

ad femifummam addita femidifferentia dat angulum ma-

jorem C z 60, a femifumma fubtra&a femidifferentia

.omnttl&

,dat angulum minoremi 4 — 37? 30'; cum jam habeatuf

oppofitio, fiat ut 4: BCzz B; AC. vel C B e B : AC

sius €

Q gltert

Metsb. Maih, P. LIT

C

BC

jatet)

2&

CAPBPDT IV

BC 2—3.350442

AB z 3-505519

—æä::g 78444']

— C z9.93753I

(

—— MÀÀMÀ—ÀÀ — —À

PU

3.565995

8. 565987

- 5 — 9.996268

L

btt

-L-5z9.996268

meM

us iel

AC z 3.562263 & 3649

3* 562255

3646

dat n

1

1U0tia

PROBLEMA VI

ieu

feuq

6^5. Datis tribus lateribus 44C —

tipli

5650, DBCzz

eft q

2198, AB — 2241. invenire ángulos. (Fzg. 24.)

«t

D

RESOLU

IO

t

Tumr

AC latus maximum

:

naccucs

Alg

as

562293

AB -- BC L 5429

—

3: 734719

ph ;

BC — A4B —

r-s.efnfæ,

addantur

FOH

Ddq "--

2.9

(-3

49

Dr

comp

6 711068

funn

3. 562293

3. 148775 2 1408 ——x

33

ergo AC — x z 3650 — 1408 — 2242, quod divi-

fum per 2 dat rr21r bafin primi trianguli rectanguli &

iden

Tatii

$650 — 1121 — 2529 dat bafin fecundi, unde fequw«

labo

tes due pïopomones

10n

AB :

II2I] —

z íin. Tot. cof.4; & BC: 2529 z Íin.

loesi

Tbot: cof. C

3. 049606

3. 402949

hx.».m

1O

IO

g

13.049606

I3. 402949

A0tt

3. 350442

3. 503518

9. 699164

cof. 59*, CO

9. 899431 — cof.3

831

A

cum ergo A——äg, C ::."T*Pi"'fïl erit B-——-OS 29

$CHO-

&

2

RESOLUTIO PRACTICA TRIANGUL.

E

ph

SCI-iO'LION.

64. Nonnull in his operationibus logarithmicis, ordinarii

logarithmi loco complementum arithmeticum adhibent, per quod

eft enim comple-

fubtractio logarithimi in àdditionem mutatur ;

mentum arithmeticum differentia logarithmi a logarithmo unita-

tis feu radii id eft 10. 0COCCO ; Obtinetur igitur fi logarithmi

3549

dati notz omnes novem prxeter primam ad dextram,, qu a IO

fubtrahi

debet,

à 9 fubtrahantur,

Ita complementum arith-

ineticum logarithmi 3. 350442 eft 6,649558 ; jam cum idem ^t

-

H

feu quantitas v.9. 3 per 9 dividatur, feu quantitas 2 per

^

mul-

tiplicetur, ut ex arithmetica patet, (fa&um enim hoc 7 äequale

E

2

eft quoto priori

) igitur divifioni fuccedit in tali cafu multipli-

)

catio, Íi divifor fiat denominator fractionis,

cujus numerator

eft rt, fed talis fra&ionis logarithmus eft complementum arithme-

ticum; habetur enim logarithmus hujus fra&ionis fi a logarithmo

numeratoris I fubtrahatur logarithmus denomiratoris ( $. 225.

Algebre), ergo in integris numeris divifüoni fubftitu'tur multi-

plicatio, in logarithmis igitur fubtracioni additio; $i igitur lo-

at

garithmus fit fubtrahendus,

qualis

femper eft primus terminug

in proportione pro inveniendo quarto, accipiatur ejus logarithmi

complementum

arithmeticum ,

& fumms mediorum addatur,

Q*

"

fumma totalis erit quartus terminu

9

utiin ultimo exemp!o

*

ad 13.0496c6

feu ad 13. 402949

. €

Or

—

addatur comple

ith. 6. 649558

add. comp. arith. 6, 4964C2

.

ETRU PL UNES

pos p

divi*

9.699164.

9. 89943t

Poterit erzo ope-

J &

idem in utroque cafu logarithmus qui prior,

ratio vel hac vel priori methodo inítitui;

ego certo magnurnm

laboris compendium inde haberi, ant periculum errandi evitari

;uefl'

non arbitror, ni fors adíit focius, qui e canone logarithmorum,

fa.

logarithmi quzfiti loco, complementum arithmeticum dicet, folus

enim íi labori infudes,

cavendum fedulo habebis,

ne falfam

unam alteramve logarithmi notam exfcribas; atque hac ratjo-

ne faciendo fimul complementum arithmceticum la!'orem pot'ug

augeas quam imminuas, quod de compendiis omnibus univeríum

AOtatum velim,

E m a

!*

T

H

ELE-

Q

C 32

&.

46

CNPUT-^I

e

]

Qu

[vl

I I MU IN Z

Gäüïwäï WL PRACTICAE

fion

W——;ä

16 (

6*"1

PT

?

(OT

ONES

;ENERALES

TM

D'ETIAITTO 1

dic

J

i

comelria pratlica eft fcientia principia theoretica

p

G

geometrie in praxim reducendi, & dlmc—.nüom—

duo

bus in campo fa&is applicandi.

Cite

tran

COROLLZATHBRIU !?MM.

aq

9. Cum diver fvïs

euend:, finis effe poffit, diverfe quo-

Q

que prmcmlurum appii

Cd

;,.

iones & methodi habentur ;

vel enim

fitus locorum

M

refpecive ad plagas mundi determmatur'

vel

diftantia 1o

rum a fe invicem cum figura fylvarum, fluviorum,

tur

viarum &c

denotaturs;

vel quantitates ipfze rerum dimenfarum

fifig

V. £. quantitas campi, aut fylvze &c. in hexapedis affügnatur; vel

tdf

preter predica, loci qvoque aptitudo refpe&ive ad actiones mi-

Mitares, ad infidias ftruendas,

ad caftra ponenda ,

ad trajicien-

fen

dos fuvios &c. defcribitur;

dle

veldenique figure & fitus folfarum

WV.

fub terra conftru&arum examinantur

que diverfe methodi has

Ti

dant definitiones

jl

l

DEPINIIIO 9IL

lem

3. Dimenffo geographica eft: determinatio loci refpectu

Ter

duorum circulorum maximorum, quos in globo terreftri

g«

i

fingimus, & quorum ope cujuslibet. loci & fitum abfo-

H

I

lutum & refpe&ivum ad alia loca determinamus ( $. 7

eu

fe&

ial

L

H

a

NOTIONE

3

GENERALES.,

al

P

u$ REl

ï;e&, conic.) $Si tantum inquiratuf in longitudinem gra-

53

dus talis circuli maximi, dímesfío gradus weridiam wo-

COROLLARIUM.

ho

4. Ad hoc igitur notitia quedam primorum faltem princi-

piorum geozraphizm & aftronomis requiritur, ex quibus definitio-

nes quasdam magis neceffarias infinuamus.

DEFINIITIIO

IIT.

VSO

5. Linea /MS, (Figura 23.) circà quam globus ter-

raqueus rofatut, eft ax;; terre , ejus extremitates pol;

dicuntur, fuperior quidem ;5 féptentrtonalir, inferior

JM mevidionalis vel auflalis.

Circulus maximus 4B a

oretict

polis 9O gradibus diftans, & globum terraqueum in

níioui-

duo hemifpheria biíffecans, dicitur eguator, ad quem

Cireulus maximus alter perpendicularis per locum «

tranfiens Sc/M dicitur meridiazus illius loci; unus igitur

equator, at indefiaiti meridiani per omnia circa cir-

i que

cum loca tranfeuntes fünt, qui omnes in,polis $, &

| enima

M convergunt; Meridiani etiam circuli. verticales dicun-

t vel

tur; fuperius hemifpho»rium fupra sequatorem /épteutrto-

riorm,

"sle, inferius vero feu infra xquatorem auflrale feu ime-

farum

üt; vd

yzidionale vocatut.

Pun&um A & totam plagam, que ob.

nes m«

fervatori, facie feptentrioni feu.pun&o .5 obverfa, ad

ajicien'

dextram jacet, dicimus ortum, 4 vero & plagam fini-

plfarum

dj hat

ftram oecafum vocamus; qus direcio cuilibet quoque

maipp» geographies dari folet, ut feptentrio fummam,

metidies immam, ortus dextram, occafus finitram par-

tem occupet. . Circulus ms parallelus ad squatorem,

—

per locum datum tranfiens eft circulus. parallelus , cujus

pett

generis infiniti inter zquatorem & utrumque polum duci

reltrl

poffunt.,

Meridianum primum, ex cujus interfe&ione

abfo-

"n

cum squatore numeratio per 360 gtadus initium capit,

U

C 3

geo-

oUb

d

[*

CAPUT L

55

inde reli-

geographi per infulam. ferri ducunt, atque

quorum diftantias computant.

COROLLARIUM.

6. Hos duos circulos, ad quos locum datum referimus, pre

determinando fitu immediato locorum fufficere diximus in geo-

u

metria

curvarum $,

r

ita fi per f tranfeat primus Meridianus,

diftantiam fd'oci c a frimo Meridiano in equatore acceptam di-

cimus Jongitudinem loci; diitantiam dc ab xquatore in Meridiano ac-

YO

ceptam dicimus Zatifudinem loci:

Fac&a hujus utriusque proprie-

tatis obfervatione aftronomica,

ilculis definitur pun&um c ,. at-

üi

fi

que hic feu quivis locus, ubi obfervatio pera&a eít, in fphera

terreftri determinatur, & dimenfio eeographica abfolvitur.

Quin

unum tantum locum hac ratione determinare ,.aut jam -determi-

natum aíffummere fufficit ,

atque inventis

dein per operationes

trivonometricas aliorum locorum ab hoc determinato loco diftan-

tis reliqua quoque loca ad hunc referri, & eorum longitudo &

latitudo figi & determinari pot?rit.

SCHOLIOdMX.

i

7. Hec methodus & determinatio locorum fuffcit,. dum

(t

£eosraphica tantum, dimenfio regionis cujusdam petitur,

fixis

Q

enim,

ope talis operationis & calculorum,

primariis

pun&is

lu

exiguo labore & citra omne errandi periculum ea, qus inter ta-

feu fluvii,

feu

lia pun&a continentur, feu fylve, feu lacus,

agri, feu loca minora fint, inferl, & tota regio depingi poterit,

ut gzeographice eruditioni aut. ufui civili inferviat ;

atque eam

methodum nos quoque obfervavimus,

dum Cl, Liesganigio Au-

guftifüimorum voluntate dimenfio geographica regnorum Galliciz &

t

Lodomirie demandata eft,

Septem obfervatores quadrantibus

.aüronomicis inflru&i & in feptem territoria diftributi, loca pre-

cipua tanquam totidem fixa pun&a determinavimus,

inter que

H

focii,

partim offciales partim tirones militie,

loca interme-

[U

dia cum 1is,

que ad illa pertinent,

inferebant,

atque ita im

vafta hac regione, mille & ducentas leucas quadraticas fuperante

5

laborem difficillimum ( lingue fcilicet ignari, omnibusque com-

n

4^

»

ries

Jaborare in

L

modis de(/tuti ) tredecim, ^quibus aerls tem

pc

campo concedebat,

menfium ífpatio abfolvimus.

Qua ratione

opus aegereífi (imus, -& ad finem perduxerimus, ordine fuis in

Yocis anne&am, ubl regulas praxeos premifero.

*

SCHO-

s

TOR

NOTIONES GENERALES

39

S C H

LION

H

- feli

$. Situm linez, aut loci refpe&u ad Meridianum deteimp

xamus vel obfervatione aftronomica,

vel acu magnetica prius

rite ex

min QLW, de qua inferius.

5, pte

1 260«

DEFINITIO IV

nvxuS,

n ül-

o. Dimenfio reomeirica, qus & praxis ÉwufAymetrica

NU

vocatur, eft methodus determinandi diftantias objecto-

oprie«

C .. 8L

rum quorumvis feu acceífibilium feu non acceffibilium,

phara

fitumque five Aorizontalem five verticalem verum & relati-

Qu

vum inveniendi, illa in fpecie lonzimetria, h»c aliimetria

'eHie

dicitur

t10nes

L'uM'

xo &

D.EEFINITIO V

IO. Ferticale dicimus , quod ita fuperficiei felluris

infiflit, ut prolongata linea ab altitudine objecti per

duf

centrum telluris tranfeat, ( Fig.26.) ut 4B, DE, ad

IXif

quam verticalem in ipfa fuperficie erecta pPïPendICUld-

wif

ri$ FG vel GI eft liuea Aorizontalis,

p (

feu

|

COROLLARIUM

tetity

eam

II

Cum fup

jes terre fphwr

x;1"7

fit,

& ad rotundi-

À*

tatem circularem accedat,

fequitur,

lineam horizontalem tan-

i&

in quo verticalis fuperfi-

gentem effe ad tellurem in. eo pun&

,

1005

ciei infiftit,

.-.

roinde nonnif fpcmxn aliquod cum /peripher!a

pre

commune h'».be e, ut de tangente in geometria diximus, quod

qw

tamen fpatium ob masnitudinem ambitus telluris ad longiorem

:me

nonnihil d'ftantiam pro re&a.haberi pot:

eft.

adeoque cum tan-

m

horizóntalis FG re-

£ente colncidet

jn maenis vero diftantiis

Ante

ipfa inclinata eft ad arcum , qui fuperficiem telluris reprefentat,

om-

adeoque ad hanc reducenda foret,

ut parallela ad horizontem &

n

triangulorum inde ef'ormandorum bïüs teret ;

linea horizontalis

L0nt

etiam 1196&. ad libellam du&a, feu Enea Zzl«elæ dicitur

——

j

C 4

DEFI-

»

,(

p

CAPUTIL

AO

DLEFINMITTIO YL

I2. Libela eft inftrumentum , quo utimur ad de-

terminandam lineam horizontalem; /Jibellare autem dici-

£ttun

mur, dum ope linez horizontalis differentiam altitudi-

detet

nis duorum locorum determinamus.

»C HOLIOTN.TL

terea

13. Libelle varie efformantur: notifima eft, qu» ab ar-

ehitectis,

murariisque adhibetur,

eftque triangulum Ifofceles

as 4

ABC ( Fig, 27.) ligneum, e cujus vertice ad bafin pondus plum«

puatu

beum a filo tenui fufpenfum lipere dependet ;

in bafi pund&um

m

biffe&ionis c accurate determinatum eft; dum igitur petpendicu-

dete

lum cum pun&o biffe&ionis congruit, filum ad ba(n verticale

fier,

eft; (geom. S. 137.), igitur bafis linea horizontalis eft ex def,

Vi. fi igitur hoc triangulum ad perticam ligneam aut aliud pla-

num ponatur, hoóc horizontalem fíitum habebit, (i pondufculum

accurate cum divifione congruat, adeoque filum ad bafin pere

í

pendiculare fit,

dua

scÓütotrow x

tum

I4. Accuratiores funt, quz ex tubo vitreo parantur, (Frg, 28,)

qui fpiritu vin! non colorato ita repletur, ut bullaaeris in eo re«

linquatur ; tubus utrinque hermetice, ideft, liquato ad lammam

vitro clauditur, & aurichalcinze regule affisitur; dum libella in

fitu horizontalt eft, bulla aerea in medio ac manet; cum enim

femper aer fummitatem petat, adeoque in hoc cafu nec ad dex-

?

£a

tram- nec ad finifftram afcendat, fignum eft, nullam extremita-

tem magis quam alteram elevatam effe,

fitum proinde horizon-

16

talem totam re$ulam habere; tubulus vitreus interne bene po-

ln

liri dehet.

Pfolibellatione dioptris inítrui folet, aut alia ad-

Vl

hiberi, duobus tubu'is vitreis inftru&a , quam inferius, dum de

libellatione acturi fumus, dabimus.

D

u

DEFINITIO.VII

ài

Ti

I*. Bafi dimenfouir dicitur linea re&a, in campo

l

menfurata, ad quam omnes relique longitudines refe-

runtur, & diftanti» determinantur; eft igitur totius di-

menfionis fund

ais

)fTt!'

entum.

(

DEFI-

NOTIONÉS GENERALES.

AX

DEFINÍTIO

IX,

de.

16. Dimeu/ío oeconomica eft determinatio quantitatis

üici«

rerum dimenfarum, in menfura quadam aífumta, jam

JAdi

determinata in hexapedis, decempedis &c.

COROLTVJARIUM.

I7. Non differt igitur a priori dimenfione , nifi quod 'pre-

o

terea adhuc requiratur, ut preter figuram ipfa quoque quanti-

teleg

tas aífhgnetur, atque hac ratione ipfius fundi

?

feu ager,

feu

J

pratum , feu fylva, feu ftagnum fit, pretium haberi , & poffeffo-

um

ris facultates xftimari poffint;

geometrica ergo dimenfio prece-

[lcue

dere debet, ut oeconemica rite inftitui, & fi opus fit, gmodefia

cale

fieri queat,

üe

pla

h FgNIZXI.

lum

x

I8. Gaodefa autem eft methodus dividendi, &

diftribuendi planum in ratione quadam determinata, in

duas v. g. tres, quinque xquales aut proportionales tan-

tum partes. (Geomf $.275.)

))

te*

D.EFINITIO XL

m

n

I9. Dimenfo militarit pretet geometricam, quie fem-

nl

per precedere debet, id quoque requirit, ut omnia

ex:

tà

ea cum fumma accuratione annotentur, qus ad naturam

)

regionis pertinent, num inundatio a repentina pluvia

007

timenda, num rivi femper permeabiles, quz litora flu-

|de

viorum & ubi fint elevatiora, ubi infidie ab hoftibus

]je

ingruere poílint, qui colles regioni fubje&» dominen-

tur, qus» vie vel toti exercitui, vel particularibus

tantum expeditionibus ufui eífs poffint, ubi caftra po-

[

nenda fint, uno denique verbo, ut ad omnia illa refle&a-

tur, quzxe fummo armorum Principi accuratam loci ideam

faciunt, unde nonnunguam felix armorum exitus repe-

ln

tendus efít,

M:

DEFI

CAPUTI

45

DEI TTJINI-TIO XIL

Tl

gO

H

Dimenfo fubierranea eft methodus in' fodinis

alett

longitudinem & declivitatem platearum aut cunicuiorum

zfguu

determinandi; fpecialia ad hoc adhibentur inftrumenta,

& te

qua in ggometria fubterranea defcribuntur.

mini

fetti

D EFLXINITIO XïH.

ero

dex

» t

lchwogvaphia eft delineatio regionis, provin-

exhi

cie, territorii, dominii, civitatis &c. qus harum rerum

figuram in ea proportione in minore fcala reprafentat,

quam reipfa in magno hibent.

COROLLARIUM L

p

tt'al

22

Hxc fcala. eo major effe debet,' & eo minores partes

Y,.t

exprimcre, quo accuratior minutarum etiam rerum delineatio

eoch

requiritur, ita digitus vel etiam linea vienneníis mille aut cen-

&q

ftum, aut unam hexapedam pro diverfütate obje&i & laboris de-

abf

fignare poteft.

Geometria elementaris & trigonometria plana in-

A4il

£411

ib

2

1

p

les legzes tam pro dimenfione quam hac delineatione tra-

o2

dunt, quas cum ex praecedentibus -notas eífe fupponamus, foli

applicationi & praxi in(itemus

cujus fündamentum (C$. I5.))

feu bafis eft accurata lines re&z dimenfio, de qua fequens ca-

put tractat.

€ O.f OrLELGARIDAM O

M

lit

23. Neceffarium igitur eft, ut ante omnia fcala determine-

tur,

juxta quam ichnographia facienda fit, ut hac ratione vel

Qc

&i

linea menfurata pro bafi ftatim in.menfulam transferri, & labor

pot

continuari: velü res calculo trigonometrico peragatur, figura

riangulorum efformari poffit.

Hec fcala aut in charta craffiore,

vud

aut lizno .duro aut aurichalco fit, aut jam parata in pedes, di-

gitos & lineas divifa affumitur, & valor ad libitum pro quan-

titate laberis, & accuratione requifita determinatur, ad majores

x*nenfuras. fcala ope circini micrometrici transfertur.

D.EFJINITIO XIV.

24

Zyc.e

Circulus micrometricus &Fig, 29.) eft pertica

Í

M

lignea re

&angularis, in qua cufpis una 4 libere move-

ri,

^

NOTIONES GENERALES.

43

ri, aut in quocunque loco figi poteft, qüod ope curfo-

Ihi

ris ? quadrangularis aurichalcini & cochles c praftatur:

[

l

ul

altera cufpis e in fuo curfore quadrangulari matrici d af-

]t,

fipitur, qu» ope cochlee tenuis // tam leniter adduci

& removeri poteít, ut motus vix advertatur, atque ita

minima etiam fpatiorum differentiz examinari, & trans-

ferri p [Iint.

Extremitati cochfeæ À affixus eft in-

u-

dex i, cujus motus in orbiculo x fimul motum cochlex

utt

exhioet.

at,

COROLLARIU M.

95. Motus hujus quantitas ita determinari poterit, acci-

piatur regula aurichalcina, & in eam longitudo digiti viennenfig

transferatur v. g. cyþ"i ponatur cufpis altera in quocunque loco,

ttej

v..g. &, & adducatur cufpis prima ad B; moveatur dein ope

tio

cochle:s ufque in . & annotetur, quoties index totum circulum,

£f«

& quot preterea partes centefimas (fi diviio in IOO partes fit)

de

abfolverit, tot revolutiones integre , & partes centefimz motui

H

per unum digitum competent, atque inde per divi&onem ad li-

n-

neas applicatio fieri poterit,

ol

*)

SCHOLION,

a

a6, Ante omnem operationem igitur finis laboris in con-

fderationem adducatur, an geographico, «cconomico, an mi-

Jitari ufui dimenfio fervire debeat, & pro ratione accurationis

(

vel

requifite methodus, inítrumenta, 'f;cala proportionalis. feligantur,

& determinentur, fpatium quoque menfurandum totum, Ífi fieri

yot

poteft, prius luftetur, ut jam ideg quepiam fegionis faltem

it]

zxudis habeatur,

$

i*

E

Z

L

C A-

c€

44.

OXTUT

T.

— ——

&e

&U

M

C APU AL

[j

]i

il

t

tradi

DE DIMENSIONE BASEOS.

fetre

gieti

DERINITIO

I.

excir

prior

27. Metiri bafin ( G. 15.) eft longitudinem alicu-

tenaà

jus linez recte in campo fumma cura & accuratioae de-

piat

terminare.

in e

ad f

CO R.O,.LIA RLU-M

1.

quo

fun

28. Cum hzc bafis totius dimenfionis, fundamentum fit, a

tüta

qua omnium aliarum linearum 1longitudo dependet, facile qui-

Tun

vis intelliget, fummam in illa menfuranda diligentiam adhiben-

in j

dam eíffe, ne error dein per omnia latera multiplicatus ad in-

31,

gentem quantitatem excrefcat,

& dum labor claudendus

eft ;

ultima primis non cohzreant; continget enim, ut vel ob jufto

majorem longitudinem pars parti incumbat, vel ob brevitatem,

fpatium quoddam inter initium & finem, dum ad prinium redi-

tur punctum ,

remaneat; quod utrumque delineationem totam,

2

At

Oi

precipue fi longior fuerit, erroneam

[S4

inutilem reddet.

10

V

COROLIILIARIUM It

tt

29. Laborem inchoaturus fpatium vel intra perimetrüm &

üm

fieri poteft, vel extra limites, ubi fylva, lacus, ftagnum non

TY

permeabile occurrit, elige, quod folum planum & zquabile ha-

Qw

etri

beat, in eo dire&ionem linez baculis verticaliter erectis ( Fig.

t101

30.) & a fe invicem remotis, ut in seometria ( $. 1-3'.) dixi-

Ret

mus, deügna, vel dum linea menfuranda brevior eft, in ex-

tremitatibus tantum linezs baculos nonnihil altiores & craífiores

BR

erige; interquos catena ferrea (Fig. 3I-) vel fune in orgyas aut

Qw

decempedas divifa fpatium menfura hac methodo:

primo & ul-

timo annulo catenz menforie baculus altus inferius cufpide fer-

rea armatus inimittitur, quibus duo geometre catenam ita ten-

dunt, ut, quantum fieri poteft, ad lineam re&am accedat, quod

facilius efücitur, Íi catena terre incumbat; ne autem penitus

(h

decidat annulus craffior baculo (Fig. 32.) inferius.circumlari aut

clavus major incuti poteft, cui extremitates catene incumbant ;

i?:

dein

t

*

L

45

.DE DIMENSIONE BASEOS.

deinfefe invicem (Fig 33.) ita dextrorfum & finitrorfum dirigunt, ut

cuivis a focii baculo alter baculus major in extremitate totius bafeos

erecus, cui faciem obvertit, tegatur, fignum id erit, utrumque

in linea re&a procedere.

Abfoluta hac prima operatione ex-

[2 0

tracisque baculis primus foramini, in quo baculus erat, clavum

ferreum aut ligneum numero I fiznatum imponit, & ulterius pro-

greditur, dum alter ad idem foramem perveniat,

qui clavum

excipit, & ad faccum reponit (aut dum rurfus abeundum eft,

priori foramini ( Fig. 84.) reftituit,) baculum vero fuum cum ca-

N

tena eidem immittit, ut jam .fecunda longitudo catenz ibi inci-

Je d&

piat, ubi prima defiit, primumque ita dirigit, ut cum . baculo

in extremitate ereco in eadem re&a fit, atque hoc modo ufque

ad finem labor continuatur; bacilli aut clavi minores exhibent,

quoties totius catenz longitudo accepta fit, qui proinde addendi

funt, ut totius linee menfurite longitudo habeatur.

Si v. g.

o

fit,

tota longitudo catene ro hexapedas habeat, & bacilli parvi 12

le qui«

numerentur, linea menfurata 120 hexapedas continebit, que de-

hibene

in juxta fcalam geometricam in chartam pro ichnographia ( $.

al it-

21, 92.) transferenda eft.

$ &l

) juíto

COROLLARIUPM i.

fl æm,

fedi-

20. Hec dimenfio fuffücit in geometricis ordinariis dimen-

QU ,

fionibus, ubi fepius fub ipfa menfuratione dimeníio linee re&z

repeti, aut diftlantia duorum locorum examinari catena poterit,

& error, (i'quisquam fa&us eft, detegi; at dum major requiri-

-

tur accuratio , uti dum pun&a fixa pro geographica aut militari

dimenfione confiituenda funt, que dein pro norma ad determi-

um f

nandas res intermedias ferviant, majori opus eft cura, cum enim

n non

diverfimode tendi catena poffit, fi f»pius applicanda fit, exigui

1 H

e -

errores àdditi fenfibilem majorem efficient; hanc vero accura-

(f

lg'

tionem menfuratione per perticas obtineri praxis docet.

Quam

dixk

methodum in dimen(íione gcographica regni Gallicie adhibueri-

| ex-

mus, quove ordine procefferimus, referam fequentibus fcholiis,

(010$

31

u

t

quod fimul regulas obfervándas declar?bit.

U.

3CHOLTON RL

Iet-

tn-

g3r. Planities pro dimenfione bafeos a me & L, B, de Rain

1uod

(tunc in Academia Lincenfi, modo in univerfitate Leopolitana

1

H

mathematum Profeiïore) eleca eft penes civitatem Rzeszovien-

aut

planioribus in locis,

fegetibus

fem, eo quod maturis jam,

At s

damnum inferre follicite tum caveremus, ho; autem folum pafcuig

H

us

deftina-

46

OXATPUT J

dumetis licet hinc inde veftitum , fine damno per-

g

deftinatum ,

meari & conculcari poffet.

Perticas quantym fieri p

Utc

ofte

rat rectas,

39

A

quatuor hexapedas longas, octo fieri curavimus; quo in negotio

szagno nobis adjumento erat R, P. Bieleckie fcholis piis, quo

«0

interprete ufi fumus, & cujus nunquam fine grati animi fignifi-

"T

catione memor ero.

IÍn earum una, in diftantia dimidie circi-

milit

ter hexapeda,, exiguas lamellas aurichalcinas firmavimus, omnium

at

W

vero extremitates planis ferreis exiguisad angu!um re&um ere&is

pe

(Fig.35.) munivimus, que malleo in perticam adigi, aut ab ea remo-

feque

Dun

veri poffent. Ad primam perticam ope circini micrometrici dimi-

ticat

dias hexapedas ad lamellaé aurichalcinas, ex alia menfura Vienna

snell

hobifcum allata,

tranftulimus ,

tenfo prius per totam perticam

Jgna

filo, quod lineam recam exhiberet, ultimam vero ex utraque

du

parte in pláno ferreo determinavimus; adaco fcilicet ferro tam-

vti f

diu in perticam , dum a cufpide circini, cujus alterum crus in

acies ferri tangeretur,

Huic,

que pro

$

proxima lamella erat,

norma ferviebat, quamvis aliam feparatim impofuimus ita, ut

Per

plana ferrea fuperioris, inferioribus accurate incumberent, at-

aj

que ita aptata funt, ut extremitates fuperioris inferioribus ad

aCl

amufüm refponderent, & longitudo omnibus omnino eadem da-

UT

[

retur, quatuor ex his Rainio date,

quatuor mihi fervate, eas

01

in utraque extremitate literis .4, B. C, D. &c. fignavimus

Ut 1

Tl

ag

His preparatis lineam in campo fixis poft quinqu

u

inta.cir-

M

10'n

Vttaq

eiter quafvis hexapedas baculis ($. 2

9.) def

^o

avimus, qui omnes

AX

verfus, proceram arborem valde diffitam dirigebantur, . quod ad

ducendam re&am multum juvabat,

Dein ipfam dimenfionem

Tej

€go cum Kiffo, Rainius cum Langero, qui uterque jam, inter

fummo mane agegreffüi fumus;

hum

officiales bellicos adfcriptus eft,

E

ad quamvis perticam duo milites conítituti erant, duo prelon-

ls,

eum funem juxta du&um baculorum tendebant, duo afferculos

T

atque alia frufta, perticis ad procurandum fitum horizontalem

ACIe

Pertica AB prima ponebatur, atque

fupponenda, geftabant.

h

juxta aciem primi ferrei plani perpendiculum demittebatur, quod

u

H

n

pun&um primum bafeosin terra correfpondens oftendebat, quodque

Prt

L

incuffo palo, plano orichalcino munito, in terra (Fig. 37.) fignabatur.

b

m

NÀ

Ut fitus horizontalis pertice examinetur, libella utrique extre-

d

mitati a me & focio applicata eft, & per frufta fuppofita debie

&

20

tus fitus procuratus,

Ad primam admovebatur altera CD, quz

cum ob inzquale folum aciem ferri primz non contingeret, fed

fupra vel infra eam . jaceret, ope perpendiculi in tenui filo fufpenü

errum. filum

(Fi. 26) ad eam ita promota eft, ut utriusque perticz fi

H

tangeret atque ita cum pertica EX & GH adtum exat; dum quarta

1

aj

^

&

inige-

T.

—— —

—

m

DE DIMENSIONE BASEOS.

47

0 pte

éirigeretur,

prima, retrahendo ne in alteram impingeret, loco

'e(bat,

mota, & ad hanc translata fuit, militibus duobus ad eam defti-

erotio

natis clara voce numerum ]1 fíeu primam pertice prime A4B

; Quà

translationem id eft IG hexapedas abfolutas effe indicantibus,

ego hunc numerum in ipfá pertica

linea fignabam, quod ufque

enif-

Circie

ad finem accurate obfervatum e

ies5

ítz

numerus linearum cum voce

mum

militum concordans ,

du&us in I6

omnes dimeníz lines hexaes

redis

pedas dabat.

Nonnulló intervallo Rainius cum quatuor perticis

emos-

fequebatur, ut adeo totius bafeos dimen(ionem bis obtineremus.

M

dl

Dum no&e ingruente ceffandum erat, demiilo poft ultimam per-

]ennz

ticam perpendiculo rurfus pun&um correfpondens in terra in la-

ticart

mella, longo clavo ligneo affixa & terre immiffa, tenui inciffura

fignabatur, unde altero mane labor inchoatus & ad finem per-

Tajue

tarme

du&us eft,

Translationes pertice 4B & reliquarum trium , pro-

"s i

uti figna faca & voces militum indicabant, 75 fa&z funt, que

p pto

du&e in I6 dabanttotius bafeos loneitudinem 1200 hexapedarum,

"a

Perpendiculum ad omnem fere perticam (Fig. 38.) applicari—debuït,

eujus filum duas fere linex decimas wequabat, hinc pro fumma

at

accuratione 5 adhuc pollices feu O,

O5 addende funt,

Dunm

us ad

Rainius ad metam bafeos extremam perveniret, deprehenfutm eft

[ d2«

uno tantum & altero pollice bafim utramque differre, quod igi-

, 98

tur fignum certifimum erat, in dimenfione errorem nullum com-

ln

miffum effe, & methodum hanc (imul multum commendabat.

|OMte

ütraque meta figna prealta conjun&is ad fieuram pyramidalem af-

Imnes

feribus ere:a funt, quorum vertex punc&o bafeos immineret,

j

a

Finito labore rurfus diviüones prime pertice examinate &

Qnemt

relique ad- eam applicate funt, at nullam vel a calore vel ab

1ntet

hunore induc&am effé mutationem deprehenfum,

licet thermo-

ATUM

znetrum 22 gradum fupra pun&um elaciei o(tendiffet, quod cum

vlone

iis, qux Cl. Liesganieius in fuo opere de dimenfo gradu refert,

f IQE

mirifice confentit. Reduc&ionem ad verum horizontem ( $. II.)

alem

faciendam non effe, brevitas bafeo$ oftendit, cum Liesganigius

itque

in íua 6410 hexapedarum,

in qua meta auftralis boreali

jod

multo altior erat, nec unius pedis differentiam deprehendiffet.

4111

Preter hanc noftram bafin aliam quoque Liesganigius cum Güff-

atuf

mano & fociis ad Leopolin, & tertiam D. Capitaneus de Verité

xtte

ahl-

ad fines Silefie dimenfusr wíi, quarum accurationem confenfu&

làboris comprobavit,

ut

fed

$ CCHOLION IL

eni

Cum inter operandum honnunquam ad colliculos aut foiï'ass

d

autmolle folum & palufre deveniatur, ubi pertioa fequens auf

jarta

multe

*

"ag

DE ^

48

C A POUT JH

multo elevatior aut demiffior præceäente ponenda eft, fulerum va-

riis incifuris inftru&um, ut Frg, 39. exhibet, adhibetur, ad quas

funiculo fubftaculum parvum a,

cui una pertice extremitas in-

s.

'

cumbat, fufpenditur, & ita ad.varias altitudines elevatur, donec

quo jm

cum altera extremitate in eadem linea horizontali fit, perpen-

fei circu

diculum inter precedentem & fequentem perticam fufpenfum at«

culus Fi

que utriusque thremïtatem tangens fitum determinat, ( $. 31.)

fiiten fe

,

$

CHOLION

il

duo, ex

duatut ,

23. De aliis modis menfurandi bafes* ut dum pertica circa

tur, zel

extremitatem in femicirculum rotatur ; dum obeundo lineam paf-

fus feu greffus proprii numerantur &c,- nil addam , eo quod mul-

fcila gec

hàc at

tis errandi periculis obnoxii' geometrice operationi fervire ne-

queant, mifi fors rudis tant&m &ítimatio petatur, aut figura cir-

IO ad |

citer efformanda fit.

De curvis lineis imenfurandis uti fluvio-

etipantü

fum, viarum &c. fuo loco agemus.

CCutate

fionis t

Supra 1

T

dioptris

CUPU

I

r

,

trum fem

filis diop

DE ANGULIS, ET INSTRUMENTIS

f mobit

Ornnes (

AD EOS ACCIPIENDOS NECESSARHQS.

fila &in

extremis

DESNFINITIO £Lk.

3d dinid

fionum 1

34. Angulos hic fumimus inclinationem duarum

firmato

linearum, qux a duobus obje&is duécte, ad idem pun-

&um concarrunt.

Yel totu|

^0ntalibi

COROLLARIU M.

tem mp

poit,

35. W;el quäntitas tantum talis inclinationis, vel ipfa fisura

efformanda fine quantitatis determinatione peti poteft; utrumque

propriis preítatur infirumentis, quorum jam explicationem &

ufum dabimus.

iTA

Wk» &

N gg

DEFI

i

P

"^

r

j

E DE ANGUL. ET INSTRUM. AD EOS &c. 49

fulerm v

DEFINITIO d

1 , ad quas

temitas 1n«

46. Goniometricum , quodï& affrolabium vocamus, de

'ïï

K, donec

quo jam in geometrii $. 36. mentionem fecimus, eft,

, penpene

feu circulus Fg. 32. feu quadrans fig. 33. feu femicir-

yenfum at«

culus Fz. 34. aurichalcinus nonnihil latior, fupra quem

(*3L)

faltem feptem circuli concentrici ducuntur, ex quibus

:

duo, extimus, & intimus in 180 gradus aquales divi-

duntur, qui per transverfales lineas re&tas conjungun-

Srtica Cites

tur,

ineam paf-

Zerus cum I, r cum 2, 2 cum 3, Gc. ut in

quod mu-

fcala geometrica ($. 198. Geom. ? fecimus; obtinentur

(erure ne«

hac ratione faltein. fex partium fubdivifiones, id eft a

figura cit

IO ad ro minuta»,

Supra diametrum horum circulorum

yi f

erigantur verticaliter dux dioptre, quarum igitur fila

accurate per planum diametri, & initium & finem divi-

fionis tranfeunt, qus» femper fixa & immobilia manent,

Supra planum, diametri alia regula non nihil brevior

dioptris verticalibus inftru&a penitur, qus ciírca cen-

trum femicirculi mobilis fit, & cujus lineasmedia cum

filis dioptrarum accurate priori linee ita refpondeat, ut

NTÉS

fi mobiles dioptre cum fua regula inter fixas pouant.r,

omnes quatuor tranfpicienti unius inflar apparere, &

uis.

fila & incifure fefe mutuo accurate tegere debeant. Ab

extremis linez mobilis 4 (Fig. 34.) pars quedam ufque

ad dimidium demitur, ut accurate partes in liinbo divi-

fionum refecte notari queant; totum pedi tribus fulcris

duarum

firmato imponitur, qui vel nucem, ut vocant, gerit,

j p

vel rotulis variis dentatis, tam verticalibus quam hoti-

zontalibus inítruitur, ut iníftrumentum in omnem par-

)

tem moveri,

& fitus etiam

verticalis eidem tribui

polit,

4

(s fiutt

COROLLARIUM. ;

rumque

jnem X

57. Duo igitur precipue, ut ufuüi hoc inftrumentum effe

pofüt, cavenda funt, ne error in fixis dioptris & in diviüione:

ne in mobilibus & in centro motus (it.

'Cum radius circuli fie

AMetzb. Math, P. 11,

D

2g9us-

p

CAPUT

AH.

DE.

SO

[

equalis lateri hexagoni, ($.153.geom.) capiatur circino micrometri-

pen 1

co diftantia inter 6O gradus, & ex diverfis gradibus, hac quantitate

apules

chordz 60 graduum, fiant incentro plures arcus feu interfe&iones

LO

exigus, quz fi fe omnes in eodem pun&o interfecent, centrum &

ityales

omnes femicirculi rite determinati funt; fecus ( fiat, corre&io

ferius 1

inftituenda erit.

Dein eadem, aut dimidia etiam quantitas 3O

in6,1

fcilicet graduum, diverfis aut omnium divifionum gradibus appli-

quo g

cetur v. £. T & 31, 5& 35, 6 & 35, IO & 40, 78 & 108,

$2 & II2 &c. &c. fi multum varient, annotetur, ubi notabilis

differentia deprehenditur, & tabula correctionis pro sgradibus

erroneis fiat,

$i non notabiliter differant, relinqui poteít, ri-

40

diculum enim efífet corre&ionis tabulas confücere, ubi ob exigui-

parte pai

tatem infirumenti aliquot minutorum error evitari nequit,

fuprt par

Centrum motus a centro inftrumenti differre non debet, ut

POR

ficut linea mobilis per totum femicirculum gyratur, ita punétum

quinas

centri loco non moveatut.

Id examinari poterit, íi in extre-

minator

mitate lines mobilis, dum accurate in O & I80 jacet, fignum

iüperiori

?er

aliquod acicula aut colore fiat, quod fi per totum gyrum cum

labebit .

arcu, qui ex centro du&us eít, congruat, centrum motus idem

tlus vl

cum centro inftrumenti erit; fialium arcum defcribat, (Fig. 40.)

que aqu

corréctione opus eri , ut error in obfervatione evitetur.

CU q ,,

jor erit.

COROLPLARIUIV[ li.

1

[i

(

38. Cum arcus verfus limbum interiorem minor fit quam

d yon

verfus exteriorem fequitur, divifionem per transverfales geome-

titate d;

tricam accurstionem non habere; pars enim exterior CD ( Fg.

igk

41.) major eft quam interior 4B, hinc AC & BD parallele non

jj COng

3

(

funt, & proportionales partes diagonalis 4D^ abfcindere nequit.

j

Ynterim tamen pro inftrumentis ordinariis geometricis fufficit, in

€ moti

quibus alioquin ob radium exiguum accuratio ad unum alterum-

pat

vwe minutum fruftra exfpectatur.

Si major accuratio requiratur,

1021

120

transverfalium loco alia divifio, quam Vonnium appellamus , ad«

tota

hibetur, qüa non tantum de minutis etiam primis, verum de

SOtU e

partibus quoque illius fecuri reddimur,

DEFINITIO

1.

4l

'*J*":Tf-l'üm

30. AIMonntus vocantur duo paraï]elogramma'auricvhaï«

per'frad

cina A4B & ab (Frg. 42.) accurate zequalia, quorum acies

LURT

hw

fefe per totam longitudinem contingunt, atque quorurm

lum

unum-:immoto altero ope cochlea (ecundum longitudi-

äeï&m )

&i

nem

i

Y

DE ANGUL. ET INSTRUM. AD EOS &c. x

lictometrie

nem retrudi vel promoveri poteft, 1íixum in partes

quantitate

erfe&iones

aquales ad libitum dividitur, mobile itidem in partes

centtum &

equales inter fe, fed qua unitate numerum partium al-

Cotredio

terius regule fuperent, ut fi illa divifa fit in 5, hzc

antitas 30

in.6,fiilain 20, hec in 21 &c. &c. dividatur, ex

bus appli

quo jam fequentia deducuntur.

& 10f

votabilig

COROLILARIUM TY:.L

gb

M

otef, ri-

40O, Quslibet pars parallelogrammi A4B major eft qualibet

l exigul*

parte parallelozrammi ab, & exceifus partis cujuscunque in .4B

lit,

fupra partem quamcunque in 4b femper idem erit, quiscunque

debet, ut.

partium numerus ponatur,

fü enim in noftra divifione partes

H

| pun&tum.

quintas fuperioris

ï?

& fextas inferioris $ , ad communem denos

ln exttee

minatorem reducamus, quzlibet pars quinta 4..- 1 vel 1...2

, figuum

fuperioris habebit 4,& , qualibet .4.,. I vel z. ..2 inferioris

folx

mtur cunt

habebi

C

So :

quelibet ergo pars ab altera, vel j55 refpectu to-

bDro

1

n differentia ubie-

otus idem

tius vel £ refpe&u partis cujusvis differt,

bt

que equalis eft, quiscunque partium numerus ponatur.

Pretrea

t(* F. 4o.)

"

cum a ... I ab 4 ... I quantitate dc differat, 4,.. 29 jam ma-

jor erit quam a

..2, dupla illa differentia feu 2de, & 4,...3

ab a,..3 differet quantitate 3dc &c. fi igitur parallelogrammum

ab promoveatur, donec I & I congruant, promotum erit quans

G u

Ó

titate dc , fi 9 & 2 congruant, promotum erit quantitate 2dc,

e$ geomt-

cp (

fi 3 & 3. quantitate 3dc &c. fi eousque promoveatur, ut a cum

I congruat ( Fig.43.) tota parte 4i feu differentia dc toties

j

ylelie 008

Cum enim

pofita, quot partes in ab continentur, promota eft.

re nequit

per motum parallelozrammi ab omnes partes inferloris fuccefhve

fufflCifq it

cum partibus fuperioris congruant, dum 5 & 5 congruunt, pa-

, alterum-

rallelogrammum 5 dc promotum eft, dum igitura & I congtuit,

gquiïawïa

jam tota quantitate .4 1 feu 6 dc, que in.4 I continentur, pro«

m du

motum, eft.

verum «*

COROLLARIUM : I.

AI. Inde patet jam methodus determinandi quantitztem

diferentie cd, a qua valar nomnmii dependet, erit fcilicet fem-

per/fra&io, cujus denominator eft numerus partium inferioris

aurïchäï:

parallelogramuni mobilis, feu quod unitate numerum fuperiorum

m acié

M

partium fuperat.

$i divifioni nonnium applicare velis, qui

quorum

decem parte

»

a

fubdivifionis contineat, fuperioribus

Q

Pl

partibug

&aquale parallelogramm uza conitrue , iludque àn 1Q partes di.

yitud

2 a

vide,

nen

—— —

r

[p

D

5

ag

QAPTT

IFI.

MT

T

eividat

wide, erit differentia cd accurate

H

s partis fuperioris; fi in 20

st it

divifio facienda fit, fuperioribus I9 equale parallelogrammum

fac, illudque in 920 partes equales divide, erit cd — 25, &

at

fic de reliquis.

$i igitur in divifione per 1O, 4 I decem mi-

que

[emicir

nuta valeat, cd unum minutum exprimet, & dum v.g. 4 ex in-

tarum

feriore cum 4 in fuperiore congruit, 4 minutis index promo-

nulla e!

tus érit;

major 1

COROLLARIUM IL

it infü

e, qu

49. Dum a ultra primam toïam—partem A, CFig, 44-) pro-

nihil d

3

greffa eíft, evidens eft, jam numeros inferioris divifionis cum

polition

fuperioribus convenire non poffe, eandem tamen, qu» initio

sortedt

erat, proportionem femper manere, hinc ut in hoc cafu quan-

eon

titas motus & valor exprimatur, attendendum fimul erit,

poft

quot partes fuperius notatas initium diviüonis inferioris ftet , &

fimul que linea ex inferioribus cum fuperiori quadam congruat.

li

'Yot partes feu divifiones integras numerabimus, quot fuperius

Yettr

numeri precefferunt, tot vero partes minores feu fubdivifiones,

quot unitates in inferiori parallelogrammo numerus linez illi

verfüs

adfcriptus habet, qua cum fuperiore coincidit; initium igitur

fatlo 4

divifionis fimul index eft, atque propterea figno quodam arma-

miffi fie

tur, illique inferius zerus adfícribitur.

In Fig. 44. Nonnius

ge) &1

oftendit duas partes fuperioris linez & adhuc feptem fubdivifio-

tevatur

abfeind

nes inferioris accipiendas eíffe, cum index inter 2 & 3 ítet, &

Si pars quavis divi-

lines j

linea 7 cum fuperiore quadam coincidat,

fionis fuperioris decem minuta, inferior diviüo partes decimales

?tadu

hioc o

exhibeat, exhibebit l*ïonnius 20 -7 hoc eft 27.

YUm Q

tbferv;

COROLLAP?IUM IV.

petven

43. Ex his jam abunde colligitur, hanc divifionem alteri,

übfen

que per transverfales fit, longe effe preferendam, cum multo

Cextra

sccuratius & majori cum przcifione partes minores determinen-

Doteris

tur.

Accedit, eam etiam omnibus arcubus applicari poffe, dum-

Tolvas,

modo etiam arcus moznii parallele cum arcu fuperiore moveatur $

pet f

cum enim (Fig.45.) ab, in qua divifiones fiunt, & ad fuperiorem

Qu a

limbum .4B & ad inferiorem fimul pertineat, fequitur differen-

41 o

tiam ob radiorum inequalitatem nullam fieri poífe,

proinde ea

um

hic dicenda eífe, qus de lineis re&is diximus.

Goniometrico

ts di

igitur monniu$ applicabitur, fi in regula mobili longitudo quz-

Atote

dam, cujus extremitas duabus divifionum lineis in femicirculo

UM

accurate refpondeat, eligatur, (Fig. 46.) atque hec in tot partes

lg

divida-

M.

DE ANGUL, ET INSTRUM. AD EOS &c. 43

fiin 09

dividatur, qu» unitate fuperent illas in femicirculo acceptas,

rammum

atque ita angulum in minori etiam inftrumento , ad 1O, ad 6,

H

76!

&

aut ad unum etiam dimidiumve minutum obfervare & determi-

cem mie

mBare poterimus.

Siinftrumentum nonnihil majus fuerit, atque

| ex ine

femicirculi loco, ut pondus levius fit, quadrans adhibeatur, diope

| promo-

trarum loco tubuli adhiberi, & divifio ita aptari poterit, ut non«-

nulla etiam fecunda in zoum/o obfervari poffint, qua accuratione

major in praxi geometrica non requiritur,

Tubi tamen predi&i

ita inftrumento applicentur, ut fuperior tubulus mobilis nonnihil

€2, qua oculo iminet parte elevari, inferior vero feu fixus non-

BN

nihil deprimi pofüt, ut hdc ratione quadtante in horizontali

ns cum

pofitione reli&o obje&a elevatiora & depreffüiora obíervari, &

: jnitio

eorredio alias in his angulis adhibenda (infer, $. G9. & feqq.)

) qua-

evitari queat.

t, pf

fet, &

SCHOLION I

)ngtüats

44. Obfervaturus jam angulum inter duo objeca, inftru-

fuperius

1üiones,

smentum ita difpone, ut dioptre fixe ad dextram fint, & mobiles

yez 11l

verfus finiftram promovéantur, centrum vero pun&o in terra

) igitut

fa&o accurate refpondeat,

quod ope perpendiculi ex centro de-

átm

[

"L

miffi fieri poteft.

Dioptras fixas verfus objectum quoddam diri-

ge, & mobiles tamdiu verte, donec alterum obje&um a filis ob-

onnius

tegatur; dein vide, quam,diviüonis partem acies linez mobilis

iiciio-

abfeindat, id eft, ad quod interfe&ionis pun&um a tranfverfali

pt, &

linea in circulis intermediis, vel & Nonnio fa&um cadat, & tot

1 al'd-'

£raduum minitorumque angulus obfervatus erit,

Determinato

cimales

hoc obje&o, mobilem regzulam ultesius promove, dum in no-

vum obje&um incurrat, atque hanc quoque magnitudinem aneguli

obfervatam in libello nota;

dum jam ad obtufos valde angulos

pervenitur, fixas dioptras cum femicirculo move, &ad ultimum

alieti,

Obfervatum obje&um dirige, per mobiles alia obje&a fequentia

julto

dextram verfus determinaturus, atque ita laborem continuare

poteris,

donec totum circulum & omnem circumferentiam ab-

ninen-

jl

dum.

folvas.:. In libellum angulos relaturus, ita fcribe, ut obje&um

per fixas determinatum, primum locum obtineat, reliqua dein,

atl y

que ad dextram fequuntur, v. g. turris (addendo fi du fuerint,

riorenm

an occidentalior, an orientalior) cufpis arcis, capellm frontifpie

feren-

cium aut crux, domus altioris caminus, aut ornamentum mae

de eà

gïs difin&um, patibulum, mola, crux, ftatua, arbor eminens &c.

etricó

annotentür, & anguli quantitas diftin&e fcribatur , ne confufio

que

timenda fit,

Perfe&o circulo videatur, num anguli accepti

|

À

irculo

multum a 360 gradibus differant, quod fignum erroris foret, &

yuté

COr«

D3

jj

Al

ä

p

&. du ,

"

54

CA.P

U

]T. X

Dum

corre&io in tempore adhuc adhiberi poffet. vide $. 47..

Ti

dioptrarum loco tubi adhibentur, ante omnem oOpetrationem vi-

dendum eít, ut filum fixum tubi mobilis accurate cum filo fixo

M

tubi fixi correfpondeat, & dein primo anguli capiantur,

j

S.UHOLIOTM

Yt.

À45. Si altitudo per obfervationem determinanda 6t, inftru-

mentum verticaliter erigi debet ita, ut dioptre fixe accurate in

Hinea horizontali jaceant femicirculus vero fundum refpiciat,

quod obtinetur demiffo ex centro perpendiculo, ut gradum no-

nagefimum abfcindat, perpendiculum enim, lineam verticalem per

2353

)'f |

eentrum telluris tranfeuntem reprzfentat,

que cum horizontali

Glo

femper angulum rec&um facere debet, & cum.inu noftro cafu per-

pendiculum cum diametro, feu cum dioptris fixis angulum rectum

i

1

h1

faciat, quod nonagefimus gradus oítendit, fequitur, dioptras

linea horizontali eíffe ; mobiles deinde ita dirigantur,ut per eas

apex obje&i confpiciatur, & obtinebitur angulus inter horizon-

tem & apicem quzítus.

Pol

$.C-H O L rO-N .HlI.

4

46. Aítronomi, qui fummam accurationem fe&antur , mi-

fores fubdiviüiones micrometro vel interno . filari,

vel externo

V

Obfervant, quo angulorum quantitates etiam dd minuta fecunda

uUfque determinant, cujus conftruáio in laudato jam fzpius opere

Cl. Liesganigii videri poteft.

In Gallicia ad manus mihi

erat

quadrans circuli, cujus radius pedem zquabat, tubis & micro-

metro filari inftru&yus, quibus angulorum quantitates determina-

bam.

Ex nulla fere ftatjione abivi, quin omnia, qus videri

$goterant, loca tubo caperem, &.totum circulum per angulos ac-

eeptos complerem.,

Obtinui per hoc, ut certus effem , in quan-

titate angulorum errorem commiffum non eífe, fi fcilicet (umma

?

angulorum, qui totum gyrum horizontis efüciebant, 360 gradi-

Dsí

j

bus quam proxime sequalis foret.

Angulos in librum retuli, qui

pro calculo aut ichnographia dein fervirent,

quemadmodum fe-

quens tabula duarum ítationum angulos exhibet.

Primum ob-

je&um per tubum fixum, fecundum & omnia, quz particulam

& prefixam habent, per mobilem videbantur, hinc ifta in ich-

;

nographia femper ad dextram ponenda funt.

ï

'

47. Sfa-

F

-

n

f

Ed

&

5&C. 5

*.

]

DELANGUL.ETïNSTRUk&AD!ä?

Dit

A47. Statio in turri Paroch, Rxefzov.

ll Yl«

Tubus mobilis.

Tubus fixus.

) Ào

Meta Septen

tr. &

tis.

S. Magdalena: turris — — 26^ 437

27

&

S, Rochus turrisk-— — 44

28

35

&

Meta Meridion

.

LI

.

L

—^

..—./'755, HI'

—— 009

Meta Merid.

& Zalezie Turiis

5

48

49

lx d& RO

49

31

& Tyczin Turris

:

m

f'üe

&

Drabinianka I^ dom. — 24

44

ER!

.......

lO

n

& Rutka ultima dom,

&Svwi

Valv

aonn

zizatabernecaminus 62

59

dat,

nn

& Bozuchwala Tutris-— —— —

2 B

6 26

d

— AÓ 37 3í

j

ne

Pvw

Boruchwala &.æ ad Staroniwa. ——

FH & turris P, P. Miffion. in Glogow —

—

— $83 14 48

ontali

—

II

8

Glogow & Zazernie domus fola

& Staramiefcie turris

—

r

20

2I

20

& Stobernie domus quedam major

37

36

$3

& Jafionka dom. major

45

RY

9

t ea

—— — — — —

& Lonka tu,rris

67

ao

13

TE

& Pobitna taberne camin. ——

—

——906 37 33

Pobitna & Bas. meta Septentr.

Ed

Ll

—

a

9 18 43

——

Ll

—

rÁ

—

- "Totus horizon

——

——

360 0O 32

Statio &4 ad Staroniwa.

nt

p

terü

Glogow & Zazernie domus fola

-e 99 239 28

& Przybyízowka aule caminus

8

23

ae—À—

33

& Lonka Eccl.

——

——

53

13

55

& Staramiefcie Eccl.

——

—

56 2I

en

24

& Pobitna diverforii camin.

68

5I

al

& Lanchut arcis turris Merid,

69

3I

4

& Rzeszow Eccl. turris

nl

72

20

49

——

Ausueuo

& S. Magdal. turris

87

2

9

d

& Drabinianka I^ domus

—

36

:

at«

& Signum Mer. Baf.

—

—

d

......93

8 5I

nmi

FEPEIEBEEISI

I

l:;-

Baüs Meta Merid, & S. Rochus turris

17

24

&Swiencziza tabernz cam,.32

48

28

& Ticzin Eccl. turris — 34

49

& Boruchwala Eccl. turris —

— 64 43 31

0

— À

cmuium — À

Bosuchwala & Zlobien Eccl. tutrris

— Oa

2 oO

lam

Zlobien &

S

m

P SEL EE M d

"

ienum fequens

— 53 48 26

Signum- & Glogow

.-——;.;.—-———-—,,,.,—651523

— M

O II

Gyrus horizontis

4.,.,..-...'360

TY

i.

,

Á4.

SCHO-

?

"od

f'ï;/

£Ó

D

CAJZBDT IU.,

S.C HiOLION JIVv.

«ione

3lla f

48. Dum inftrumentum ad centrum ftationis poni non poteft,

PU

eorrecio in angulis adhibenda eft, & centrum inftrumenti, in

de

yiari

quo anguli capiuntur, ad centrum ftationis reducendum,

p )

qua corre/tione dein agemus, nunc alia inftrumenta , quibus an«

hc m

guli determinari poifunt, acum magneticam ícilicet , & menfulam

pretorianam xeferamus,

partes

ita f

DEFINITIO

TIT.

exteri

dioptt

4Q. us magnetica (Fig. 47.) eft filum aut fruftillum

üt,

ferreum, nonnihil latius in medio, in cufpidem autem ex

utraque parte definens, hec fifepra magnetem armatura

ferrea infiruCum aliquoties, fed femper verfus eandem

partem, trahztur, virtutem magnetis imbibet, & fi in

(um

praci

medio, conulo exiguo inítruatur, & cufpidi, ut libere

üivat

moveri poffit, imponatur, unam extremitatem fepten-

ttiam

trioni, alteram meridiei obvertet, & hac ratione lineam

tedu

C0tre

Meridianam defignabit, de qua füperius egimuss extre-

mitas feptentrioni obverfa aut figura, aut colore ab

angul

altera diflingui debet, ut fciatür, qus plaga fepten-

trionalis fit.

COROLLARIUM I

5O. Si igitur acus intra peripheriam circuli, in 360 sradus

tis,

divifi, diametri inftrar ponatur, (Fig, 48.) & peripheria cum dioptris

d

circumrotetur ,

cum dire&io acus verfus feptentrionem immota

Con

maneat, quivis gradus ab ea defignari,

atque ita pro quovis

tab

angulo determinando adhiberi poterit ;

fi autem ad regzulam

itve

dioptris inftru&am ponatur ita, ut medium regule accurate per

O & 180 tranfeat, & ita fulcro feu pedi imponatur, ut liberze cir-

coc]

«umagi queat, ( F/g, 49.) dioptris verfus diverfa objeda dire&is acus

Tich

diverfos quoque gradus in fua peripheria oftendet, quorum diffe-

füm

rentia quantitatem anguli dabit.

Oftendat enim in primo fitu

Bl n a

acus 47^, in altero I I4^, erit angulus inter duo objecta compre-

libe

henfus —— 67?.

Mi

COROLLARIUM II

üe]

£Y, Facile quivis advertet, cum h:* periplterie magne effe

P

non pofünt, gradus quoque valde exiguos fieri, itaque accura»

&

tionem

w

dl

E

h

DE ANGUL. ET INSTRUM. AD EOS &c. $5

tionem magnam expectandam non effe,

Accuratiora nonnihii

Y potett

illa funt ,

qua ita conftruuntur

ut annulus aurichalcinus ,

quí

leftr, 1f

acum includit, fupra aliud planum circulare, in 360" divifum

wm, de

xotarl queat.

Ipfe annulus fuperius dioptris, inferius ad pla-

ibus at

»um Nonnio inftruitur, cujus index filis dioptrarum refpondeat

mivfulo

hac methodo fubdivifiones plures v. g. quartz aut minores adhuc

partes gradus haberi poterunt.

In obfervatione inftrumentum.

ita firmatur, ut acus in eadem recda cum O & 180 peripherisze

exterioris fit,

annulus dein

rotatut ,

ut diverfa obje&a per

dioptras confpiciantur, differentia graduum, quos Nonnius oücnw

Villum

dit, anguli quantitatem dabit,

te ex

COROLLARIU M Ill

TOaCUIa

3nden

52, Frequentior acus magnetice vuífus eft in determinanda

&i

eurvatura viarum ,

curfus fluminum ,

tramitum Sylveftrium &c.

libera

precipue autem ad determinandam lineam Meridianam & refpe-

nonnulli

éivam ad hanc lineam poíitionem locorum adhibetur ;

fepten

etiam menfule przetorianz

5

per folam applicationem acus ad

lineam

xecam in eadem ductam, fitum verum dant, & punc&um in terra

extte»

correfpondens in eadem exprimunt, atque ita etiam quantitatem

angulorum determinant

fe 20

de utroque fuo loco dicemus.

epten«

DEFINITIO

IV

S3. JMenfala pratoriana eft tabula lignea. re&angula-

aradus

ris, fulcro ita infiftens, ut libere circumagi poffit, quod

jjoptris

ope axis aurichalcini (F7g. 51.), quo tabula cum fulcro

ymmota

Facilius id efficitur

conne&itur, przftatur.

fi major

qJO'B

tabula minoti per duas incifuras ligneas, quz in tabula

egülam

t pet

inverfà in (F/g, 59.) exhibetur, conjungatur, '& ope

cochleze firmetur

Pro fitu horizontali tres cochlez au-

12diyl

15 aCUS

richalcinz ( Fir. 53.) per fulerum tranfeunt, qus fur-

. dife-

fum trufe vel retrufe menfulam quoque in quacunque

qo fit

libet parte elevant, & fitum horizontalem, quod libella

p

indicat, procutant. Ad hanc regula, dioptris fuis vertica-

liter ere&is, (Fig. 54.) inftru&a (de qua in geometria $.76

egimus) ponitur, in qua acies, juxta quam linez du-

g t

€enda funt, in eodem plano cum filo & incifura iu

ecura

D $5

dioptris

u

P

MO i r EE

CA P3UVE . IH :

p

s

jaxis,

äioptris fa&a effe debet, — H jam dioptre -eäciemx in

eculus.

menfula pun&to applicate, fi ad diverfa objeda diri-

quocf

gantur, & linez re& juxta eam ad idem pun&um con-

$0/ Cf

vergentes ducantur, angulos per totam circumferen-

*

tiam dabunt.

COROLLARIUM. I.

"6

4ico in

5$4. Obfervaturus l]gitur angulos pun&um in menfula , :cor-

nobilis

refpondens centro ftationis v. g. initio bafeos dimeníe,' deter-

gnam

mina,- quod fit ope perpendiculi in ferro, forcipis inftar effor-

fegu

mato, fufpenfi, (Fig., 81.) cujus una extremitas cufpidata fora-

fieri p

mello alterius extremitatis accurate refpondeat,

ita enim per-

vitri (

pendiculum in terra & cufpis forcipis in menfula in extremita-

plano

tibus ejusdem lines verticaUis erunt, & ita pun&um ad cufpi-

quam í

d?m efformatum ex meníula ad terram ,

aut efformandum ex

ETE

terra in menfulam transfertur.

Situm dein horizontalem ope

Q er

libelle menfule procura.

In pun&o figitur acicula , cujus caput

Vr

cera hifpanica obducitur, ut fortiter apprimi queat, atque illi

flum (

*egula ita applicatur, ut objecum determinandum alterum, v. g.

tíone o

puncum bafeos, afpicienti a filo obtegatur,

Du&a juxta aciens

it de i

regule linea dire&ionem exhibet, quz ex fcala geometrica de-

terminatur, & lineam bafeos dat.

]ta per totam circumferen-

tjam procedendum,

dirigendo dioptras verfus

omnia objeca

circumfita ,

x

veigat,

& adfcriþendo ubivis verfus quod obje&um lines

tris a

COROTLZLGLARIONM TL

adhibe

libris

$5. Cum nonnunquam ad objecw elevatiora aut depreffio-

Dant,

r3, quam ftatio fit, collimari debeat, dioptra prima duabus mi-

I g

noribus dioptris inftruitur, qux depofita ultima dioptra & inclinata

2€0, Q

b

m

nonnihil prima (Fzz, 55.) eriguntur, & verfus objectum diriguntur,

hac ratione linea dire&ionis in eodem plano.manet, & accurate

verfus obje&um feu dépreffum feu elevatum tendit,

Harum loce

etiam filum per medium fummitatis utriusque dioptrz tendi poteft,

i

quod cum incifura & altero filo verticali in eodem plano erit,

*unt, 1

igitur fi per hoc filum obje&um altius per incifuram vifum ob-

Wntur

tegatur, aut.fi hoc filum cum verticali & obje&o depreffiore in

D

eadem reca fit, linea vera directionis juxta aciem regule duci

9»Lyi

b

t

poterit, & anguli determinari,

Nonnulli incifure loco in pri-

i

ut hinc & inde immota re-

ma dioptra etiam filum adhibent

11 h

evitatur quidem per i

d paral-

$u!la obfervatio inflitui poffit;

hs e

laxis,

mlb us

DE ANGUL. ET INSTRUM. AD £OS &c. 59

dem ia

laxis, (feu varietas ejufdem anguli obfervati) qus» oritur, f

O6culus incifure apprimatur, at ea rurfus enaícitur diffcultas,;

2 i.

quod recedi debeat a menfula, ut filum utrumque videatur, ad-

11] COl«

ioto enim nimis oculo filum vicinum difp'aret,

ierelt

COROLLARIUM IiI.

56, Dum ad dv.::intias majores collimandum eft, dioptrarum

loco in medio regule figitur tubus, (Fzy. $6.) qui in pun&to c ita

|t, C0f«

mobilis eft, ut furfum vel deorfum afpicienti ferviat; quod ma-

, Qetet-

gnam valde commoditatem, adfert, eo, quod menfula una cum

at effot-

regula in fitu horizontali femper maneat, & ob(ervatio diftin&a

t fong

fierl poffit,

Id autem hic

cavendum

eft, utfilum , quod in foco

i pete

vitri ocularis

verticaliter

firmátur ,

in eodem cum acie regule

ttemitae

plano fit, quod hac ratione praeftatur :

In meníula in diftantia,

d cpi-

quam fieri poteft, remotifüma erisantur perpendiculariter due

dum exX

acicula , & ita verfus quoddam obje&um diffitum dirigantur, ut

em ope

cum eodem in eadem recta

fint, dein his aciculis regula admo-

us caput

f

;—.

-.

um

veatur, & fi obje&tum per

verticale fecetur, fignum erit,

que illi

Hac ra-

filum & aciem regule in eodem plano verticali effe,

n V&

tione obfervationes fupra menfulam ads diffitas valde diftantias,

j aciem

ut de inftrumento goniometrico diximus

fieri poffunt.

"

?

«2 de-

nferene

SCHOLION I

objeta

; jne

57. Hiec funt inftrumenta, qu» a mäthematicis & seomes

tris ad determinandos feu in quantitate feu in figura angulos

adhibentur, & ad qux» omnia ea reduci poffunt, quz alii in fuig

libris depredicant, dum hinc inde varia addunt, aut combie

nant, quibus tamen ufum implicatiorem reddunt,

& accuratio-

spreito

AT

nem minime auge

D

nt;

id certo e

A

longiore praxi perfuafum ha-

hus mi*

ydinta

etiam inftrumentum effe,

beo, quo fimplieius, eo accuratius

jnt

]

l

P

ï

Q

Wa

SCHOLION

IT,

g loct

48. Ut autem ea, que de his infütrumentis hacenus di&a

ooteR

funt, applicari poffint angulis, qui in centro ftationis effe fin-

TR2

£untur, neceífarium eft, ut etiam centrum inítrumenti, aut ver-

^m 00

"

tex anguli in menfula efformandi fupra centrum fítatior

i

is fit.

*ot

e M

Si igitur inftrumentum fupra hoc pun&um poni nequeat, ut con-

& u

üd

tingit Ín »edificio, in turri, mola, figno aliquo ere&o amplia-

n

rl &c, angulus per illud acceptus ad centrum ftationis reducen-

9t fe*

dvs eft, ut vera ejus quantitas habeatur; Item f unum, ob-

a

jedum

105

L

DCAPUT. .

DE!

60

je&um in horizonte, & aliud infra velr fupra horizontem fit,

ità ut infrumentum in angulo obfervando fitum inclinatum acqui-

rat

$

angulus quoque immutari & ad horizontem reduci debet,

quod utrumque fequens caput declarat.

ät')

gO

[X DOXRIIIURUNIE YO M Oron n

E

Vatus.

CAPUT IY.

3b 00

pof

DE CORRECTIONE ANGULORUM

RELATE AD CENTRUM STATIONIS,

Qbíer

ET HORIZONTEM.

PROBLEMSA L

dexti

$9. Corrigere angulos extra centrum ftationis

obíervatos, & ad illud reducere,

CASUS L

fn'ang

Dum obfervator in eadem cum centro directios

m

ne pofitus eft ante vel poft centrum v. g. in 9 re-

fetvat

tür m

fpectu £.

addat

RESOLUTIO & DEMONSTRATIO.

el: :

CFrg. 57. N. 1-) cum;angulus obfervatus o refpectu

pate

trianguli woc fit externus, erit o —— m -- c, igitur ma-

jor angulo vero in c quantitate m & o — m z c. Dum

igitur angulus o inftrumento accipitur, imminuendus eft

f

quantitate $ ,' ut habentur c.,

Dum (N.2.) vero in-

m&y

ftrumentum in o poft centrum ftat , erit ob « —o -- ,

aol

d

ad angulum o addendus angulus m, ut habeatur c.

ne h

U An0nus3 AL

in?ulo

"!

60. Dum obfervator ante vel poft centrum

fïïe ür

& in 0 pofitus eft.

g

RESO-

DE CORRECT. ANGUL. RELATE AD &c

'ontem fit

RESOLUTIO & DEMONSTRATIO

tum aequ

ul debet

(Fig. 58. N. r.) Cum refpe&u utriusque trianguli mco

& nco angulus o fit externus, erit o £—m -- c, &o—— n-t- c,

ergo 20 z m--n--2c, fed 20

O & 2cz C, ieitur

O z—me-254-C, & O— g—nuzzC. ut igitur obfer-

vatus angulus O zqualis fiat vero in C, uterque m &m

ab eo fübtrahi debet

Dum ( N. 2

.) obfervator eft in

RUM

O poft centrum erit ob C —

.O - m - 8 ad angulum

ND

obfervatum addendus angulus m & s

C A:SUS Il

61. Dum obfervator in. directione obliqua ad

dationis

dextram vel finiftram eft.. ( Fzg. 59.)

9.

RESOLUTIO

A

DEMONSTRATIO.

Angulus x (N.1.) externus refpe&u utriusque

trianguli Cxm, & uxO eft equalis C-1- m & O-4- 1 ergo

jredtio

C-r- m z O0--52, ergo C z O-t- 5— m. Igitur ad ob-

1n 0 I'D:

fervatum O addatur angulus ad dextram s & fübtrahae

tur , ut habeatur C

In altero autem cafu ( N, 9. )

addatur m, qui eft ad finiftram, & fübtrahatur z; hoc

IIO

eft: angulus füperior ex parte obfervatoris addatur, ex

yefpeäu

parte centri fubtrahatur.

itur Dà*

COROLLARIUM I

. pun

us &l

62, Ex his jam patet, totum eo devolvi, ut angulorum

m & n quantitas inveniatur, quorum fubtra&ione aut additione

'ero il

angulus obfervatus corrigitur

Varie ad hoc adhibentur metho-

s;

di ex principiis trigonometriz petitz.

Noto enim quam proxi-

(5

me latere, & diftantia loci obfervatoris a centro ftationis, &

angulo, quem hzc diítantia cum latere facit, trià nota funt,

unde reliqua inveniuntur,

Si autem, a loco obfervatoris ad

uUtrumque latus, quod ab ebje&o ad centrurn ftationis duc&tum

entrürt

effe finritur, perpendiculum demittatur, refolvendum erit trian-

gulum re&dangulum ex notis duobus lateribus org & xy. (Fiz, 60.)

$0

COROL-

63

ÜQAPUTJIV.

DE

COmROTZdLERIUSIAI.d.

62. Angulo igitur Ínter duo obje&a obfervato alium quo-

Mafie

que angulum x inter ftationem & centrum cape, & difítantiam

[

centri e ftatione oc meníura

?

utrumque in librum interea an-

/1à

i

gulo obfervato adjice.

Calculato dein triangulo omm tam ex

Ites

precedentibus obfervationibus, quam ex novo obfervato angus

liq

D

loo,innotefcit quam proximelatus om & on, unde rurfus per regu-

ntepru

lam $.62, in trigonometria propofitam, dum angulus intercept s

r;

33 &C,

x cum duobus lateribus om, xc datur, invenitur angulus omc,

La

tüt

Grivius in fua trigonometria tabulas hujus corre&ionis proponit,

in quibus latus z:0 a 10O ad 16000 hexapedarum in fronte ta-

ia g

bule,

diiantia a centro ab I ad 129 pedes in fummiate co-

qui 3

lumnarum affumitur, anguli vero inter centrum & objesum quan-

à u

titas in prima columna & 5? ad 90? ponitur, (in ootufis enim

complementum ad duos re&os fumitur,) unde apparet, hanc cor-

[/

re&ionem in parvis diftantiis magnam, adeoque neceffariam effe,

in magnis vero errorem ad aliquot tantum fecunda affurgere,

adeoque negligi poífe ,;g cum de fecundis in inftrumento minore

alioquin certi non fimus ;, exemplum tabularum in fine annedte-

3us.

COROLLARIU MM- .

64. Altera methodo fi uti velis, filo latiori aut pertica ia

pedes & digitos divifa diftantiam perpendicularem a centro in-

ftrumenti

ad utrumque latus ab obje&o

ad ftationem

effictum

menfura, & cum figura, qus eX predicis tribus cafibus hic ap-

6

pit

E

plicanda

eu

$t

in libellum transfer, erit

E;

em

triang.'um

ï

)

recangulum xmy refolvendum, & proportio conftan

'i

ut latus om

noi

adfinum totum, ita perpendiculum xy ad a

H

u

lum ?5,

In hocergo

1De

cafu chara&eriftica logarithmi diftantie perpendicularis rO uni-

àj

tatibus augetur, cum finus totus habeat logarithmum IC, 000C00,

& logarithmus lateris ab eo fubtrahitur, fi hec operatio pro

utroque latere in fecundo & tertio cafu, accepta utraque perpen-

diculari, infüituatur, erit differentia utraque finus angulorum z

vel n,

SCHOLION

"

65. Clarum eft, hanc operationem ricorem geometricum

non habere, «eo quod laterjis om vel or magnitudo ex angulo e

incorrec&o calculetur, ihterea error, qui inde exurgit, tam exi-

PUPIUC MA

guus eít, ut fenübilem differentiam in angulos inveniendos &

calculanda dein ex ii$ latera vera non inducat,

SCHO.

DE CORRECT. ANGUL. RELATE AD &c. 65

SCHOLIO'F H.

66.

Cum latera majora femper in hexaped!

dentur, di-

I qu.

tantlam

flantis autem a centro in pedibus & digitis accipiantur, opor-

feà al

tebit quoque hoc ad menfuràm hexapedalem in fra&ionibus de-

cimalibus reducere, ut correfpondentes logarithmi accipi queant,

tum ex

Y atf

partes unius hexapede feu 6 pedum fequens tabula exhibet, pro

cum enim 6 pedes femper T

érrezu

reliquis, facile inde habebuntur,

integrum feu hexapedam, 12 pedes 9 integra feu duas hexape-

rcep:r i

m

das &c, faciant ; quotus qui ex divifione numeri pedum per 6 pro-

*

'Opotit,

dit, integra oftendit, partes remanentes in tabula quaruntur,

ita 9 ped. 3 poll. dant I, 542 funt enim I hex, 3 ped. 3 poll.

yte («

qui 3 ped, & 3 poll, dant O, 542 quz ad unam hexapedam, feu

ate c0-

1 quàn

ad unum integrum addita faciunt 1, 542.

is enum

Ped, Poll.

Decim,

&

c

1

m

Óic COf*

Ped, Pol | D

i

Ped, Poll. ) Decim,

O;

O, 014

l;

0,18

jI

m efte,

25

0, 347

)tzet

1

0, 028

I95

O, 361

O

—

mj

jl

0re

O, O42

O, 209

-

7;!

)

Dneaee

9, 056

0, 223

O, 389

O, 069

O.2

27

Q, 403

O, 083

O, 251

O, 417

hasss

O, 265

0, 097

O, 43l

t H

O, I 1I

279

O, 445

tro t

0, 125

WO 6673 OVCA -& 02 D -

o, 293

O, 459

O 0 0 dU I«

Yo

8&000'*! OC d- 05 p «

efium.

0, 139

O. 307

0,473

-

O, 153

ET

O, 320

11

o, 487

hic p*

pg u

I

[9]

O, 167

2;

O

O, 232

O, 500

atus on

Ped: Pulh

Decim.

Ped, Poll.|

Decim.

Ped, Poll.

J

Deg im,

PAMUATCIKAN

üt

er2o

[o Ul

AT

T

O5514.

4»

O, 681

3

847

O, 528

O, 695

ïï

3

[52

^n^

21

D

O, 542

O,709

L9]

5

bQPLJ

erpen

0, 556

0,723

u-

O) ö(:

Q, 570

Q, 90

D)

ais

"um f

O, 736

583

O,750

O, 917

O, 597

Q, 764

O, 931

Q OTT

O, 778

0, 945

0, 625

ä

0,792

0, 958

ericttl

O, 639

lo

O, 806

O, 97

$

.gu'iü $

0,653

II

( Ef(i*

O, 819

Ls

O,

209

4,

G:S*OOOW O n . UO H m

O, 667

5

Q

2

Q - O2 €660^1 OUn . 05 !0

Q, OOO

"

»d05 64

*

CHO

MA

£,ATDUT

IV

DE

64

SCHOLION Il

67, Logarithmi harum partium lidem accipiuntur, qui im

omiífa interea cha-

$£abulis refpondent

his numeris integris

BI

ra&eritica, ita pro 2' 6 feu 417 accipitur logarithmus nume-

114175 pro oO 5

feu 069 ef logarithmus numeri 695 pro

Lont

dum dux note decimales

characeriftica autem fequentia nota:

prefixo zero occurrunt

id eft

dum centelim* & millefimze fine

dum tres de-

decimis dantur ,

chara&eriftica logarithmi eft $,

c«imales, feu prseter centefimas etiam decimez adíunt fine inte-

gro, characeriftica eft 9, dum ad 6 pedes,

feu ad integzram

infin

hexapedam pervenitur, characteriítica eft 0, ufque ad novem

hexapedas; dum 10, zi feu duarum notarum hexapedz occur-

eley

ceunt, chara&erifüca eft 1, ut in integris diximus,

f ;

d»lm.

CHOLION

IV.

D

etit

[

68. Exemplum operationis

primo triangulo ad Rzefzow

ebfervato adferam

Quadrans ad finifiram centri turris ftabat,

ut in kertio cafu propofuimus, perpendicularis ad diredionem

verfus Metam feptentrionalem erat 2 pedum,ad directionem ver-

fus Meridionalem 9 pedum,

illud ex tabula eft — o, 333, hoc

iiffat

eít — 1,500; primum latus inventum erat 577504; alterum

i

1300,6 hmc fequens erat proportio facienda, ut

—

lut

latus 577, 04: fin, tot

perp.

333 : fm

L:

16

l.

fin, tot

p

zi

perp,.1, 500 : Jf

let

I' Log.perp. O, 333 —:9, 522444

Vato

-- IO.

Pam

Log. lat, 577, 04 — 2, 761200

01

761244wín,1,59 zum-4-

h

A' Log.perp, I, 500—0,17609I

effe.

-- IO.

00 p

Log. lat, 1300, 6 —32, t 14145

fii

7,061946 —:1in,5,57 —m—

fi v

;mtur ab angulo obfervato m turri 67" 5 31

fubtrahendum erat

pfo]

3 "7 & addendum 1'59", hoc eft,

a toto fubtrahi debent,

nngu

15 58 , & erit angulus corredus 57 2'33", & fic de reli-

quis,

up

W

ten

P RO-

di

DE CORRECT. ANGUL, RELATA AD &c 65

PROBLEMA IL

t, qui

eteà Chie

| 69. Reducere ad planum obfervatoris angulum,

H

lu$ numee

qui inter duo *objecta sequaliter elevata fupra hori-

695 pro

decimales

zontem aut depreffa infra eundem capiuntur,

fima fine

ttes de«

RESOLUTIO.

je intee

T. Obfervato anguloo (Fig. 62.) obfervetur quoque

integum

jd novem

inftrumerto verticaliter ere&o angulus altitudinis, feu

( OCCf

elevationls m vels. IL. Fiat proportio: ut Cof. altitudi-

L

Wif fim. tobzz- Cof. L angult obfervati: ad fim,

2

anguli v*

i

duciz,

L. Duplicetur hic inventus angulus, & duplum

erit angulus correctus.

! Mefrov

s flabaty

DEMONSTRATIO,.

redtionen

Sint « & P duo obje&a elevata, ad qus ex o col-

em vere

limatur, c & d fint puncta in horizonte, ad que de-

35 hoc

alterum

miffa ex a & » perpendiculatis cadit, certum eft angu-

lum aob obfervatum majorem eífe quam cod, qui in. hori-

) :

zonte exifere deberet, & ad quem prior reducendus

$o: 5

eífet.

Si enim per obje&a dc, ja, & o locum óbfer-

vatoris düo plana in o coeuntia ( Fig. ÓT.) tranfire fin»

gamus, patet angulos in o efformatos eof, dob, goh

eo magis ícmper decrefcere, quo propius ad verticem

i :m*

latera ducuntur, & angulum in horizcnte coz maximum

eífe

Certum praterea eft triangula 2do, aco, ( Fig. 52,)

ob perpendicula ac & ?d effe ín c & d re&angula, hinec

fi in o verticaliter ere&o inftrumento capiatur engulus

g

$? Vel ?, qui in hoc cafu zquales effe ponuntur, hoc

ipfo habentur etiam anguli? & r, qui funt complementa

adum etit

j debe

angulorum altitudinis m & z; & trlangulum quodvis,

de xcïi'

ut patet ex trigonometria fequentem dat analogiam $

ut fin. tot. :lat. ao z Cof. m ( feu fin, i j :1at. oc,. & al-

ternando: oe: ao z Cof. altitud, : in, tot, Du&a ad 4b pere

X

pendi-

Ü

Metsb, Math, D. 11 :

——

72

ul UP ii

CAPUT

lV.

DE

66

N ?

endiculari of cum o« & ob squales ponantur, divide-

emiá

tur linea 4 & angulus aob in duas partes zquales ( $.

gir

157. Geom.), du&o arcu ce, radio oc, fiet oc o5 &

j tria

Áuplur

eg, que eft perpendicularis ad fo & parallela ad af,

erit finus anguli aef, feu dimidii aob obfervati, ficut ck

ob re&um in A finus eft anguli cof feu dinüdli cod am-

guli corredi ; Jam ob triangula oeg & oaf fimilia erit oe:

fenpe

og z 6eg: uf: velob oezm oc & af zmz Chberitoocs or cum

—

x

tvl

eg: ch: hoc eft: Jatus mimus ad latus majus , ficut. frnus

X

aob ad fraum

——

2

cod, combinata hoc analogia cum priore,

habebimus:-

oc: oa z cof. a

?titud. : ffti.. tot.

oc : 00 — fin, X ang. obfero, : frn, & vedutli s

1

Ü

igitut omiífa prima ratione zquali erit. cof. altitud, : fu.

H

200

L

tot —— fin.

s

I

amngul obférvati ; — fin.

——

2

redu&i, inventus an-

DW

gulus fi duplicetur dat angulum ad horizontem redu-

À

&um. Q, E, F. & D.

COROLLARIUM EL

/

tioti

7o. Uthoc exemplo illufremus: fit angulus o obfervatus

e

j

66? , fit angulus elevationis m & n —4^, complementum ergo

eorum adeoque cofinus z feu finus 2 —— $6^5 accipiatur ergo an-

angu

fin.33^ — 9,736109

guli obfervati dimidium , —:33, ad-

18

datur ad logarithmum finus 33^, fi-

fn, tog.—: IO, OOOCCO

& a fumma fubtrahatue

nus totus,

19,736109

logarithmus complementi feu $6^, re-

fiduum eft finus dimidii anguli cor-

fin. 86* 9,99894t

reci, quod refiduum in tabulis loga-

m

9,737168

rithmorum inter 33? 5'& 33^ 6' re-

Ilüs

peritur: fiat ergo ( $. 54. Trigon,) ut differentia tabularis — 194,

ad differentiam minoris logerithmi a meo —2 88, ita G0" ad 27" , €r-

QUj

üq

ld

£o dimidius corre&us —— 43? 5' 97", & totus— 66? 10 54 .

(&

COROLLARIUM I.

bai

Clb

71. ldem poífet ebtineri, fi ex latere uno noto vel affum-

to triangula calculemus,

Sit latus ao ex praecedenti calculo no-

d

N

twm, cuim aob fit trjangulum l(ofceles, ebfervato o invenitur qui-

vig

— —

—

— -

m

DE CORRECT. ANGUL. RELATA AD &c. 67

divide

vis ad bafin s, & £, & refolvitur totum triangulum, habetur

2)

j

ergo latus ab & illius dimidium ef;, quod eít zquale ch,

In trians

u

&

gulo re&angulo eoa ex noto aó &angulo m invenitur cü, tándem

in tfiangulo rec&angulo coh ex noto co & ch invenitur coh, cujus

d :1f,

duplum erit angulus cod correctus,

cut ck

5

d 3l

SCHOLION,

It oe

72, In hoc cáfu bene notandum eft, angulum Cöï*?-e&'ufä

u-

femper majorem fieri obfervato, hinc corre&io femper eít addis

If

tiva, feu obje&a fupra, feu infra lhorizontem ponahtur,

J[1016,

PROBLEMA Ill

| 75. Reducere ad þlanum obfervatoris angulum,

qui inter duo objeca, quorum unum eft in hori-

m

zonte & alterum fupra vel infra horizontem, for-

J5 at

matur. ( Fiz. 63.)

Tedi-

RESOLUTIPO

l. Obfervato angulo, obfervetur angulus eleva-

ï;

tionis, feu altitudinis objecti.

Il. Fiat proportio: ut

tvatus

j etgo

cof. altitudinis: fiü. tot —— cof. anguli obfervati: cof.

anguli redu&i.

0 u

Angulus ita inventus eft angulus cor:

^ ad'

2

rec&us.

Dü.

DEMONSTRATIO,

ihatue

y

It

Sit obfervator in a, qui inftrumento capiat angue

| Cor

[oga*

lum ca?, obje&um » fit in horizonte, c fit apex alte-

j t

rius obje&i, cujus altitudo fupra horizontem fit cd, &

195

cujus menfura eft angulus cad feu w.

Habetur in hoe

e

[

triangulo acd re&tangulo hac proportio; ad: ae z /jm, $

r B

(feu cof: ) : /fis. tot,

Du&aex c, perpendiculari ce ad

bafin a^, fupponamus verticem v feu lineam ed in d in«

clinarti ad planum ab, fiet triangulum «cd obtufangulum

ffont

in r, & erit proportio ad * ac x Jin. i : . v,. fed finug £,

) 00

um ace teGangulum in £, efi

qii

fi couüdereuägs triangul

j

av

Gol

—— Á

DE

68

Q AIEDT- IX

inp

eonty

complementum anguli cae, feu cab obfervati; finus r,

quzr.':ta

utpote obtufus squalem habet finum cum acuto d con-

jrele

|

tiguo, qui d in triangulo «de re&angulo in e eft com-

plementum anguli dae feu dab correcti

ergo fübftitutis

his valoribus pro 7 & 7 erit proportio ad: ac zz cof. ati«

guli obfervati : cof. anguli reduti.

Combinata jam hac

Vr

proportione cum prima fuperius allata

ad: ac z cof. t ; frtt. tot,

hu

adiac m cof. ang. obférvati : ad. cof. aug. corretit,

omiffa prima ratione, zquali ubivis, habetur cof. m,

d

Íeü cof. altitudinir ; ffu. tot z cof. ang. obfervati : — cof. ang.

correti, Q. E. F. & D.A

p

COR OLLARIUM 3^Sk

obíe

6

74. Sit altitudo obje&i feu angulusz — I^, erit ejus come

[

plementum feu angulus — 89^, angulus obfervatus fit 65^, ejus

U'tüj

ergo complementum erit 95, igitur ad logarithmum cof. anguli

fin, 25^ — 9, 625948

obfervati addatur finus totus, a fum-

"l t

ma fubtrahatur log. cof. altitudinis feu

[c4. 2—I10

icci

fin. 89? refiduum dat cof. anguli cor-

19,625948

re&i feu 64? 59' 4.5"

fn. 89*

9. 999934.

——

9, 626014

URT

COROLLARIUM

I.

et

[

75. Cum tubus mobilis femper magis removeri debeat, a

8

fxo & arcus major aperiatur, quo elevatius vel depreffius et

obje&um ,

& quo magis inftrumentum , reli&o uno tubo in fitu

horizontali, inclinatur, fequitur angulum femper vero majorem

fieri, hinc corre&ionem femper effe fubtra&tivam; diftantia enim

ab eodem obje&o horizontali femper minor eít verfus bafin,

Ynk

quam verfus apicem turris.

fs

SCHOLION.

76. Cüm hic cafus frequentior fit, atque ad illum alii quo»

que cafus, ut mox videbimus, reduci queant, wut moleftia cal«

culi evitetur,

tabulam pro hac corre&ione damus,

que in

N

gerit,

ironte angulum clevationis feu deprefüonis, a IQ ad 4^

in

—

DE CORRE?T. ANGUL. RELATA AD &c. 69

in prima vero c;)lumna ad dextram angulum obfervatum dat,

IU f,

eoncurfus columnarum deorfum & a finiitra (imul verfus dextran

| con-

quantitatem fubtrahendam e&hibet; in exemplo fuperiore angue

lus elevationis 1? & obfervatus 65? dant 15" fubtrahenda, &c.

Com

Litutlg

;

)/. lfl

PROBLE'MA IV.

XX

] àt

471. Reducere ad/þlanum obfefvatorïs angulum

;þ

H

inter duo objecta, quorum unum fupra, aliudinfra

horizontem eft,

N

)

i

f. ft,

RESOLUTiO«

. lf

l. Obfervatis angulis ( Fig. 64.) elevatiohis & de-

preffionis utriusque obje&i m & v, inter ipfaobjecta c & d

obfervanda eligatur aliquod obje&uin in horizonte v. g.

conte

€, & capiatur angulus coe & angulus eod.

II. Uterque

juxta problema pracedens applicando bis proportionem

€Jus

nouli

ut 'cof. altitud. : fu. tof zm cof. ang. obfervati : cof: ang. cor-

fütte

ve&it , cor*rigatur', vel pro utroque ex tabulis correctio

; feu

accipiatur.

NOS

SCHOLION,

: 78. Sit angulus coe 5$? & eod 32?, 30', angulus eleva-

tionis prioris obje&i — I? So', depreffionis alterius z 55 erit

correctio ex Tabula pro primo 49'", pro altero 41", qui proine

de fimul in unam fummam colle&i & fuübtra&i a toto obfervato

89? 30 , dant angulum corre&um 88* 28' 30"".

s elt

PROBLEM.A S V.

ftü

tenm

19. Reducere angulum ad planum obfervatoris

nim

;ll

inter duo obje&ta diverfam altitudinem aut deprelfio-

nem habentia.

RESOLUTIO.

j

al

L Angulis altitudinum vel depremonuxþ obferva-

d

tis (Fig. 65.) eligatur aliud obje&um in horizonte feu

'Tifp

E 3

plano

m surssieeus

—

f,/

E

"O

APU E I

D

4

|

tun i

plano Obfervatoris ,

& angulus inter màájus objecum &

-nxewwï,

hoc Ügnum, & iterum inter minus & idem ügnum ca-

II. Reducatut feu corri-

!(]w' 4

piatur ,. fcilicet coe, & doe.

jus C

gatur per analogiam prafcriptam vel per tabulam an-

(otda :

gulus i

I1l.

Minor a majore fub-

u d

uterque obíervatus.

üag,;us dat verum angulum correctum aob.

am J

ftantia 1

lum ang

OROLLARIUM I)

tum sfte

ut quo

$0. Potuiffet etiam ut in priori problemate objectum quod-

dam intermedium in horizonte accipi,

tunc fumma utrius que

dent,

Slt ane

anguli corre& i daret angulum ad hmmontem fedu&um,

sdent

gulus e,kvatmvus obje&i ac z—I?

40 ; Obje&i bd —

?Oa

coe

i

f-

12 t

alterius

fit—85?

doe fit zi ?o

erit pr1m1 ccrrpc*ho — 8

corredtlo-—- 14 , €rgo aoeM84 59 5., , & boe

9* 59 46 ,

erco aob —

55 oo' 6..

, R

&

COROLLARIUM

XT.

tv

$1. Cum hzc duo problemata immediate ad Problema IL. f

horine

ducantur, eadem , qua ibi data eft, demonftratio etiam his aps

tedent

l

m e

plicari poteft

Yatit a

SCHOLION EL

17?45 FV

tuec i

$9. Si obfervator inftrumento goniometrico inftrucus . eft,

fiotter

in quo tubi furfum & deorfum motum etlam verticalem habent;

aut dioptra! filum fuperne tenfüm gerant ; vel inclinari poffint

Vét0 (

tnt

ut $$. 43. 55. diximus;

vel denique angulus altitudinis exiguus,

& latera longjora fuerint,

eorre?tio ,

que fere in fecundis, ut

toted

ha&enus vidimus, confifíit, fine errandi, aut confufionis peri-

culo emanere poteft, & angulus obfervatus pro vero affumi ,

eum emm, ut JA]TI 1n:1nuaveram

in minoribus etfi accuratis in«

firumentis de vno alterove mmuto in angulo obfervando fecuri

non fimus,

fruftra accutationem in additione vel fubtractione

vnius, alteriusve minuti quarimus.

SCHOLION II

$3. Si lztera valde magna accipiantur, horizontalis linea

ab (Fig.66,) jam non eft parallela ad fuperficiem telluris &

verum horizontem, fed fit illius tangens, adeoque angulus verus

ad obfervas

altitudinis non eít angulus obfervatus €ab fed ead,

um

tum

—

E d

DE CORRECT. ANGUL. RELATA AD &c. *r

um d

tum igitur eft Ped addendus, aut in depreffione fubtrahendus, cuju$

lum (a-

menfura, utpote a chorda & tangente fa&i, eít dimidius arcus

"

Ü COrtl-

ad ($. 95. geom,) inter duos verticales ac & dc, in centto tel-

juris coeuntes, interceptus; quae quantitas arcus a longitudine

lam a

cehorde ad, cui proportionalis eft, aut a longitudine tangentig

Ite füb-

feu diftantiarum inter duo obje&a dependet,

Liesganigius tzbue

lam in fuo opere proponit, in qua angulus predi&us data di-

fantia in hexapedis exhibetur, unde innotefcit, dimidium il«

lum angulum in diftantia 200O hexapedarum ad £ minutum tan-

tum afcendere; quibus igitur inf?rumenta majora ad manus funt,

1 quod«

aut quorum labor fummam exigit accurationem,

ilius opus

ttiusque

adeant, ubi & minores alias corre&iones,

4 refra&ione aeris

it ate

pendentes, & redu&ionem majoris feriei triangulorum ,ad libel-

0;

lam maris inyenient, nobis hec infinuaffe fufficiat,

alterius

$ 46

SCHOLION

XII.

94. Al hanci redu&ionem per refolutionem parvi illiu$

trianguli re&anguli faciunt, quod habetur demilfa ab obje&o ad

iL fe

horizontem perpendiculari cd in Fiz. 63 Nempe latus ac ex prae

hig 40*

eedenti calculo ,

ob determinatum jam ab & obíervatum a, no-

tum effe ponitur,

n angulus elevationis obfervatur , & inde

latus ad in horizonte invenitur; eodem modo ex altera parte

latus bd, & tandem pun&um d determinatur.

At cum caïz:uïas-

hic femper à latere inclinato , & incorre&o inchoetur, fequitur,

ys -eft,

rfigorem geometricum hanc operationem non habere.

In noftro

abent:

vero cafu cum anguli initio corrigantur & ad horizontem redue

peflint

inchoari

eantur,

caleulus imediate a latere horizontali vero

ciguus,

potet, quin aliquid incorre&um , ipfum calculum inmgrediatur,

i, U

; peri-

ifümi ,

(s if

(ecuri

&tione

finea

üis k

verus

era*

E 4

CA-

m

|

1

pR(

TÀ

CAPDIPT V

ia

dendin £

tyuat

CnRTUT *

fuere !

DE

CONSTRUCTIONE

RETIS

PER

8

TRIANGULA, ET CALCULIS TRIANGU-

tUf QU

LORUM OBSERVATORUM

D ETINITIO H

]

Curatio

$. Rete vocamus feriem triangulorum inter fe con-

&

L]

À

nexorum, qui efformantur in plano menfurando, qui-

Y

bus totum quafi te&um & inclufum eífe fupponitur.

anp

hu

Vi

ilà,

LDEFINITAO.À

^

qa f

- 86. Caleulare triangula eft convenientem laterum

Inter (

l

!

;

long!tllüïnem quam diffanti» per illa reprefentata hà»

(üt j

4

bent , ope formule trigonometricze determinare.

ob

TiOtoS

COROLLARIUM

I.

omne

87. Cum ad ealculos hos quantitas angulorum mota effe

tantu,

&ebeat , patet ,

hanc methodum. tantum lotum habere, dum ans

Tabet

guli mfhumenro goniometrico e.cc:lpwntur in menfula enim figus

line:

$& folummodo habetur.

JUS,

1

COROLLARIUM — H

&in

$8. Ad hoc ereo neceffarium eft

ut meniurato uno primiz

£rianguli latere pro reliquis deinde efformandis unum fempes

l$"

notum latus ex prxcedentibus habeatur, cui novum triangulum

i

dl

fuperíruatur, quod obfervatis ad illud latus angulis verfus ob«

je&um quodpiam generstur

Scientia igitur tiigonometrica pro

Cedi

ealculando

o (

& propofitis hoc in tracatu principiis pro conftruen«

dis triangulis probe imbntus flt oportet, qui dimenfionem talem

&i

aennexis cum calculis affumere audeat, fi unum ex his neglexe-

dt]

y1lt, aut ignoraverit, non miretur, $2j, redeundo in ferie trian-

£ulorum ad aliquod jam mnotum przecedens pun&um

aut latus,

hzc ultima primis minime cohzrere experiatur, 2atque ad claus

j

H

dendam

|

DE. CONSTR. RETIS PER TRIANG. &c, 73

dendam feriem vel partes quafpiam exícindere, aut nova loca,

fylvas & campos, ut ipfe non fine ftomacho oculis ufurpaveram,

efüngere cogatur.

PROBLEMA L

PER

89. Angulos pro conftructione retis triangulo-

GU.

tum obíervationibus goniometricis capere.

RB$Od UT:O.

T. In loco apto menfuretur ea, qua requiritur ac-

e con

curatione, linea redta, qus bafis totius dimenfionis (fit:

j qUA

g. 27.. & feqq.

II. Obferventur inftrumento in utraque extremitate

h

anguli, quos circumfita obje&a, turres, cruces, pati-

bula, arbores diftindte, camini, aut ornamenta fume

ma folitarie domus &c. cum altera extremitate, aut

'erum

inter fe unum cum altero facit, & in librum referan:

j h

tur; (.$.47-) loca ex utraque parte obfervata per has

obfervationes hoc ipfo determinata funt, cum ob duos

notos angulos & unum notum latus triangulum totum &

omnes diflantix» innotefcant; ad ea vero, qu» ex una

| efto

tantum extremitate confpici poffunt, dire&io tantum

1 as

habetur, ibi enim triangulum clauditur, & longitudines

fiie

linearum abfcinduntur, hic vero innotefcit folum angue

lus, qui lateribus etiam in ipünitum prolongatis idem

& invariatus manet.

"ni

IIT. Transferatur dein inütrumentum ad unum ex

pe?

his vel determinatis , vel tantum vifis locis, &ad duo

lum

jam determinata ]oca, inter qiwe latus notum inter-

obs

cedit, collimetur, hic quoque locus feu- nova hec fta-

pro

tio ob notos duos angulos & unum latus determinatur,

e

lem

& in hac nova fiatione reliqua quoque 1oca confpicua,

xe

ut prius, angulis capiantur,

n

IV. Dum locus non occurrit in e

ä parte, verfus

If

quam dimenfionem promotam cupis, qui pro fítatione

Tu

ES5

fe-

;,ïvt f*

74

CATPUT —:V

fequenti fervire poffit, fignum in diftantia conveniente

ï*oflLIO

Q? )í

liil

V. g. arborem decorticatam , cui in fummitate tantum

aut vexil-

Iit.

quidpiam ramorum & foliorum relinquitur

lum altius &c. erigi cura, & hoc fignum determina, ad

Cm,gïm

quod dein inftrumentuim transferendum fit

ia Qu

Vo

Ita labor tamdiu

continuetut

( attendendo

ut femper ad duo loca determinata

inter que la-

tus jam notum intercedit, ex ftatione nova collimatio

W

fiat,) donec percurfo toto fpatio aut majori faltem illius

extremit

parte ad primum iterum latus perveniatur, & labor

$ (R

Claudatur.

Ad calculos jam & figure conítru&ionem

ln me

1n N

procedendum.

- 1

PROBLEMA II

$ em

fye'urü

90. Figuram retis ex calculis trigonometricis

Conftruere

4ddatt

RESOLUTIO

1j

Di

I, Cum in quovis triangulo tres anguli cum uno

lor,$

latere noti effe ponantur, femper habebitur oppofitio ,

m

($. 42. Trig.) igitur corre&is ab initio, fi opus fuerit,

ex prxcedenti capite angulis fiat proportio , //mis anguls

BXxm

bafi vel lateri woto oppofiti , ad ipfum hoc latus, ita fimus cu-

ermir

furvis alterius anguli ad latus oppofitum , qu&e proportio per

prma

omnia, hac ratione, accepta triangula

fervatur, fi

totted

Atlt

enim alia methodo determinetur triangulum, aliz in ob-

htemn

fervando ,. & calculando accxpwntur regule , quas infe-

rius in f'olutlone problematum videbimus.

H opera-

tiones cum triangulis bono ordine in chartam transferane

tur, ut longitudines laterum mventæ pro conftruenda

figura excerpi queant.

IL. Conítruatur fcala conveniens fini dimenfionis ,

(S 26.) & ex ea bafis in chartam transferatur, cujus

p

DE CONSTR. RETIS PER TRIANG. &c

i5

n Ventente

pofitio relate ad meridiem quam proxime nota eH"e fupe

te tantum

ponitur. (S. 49.)

àut vexil

II Invente diftantie ex fcala geometrica circino

tmnina, ad

capiantur, & ex utraque bafeos extremitate interfecentur,

pun&um-interfe&tionis locum defignabit

tendetdo

Que /a

SCHOLION L

Qllimatio

Ut hoc exempolo illuftremus:

tem ilhus

9ïl

fint obfervati aneuli I&

extremitate bafeos,

C$. 47.) & quidem in meta feptentnonah

& lobor

$ ( Eig. 67.) verfus metam meridionalem & Rzeízow — $6? 4,1

uCtionem

In meridionali MM verfus feptentrionalem & Rzeízow/ 26?

16,

In Rzefzow T obfervatus, & redu&ione centri corre&tus ($. 59)

57 3 Bafis $M. eft z— 19005; igitur erit

proportio fin. T':

SM —ün.

M: TS

fin. T; SM zfn. S: TM

& erzo a logarithmo SM fubtrahatur fin, T'& ad differentiam ad-

n fï lv!;

datur fin, M habetur

diftantia TS

fi ad eandem differentiam

addatur fin. $ habetur diftantia TM

Log, Baf.—: 3. O7918I

fin, T, 67? 3 9. 964187

Differentia 2— 3. 114994

3. 114994

im uno

Log.$,86 4

9. 999272

9. 645962, Log. M, 26* 16

pOJ

Log. MT — 3.114266

2.760956 Log.ST

s TUef)

MI —I30t

STzri 576,7 feu 577

u atf

EÉx metautraque, alia quoque objecta verfus omnem partem de-

fm: a

terminata funt

ut fequens tabella oftendit

in qua columna

[oper

prima nomina locorum, fecunda angulos obfervatos & ex $ 59.

corre&os, tertia logarithmos finuum horum angulorum, quarta

[tUT j

; j1.00

logarithmos laterum his angulis oppofitorum . quinta quantitatem

laterum , fexta fitum trianguli exhibet,

s infe-

ope

sferitte

enda

.

fofil

?j

eu

Nomi-

P

CATUT YX

276

DE(

Nomina locorum

Anguli

Log. fin. Log. Jat

Latera

ab eatenn

UTR

BaüsMetaSeptentrio.|$ 86

999272|3., 079181|x200 SM|:

|tat

Baüs Meta Meridion

M 926

1

64596212, 114266|1201 MT

Y

jvey

Rzeízow turr, Paroch

T 67

ää

964187!o

2.760956, $77 ST

*j2220 *jiaA|

9

Baf, Meta Septentr

8 154

21|9. 636360|3. 461780

2895 RM'

J.!ä A

mappi e

^i

|

Es

Meridion

M 15

1919. 421857 |3. 247277

1767 RG.,O

digitus un

Stáromiafte.

R 10

20|9.253761]3. 079181,1200 $14

Yipedas Y

euyos 1o

impolita

af. Meta Septentr.

$ 115

25/9. 955789 3. 164809

1461 PM

j1e4| 3093

fitw, fu

xB

-

-

Meridion

M 16

43 |9- 458848 |2. 667868|

1

465 SP|'

fertilitas

$J

Pobitna camin.diverf.|P 47

52'9.871061|3. 079181;1200 SM

la adum|

—

offclale

Meta Baf. Meridion.|j»r 47 97|9.867283|3.473325|2974 CT

fuis pun

à

Rzefzow

-

L.

[

T113 45|9.961569/3. 567611|3695 CM*

longior t

T ad btdromwa. -* -C. 18 4819-. 508214|3. 114266! 1301 MT|

Jate

J'eoao 519A[ *310

Théc n

beret, j

SCHOLION I

quot fex;

erfedtion

99. Cum verfus occafum mihi eundem effet,

elcn.crucem

fentibilis

quandam ad Staromwa, cujus angulum cum Rzefzow in Meta me-

Tientum

$íidionali bafeos jam cabpiebam, translato ad crucem'inftrumento

angulus inter Rzeízow & metam meridionalem obfervatus dedit

Tno in

TéQuten

quoque tertium angulum in triangulo, cujus anguli funt crux,

Wzefzow & meta meridionalis, in quo ob notum latus TM

triaà

"ota, & oppofitio habetur, proin diftantie -f- T& -!- M invente

funt, & crux ex fcala in chartam translata eít

Ex iisdem íta-

L

9

tionibus quoque loca, Tyczyn, Boguchwala

Glogow determina-

Y ob

bantur.

Inde fignum deorfum verfus Meridiem poni curabam,

E ïOng-

quod cum locis duobus determinatis per obfervationem anzgulo-

& amplij

tum conjungere poteram, & fimul fatagebam, ut locum quem-

pindg f

dam novum determinarem, qui pro conjun&ione novi iterum

Vitur;,

unius alteriusve figni ferviret , quam methodum per totam di-

I

menfionem obfervavi

tibus q

donec deorfum ad fines ufque Ungariw

continuatis triangulis, atque inde in conveniente diftantia fure

bebint

fum verfus Viftulam

iterumque deorfum ad latus ex primis de-

t£erminationibus notum

laborem perduxiffem, ubi inventa exi-

A fine

U .

£ua omnino differentia

ealculi

exacitudinem fimul

& verita-

bt &

t£emó compróbabat,

$8 nün

Reinius verfus ortum ibat, fuaque triangula

ab

VA.

DE CONSTR. RETIS PER TRIANG. &c. 77

Ktera, |

—

—— J—

2b eadem bafi verfus S. Mariam Magdalenam eadem methodo

9 $M

| "

eontinuabat , donec ad notum latus verfus Glogow & Rzeízow

"h

) MT

j9

rediret,

7 $T.

^

SCHOLION.

Il.

EEj lts

&e

in

[4

93. Pro triangulisad chartam transferendis, & conftruenda

5 R5

$nappa elegirnus fecalam pedi viennenfi proportionalem , ut fcilice&

1 RG,

digitus unus IoocOo hexapedas, linea ergo decimalis centum he-

0 )l

xapedas valeret, quod fuffcientem omnino charte geographice,

:

——

cujus folum , & non militaris aüt ceconomice conftru&io nobis

|

impofita erat, magnitudinem daret, ut loca fingula , fylvarum

3 PM

ä

8P.

fitus, fluviorum curfus , agrorum. fimul etiam in ipíüs coliibug

)

^0

fertilitas exhiberi poffüt, quarum rerum inter ipía jam triangu-

0 5

;

la adumbratio ad militares qui nobifcum laborem partiti funt,

l

4

officiales pertinebat.

Licet interfecdtio ipfa duorum laterum exia

|n

4 CT

£uis pun&ulis notaretur, umpediri tamen non potuit, quo minus

Cu

longior triangulorum feries ad dextram magis finift(ramve decli-

3

E

T M1

naret, ita ut quamvis calculus poft reditum ad eandem lineam

| g

nec unius hexapede, ut mihi quidem evenit, differentiam exhi«

beret, ipfa tamen conjundionis pun&a duo centenis etiam ali-

quot hexapedis diftarent ; ita difficulter fcilicet minima illa in-

terfecionis quantitas .notatur, in qua error initio omnino in-

Tereem

fenfibilis in magnam excrefcit differentiam ; evidens fane argu-

Meti me«

mentum, hanc methodum in conftruendis chartis majoribus om-

(rumenta

nino inutilem effe, unde ad aliam, de qua inferius fermo erit,

yus dedit

recurrendum.

unt CX

TI, ttia

S-CHOLION IV.

[ jinvent?

94. Si inftrumentum tubis inftrudum fit, ut. remotiora et-

sdem f«

iam obje&a ditin&e videri poffünt, latera triangulorum etiam

sterminie

ad longitudinem aliquot millium hexapedarum prolongari poffunt,

curabatt

& amplifima regio paucis triangulis includi, quorum., vertices

) anzulo-

punca fixa conftituunt, aliis deinde minoribus dimenfionibus fer-

n quem-

vvitura, quod ad accurationem multum confert,

- jterum.

Ita fixis a nobis ut figura 78, exhibet pun&is a fe remotio-

ribus cum fuis

menfulis domini

n d-

offidiales fuccedebant;

ha«

Ung?,ïiæ

bebant ante fe aliquot jam punda ita calculata, & ele&a ftatio-

jui; f

ne vel in ipfo tali pun&o, vel intra illa, figuram intermediam

imis de*

jam fine errandi periculo, fine novo dimenfionis labore depingee

j a

bant, & exa&um laborem comprobabant; cujus operationis le-

; veritt*

£es ad menífulam praetorianam pertinentes in fequenti problemate

damus.

qianga

jb

P R Q-

P Rs

——

H

De(

78

CAPUT Y

n

catu

PROBLEM A Ill.

1

]ing 5

;l obj

95. Conftruere mappam ope meníule Prato-

qtiuncur

tijanse,

(b trans

RESOLUTIO.

gulorum

& quor

1. Menfurata bïaüs ex confiru&a competenti fcala

minabut

ad chartam in menfula firmatam transferatur & ipía men-

fula ita in una extremitate bafeos ponatur, ut punctum

extremum bafeos in mnenfa accurate eidem pund&o ba*

961

feos in terra correfpondeat, (S. 54.) in quo pun&o

L

acicula perpendiculariter ere&a firmetur. Il. Applicetur

quadrant

landi ev.

acicule lineale cum dioptris ita; ut accurate totam ba-

tus in d

feos lineam in menfula du&tam tangat, ipfa vero men-

[ SWGN

fula tamdiu circumagatur , donec alteram " extremum

um (

bafeos per dioptrasappareat, correfpondebit hac ratio-

udas ]

&Citiof[b

ne linea bafeos in menía accurate linex bafeos in cam-

debuntut

;

po. lIl Dirigantur dioptre verfus circumfita obje&a

AtCurate -

ita tamen, ut femper acicule adjaceant, & ducantur

ücantur

tenuiffume linee fecundum aciem regule, quibus loca

tfas acte

verfus qu» hs dire&iones fiunt; adfcribantur.

IV.

ipfa deb

piam, $

'TTranslata menfula ad alteram extremitatem bafeos ita

&ionum:

rurfus ponatur, ut pun&um in menfía verticaliter cor-

&iones j

quodnam

refpondeat pun&to interra, ($. 54-) & fixa ibidemaci-

cula & applicata regula fitus talis' procuretur, ut linea

i-prim

bafeos in menfa correfpondeat linezx bafeos interra. V.

pundis,

Vend i

Dirigantur ut prius dioptre verfus circumfita objecta,

tenuiffimze linese ducte priores interfecabunt, interfe-

&ionis pun&a loca fingula determinabunt. VI. 'Trans-

9.

feratur dein menfüla ad unum ex his determinatis locis,

?ïeta 3'&3

& ibi ita omnino procedatur, ut primo & quarto dixi-

idebet,

Wt ttja

mus, uf fcilicet puncum in charta notatum accurate

pun&o ftationis refpondeat , & lineale acicule, & linez ,

ïttïïope

t T;tg?nr :

qua inter duo obje&a jam determinata du&a eft, appli-

Catuin

tig li

i

.DeCONSTR. RETIS PER TRIANG. &c. 79

catum per dioptras idem ooje&um exhibeat, erit hec

linea bafis novarum interfe&ionum, que dire&o ad

? Preto-

alia objedta lineali, & du&is juxta aciem regule lineis

oriuntur. VII, Hac methodo ad alia interfe&ionis pun-

&a translata meníula feries rurfüm &quafi catena trian-

gulorum exiítit, quz totam füperficiem chártze obtegent,

& quorum anguli totidem loca plani menfurandi deter-

ewt fealà

minabunt.

tp men.

$.

COROLLARIUM.,

.

t pundtur

Yundo bie

96. Videri poffet ex hactenus didis, hanc methodum mul-

o pundo

ta commoda przxfeferre, ob que mehfula pro goniometrico au&

quadrante eligenda foret, przter hoc enim quod moleftia calcu-

ipplicetut

totam ba

landi evitetur, ipfa flatim figura totius plani, ipfe locorum fi-

tus in charta obtinetur; at fi nos reflectamus ad ea, qus prius

Veto Imen

de interfe&ione linearum in $, 92. diximus, difücultatem hic

extremurm

au&am deprehendemus, cum enim interfe&iones per ipífas jam

hac fatio»

du&as lineas, que non omni carent latitudine, oriantur, hz im

acutioribus przecipue angulis per aliquod fpatium coincidere vi-

g in Cam-

debuntur, cujus medium utpote verum loci determinandi puncum

| objeöta

accurate vix deteges.

Accedit, quod cum lines penes regulam

dycatur

ducantur, aliqua licet exigua diftantia a radio vifuali per diop-

jbus locà

tras accepto habeatur, unde concurfus linearum in eo, ubi re-

ipfa deberet effe loco contingere non poteft,

Nec prodeft quid-

At IV'

piam, fi ex pluribus ftationibus ad idem obje&um linez interfe-

bafeos. 1f

&€ionum ducantur, nunquam enim continget, ut omnes hz interfe-

yiter Cof*

é&iones in unico fiant pun&o, hinc novum enafcetur dubium ,

quodnam verum, quodnam pro ftatione eligendum fit5 hec ;jam

bidem aC

in-primo pundo difficultas quantum in multiplicatis augebitur

| yt lined

pun&is, que inde diítantiarum differentia, que dimenüonem clau-

p V.

dendi impofüibilitas ?

! Obieäa*

COROLLARIUMWM

[

-jaterfe

, Ts

97. Alia, eaque non rminor enafcitur difüicultas, dum im-

ie Jocisy

pleta jam priore charta labor eontinuari & nova menfule indu-

ci debet,

Transferénda enim 1fuht ex priori charta pun&a duo

aut tria ad novam, & aliud jamí punc&um inveniendum , quod

vel!ope acus magnetice , vel ope obfervationis angulorum, qui per

& line?y

tranfportatorium examinantur, preftari folet, ubi nova rurfus fta-

t, 1ppl*

tio eligitur, de utraque methodo inferius ágemus; at qui muta-

tlo-

cawïü

CAPUT V.

$o

DEC

tiones, diurnas fepe, acus fagnetice nofcïtï, ipfamque fimul

snagnitudinis chart? differentiam a variaaeris temperie pendens

$em confiderat, accurationem hic non expectabit.

100

COROLLARIUM

IH.

jun conf

98. Sequitur quoque ,

cum res$, quae determinantur, ture

zes, domus, mole, arbores procere &c. effe foleant , menfulam

in his locis plane adhiberi non poffe, punc&um enim correfpons

dens verticaliter ad meníam transferri nequit, hinc penes fta-

kB

tionem ponenda foret, ut collimatio fiat, at quantus 1nde ena-

fetantur,

fcetur error, cum nulla angulorum quantitas nifi per transportae

tehtüf , (

torium: nulla diftantia nifi per fcalam determinetur, adeoque

nullum corre&ionis, aut redu&ionis adfit fundamentum , hinc fi

Il

tamen ope menfule operatio inftituenda fit, przeter obje&a reae

,v

lia aliud quoque fignum inter illa ere&o baculo conftituendum

Qum (ta

eft, quod ex utraque meta bafeos interfecari & determinari de-

bet, ad quem dein locum menfula transferri, & ad pundum exe

Ope dio

nus di(l

cepto iterum baculo accurate poni poteft, & fic multiplicatis bas

culis & ftationibus labor continuandus erit; at quo temporis die

unda r

fpendio & ob repetitas ftationes errandi periculo quis eft, qui

etiam re

non videt.

Poterit quidem Ífine ere&is fignis locus menfule in-

Il.

ter obje&a determinata pofitz in ipfa menfula exhiberi, & pune

étum fignari, ut inferius videbimus, at rigor geometricus & ace

tentis ex

curatio requifita tunc minime habebitur.

tüt; d

C OROLLARIUM IV.

tüte f

tium al

99, Ex his jam colligimus , quantum menful in conftruene

lis arti

do reti, quod fundamentum fit dimenfionis, fidendum fit, exeme

exhibas

plis enim & autoritate hic pugnandum non eft; certe qui metho«

dum cum goniometrico aut quadrante ex calculis determinandi

IV

punc&a fixa perfpiciunt, menfuülam pro hac Operatione omnino ree

transfer

3iciunt, quam ideo tantum adhiberi ab aliis vident, quod.fine

jabore obfervationis angulorum aut calculi tzedio per folam line-

aCU$ ma

arum interfe&ionem pofitionem objecorum veram obtinere fefe

Is ho pa

arbitrentur, Non tamen per hoc ufus omnis menfule rejicitur ; id

Tine &tto

tanquam principium fundamentale de utraque methodo di&um fit:

Vitate :

Pundía fixa per feries triangulorum inftrumento

Ttatims

goniometrico: puncía vero intermedia , & fingu-

la capita , ut agrorum , fylvarum , pagorum,

domuum &c. quantitas G figura, menfula prato-

1)

viana determünentur.

ele lh

PRO

Wl &

i

DL

"

Dv

TU O ETE ais

AB5s

ht

g rR

DE CONSTR. RETIS PER TRIANG. &c. gr

Que fii

It pendene

PROBLEMA IV.

ICO. Determinare punéta intermedia in reti

jam conftructo ope menfulze.

menfülam

RESOLUTIO.

'Ortéfpone

penes fia

wde efae

I. Pun&a quotquot þo(ï'unt fixa àd menfulam trans-

fansportae

ferantur. Quot obfervatores adfunt, tot menfulae pa-

, adeoque

tentur, & tota dimenfio inter eos diftribuatur.

t, hinc f

H. Menfula vel in ipfo determinato trianguli late-

jeda 1ete

Uituendum

re, vel extra, vel intra triangulum ponatur, & pun-

€um ftationis

M.

iinar] de«

uti inferius trademus, notetur, indejam

ndum exe

ope dioptrarum obje&a fingula capi, & alia figna mi-

jatis ba«

nus diflantia in colliculis determinari poffunt,quze con-

nporis di«

un&a novas bafes pro interfectionibus minutiffimarum

eft, qui

etiam rerum fuüppeditant.

sfufe ine

IIL. Spatium quodvis triangulare rebus in eo con-

, & puns

us & o

tentis expleatur, & aptis coloribus fingula adumbren-

tur; dum fluvius, aut via regia occurrit, ejus curva-

turz fpecialiter affignentur, collium quoque aut mon-

tium altitudo major, & declivitas juxta proprias regu-

'onftuer

las artis delineatoris majori umbra

& lineis pluribus

;! exeme

exhibeatur,

1l methoe

eminandi

IV. Ex his partialibus mappis fingula in majorem

nnino 16«

transferantur , vel etiam, fi in quavis debitus fitus ope

yod. finé

acus magneticz aut]lines meridianz obfervatuseít, ip-

un Jue*

ere, felé

fe he partes ope glutinis in unam compingantur, quod

fine erroris periculo facile przftari poterit, cum deter-

itut ; ió

& (tt*

minate jam & vere diftantie ex ipfo reti affumtz fine

gmenio

mutatilone habeantur,

)/ingu.

COROLLARIUM,.

oru

orato

IOI. Qui igitur ad talem laborem deftinantur, notum iis

effe debet, cui ufui mappa conftruatur, fi enim militaris foret,

nihil eorum negligendum effet, que ad artem militarem perti-

»O,

Metzb, Math. P. 1.

F

nent

l

n

pEG

E

À

e

CAPUT V.

4205 &i

nent, ü'bi caftra v. g. ponenda, infidie ftruend?, aut timend2 ,

gnterécant

hofium incurfiones precavende , &c, qu: jam $. 9. infinuavimus ;

jtdfonm

in geographica,locorum precipue, viarum, paludum & curfus

fuviorüm cura habenda eft, inter que agri & fylve interferan-

tur, De aliis partialibus dimenfionibus inferius agemus.

9?

quem acis

PR!OBLEMA V.

luic metho

satius coni

ro5. Conftruere rete triangulorum -ope aeus

ip& quoqi

jufta fahe

magneticae.

poft um

RESOZLUTIO.

tüm abert

as etiar

4, Ponatur acus magnetica, qua dioptris & annu-

angulorum

jo sürichalcino in 460 gradus divifo , inftru&a eft,

&ionibus,

fandi yeri

ad unam metam bafeos, & dioptris verfus alteram me-

transporta

tam directis, notetur gradus, quem acus oftendit.

IL Dirigantur dein dioptre verfus aliud obje&um,

notetur rurfus gradus, quem acus oftendit, & ita per

04.

98v) de n

totum gyrum horizontis obje&a & gradus ab acu nota-

& fv de

ti in libellum transferantur,

iom edoy

ticà comm

IH. Translato infitumento ad valteräm "bafeos me-

gunt, &

tam, & dire&is dioptris verfus priorem metam, idem

fium & $

qui prius gradus ab acu notabitur, dein dioptris ad

priora obje&a direétis notentur rurfus gradus, & in li-

bello adícribantur.

IV. Translato inftrumento ad alias ftationes, no-

tentur iterum gradus inter fingula obje&ta , & adícriban-

tur, atque ita procedendum eíft, donec ad unum ex

nofis jam obje&is tedeatur, & planum totum percurra-

tur.

V. Figuram in charta delineaturus angulos ex li-

'bello excerpe, quos ex differentia graduum obferva-

torum facile determinabis; & eos initio in utraque ba-

feos

DE CONSTR. RETIS PER TRIANG. &c. 85

eos extremitate dein ope ttanfportatoris in lineis fefe

: (imenda ,

inuavimus;

interfecantibus feu in flationum pun&is conftrue, lines

| & curfus

ita efformate rete quam proxime dabunt.

intetferite

COROLLARIUM I.

Dus,

103. Neminem fore arbitror, qui

cognita annuli, inter

quem acus clauditur, & graduum diviüonis exilitate quidpiam

huic methodo tribuendum effe cenfeat, dum rete aliquod accu-

ratius conftruendum foret; Nam preter obfervationem dubiam

E

ipfa quoque conítrucio angulorum ope transportatoris, ni hoc

jufte fahem magnitudinis affüumatur, in errorem inducet, ut jam

poft unam alteramve interfe&ionem figura in charta a vera mul-

tum aberret,

Accedit, quod ipfa declinatio acus magnetice di-

urnas etiam & horarias nonnunquam variationes fubeat, quibus

(p annu

angulorum etiam maghitudines vitiantur, verbo, fi in interfe-

Kl eft,

&ionibus, qus calculatis lateribus fiunt, tantum jam adeft er-

eram m

randi periculum.

$. 92. quantum erit in lateribus, qua ope

Wi

transportatoris triangulum claudunt.

COROLLARIUM

II.

,j'eäums

a per

IO4. Id tamen hic rurfus di&um volo, quod fuperius ($.

U Dot2"

98.) de menfula pretoriana infinuavi, fi de obje&is inter pun-

&a fixa determinandis agitur, ubi error extra quodvis triangu-

lum calculatum propagari non poteít, obfervationes acu magnee

tica commodiffime inftituuntur ,magnum enim apparatum non exi-

208 1De-

gunt, & citifme abfolvuntur; optime autem in curvaturis via-

; ide

rium & fluviorum adhibetur, de quo inferius agemus,

ptris i

& in l

Qäþ

54 no

"ribatl*

um ex

peutta

exlr

b['erv a-

EF

C.A-

Que p

feos

[l

t

84

pE

CAPU T;.Vï,

N MOR (v

ynato t i

qed, perpt

py:bfematïs

minda Ms

GAPUT V

iime verfus

DE CALCULIS RESPECTU LINEJE

9 fnguh .

jn qui femg

MERIDIANZE ET PERPENDICULARIS, ET

tot, atquims

CONSTRUCTIONE MAPP/AE PER PUN-

inde dertet

iaf

CTA FIXA.

tüm ob

1

NU

dent, Qa

DEZFINITIO IX

Unterbern

cales diy

Yo5. Linea Meridiana eft pars circuli meridiani per

eonteripfi

gonometri

locum transeuntis, ( $. $ ) feu eft illa linea, ad quam

blema ex;

dum fol pertingit, ad maximam fimul quam eo die ha-

Omnino ne

bere poteft, altitudinem pervenit, adeoque.eft mecium

pun&um inter ortum & occafum ; dire&e illi oppofitum

pun&um eft pun&um feptentrionale.

] Og'

£tetice h;

COROLLARIUM.

& Yerf:

N

IO6. Du&a ergo per locum datum linea re&a, tranfibunt

loc pun&

fujus extrema per feptentrionem fuperius, & meridiem inferius,

tiverto te

& ereca ad hanc, in ipfo loco, perpendiculari hujus extremá

tutls obf

?

per occafum a finifris & per ortum a dextris tranfibunt, qua

perpendicularis fimul pars circuli paralleli (S 5.) et. Cum hzc

proprietas de quovis loco dici debeat, omnes lineas meridianas,

mn,op, niüi ad magnam longitudinem excrefcant, (F'g. 63.) per

parallelas exprimimus, fi enim longiores affumantur, non linee

k

fed circuli eruntin polis convergentes ut $. 5, diximus.

(?G";füä

SCHQLION'I.

107. Demonítratum eít in geometria curvarum $. 7. ad

determinandum fitum pundi duabus tantum lineis fefe interfe-

In

eantibus opus eíle, & ex nota diftantia ab una & ab altera li-

nea & fimul nota plaga dextima vel finiftima , aut fuperiore vel

it CU

inferiore pun&um figi. Affumtis igitur his duabus lineis feu du&a

HiS) u

perpendiculari & meridiana, fi diftantie perpendiculares loci ab his

flrcumi

"«duabus lineis innotefcant, dependenter ab illis omnia reliqua

Iïl

3

loca determinari, & &ne depica triangulorum íerie, aut effor-

üteot

ina-

w

h1

DE CALC. RESPECTU LINE/JE &c. $

xnato reti in mappam referri poffunt,

Patet etiam quasvis lineas

fe&as, perpendiculariter fe interfecan

foc

to»9

(Fig. 68.), refolutioni

problematis fatisfacere, hinc magna valde accuratione'in deter-

minanda Meridiana opus non eífe, fed fufficere, &i quam pro-

xime verfus Meridiem & Septentrionem dirigatur, ut veram. lo-

INLE

ca fingula a fe invicem diftantiam & confuetum mappa fitum

JS, ET

in qua femper pun&um feptentrionale direc&te fuperius effe poni-

£ur, acquirat.

PUX-

Obvias dabo inveniende Meridians methodos, &

inde derterminandi angulum , quem pzimum ex eo loco obferva-

tum obje&um cum hac meridiana facit; qui accuratiorem deí-

derat,'v(l]ar. Liesganigii, fzxpius laudatum opus, aut Celeb.

Unterbergeri, qui confpicua inter primotes.rei tormentarie offü-

ciales dignitate floret, geometriam prac&icam, quam pto fuis

Idiani per

sonfcripfit tironibus, adeat, cum enim neceffario principia tri-

Udili

nm

gonometrie fpherice, & quedam etiam aítronomis ad hoc pro»

adq

9 die h-

blema exace folvendum requirantur, neque exac&ifima folutio

t mecium

omnino neceffaria fit, me illis fuperfedere poffe arbitror,

yppolitum

SCHOLIO,N. [

i1O8, Facile fitus feu dire&io linez mezidians opeacus ma-

gnetice haberi poffe videtur, cumex przcedentibusjam notum fit,

eam veríus feptentrionem femper dirigi; quia tamen non nihil ab

&ranibunt

hoc pun&o declinat, & ipfe hz declinationes diveríis in locis &

1 jnfermus,

diverfo tempore variw funt, hzc declinatio, du&a prius ex infti«

ij exttema

tutis obfervationibus linea meridiana, examinari debet,

junt, qu*

, Cum 1ec

PROBLEMaA r

i eridianïs.

, 63.) pet

Xco9. Ducere lineam meridianam & eïxaxmïnare

non li[leæ

d5

declinationem acus magneticze, (F7g. 69.)

RESOLETDEMONST,

6T d

Inligno duriore, & bene polito ducatur circulus,

fe inzetftï'

To

in cujus centro c ftylus tenuis ce erigatur ejus altitudi-

vriore v*

nis, ut intra nonam & deciman matutinam ejus umbra

; fet duda

circulum raàdat, quod pun&um s notetur,

foci ab hi

1I. Immoto affere, cum afcendente fole umbra de-

in reliqia

' gut eïï)r'

Crefcat, & defícendente rurfus erefcat, ob(grv etut, dum

"

g«

k

^

ins

86

CA'POT WL

iptra horam fecundam & tertiam rurfus extremitas um-

bre ftyli eundem circulum in 7 radat.

IT

IIL. Conjungantur pun&a m, z, & biffecetur bxc

j Quf

chorda m», linea per centrum & biffe&ionis pun&um

du&a, eít meridiana; eft enim in medio inter aquales a

horizonte altitudines , quod ipfum medium eft meridies,

& directio linex meridianz.

Ii]

fetvetut

IV. Applicetur ad hanc lineam acus magnetica a

Qam o

cum circulo fuo in 460* divifío, & videatur, quantum

a zero feptentrioni obverfo aberret, quantitas aberra-

puuto

duum ;

tionis efít declinatio acus magnetice.

[ :

gulus 1

C OR OLTARIEUM:kL

U

jnter at

XIO. Patet, fi plutes ex c concentrici circuli

ducantur ,

facit o

& appulfus umbre mane dum decrefcit, & rurfus poft meridiem ,

dum accrefcit, a& quemvis circulum notetur, & pun&a correfpon-

Jabetu:

dentium arcuum chordis conjungantur, lineam ex centro du&am

per omnes biffe&iones tranfituram , quod argumentum, & confir-

matio funul erit obfervationis rite inftituta.

IU

4

COROLLARIUM IL

hihetr,

inw. Xte

III. Si jam menfule dire&ionem verfus feptentrionem dare

£,

velis, lateri menfulae planum, (J7g. 70.) cuicum fuo annuloacus

Puntum

infiftit , ita applica , ut ab, quod ad SM parallelum eft, accurate cum

illocongruat, dein menfulam tamdiu move, dum acus competentem

Ix

i

f

1$ verfu

declinationis gradum exhibeat, hoc ipfo menfula verfus fepten-

trionem dire&a erit. Si vero in campo lineam meridianam defi-

D feu üch

d lioc eft

gnare velis, applicatis dioptrislinee meridiane in menfula prius

^r

——

efformate jubefigi baculum in ea dire&ione, quam dioptre ha-

L

31

brom lz

bent, & linea per ftationem & illum baculum duc&a meridiana

erit, angulus vero inter illum baculum & primum puncum fxum

Exem

erit angulus quafitus, quem primvin pun&um cum meridiana facit.

Pro his cafibus, dum dire&io folum lateris cujusdam , aut fitus

fulus cum

menfule quzritur, fufficit apparatum acus ita inftru&um habere,

platur qux

ut arcum & zero hinc inde ad aliquot folum gradus produ&um ex-

'::540 164

hibeat, ni! enim aliud quam directio verfus feptentrionem aut de»

um "t

[ immed

clinatio ab eo pun&o inquiritur. (Fig, ro.)

ET

PRO.

DE CALC. RESPECTU LINE/E &c. 87

PROBLEMA H.

4$ Ull-

t

rir2. Determinare meridianam per obfervatio-

It hec

um

ges Cum quadrante aut goniometrico tubis inftructo.

les a

RESOLUTIO.

idies,

L. Inftrumento horizontaliter pofito (Fzg. 71.)ob-

fervetur angulus inter orientem folem a & aliud quod«

Y

Janturm

dam obje&um fixum 5, & rurfus in eodem omnino

abertae

pun&o angulus inter idem obje&um 5 & folem occi.

duum d; qua obfervatio peraliquot dies repetatur, an-

gulus hic biffe&us in e dat directionem linez meridians

«6, & fimul differentiam inter angulum acb & ace vel

inter ahgulum bed & ecd, verbo dat angulum bce , quem

icantut ,

facit obje&um prius affumtum cum meridiana, proin

ridiem ,

^

habetur vera directio meridianz refpe&u pun&i fixi ».

vrelpon-

du&am.

COROLLARIUM,

- confite

II2, Si bfit ipfum obje&um, & quo calculos incipere velis,

habetur, quod quzritur; fi autem penesillud, primum puncum

fixum jacest v.g.in f, queratur inter iftud & illud obje&um an-

em date

£ulus, differentia dabit angulum inter meridianum & primum

pun&um fixum,

ulo as

'te cum

Exemp. Sit in- C feptentrionalis meta bafeos, dire&tio ba«

etentem

fis verfus arborem 5 tendat, angulus inter ortum feu intera &

feptete

L feuacb obfervatus fit 86? 4', inter b & occafum feu inter ^ &

m de&'

d hoc e& Lcd fit 89?6", erit totus acd — 175^ IO', & dimidius

m

97?^ 35, hujus differentia a primo & altero obfervato dat 5ce,

gg hi

proin bcec— " gï',

ridiana

Exemp. II. Sit caleulatio a pun&o f incipienda , cujusan-

l fu.

eulus cum meridiana quaritur; obfervatione ut prius facta, .ca«

" fZCif:

u fitus

piatur quoque angulus 5f , qui fit — 54? IG', erit zecf——bcf — bce,

&«

——54? 16 — 1?. 31* — 52? 45 , igitur angulus inter f &meri-^

haberes

dianam verfus occafum & meridiem eft 52? 45'.

Seu qu&rana

um ex

tur immediate acf & fcd, & fiat operatio ut prius,

aut d*

F A

C O-

Q

1

88

CAPUT VI

COROLLARIUM I

e

ft,1

ib5

II4. Si exftatione ambo pun&a horizontalia ortus vel occa-

tur j

fus obfervari nequirent, quadrante verticaliter ere&o obferven-

Toer/i

tur mane & vefperi du altitudines folis zquales v.e, dum ad

qui tt

10 mum aut 2O mum gradum mane afcendit, & rurfus poft meridiem

ln

ad eumdem gradum defcendit, & fimul altero inftrumento a fo-

cio angulos inter objecum fixum & hzc duo loca folis obfervari

lp

ii om

cura; biffe&us totus angulus dat lineam meridianam, & diffe-

rentia dat angulum inter meridianum & illud pundum fixum ut

fuppor

prius.

Verum cum hic duo inftrumenta & duo obfervatores re-

tione

quiramtur, & anguli nonniü inclinato altero inftrumento accipi

dire&

debeant, qui ad horizontem $. 73. reducendi funt, fequitur ,

magnam difficultatem hanc obfervationem involvere; cum igitur

notu

idem omnino fit, a quocunque loco calculum ordiamur, tale pun&um

:

fixum pro obfervatione eligendum erit, unde profpe&us horizon-

talis verfus ortum occafumque habetur,

7

ï

COROLLARIUM IL

IIS. Ex principiis aftronomicis conftat, folem preter mo«

&npul

tum circa tellurem apparentem, motum alium quoque verfus

tropicos habere, hinc femicirculus matutinus a pomeridiano ea

tidian:

differt quantitate, qua fol verfus tropicum alterutrum quotidie

[l

movetur, & hinc angulus quoque ea quantitate corrigendus eft,

tl

dum exaciífima omnino linea meridiana aut per ftylum, aut per

1095 (

goniometricum quaritur. Cum tamen hzc differentia fole in tro-

E

|

picis aut in zquatore verfante nulla fit, exa&a quoque fine omni

corredione erit obfervatio, fi circa hoc tempus inftituatur.

aïtersi

4

PROBLEMA IIL

& du

dabjt

Ir6. Ex noto angulo, quem linea , ab uno ob-

tum,

jecto veríus aliud objectum ducta, cum meridiana

lit p

objecti primi facit, determinare loca fingula rela-

te ad diftantiam a meridiana, & perpendiculari.

RESOLUTIO.

)

)

KUCC

us

I. kUt determinetur fitus pun&i 5;du&a pet locum

ültty

j

& ( Hy.72.) ubi obfervatio lines meridianz inftituta

*1

ET

eft,

DE CALC, RESPECTU LINEJE &c.

89

ft, linea re&a verticali, & erecta horizontali, habe-

[ otts-

tur per pracedens problema angulus z, quem /a cum

fetvene

meridiana verfus feptentrionem facit, & fimul ex cal-

üm ad

culis triangulorum pracedentibus notum eft Jatus a?.

tidiet

1 fo-

Ii. Du&a perhoc novum punétum?'ad 2« paralle-

la ct&, erit hec linea. meridiana hujus loci , quam ab a

Ile

dj

ad omnia confequenter objecta parallele translatam effe

Wm ut

C o5vnv

fupponimus, & o alternu

5 Ci

it zx7, obtinetur hac ra-

ts f*

! accipl

tione triangulum J(2a, in quo notus angulus o, ex

Quitur,

dire&ione lateris verfus meridiem cum meridiana, &

ipitur

notus.ms prioris complementum, & etiam notum latus

un&um

ba. igitur invenitur [2a — ia diftantia a meridiana, &

tizote

hD diflantia a " perpendiculari per fequentes analogias

Sin, tot. : latus notum z /iz, o : (9a

ex fm. m:l[o.

Ad logarithmum igitur lateris noti ab addatur finus

r f0*

verfus

anguli noti o, fumma eft logarithmus diflantie loci a me-

10 €à

ridiana 4[9 z P«.

otidie

H. Addsztur ad eundem logarithmum lateris finus

set,

complementi prioris anguli feu fin. m, (üumma eft logatith.

U

f (tü-

mus diftantie a perpendiculari D a,

omni

HL Prima diflantia ex a ad perpendicularem in,

altera ad meridianam in « transferatur.

IV. Per B du&a parallela ad meridianam f2

, pet

& du&a parallelaa? ad perpe

ndicularem , interfectio in P

dabit fitum verum loci ». Aut apertura ax duc ex B ar-

) Q

cum, & apertura af ex & eum interfeca, interfe&io da-

2114

bit pun&um ».,

6 |

2s

COROLLARIUM,

II7. Hac jam methodo labor per totum

73.) Procedendum $it ad

continuatur. (Fig,

proximum' puncum v. g, c, quod ver-

fus occafum Jacet.

alternus

"

A9

Du&a per c meridiana 7LÀ, quaerendus eft

u

feu bco' verfus feptentrionem, qui eft equalis alterno

* feuCr(u, verfus meridiemi, hi

v

n

autem ob notum ex triangulorum

h

Wa

ferie angulum ch

feu r -j o

ex

precedenti

obfervatione

F3

etiam

9^

CAPUT VL

etiam innotefcit , proinde etiam x habetur; isitur in triangu-

1o cob, ob notum x & complementum s cum latere cb (ex tri-

angulorum: ferie) erit iterum proportio fin. tot. "ob u—xr04.

J

$40 0^.

q

Addito igïtm: ad logarithmum lateris moti bc finu x, & addito

rüe

ad eundem logatithmum finu s, dabit prior fumma latus bo —

5

ïï

yr, altera dabitlatus có' z— Gy -i- yc z &ty, cum jam c. magis vet-

füs occafum jaceat quam b, ad diflantiam priorem b& feu ta ad-

dendaeft bv , uthabeatur diftantia a primo meridiano &« feu cgy

Es d.

Cum vero oc velilli equale ayy majusfit quam oy, quod

eft— Ls aut a& diftantie verfus feptentrionem pun&i b, patet,

quotur

ab hac diftantia aqj fubtrahendam eífe diftantiam priorem «a &

detern

linea

zefiduum acy fore diftantiam pun&i c a perpendiculari veríus meri-

iem. Etita rurfus cum aliis pun&is procedendum * quzratur fci-

primu

licet determinati jam obje&i angulus, quem dire&io cum pun&o fe-

metlio

fw

quente verífus feptentrionem aut meridiem facit , ic angulus fem-

per erit alternus refpe&u anguli, quem objectum determinandum

cum meridiana facit:

$i determinati obje&i angulusa (Fig. 74.)

P'x

fuerit verfus feptentrionem o' alterius determinandi b erit verfus

ad r

gmeridiem j,: fi ille fuerit verfus j (Fig. 75) hic erit verfusc.

avi ?

L

In hoc ergo anguloex aliis circa idem obje&um obfervatis eru-

endo totus labor eft, qui autem conítru&a femper angulorum

obfervatorum figura. facillimus evadit. Anguli hujus finus. & cofi«

QU2 t2

Ínventz diftantios

pnus addatur logarithmo diftantize objectorum.

a,

fi verfus eandem plagam labor continuatur, prioribus diftantiis

fibuf

àddantur, fi in plagam oppofitam, a prioribus fubtrahantur, &

wpt r

ita remotifimorum obje&orumab iifdem duabus lineis diftantie

min

ad muita hexapedarum millia continuantur, & obje&a determi-

tm ;

nhantur,

Tur(i

totum

C OA OT,LARIUSM i.

üt di

ceden

II8 Difüciliimum omnino foret ex eodem pun&o a omnes

90 refj

diftantias ad millia aliquot hexapedarum ufque transferre, quin

aCtede

in majori fcala vix non impoffibile foret: occurritur huic incom-

brion

modo, fi poft determinatas quasvis diftantias 500 , aut 2000 aut

igura

4000, aut 6000 hexapedarum parallela verfus omnem partem

ducantur, quibus fuperne & ex latere numerus hexapedarum pra-

V'Pl

'Dm,

cedentium adfcribatur, (Fiz, 76'.) obtinetur hac ratione, ut

femper diftantix» ab ultima parallela tantum accipiantur, nec ul«

tra fequentem producantur, & ftatim inventis ex calculo lateri-

bus appareat, in quo quadrato obje&um detc—rminatgm fit collo-

candum.

en

C QO-

sU

E

DE CALC. RESPECTU LINEJE &c.

9

trittt-

COROLLARLIUM II

(ex tti-

2

Ü

h

C0

II9. Optimum erit, fi calculi, uti prius de triangulorum

; addito

ferie diximus, in tabulam referantur

?

poterunt inde dum ma^pa

N-

conftruenda eft, excerpi, & fine confufionis periculo in mappam

iis vet«

tràansferri. (vid. pag. 93.)

ll &L Md

SCHOLION L

feu ay

j quod

120 Hac jam ratione loca regnorum Gallicie & Lodomiris;

pattt,

quorum diftantiv* per feriem triangulorum calculate innotuerant,

m &

Du&a per Leopolim

determinavimus , & mappam conftruximus,

]s fteri-

linea meridiana, & ereca in eadem perpendiculari angulum, quem

itür fci«

primum pun&um fixum cum hac meridiana facit, Liesganigius

ndpo fe-

methodo aftronomica

cum azimutho folis obfervabat, inde ver-

us feme

fus omnes quatuor plagas fupputatis7 ut prius exhibui, angulis Güfs-

mandunm

manus mecum loca precipua in chartam transtulit, ductis prius

i 14)

poftíex quevis milliaria parallelis ad occafum quidem ad 198000;,

vetfüs

ad ortum vero ad 102000 hexapedarum: verfus feptentrionem

rfus ,

ad 78000, verfus meridiem ad 128000; ut totalongitudo tercen-

ZiS etüi-

tis millibus hexapedarum , latitudo 206000 clauderetur , per quod

uorüm

obtinuimus, ut mappatota inquafcunque partes minores feparari,

& cofie

quae tamen accurate femper cohererent: ipfa etiam fcala in ma-

(lantite

jorem , aut minorem formam. tranfmutari fine novis calculi amba-

kantlie

gibus pofht;

du&is enim in minori jam diftantia parallelis, in-

tr, &

ter eas loca

fingula juxta adícriptos

calculos ex majori vel

flantia

minori fcala inferuntur, quin (ingula etiam talia quadrata fepara-

:teHie

tim accipi, 6 inter diverífos delineatores diftribui poífunt, qu»

rurfus labore perfe&o fine erroris aut difcriminis periculo in idem.

totum coalefcunt.,

ciY

erv

avimus,

Íd preterea inter calculandum obf

ut dum obje&um novum magic a meridiana vel perpendiculari re-

cederet, diftantie invente ítatim fignum -- adjiceretur, cum ve-

o refpe&u prioris obje&i ad meridianum vel perpendicularem

OMICS

quif

accederet, fignum —— preponeretur, quod exhibeat diftantiam

1coft"

priori vel addendam', vel ab ea fubtrahendam eífe , utrumque ex

yo at

figura aut quantitate anguli facile agnofcitur? In fáne adjicieba-

artem

tur plaga verfus quam latus tenderet: N, Nord, vel feptentrio-

1prg-

nem: W, Wiftoccafum: $S,Sud, meridiem: O, Ojf ortum fgnifi-

,

u

€at.

W

SCHOLION Il.

atetie

.:oïïü'"

r2r Pro exemplo partem retis verfus Cracoviam accipiam,

quod Fig.78. exhibet; ífcalam quadruplo minorem, ac nos ufi

fus

9

9

-

APUT

VI

fumus, (Fig, v7.) felegi, ut fcilicet digitus Viennen(is milliare

hu

integrum, feu 4OO0O hexapedarum, daret

Calculus a civitate

ls;

onérti

Leopolitana ex determinato , ut prius dixi, angulo incipiebat, inde

Güísmanus loca obfervata Iaroslaviam usque, & ad $0000 hexa-

& ss

,pedarum calculavit, & laborem fuum cum Rainio conjunxit, qui

[

; t

Rzefzoviam ufque progreífus jam ultra 76000 Leopoli diftabat.

u)

[2

Inde jam mea calculanda venerant pun&a; conjun&a baü nofra

p

hac $2tione cum reliquorum |labore advertimus

eum lineam

l Satua,

meridianam refpe&u bafeos habere fitum , quem nosexacu mag-

netica eidem dedimus.

Inveneram ex produ&is ulterius calculis

s

gSm

puncum z (I"z"f. 78.) ditare & meridiana Leopolitana occafum

S»a o

ä

verfus 14395t Àex.;a perpendiculari 14815 Rex. feptentrionem,

;

$

IM?

verfus, angulum vero, quem hujus loci meridianus cum fequen-

|

o

n

p.ó

$94

"

ti ftatua, quod jam pun&um determinandum veniebat

Ineris

i 44

diem verfus, id eft Smu, faciebat,

fle 53^ $54 , quiigitur xqua-

Cnx

lis eft angulo alterno o'$m feu ftatue verfus feptentrionem , cujus

complementum eft 236" G, di(tantia vero Sm ex calculis triangue

SADN

lorum erat 2522 hex.

u

Ad cujus logarithmum additus finus 53

en x

$4

datlogar: numeri 2038 dittantiam a priori meridiano ,qua

Wa

[

ad pun&i m diftantiam addenda eít: & ad eundem lateris loga-

Conp.a

, dat loganthmum nus-

;

rithmum finus anguli complementi 36?

"s

meri I486 diftantiam a perpendiculari puorl, qu2 à perpendi-

Signum

culari diftantia pun&i m fubtrahenda eft

igitur vera diftantia

*

ftatue a meridiano Leopolitano eft 134989, W'; a perpendi-

"kióe

culari 13329 , NN; calculus in tabula inferius primo loco exhi-

m

betur.

n Y

pE a

Acmplatur Jam crux ante Wieliczk4: Angulus in ftatua inter

Omp

Ha& meft g1?

4O. Sed o'$m inventus ef 53^

54 , igitur eft

nk ]

€SHB. feu angulus in ftatua inter F& & meridianam differentia pri-

Xr

orum — 27^? 46', qui æquahs alterno PHBS verfus meridiem , cu-

jus complementum eft 62?

14', utriusque finus additus ad loga-

KM i

rithmum laterisi4S

2744 hex., dat diftantias prioribus con-

L

jundas 147267, W

pX 9

& I5757, N; ut in tabula fecundo loco

exhibetur,

aM D

Mu 8

Angulus in y»B inter p & S eft r157?

zo, fedpä*

C(an

27? 46' ex praecedentibus, 1g1tur/.c HRnet I129? 44', qui iterum.

X 4

zqualis eft altetno o'n ecH , qui cum fit obtufus, ejus differentia a.

Mog)

180 acc1plenda eft ($. 1I. Trigon,) — 50' 16 cujus complementum

diftantia,

eft 39? 44'; calculo ut prius inftituto inveniuntur

Pun&um quogue Cracau, & Mogila fequens tabula exhibet

I(

BI

*l &

SCHO-

DE CALC. RESPECTU LINEJE. &c. 93

nilliare

civitate

Anguli

Solutio per

Diftantie

Solutio per

Diftantiz a

obfervati.

proportionem,

a meridiana.

t, inde

proportionem,

per pendicul

hexae

pm$ $3954' m5

3.401917

mS

8.401917

it, qui

m$5 $3 54 m$c

itabat,

Comp.26 6

9.907406 m 143051

Comp. 9.770260|m 14815

nodta

m$ 2523

3.306650 - - *-2038

8172177|: : —-1486

Statua.

$. 145989 W

18329 N

Lineam

| ag*

AE

PESm 8gr 40[S&E.

3-438384

Sria 3-438384

ceafum

o5m 53 54ls MA

9.66826€/s 1459g9 |Comp. 9.946370|$ 13329

tjone

cSM 27 46

8-106650|- - - 1278

3-385254| * - 2428

$i 97 46

Pa

dxara6z w

iequett

Comp.62 14

15797 N

meris

SrE. 2744

E

Crux

; Cs

tiongye

$9bn157 30|Han

3.59328€

[rEn

3.593286

pn199 A4|un

HH 9.585942 |4147267

Comp. 9.505647 J 15797

5j

cn* 129 44

0,qui

pnd 50 16

3.479228 |- : *-3014

3-398932|: * 7- 2505

; lopi-

Comp.39 44

n I150281W

18262 N

rBna 3920

I n«

Signum n

yendi«

—

fantia

Kn-R165 35lux

3.103804

nKÉ

3.105804

endi«

cnri 129 44 |I.n

exhi«

Kno 35 5i

767649|n 150281

Comp. 0:9097 83 n

18269

EKn 85 $i

2-971453 - * -t* 744

3-012585|- - 1029

Comp.54 9

|K 151025W!

19291 N

j Mfet

ur ef

nK 1970

Krakow.

a pri-

i

2,C-

KM

lore

KaAM 55 27|KM

3.637660

3.637690

' con*

MKn 35 S1|J.MK

9-999358 |K. 151025

Comp. 8.355783|K. 19291

1p^KM 91 18

| ]oco

cKM $89 42j

3.637578|— - 4941

1.993473| -7 "98 -

FMK 88 42

M146684W

19889 N

Comp. 1 18

Sef

KM 4342

r

Mogyla.

tí &

pturma

SCHOLION

ntï&'.

IlI,

L]

192. Translationis

modum & partem Fig, 76. exhibet,

parallele hz inter 100O hexapedas inde a Leopolitana merici-

$na, & perpendiculari du&e cífe fupponuntur,

Cum igitur Sta-

fua

Oe

CAP. VI. DECALC. RESP, LIN. &c.

94

tua 145989 verfus occafum remota fit, pof meridianam, cui

pl

adfcripti funt 145000, ponenda erit, & cum a perpendiculari

13329 diítet, fupra horizontalem 130O0O poni debet, accipian-

tur itaque ex ícala hexapede 9$9 '(quæ funt exceífus diftantiz

*

fupra 145000 ) & figatur crus unum in

in interíe&ione przdi-

&arum di

fantiarum , & £fiat fignum in linea verfus finiftram 5

TSC

accipiatur & altera ditantia 329, & fiat fignum in altera linea

LA

furfum verfus feptentrionem, concurfus parallelarum , qua per

hec duo figna prioribus parallelis ducuntur, ut $. 1I16. diximus,

IRAX

determinabit pun&um verum ftatum ; ita de reliquis.

DEÀ

IV.

SCHOLJILDA

l

122. Supere& udhuc , ut de quibusdam cafibus, qüi'fub ipfa

menfuratione occurrere poffunt, dum vel ad loca acceffus non

vel alitudo inquirenda fit, vel ignotum pun&um ex

pateat,

aliis notis determinandum fit, agamus, qua problemata fequens

In memoriam autem hic ea revocanda funt ,

caput refolvet.

a

que fuperius $$- 445 45» 54* 95, de methodo accipiendi angu-

los diximus, & quod Fig.79. & feqq. exhibetur; In Fig. quidem

79 eft aftrolabium in fitu horizontali ad figna diverfa hinc inde, in

eollibus etiam, difpofita dire&um ; in ng, 80, autem in fitu

7

vertica

li e& ad altitudinem obje&di determinandam, ubi perpen-

diculum a fitum verum inftrumenti oftendit,

In Fig. 82. ef

tut dater

menfula in fitu horizontali, in qua pun&um a« ex terra in menífu-

hteri a

lam in c ope forcipis (&ig. S1.) translatum eft, & acus b in

ilvenih

eodem f£rmata; linem verfus figna, & objeda, ope dioptrarum

ad acum in c applicatarum duc&z in menfula exhibentur, In

M

Fiy. 83. eítütus menfule

verticalis, pun&um correfpondens per«

&8-, t

pendiculo determinatur, & ad acum ibidem fixam dioptre ap-

& X

plicantur, ut angulus inter horizontem & altitudinem determi-

fea]a d.

netur, de quo fuo loco agemus.

iL geor

122

at

hL

Vituta [

Uf

DA«

E

J

*X KS

Mb

d

GA

95

1, CM

li eulari

Clptan

f'antie

CAPUT-' Vi

pr&di

kram ;

lineà

RESOLVUNTUR VARIA PROBLE-

2 p

AUs

MATA TRIGONOMETRICA , QU/E IN IPSA

PRAXI EUTHYMETRICA ($. 8.) ET QUI-

DEMIN DIMENSIONE HORIZONTALI OC-

CURRERE POSSUNT

b

non

PROBLEMZA I

[ ex

queni

124. Metiri diftantiam duorum locorum 44 &

funt

angu

ÀB acceffibilium. (Pzg. 94.)

dem

REÉESOLUIIOS

le, il

fitu

Goniomelrico

rpen

Obfervetur angulus C & menfuren-

eft

tur latera. CA4, CB, habetur triangulum, in quo duo

enu

latera cum angulo intercepto nota funt, proin tertium

jin

arum

invenitur $. 62. Trigon. ,

if

AMenfula. 3n puncto C dire&ione verfus 4 & Bdu-

(ta, transferantur ex fcala longitudines menfurate J4C

; per

p

& BC, conjungantur pun&a 44 & B, longitudo 4B in

tnr

ífcala dat veram longitudinem diftantis objeCtorum, ut

in geometria de proportionalibus lineis diximus

SCHOLION

1252, Quod goniometrico fit, etiam acu magnetica dioptris

inftru&a fieri poteft $. 49. hinc fpeciatim de hac non agemus

PROBLEMA II

126,. Invenire diftantiam duorum locotum, quo-

rum unus tantum accedi poteft. P?7». 95

R E

m

Q

R

CAPUT VIL

RESOLUTIO.

Gontomtetrico, Obfervatis in 4 & B angulis & men-

ff

furato latere .4B, habetur etiam tertius C, & ex oppo-

fi

m i

fitione refolvitur triangulum, & latus BC invenitpr $- 59.

Trigon.

z.enf-x

n mte

AMenfila. Du&is in 4 dire&ionibus 4B & Z4C,

Qm enes

quafit

furetur longitudo 4B, & in menfula in linea 4B abícin-

datur, deintransferatur menfula in B , unde, procurato

prius menfule convenienti fitu, du&a dire&ione verfus

C clauditur triangulum, & determinatur pun&um C.

per !

a2neu

PROBLEMIA lL

int

€ata

4AD.

12*. Invenire diftantiam duorum locorum,

quorum neuter accedi poteft. (Fze. g6.)

RESOLIUITI O.

Pofito inftrumento in C, obferventur anguli inter

fin

A, B, & D, ubi fignum quodpiam ponitur, menfure-

&

tur CD & translato inftrumento in D obferventut an-

Íun

guli inter C, 4, & B, erit in triangulo Z4CD notus

angulus C & D, proin innotefcit tertius in 4, & ex da-

to D ob oppofitionem invenitur 4C & AD,

In trian»

gulo CBD ob notum C & D, habetur B, & ex dato

CD ob oppofitionem invenitur BD & CB. .Jam in tri-

lus .

lus 4

angulo .4CB vel A4DB habentur duo latera nota cum

de o

angulo intercepto,, invenitur ergo tertium .4B Diftan-

tia quefita $. 65. Trigon.

quodl

Menfüla pofita in C ducantur dire&iones verfus ,

B, & D fignum. Menfurata bafis CD in dire&ionis li-

Ver(i

nea ex c in d abfcindatur; ponatur menfula in d ita,

tegt.

ut Cd accurate linee CD refpondeat, ex 4 du&e di-

Wetu

rec&iones /d, B4, pun&a a & b abícindent, &— dis

ftantiam 4b ex fcala determinabunt.

C O-

l

a

—

RESOLV. VARIA PROBL. TRIGON. &c. 97

COROLLARIUM T!,I

& mene

128, Patet ex folutione eodem modo diftantiam inveniri poffe,

fi verfus obje&a linea bafeos menfuranda fit, pofita enim men-

[ Oppo

fula (Fig. 87.) in .4 collimetur verfus C & D & alteram faatio-

1.5

em B, translata huc menfula, & ex menfurata bafi AB fixo in

menfula pun&o b, collimeturrurfus verfus C & D , abfcindentu£

, [Iet

in menfula pun&a e & d, que conjun&a dant ex fcala diftantiam

quefitam

a(cine

TOCUTACO

COROLLARIUM ÁYI

? Vetfiüs

129. Poteft etiant linea CD

feu bafis, ita accipi, ut

&un (

per lineam .AB tranfeat

(Fig. 88.) obfervatis enim in C & D

angulis, innotefcunt A4D, 4C, CB, DB , unde etiam latus .4B

in triangulo ACB vel ADB invenitur,

Menfula in. D &C collo-

cata obtinenturex utraque parte interfe&tiones in 4 & B, proin

COTum,

A4B ex fcala haberi poteft,

PROBLEMA IV

150. Ex nota linea Z4D,

ad cujus unam fane

]j jnter

tum extremitatem 44 acceffus patet,

invenire pun-

exfure

Cum novum, & triangulum determinare, unde men-

ntuf an

furatio continuari poteft. (FZe. 89.)

) notus

RESOLUTIO

T0

[n £T1àD»

Fixo ad libitum figno in C, capiatur in 4 angu-

24 (.0

lus B.4C, translato mftr:xmento in C, capiatur angu-

m

lus ACB, habetur inde etiam tertius mB feu 4BC, un»

gt Cumm

de ob oppofxttonem latera innotefcunt.

Difai

AMenfüla in 4 collocata juxta dire&ionetm lateris AB

quod jam in menfula effe fupponitur; ducatur directio

lus Á

verfusc : pofita menfula in c, ut ac linez AC refpon-

onis li«

deat, applicentur dioptre punc&to ?, & verfus B colli-

TEUM

metur, abfcindetur iülinea 4e pun&um e quafitum.

& di

AMetsb. IMath. P. 1.

G

CO«

€2

c

—

95

C A JPEDST

I

R

C O.R;0/E L

ARIUIM

que

12I. Inde jam ad alia pun&a determinanda menfuratio

d l

C obferva-

continuari poteft ; eft enim CB (Fig. 90.) novà bafis; In C

Á

tio inftituitur verfus novum ele&um pun&um D atque translato

ad hoc puncum

determinatur eadem

nfirumento vel menfula,

jiethodo pun&um D aut B, & inveniuntur latera CD, CE, que

ruríu

bafes novorum triangulorum fiunt

datam

mus

PROOBLEM.JA V

tatur(

ur

Ex nota bafi 4B , quze nullatenus accedi po-

i

teft. determinare punctum novum cum trianeulo

une

NeTI

dP dimenfio continuetur

fine meníuratione novae

bafeos. (Pig. 91.)

1

h, (

RESOLSUTIO

iut

8

Inftrumento goniometrico in C'collocato aliud pun.

tür

)

&um in Dfigno notetur, verfus quod & utrumque ob-

n

je&um collimatio fiat : t

ácd

Ta

slato infirumento in D capi-

dpplic

antur quo que anguli inter C

4, & B, ut in Proble-

Yetius

Vs

mate T1I.

diximus.

Hist

jeractis affumatur CD ad libi-

$Qs

tum 1CO,

10co OY, ho

P

xa

pedarum, & ex hac affumta

Qhw

bafi LdlCUiGP'""I triangula,ut in eodem problemate di-

cede

é&um eft,

& qu&ratur 4£D falfüm quidem , fed proportio-

de

natum aíffumto lateri CD ; fi

proportio, ut falfa linea

AAB ad veram AB , itafalfafeu affumta CD ad veram CD

AVenfila in C collocata ncte

entur dire&iones verfus 4, B,

Dunc

& electum feu affumtum quoddám fignum D, transiata

menfula ia D

affumatur quoddam pun&um in linea

cd, v. g. d, & in eo verfus C, 4, B, rurfus collimatio

fiat , eritfigura f?cd fimilis figurze 4BCD ob zqualitatem

Yel l

angulorum, & ? erit propoitionalis linez .4B , fi ergo

ipfa longitudo .4B in f? transferatur , ut ab infcala fit

zqualis 44B . not

du&a parallela ad z

dum occurrat

diagonali «b, poterit etiam xy ad 4b parallele duci,

qua

RESOLY. VARIA PROBL. TRIGON. &c- 95

quz in fcala dabit longitadinem CD, Eftenim hxc par-

Toenfüratia

Cobfetvae

va figura axy fimilis priorzi, & magne, proin ficut aj

z dB,ita xy — CD in proportionata fcilicet fcala.

w translato

latür eadem

QOROLLARIUM

, (E, que

133. Poterit

etiam pun&um D ita affumi, ut CD lineani

datam .4B íecet , uti $$. 127. & I29. & Frg. 88. exhibui-

mus, qui fitus calculum non mutat, fed ex affumta baü CD quz-

ratur CA, & CB, ex quibus, ob inventos jam 4, & B, quzra-

tur in triangulo .4CB latus .H proportionale affumte baft C/) ;

(Ccedi po»

fiat rurfus ut AB falfum ad CD ífalfum, ita AB verum ad CD

ouJo, We

Vverumn.

me nov&

COROGOUTYTLARITUM i

134. Si linea AB jam in menfula exiftat una cum dire&io-

né, quam acus magnetica tum habuit, cum héc linea ab notat&

fuit, (Fig. 92.) poterit quevis tova affumta aut data ftatio v. g,

& in meníula notari, Rotetur enim menfula, que in C effe poni-

III'UC' PUR"

tur,

tamdiu,

dum . acus magnetica rurfus

eum quem prius

ha»uit fitum acquirat, quem linea mm exhibet. Dioptre ad 4

amque oU«

àpplicáte verfus .4 dirigantur, & ducitur ac; aàd b applicatse

t Dapk

verfus B dirigantur, du&a 5c priorem in c in pun&o

ftas

j Proble

tionls interfecabit,

Eft enim .4B ad ab paralléla ,

quod

D ad iibi

acus magnetica exhibet, proin triangula .4BC & abc [(imilia &

;c afunita

puncum Cutrique commune. Inde aliud jam quodcunqueex prz«

cedentibus regulis decerminari potett,

)ïfm.aïe dl'

EBROBE-JEMA VIO

proport

falfoTinea

verat U^

135. Determinare punctum ftationis, unde tria

erfüs Áj

puncta trianguli cogniti videri poffünt.

; g a

RESOLSUTIO,

, ín ine

Vel ftatio aífumitur extra vel intra triangulum

commäïï'o

vel in ipfa dire&ione lateris trianguli,

CASUS

I.

MEG

jpicala t

Dum extra triangulum ftatio x affumitur,

j occurmt

!eïe d[fCii

qüï

G 3

—

E,;H

Ts

REX

1Cco

CA DPAUFT- VH.

RES$)0 L'UTIO.

ett, d dt

dexan cd

Per pur&um flationis x , & extremitates ateris 4B,

d Mtet

cppofiti flationi x finge tranfire circulum, qui vel

totum triangulum , ut Jiz. 05: vel unum tantum la-

1.

tus ut Piz, 94. exhibet, includet ; ex xducantur linez

at effo

quarmdam

verfus tria pun&a obfervata 4x, bx, & Cx, quae ufque

in D prolongetur, erunt hz linez ax, bx, dx fimul chor-

tuntimet

& detemir

dz circuli , ducantur etiam chordaze 4d & Bd: eranta & b

tere deterr

anguli obfervati, quorum menfüura eft 1 4d& X 4B, fed

catas ob

etiam anguli v menfura eft £ 44, & anguli m menfura

ert, M

mova ad 1

eft X D)

ergoa z g, & b-—m.lgitur in triangulo

griore me

ABd eft notus m & , cum latere .4B, ergo inveni-

V contit

tur /d & dB , & fimul inpotefcit tertius angulus in d,

feu r --5 in E;g. 93. & o in F7g. 94. Jam in triangu-

lo 4dC notum eft.4C ex hypothefi & 4d ex praceden-

1

tibus cum angulo intercepto 4 -i- min Fig.93. & 4——m

lm ABC,

AI/, Bez-

in Fig. 94. ergo refolvitur triangulum , & invenitur an-

litt. angu]

gulus r & eadem methodo in triangulo .5C4 angulus s,

!ülus d«

Denique in triangulo 4dx in Fzg. 94. vel.4Cx in Fig.94.

t comple

notum efít 4 ex obfervatione , r ex pracedentibus cum

latere 4d in. Pig. 9. vel A4C in Fig. 94. proin habe-

4B, z5

tur /x ex una parte, & eadem methodo Bx ex altera,

N

quarum interfe&io punéum x determinat.

£

COROLLARIUM 1.

:r,?nl n' :1

156. Menjþl&, fi hec operetio peragi debeat, transportato-

Pro trjap,

rfium ad manum effe debet, quibus anguli & menfurentur, &ef-

teptus 4

formentur, directis fcilicet dioptrisin pun&to quocunque x (Fig.95.)

J lter .

verfus tria jam determinata obje&a a, b,c ducantur in charta fepa-

L Cifetens

rata direciones aX , bx, cx, que transportatorio examinentur

ut m & & innotefcat; applicato deintransportatorio in ipfa men-

fula (f£ig. 96.) ad lineam ac, que pun&o affumto opponitur, fiat

in pun&o dextro c angulus e» ad finifram obfervatus, & inpun—:

"to finiteriore a angblus m ad dextràm obfervatus, per inter-

fedienem linearum in d, & angulum tertium ducatur linea, in-

7

qgua pundum * effedebet, trans(eratur dein adx ,qui ad finiftram

(n!ï*

N

A

RESOLV.VARIA PROBL. TRIGON. &c. zor

eit, ad dextram in c deorfum, ut fiat acxe — adx:

& alter ad

dextram cd« transferatur ad finitram in a deorfum jut fit cax —

$4D.

cdx, interfe&io in linea dx pun&um x

v determinat,

ui vel

COROLLARIUM Il

un L

7. Cum fepius hic transportatorium adhiberi debeat,

linez

nguli efformari, nemo non videt, magnam in iis accurationem,

quaerendam non effe,

recure

! Ulque

Qui dimenfiones menfula peragunt

1chore

runtad methodum, quam fuperius dedimus de acu maenetica q r33

"T

& determinant pun&um x, uti prius pun&um . qux(ivimus vifo la-

re determinato qc

fi dein per dioptras ad pun&um x & b appli-

iB, (ed

catas Objectum tertium B confpiciatur, pun&um rite determinatum

menfüta

erit

Magnum hoc adfert commodum, dum impleta jam charta,

novaad menfulam ponendaeft,

tmnslat's enim tribus punctis ex

fangu

Inven

priore menfula ad novam cnzutc.m, quodcunque pun&um eligi & la«

bor continuari poteft.

M lnd

SCHOLION

Tiang

:Cegelle

138. Exempl. Babeatur ex pxæcedenubus calculis trlangie

lum ABC, (Fig, 93.) in quo 4 —

47?35', B z 83^ 54 , C —

:48^

—f

Al, BC 1717, AC

2307 , AB — 1738

In x obfervati

(f at

funt anguli a — IO 37, b —'I/ 59 facto triangulo ADB eft

lus 5

angulus Z £ r7^ "9 & n — 10^37, proin o — I51 944 , CU*

T0

jus complemeatum 28?16/; fiat ergo

ut fm. o

AC-z fm,m: 44

i Curm

Z

u

£

Bd

; hobe-

zAB, ——

40089

o

alielt,

Jn

—

/75389

3 564661

58. $6466I1

Jin.

9. 26$277

i. d zm 9. 48!731

2. 83c038, 4d4—676

8- 04539'

Bd z3 1113

Pro triangulo C4d eft AC-.—: 2307, 4Ad z 676;

anzuius intera

port*tc'"

ceptus 4 —iz47^35 — 17 39-——29 ?$6' ;flat ergo ,

ut fums

, &e

2170j

p

ima later,: different. lat, z— tangens F fummæ angulorum tang,

o ,4

g fepee

$ different, hoc

et, 2983

ïö&ï»m

y?

—Q—ffpu 32*??

d

inen

1631

peum

—

(2 man

DC—

3. 212454

2984

poweed

3. 47465 ;S

üt, 4«

]n

pur

9. 737*301

i)

r 1ntei

15 2

I9. 572959

xeanl!

19. 310760 —5 63^57 z3

differ,

nl(lxï'ï

E

elt

S $

CAJIET. VI

YMN

102

Wi

L.M

Ergo angulus major — 75 2 -- 63 57 —138 59

aneulus r minor

75 9 — 63 57 z—

lI

«

lv—-—lr

Pro triangulo CBD eft BC — 1717 Bd ——1112;2ngulus intere

[9]

Fin

ceptus B —n—83 54 — I6 39.——79

I7 igitur, ut fum.

juscun

: differ. lat

106^43

*

lat,

T : — (ei 53 of 30.

qu&is ch

WeDdm ;

605

2. 781755

—

ti 1 fed

2829

3. 451633

atei, il

"

32090122

T.53?21'30 ——-10.103947

TO ui

9. 458669 —— T6 L25 r

differ,

: obír

I

Ereo ansulus major

—

peuumi

53 21 -- 16^1'28'-— 69?22 58

—

habrr

minor

feu $

—

$3^91'

$ 16 1 98

237 20 2

'6fandem. in triangulo x4C Obfervatus a —— . 1O 37 , inventus F

Ln [ ,(4

——

n

g o hp5

cum latere .4C

[ssmued

2307 , dat hanc proportionem ?

& lic

J

latus AC

Jin,.a

Jun T o:datus xAÀ

h f

Lat. AC —5 3. 363048

p /zn

a

—

—

Venity

9 265 277

betur

4. 097671

itvenitt

- n

T

9 20 w8

L

J tj

3- 581507 m lateiteMz 2407

In triangulo xCB obfervatus 5 ——

-17/89,inventus 5 2 37 ^2o0 2"

ebus

cum latere BC z— 1717, datífequentem proportionem :

hulg

Sin, b

lat. BC z Jim s: lat. xB,

ORE

Ti ero

Lat. BC

—

3. 234770

]

e— jln.

b

—

dul

9. 481731

later

8: 7538039

Cepta

—

- Jn,

$

9. 7828c2

38.535841 — lateri xB — 3434

His duobus lateribus in fcala acceptis fiat interfetio ex puncig

1j0.

A & B, & habebitur x quefitum ftationis pun&um cum dimenfio-

Chütta fep

ne reliqua connexum.

Tinent

l'lï'hur CO*

C A/SBUS H

N 0bfer

** QCXttan

159. Dum ftatio x intra triangulum affumitur.

f-—atr :

51;1.&

(Fig. 9] )

2Xam;

kï'l

D

le

f !h??n?

R E-

i

o

I

:

iSOLV. VARIA PROBL. TRIGON. &c

O05

HRGSUOLUTTTSO

f

) iffetv

x & duas exttremitatos

ut 101

Fingatur duci cxrmzlas per x

cujuscunque lateris 4 &

& B, producatur cx utque in d,

ductis chordi

x

Bx, item 4d,

Bd; erit » comple-

mentum anguli obfervati «, & : complementnm ob*erva-

ti ^, fed m z e, & n—::, cum ad peripheriain eidem

hi arcui

(7

infiftant ,

er-

arcui, illi quidem arcui 247

IETA

EC

e eft complementum obmvm a & angu

ïlï

À L N

g£o a*xgm

j

1 obferv

Lvy

DE

IN

15

lll. d

i b, ergo etiam in triangulo Bd tert

90

habetur cum latere AB;

ieitur Qï

os-omtmmm erit

Q2

fin. d: ÁAÁB

fia

e

& fin

.[l

Á

F

E m ll[le

: Bd

ventus f

& hinc invenitur d & Bd,

tonem

Iu triangulo 4Cd ob notum 4 ex hypoihvfi, &

juventum per pr

cedentem operationem angulum 7, ha-

betur latus 4 cum duobus lateribás AC & d

igitur

invenitur anegulus 7.

ob notum

Et in triangulo CBd

B in triangulo J4BC & inventum e habetur totus B.a

40f

"

lateribus notis interceptus, invenitur ergo angulus s

In triangulo x4d notus eft m & v cum latere 4d, inve-

nitur ergo Zx , & in triangulo x Bd notus eft & & s cum

Jatere . 5d

invenitur ergo Dx.

Ax & Bx in fcala ac-

cepte dant interfe&ionem in x , & pun&um ftationis,

COROLLARIUM

H

Obferventur in

140. Menfula inter triangulum collocata,

1 Puxxälg

charta feparata anguli per dioptras, & ope transportatoris exa«-

lienlo

minentur

menfula deinde ad pun&um ftationis pofta, transfe-

fantur conplementaà angulorum obfervatorum ad latus medio an-

gulo obfervato oppofitum AB ita

ut complementum fini(tri anguli

ad dextram veniat,

quod vero ad dextram eft, ad finiftram trans«

feratur,

uti fuperius oftendimus, id eftm ad e & aj i; inter

ynitab

&io in d conjungitur cum angulo medio C , erit in Iineacd punctun

x. Examinentur deintransportatorio a.w)wx ad d,fcilicet r& $, &

ad latu

]5 idem .A4B fed introrfum transíerantur,

ut jterum is ad

finiüram r ad dextram in B,

& alter $ ad iniftram in .4 veniat

intejfedio pun&dum x

daterx*ïimt

quod fationi in terra accurate

LE

G 4

$w

YOsa

s Waw&;**-—v'

1O4

CAP VT VH

RES

refpondere debet,

Calculus eodem , quo in priori exemplo ufi

It

fumus , ordine inftituitur,

Jatejus

exato !

CASUS Il

vajtut .

r41T. Dum ftatio in ipfo latere, vel in directio-

Tan

ne lateris affumitur (F?e, 99.)

A.

üt laten

HESOTLSOUTTO.

&iam ol

Nullam hee refolutio difficultatem habet, obferva-

to enim aogulo m in x, velz in y, femper habentur

V

tres arguli cum unolatere, cum B & 4 notum eífe po-

1)3.', CJ', [

natur, unde triangula, & fegmenta baíeos quafita in»

Ynttrfedio

notelcunt.

EHrit enim

V rnde

fin. g

C5

—

l4

9

[]

x B,

(292"-:5 4

fin.s2 : CB

—

"E Ey,

COROLLARIU M.

143

142. Menfula & adhibeatur, cum pun&um in ipfa linea

sut faltem in eaprolongata effe ponatur, tantum opus erit, di-

fudum

optras ad pun&um c (Fig. 99.) menfule applicare, & ita diri-

Gdj

gere,.ut objecum C per illas appareat, duc&a juxta regulam li-

»ea, pun&um x, vel y abfcindet, dummodo prius &b in menfula

accurate line2 A4B, que in terra efformata effe fupponitur, cone

gruat, quod femper obfervatum eífe fupponitur,

Elin

fitur 4

PROHBLEMS./, —WIL

n j

I45. Determinare punctum ftationis, dum ob-

0 tvent.

jecta tria nota, quie obfervantur, in eadem recta

'ddd" 2

"19 p obr

jacent. (Fzg. ro1.)

Í

M in

RESOLIUTTIO.

*! QUadie

Cum afit z m, & b z n erunt in. triangulo A44C

üd fign,

Pb: g

noti anguli cum latere 44C, invenitur ergo d & de,

l. f

In

d

RESOLV. VARIA PROBL. TRIGON. &c. 10

| tl

In triangulo .4B4 ex noto v intercepto, & duobus

Jateribus d & .4B invenitur r.. In triangulo vero. 4C4

ex noto m intercepto cum duobus lateribus cd, cB: in-

venitur f

i

Tandem in triangulis Zdx & xdc ob obfervatos 4

& b, &inventos r & * cum lateribus A42, dc inveniun-

tur latera Zx, xc uti in primo cafu diximus, quo modo

etiam calculus inftituitur.

IVà«

COROLLARIUM.

ntu?

Y4. Menfula: du&is in charta feparata dire&ionibus a* 3

jo

Dx, cx, transferantur anguli ad alterum punéum gin m, b in m,

| d

interfe&io dabit pun&um.d, per B & d ducatur re&a, & angulir

& s in interfe&ione d fa&i transferantur ad 4 & C deorfum, ut

fit z $,c—r, interfe&io dabit pun&um x,

PROBLEMA VIL

145. Dato latere ab, &angulo a determinare

inea

punctum trianguli c, quod ex 5 ob impedimentum

)

d

cd videri non poteft, (F?g. 100.)

(þ'fi«

m li«

nfult

RESOLUTIO.

o

Eligatur objectum aliud2, ex » & c viübilä & ca-

piantur anguli in ? & c, abe & acd, Jam cum tres an-

guli.4, B, C, debeant xquales effe r9O?^, fumma, mo-

do inventa,major erit, qui exceffus interea notetur; fi

ob

bd & dc zxquales effent, dimidius exceffus ab utroque

(a

angulo obfíervato fubtra&us dabit verum angulum ? &

e

Si inzquales, quantitas ab utroque fubtrahendo

ita quxritur: in exigua a pun&o d diíantia erigatur

aliud fignum e & obferventur ebd, ecd; fiat dein pro-

AK

portio: ut /ümma horum angulovum , ad eorum differentiam ,

I

ita exceffus. i0tur priur imventus feu fumma utriufque anguli ad

differeutiam umius ab alievo, qua diíferentia a toto excef»

G 5

fu

íi

zo6 CAP. VIL RESOLV. VAR. PROBL. &c.

fu fubtra&a dat angulum majorem, & ipía differentia

minorem, utraque a fuo adjacente angulo fubtrahatur ,

& habebitur verus ? & «.

T

SCHOLION

I,

)

J

146. Sit a — 78, abd — 66 , acd — $4: quorum fumma —

198, jufo itaque major 18* , obfervatus ebd fit. —— 4, & ecd

z—$,fit, ut fumma I2 ad differentiam 4, ita exceffus totus

19 , ad differentiam 6, que differentia 6 ab 18 fubtra&a dat Y2 5

$gitur ab angulo majori fubtrahi. debet I2, a minori vero .6.

proin angulus abd feu 66? imminuitur 12^, & alter acd feu 54?

imminuitur 6?, & erita — 78?, b — 54, c— 48 . quorum fumma

z I190, & inde jam latus ac & bc quaitum invenitur.

SCHOLION

ll

147. Plura hlc adferri poffent problemata , quibus varii

qui emergeré poffent cafus tefolvautur, fed cum .totum femper

C

eo reducatur, ut tres trianguli partes innotefcant,'inter quas una

V

faltem pars latus fit, facile erit, quemadmodum in his problema-

tis exhibulmus, pro novis triangulis novos angulos Obfervare , qua

j

ii

cum prioribus conjungantur, & unde novi quogue pun&i ftatio

habeatur. Sit v. g. (Fig. 102.) €um noto triangulo:.4BC pune

n

Si D cum .AB in una re&a jacet, ob

é&um D. conjungendum,

fin

notum b erit notus contiguus d, e obfervatur, inde moto tertio

$i non in dire&um jaceat

D & latere BC habetur CD & bD.

fed in E, d incognitum manet, hinc obfervato e menfurandum

Qu

«uoque eft CE, ut ex duobus lateribus, cum angulo intercepto

notis, reliqua cum pun&o E innotefcant.

Ità quoque in aliisfi-

ter

yailibus procedendum eft caübus.

tizc

(ein

ïefp

h

Y

7

bifi

€,

g

Pf.m

Qul

i

N

C A-

d

fs

, &t

o

N

CRÀ

1c7

feten

'I&hatüf,

C^A P.UJT- VHl

RESOLVUNTUR VARLIA PROBLE

MATA, QUJE AD ALTIMETRIAM SHU DE-

fummez

A,& d

TERMiINANDAM ALTITUDINEM OB-

fs totus

JECTORUM PERTINENT

PARUM

vetüo. 6

PROBLEMSZA 9I

d feu 54

um fummaá

148. lnvenire altitudinem acceffibilem £C

RESOLUTIO

UWS vatt

Inftrumento verticaliter ere&o (S$. 45.)& in 4,

5 femper

(Fzg 80.) quod ad libitum affumitur, collocato obfer-

et quaS una

vetur angulus «; in B, quia altitudines femper per-

sprobiema

rrare , que

PeDdICUIQFSb effe debent, eít recus, ereo etiam ter-

ii

wl fatio

tius c notus erit; menfuretur DE, habebitur illi zquale

ApC pur

latus.A4B, & in tiiangulo re&angulo erit fin. C: A4B —

mre&, Qb

fin

4

CB. Ad CB addatur altitudo inftrumenti A4D

noto tem

—— DBE, erit CB -4- DE

& Jaceat

CE, proin CE altitudo

enfurancum

quafita

AMenfila id fi peragere velis, ea quoque verticali-

)l

2 fcepto.

g i u

ter erigenda erit, (F/z. 93.) & dioptra inprimis ho-

rizontaliter applicanda & linea indefinita cd ducenda ,

dein fixa in c, quod ftationis pun&o perpendiculariter

refpondeat acicula , applicentur ad illam dioptrze, do-

nec cufpis B appareat

& ducatur re&ta jc , menfurata

bafis CA ex [fcala ad cd transferatur

in d ere&ta per-

peudicularis d5 , in fcala determinabit altitudinem BD

cui addenda eft altitudo menfule cC z DA, & habe-

bitur tota altitudo quafita 4B

G O«

u

CATPTUT VIL

108

COROL-LARIUM,

Y48, Nemo non videt hanc cum menfula gpärafiojlem non

exigua cum difücultate abfolvi, ob verticalem enim illius fitum

P'ï

dioptre inclinats9 fupra menfulam retinende funt , quod ob pondus

q

[

earum nom fine eravi incommodo prefiatur, hinc magna accura-

fian:

tio inrde.fperari non poteft.

Si adfint äïoPtræ in centro mobi-

t aL

les cum femicirculo , poterit menfula in fitu horizontali relinqui,

toteft

dioptris vel tubo inclinato (Fig. 103.) dum fummitas appareat, in

femicirculo .4 angulus altitudinis notabitur; fiat ergo re&a in-

at

definita, (Fig 1043 & in ea abícindatur bafis menfurata ,- & in

uno extremo d erigatur perpendicularis de, in altero c conítrua-

tur angulus obfervatus, & ducatur cb, erit bd altütudo quafita,

At qui accurationem amat, altitudines folo quadrante aut gonio-

metrico obfervabit.

01j

PROBLEMA I.

t3

k

g'ï

v &

M

I49. I

nire altitudinem. inacceffibilem CE

(Bo. x05.)

?

Vl

RESOLUTIO

(E

Infitumento in 4 verticaliter collocato obfetvetur

10 [n

n

angulus « verfus D pun&um horizontale, & C altitu-

ferj

dinis & in eadem re&a AE in quadam diftantia affumta

note

determinetur B,

transferatur dein inftrumentum . ad

lo

B, menfuretur 4B, & obfervetur rurfus in B angulus

t ad

b, habebitur etiam illius contiguus e.

Jam in trianga-

plem

x.m

lo aec ob notos 4, e, &latus ae(zzAB) invenitur 2C; in

triangulo 5CD re&angulo ex noto ? & bC invenitur CD,

*Ob T

ad quam addita eB — DE dat CD -- DE — CE alti-

Tdnv

Q

tudinem quefitam,

in t

i

Meufila in .A verticaliter collocata collimetur ho-

tur (

rizontaliter verfus D & apicem obje&i, & ducatur li-

nea indefinita, menfuretur 445 & in ab treusferatur,

ititu

collocata menfa in B, collimetur turfus verfus D & C,

interfecabit hzc nova collimatio priorem in c, unde

demiffa perpendicularis dabit cd.—z CD, & addita B5

dabit totam CE.

C O-

RESOLV. VARIA PROBLEM, &c,

109

'coROLLA-RIUM

Uflét non

lus fitum

Ist. Si inftrumentnum in eadem rec&a bis verfus objeétum.

poni non poffet, uti in F/g. 106, contingit, menfuretur quzcun-

b pondus

que bafis A4B, &collimetur verfusinfimum pun:tum obje&i men-

ld Qf

t mobíe

furandi D, habebitur triangulum. 4DB, & innotefcit latus AD,

& BD, obfervetur dein in .4 vel in B angulus altitudinis a, in-

! Felinqii,

»teat, 1

notefcet ob inventum BD vel .A4D & re&dum inD, etian CD

altitudo quefit2,

tedy i

n , kin

* conftrua

PROBLEMA Ir.

0 quaita,

tUt q020

152. Invenire partem altitudinis inacceffibilis

objecti v. g. in monte pofiti CD. (Fig. 10'.)

RK ESS O.LU TI1O0.

em (E

Verticaliter pofito inftrumento in 4 obfervetur an-

sulus e inter horizontem cE, & maximam altitudinem

CÉ, menfuretur quoque bafis 4B, translato inftrumen-

to in 5, obfervetur angulus o inter altitudinem quxfi-

fetvetut

O etitit-

tam CD,, & angulus m inter horizontem & pun&am in-

ferius D, feu DE, habebitur totus o -I- m, proin in-

j affumta

notefcit illius contiguus 2, & fimul in triangulo re&an-

wum d

£ulo DEn innotefcit angulus D complementum anguli

3 gnguïUS

m ad re&um , proinde illius quoque contiguus €, com-

triangte

plementum ad duos re&tos, His pofitis intriangulo Cbe

j i

ob notos angulos b & c, & latus /c invenitur 5C; in

'[tuï CD ,

triangulo CD? ob notos angulos o, & e & &C invenitur

(E o

CD quafitum; tandem fi tota quoque aititudo petatur,

in triangulo CE» ob notum o -- m, & latus Có, habe-

r o*

tur CE, aut in triangulo DEb habetur DE, cui rurfüs

catut li

altitudo infirumenti ^2 zx EF addenda eft,

ferttut »

)D& C)

I lmde

dita )

.eU

1

[

T1IO

CAPUT VIIL

nit *

z 49?230';'0

I5?20, m z— 40* proin o

ïx:(xbe

o

O Meememps

t$/

-- m ——

$5? 20 erit )

I90*?— 55? 20 22 I22"

Ed

m

$8?50'; D z 9o — 40

AO , &n

a 50 igituï.—g

z 130; AB —— bcíütz60hex.

fiat fu. n : bc — fin. c : Cb

ipus e!

Jfa.. e : Cb z fm. o : CD.

mlo

(De 0|

CD,

COROLLARIU fM.

153. Menfula in A pofita collimetur verfus horizontem G

& altitudinem C, menfuretur AB; translata menfula in B, col-

|

limetur verfus idem pun&um G, & altitudinem C, & fimul ver-

fus D pun&um infimum obje&i menfurandi , directio verfus €',

hori

ex utraque ftatione fe interfecabit, ex interfect!onis puncto demiffa

decl

perpendicularis erit tota altitudo , & ubi dire&io verfus D hanc

perpendicularem in d interfecabit, erit altitudo quefita cd,

inum

(Fi

PROBLEMA IV.

154. Determinare altitedirnem objecti, dunt

À

infra flationis horizontem eft. (Fzg. 1c8.)

À

44

ll

KESOLSUH!IU

tet fj

to in

Eligantur ut prius dus ftationes //.& B , quarum

angu

diftantia menfuretur; in 44 obfervetur angulus a infra &

& D

b fupra horizontem , quem punctum fummum C & infi-

mum J£ cum horizonte D facit.

'Trans!ato inftrumen-

tllang(,

Kit qC

to in B obferventur rurfus anguli » , & , innotefcent

lib

&UU

quoque eorum contigui e, & f. Jam in triangulo /«C

&Mcl

ob notum », & obtufüm e cum latere Pe z— AB , inno-

tefcit eC; fimiliter ob notum 2 & f cum latere eodem

af innotefcit Ee. Denique in triangulo CDm ob notum

m, & re&um D cum latere Cm invenitur CD, & in tri-

Ü

Wetur

angulo DÉs invenitur ED, que CD -- DE dant

Nütüe

4

CE altitudinem quaefitam. Vel inventis Ce ,& Ee, cum

En aho

ha-

RESOLV. VARIA PROBLEM. &c

XIE

proim o

habeatur etiam ; 4t- s interceptus, invenitur ex da.

tis duobus lateribus cum angulo intercepto latus CE

— IZJr

ipitur e

COROLLARIUM

155. S1 bafis Ee menfurari poterit, unica tantum. flatione

opuserit, ob notum enim in re&angulo EDe latere Ee cum an-

gulo n & recto D invenitur ED,' & De, rurfus in re&angula

CDe ob inventum De, & obfervatum m cum re&o D invenitus

CD.,

nyuem G

P-XROOBLEMSZA .V

,B&O

fimul vere

156. Determinare altitudinem feu infra , feu fupra

H

erfüs C,

horizonte: 1 . po fiti. objecti, dum ipíe ftationes in

&o demifa

uD

declivitate Í

fumenda funt, & ex una tantum ad furn—-

à dl

mum & infimum pun&um objecti profpectus patet.

(Fig. 169.)

RESOLUTIO

j dut

Menfuretur .A4B

pofito inflrumento in B

44 erigatur pertica, alutudmem inftrumenti habens, uf

as m

4«A(it--bB& ADB

————]QÓ

ab ,

& obfervetur angulus v,in-

ter fummitatem objecti & pertice; translato inftrumen-

to in .4, eadem pertica in 4 erigatur, & obfervetur

qll&ïl

angulus; item anguius m inter totam altitudinem obje&ti C

; infn &

& D, item angulus o inter D & horizontem I

In

( & ini

triangulo Ca, ob notum 4) cum angulis s & a innote-

nfrum

fcit aC. In trlangulo a€F, ob notum o-I- $, & re&um

710["1*«9 tt

habetur tertius 7.

Igitur in triangulo CDa ob notos i,

u ^

& m cum latere Ca invenitur D & CD quzfitum.

[, inno

me cod Cm

COROLLARIUM TZXI

Ü notum

157. Sia obfervari nequeat, quod nimis obtufus fit,

Obs

fervetur angulus depreffüonis e, qui ad p re&um, & ad o -iL- m

& i tt

pE dant

additus, & a 360 fubtra&us dabit angulum a.

81 Obje&um fit

infra horizontem (Fiz.

I10.) noto in triangulo aLC latere ab

, jj, a

mene

u

XI2

CAPUT viHt.

zaenfurato , cum afgulis ? & a obfervatis habetur latus $C ; obfer-

vatis in a angulis in & z, erit in triangulo recangulo CaD nota

hypothenufa aC cum angulo m, invenitur ergo» & cathetus CD , &

il

cathetus aD ; in triangulo aDE , ob notum z & Cathetum a?

e do

habetur hypothenufa aE , & Catlietus DE, que CD -E DE dant

CE altitudinem quafitam.

CQROLLARIUM T.

158. Facile erit hec principia rmenfule applicare, cum ut

13 al

frius vidimus, du&is dire&ionibus punc&um altitudinis per interfe-

Ícit in

&ionem determinetur, inde demiíffa ad horizontem perpendicu-

lo 4,

laris, vel conjun&io fumme & infime interfe&ionis altitudinem

determinat,

altitu

PROBXLEMA VI.

159. Menfurare diftantiam nubis a terra. (Fzg.

(Fz,

111.)

RESOLUTIO.

Menfurata bafi 4B obfervetur in 44 & B angvlus

I

4 & b, innotefcet tertius c, & invenientur latera aC &

i tec

LC. jam in triangulo aCE, demiffa fcilicet ex C ad ba-

Cllly i

fin 4b perpendiculari, eft notus & cum re&o E & hy-

lüm, |

pothenufa aqC, invenitur ergo CÉ, cui altitudo in-

Cum

)

ítrumenti addita dat totam altitudinem quafitam.

fit &q

tria

COROLLARIUM

fía hor

160. Qui ad omnes has diverfas folutiones animum advere

tià obj

tit, videt femper eo rem omnem redire, ut menfurata baíi,

que difantie ftationum evqualis eft, anguli obferventur, ex quo-

rum notitia per calculostrigonometricos latus incognitum eruatur,

hinc facile quodcunque propofitum problema folvetur , dummodo

latus unum menfurari, & duo anguli, aut in re&angulo unus ob-

if

fervari queat; poffunt talia problemata quam plurima dari, &

diverüs caüibus particularibus applicari, fed quorum folutiofe-

Qul (

paratis regulis non ezeat; proponam unum alterumve exerci;ii

caufa, unde de reliquis etiam judicium formari poterit.

PRO-

Mätz

a sss

RESOLV.VARIA PROBLEM. &c.

115

obfat

PROBLEMZA VII

aD nota

(D, &

tim a

r61. Invenire altitudinem obje&di, ad quain

DE dant

ex domo profpectus datur, (Fzg. 112)

KESORLUITIOQ

Ex feneftra 44 demiífa catena ad fundum B menfu.

,v ut

ia altitudinem .4B

obfervato angulo .4 & B innote-

inteti

pentict

fcit in triangulo 4B5C, latus BC, obfervato in 5 angu-

tudinem

lo 4, ob re&um E & latus BC notum, invenitur EC

altitudo quafita.

PROBLEM

VIII

162. Determinare altitudinem ope fpeculi.

.(Fg

(Fig. 113.)

RESOLUTIO

Pone fpeculum horizontaliter ad terram, & tam-

vigolus

ïaäC&

diu recede in linea re&a, dum cufpis obje&i D in fpe-

? gd 08

culo appareat, menfura tuam altitudinem usque ad ocu-

lum, & fimul utramque diftantiam a fpeculo 4C & CA

&hy

udo ià

Cum, ut in optica demonítratur, angulus incidenti» a

fit &equalis angulo reflexionis P, erit ob re&um in 4 &

ll

B triangulum A£EC fimile triangulo CBD, igitur diftan-

tia hominis AC ad altitudinem hominis AE ficut diítan-

tia obje&i BC ad altitudinem objec&ti BD

n gd?'ef'

ü bat,

ex (

PROBLEMA

IX

i eri ltuf,

165. Determinare altitudinem objecti ope ba-

dioodo

T Ob

culi. (Fzg. 114.)

i, *

oudioft

eïefcml

Metsb, Math, P. 11I.

H

RE-

10

t

H4 CAP.VIIL RESOLV. VAR. PROBL.

hn4aXsoLUTAo.

Id

Humi proftratus baculum ad plantas pedum perpen-

diculariter applicari a focio cura , dein tamdiu recede,

aut accede, dum per fummitatem baculi C afpicienti fum-

mitas objecti D appareat, erit E altitudo hominis us-

P

que ad oculum, ad EC altitudinem baculi, ficut .4B

difantia oculi ab obje&o, ad DP altitudinem quafitam.

PROBILBEJOM T x

.164. Determinare latitudinem fluminis. (Pg.

115.)

dun

EPESOLUTIO

Latitudinem fluminis ex una ftationé invenies, fi

ere&us in litore ad aliud litus infirumento angulum 4

étiga

Capias, ex a enim noto & 4 re&o cum Catheto a in-

notefcet APE ; fufficit quacunque

| v. g. pileo, bacu-

flatio

lo &c. angulum a feu dire&ionem

*

oculo verfus aliud

hod

Vis b

litus accipere, dein non mutato fitu vel inclinatione

terte

baculi , aut pilei fefe "vertere, aliudque pun&um 5

ex (

in terra nofare, ad quod directio tendit, erit 4b lati-

tudo quefita ob zqualitatem triangulorum 4Ba & ab.

bunt

Accuratior methodus inferius ($. 175.) dabitur.

STLTHOJI115

165, Poffunt etiam tum hzc tum horizontalia problemata

ylbï*ïl(;]

sornunquam folis baculis ad varios angulos defixis refolvi, que-

& die

madmodum exemplum in &eometria $. I24. dedimus; pofífunt

& d

etiam ansuli fine soniometrico per lineam chordarum ex circule

bunty

próportionall aut confecdis ad hunc ufum tabulis determinari,

Ui d

cum radius

P

f

emper fitequalis finui/3O eraduum; ($23 Zrig.)

verum qui accurat)onem amat, his praxibus facile fuperfedebit ,

aut eX nota jam theorja & hucufque traditis principils, ubl oc-

It

calio tulerit, novas & ipfe praxesexcogitare , aut notas applica-

lr :mv&l

ye poterit,

ET

1

C A-

lt

M 7À

BL'

C

CR

I1$

ï*»

J]ï

perpete

Teced

OCX T UT IX

enti fum

Dil$ $

APPLICATIONESPRINCIPIORUM AD

cut 4B

PARTIALES DIMENSIO

NES UT AGRO-

ehtam

RUM, FLUVIORUM. SYLVARUM &c.

P

OBLEMA I.

s (Rg.

166, Dimetiri aream campi.ex una ftatione,

dum totum fpatium permeabile eft, (Zv. 11£6.)

&BsSO0LU TO

es, !

In quovis angulo circa limites, aut extremitates$

Ipu um d

o ul ín

erigantur

verticalite

baculi 4, 58, C

&c; affumatur

bacu«

ftatio in medio, & obferventur anguli inter erectos$

$ 4 xud

baculos; meníurentur omnes linez ex flatione ad qudem-

vis baculum, erunt ubique duo latera cum angulo in-

jinatlone

tercepto nota, proin invenietur tertium ; qus fingula

i ecurm

ex fcala accepta & fimul conjun&a totum fpatium da-

J atl

bunt.

T2

,

[,

COROLLARIUM

L

167. Si menfula opetatio péragatur, in. éodem puti&o o ad

lHibitum affumto, collimetür verfus omnes baculos 4, B, C, &c;

Ti Jläm 3&

& ducantuür diretiones indefinite; menfurate dittanti2 04, o B

[ni Lihe

&c, ad correfpondentes lineas ex fcala trans: eruntur

determinas

POTJM

buntut pun&ta a, b, c, &c, quz lincis re&is conjunca totum fpae

Y clrþuïa

tium dabunt,

bj

ermih 14 ,

[rzg )

COROLLARIUM

[

edebxtQ

168. Si ex uno angulo omnes reliqui videri poffnt; ponae

j

',1 0C*

tür inrumentum aut mehfula ad eum angulum v.

]ic*

c

ad rFLg H f7ä)

äfP

& accipiantur anguli, aut ducantur dirc&&ione

28

verfus 4, B 4 €;

D,E;

nïUPrLJJLAD

làteribus

AB

&..AE

cum

diagonalibus

H2

M.

:

l

II6

CAP UT IX.

AYI

fus (, (

AC, A4D, habebuntur (& goniometrico labor inftituatur) duo atera

cum angulo intercepto pro inveniendis .4B, BC, CD ; aut in

fulcii

menfula poterunt determinari pun&a a,5, c, d, e , quv lineis

ahííndat

coniun&a fpatium claudant; omnia hzc etiam przftari poffe acu

inagnetica, fi de folis angulis accipiendisfermo fit, non eft quod

15 quot

tistur &

repetamus, cum id jam fuperius fepius infinuatum fit,

tàtelinea

COROLLARIUM III.

4&D1

169. Dum fpatium permeabile noneft aut majus, quam ut

&am di

commode omnes ha diftantie men(urentur, ad primam methodum

tautum.

recurrendum efít, dum menfurata baü AB (Fig, I18.) & dupla

transfer

ftatione in .utraque extremitate accepta obje&a radiis vifualibus

in menfula expreífis intetfecantur, aut ex angulis obfervatis cal-

tut, ab

€ulo trigonometrico determinantur.

to 4 x

quals o

COROLLARIUM IV,

y

I70. Dum circumferentia fpatii menfurandi valde irregu«

gun t

Yaris eft, ut in geometria $. 279 infinuavimus, plures & vici-

minand

niores pertice erigende funt, utfigura vera habeatur, velipfo

curvature feparatim dein a pertica ad perticam determinande

docuimi

funt, ut inferius de fluviis aut viis diceznus.

tuzlib

ü

us i

PROBLEMA IL

Overfat

Diy, Q

I71. Dimetiri fpatium, quod permeabile non

minen

eft, uti fylvae, lacus, ftagni majoris, praefidii &c.

idem n

(Fig. 119.)

füperío

Wpeteni

RESOLUTIO.

l. Si immediate in ipfa circumferentía flationes af-

fumi queant, ut penes fylvam, lacum &c, contingit , tot

h

baculi erigendi funt, quot lineis rectis fine fenfibili fi-

d

gura mutatione fpatium claudi poteft, figatur menfula

119)

in ftatione 4, & vertus B & E collimetur, menfure-

tur 4B, & menía ad B transferatur, fimulque ex fcala

Sit

in menfía

ab ab(cindatur.

Menfula ita in & colloca-

d

ta ut linea a? linez 44B correfpondeat, collimetur ver-

fus^

APPLICAT. PRINCIP. AD PART. &c.

II7

olater

fus C, & menfurata BCin Lcabícindatur , translata men-

|

quin

fula in C collimetur verfus D, & menfürata CD in 4

lineig

Ife acu

abfcindatur, fic ulterius procedendum per omnes ftatio-

nes quot quot fuerint , donec ad ultimum punctum perve-

f quod

niatur & figura claudatur ; i menfula ad É translata accu.

ratelinee DE, & E refpondeat, ita ut dire&iones verfus

A & D lineis de & ea incumbant , operationem rite fa-

lI Wt

étam effe argumento erit.

Cum goniometrico anguli

hodum

dupha

tantum accipi poffnt , ii tranfportatorio ad chartam

alibus

transferendi füunt, & latera, quz menfurata effe ponun-

is a«

tur, abícindendi, defcribatur fcilicet angulus obferva-

to 4 »qualis, & ex fcala accipiatur .4B, in B fiat x-

qualis obfervato B, & BC abfcindatur, & fic ulterius.

I72. II. Si immediate accedi x on pollit, aut ipfa fi-

|trerte

gura fpatii menfurandi ex remotioribus ftationibus deter-

( vicie

minanda fit, (Eig. 120.) operatio procedit uti fuperius

)] infe

docuimus.

Ad hoc unice fcilicet advertendum eft , ut

?

quzlibet pun&a , aut figna, autanguli obje&orum a du-

abus ftationibus 44, B, videantur, & vel calculo per

obverfatos angulos, & menfurata bafi, vel interfe&tio-

nibus, du&a ab utroque ftationis puncto direione, deter-

n0n

minentur, quod progrediendo circa circum, donec ad

j

idem redeatur punc&um, facile obtinetur, dequo, cum

fuperioribus paragraphis jam egerimus, e€a hic rurfus

repetenda non effe cenfeo.

PROBLEM A IIl

s af

tot

3

175. Ducere lineam rectam. per fylvam. (Fig.

jife

119.)

3[][[113

RESOLUTIO,.

fure

Sitlinea re&ta perfylvam v. g, ab Mad N du-

(cala

cenda, & arbores exícindende; 4a&a ichnographia

|Jocà*

] VEI'

conjungantur /M & JN Enea re&a ; pun&um , ubi

H

lo

3

linea

U)A

Ace i

od

IIÓ

CAPHT

IX,

LY

linea occurrit fylve , ex menfula accuratione fumma

tut &

in plano notetur, atque menfula illuc traasferatur ita ut

qui i

linea 7 m linez in menfula efformate refpondeat, re-

be

gula cum dioptris linee /mapplicata dabit dire&ionem

feicit

verfus JV

& oftendet per eas afpicientiarborem quam-

dat

vis renjovendam; quia fi duo tantum baculi in hac di-

re&ione figantur, linea re&a per totam fylvam conti-

nuari poterit, qua dire&e verfus /V tendat

Acu mag-

netica idem praftari poffe facile patet, cum enim fem-

Fip

per verfusidem puuetum tendat, adeoque eundem gra-

dum dire&is femel.dioptris ver/us JV abícindat, diop-

tis 3 linea re&a defle&ere non poffunt, adeoque di-

re&e cum acu mag)ne-*:ca ab s ad 5 pxocedl & arbores

exíicindi aut baculi fi:

! poterunt.

& ob

COR-O0LLASfT&TWI

tectir

Tentui

174. Hac ratione in qualibet oeconomica, militari aut qua-

vt

cunque alia dimenfione, fylve, lacus, foffe, xdificia, aut alia

fpatia , que permeari

ut accedi non poffunt, in mappam trans-

p

tedi

feruntur, obeundo fcilicet hxc obje&a

atque nonnulla circum-

ferentiarum pun&a, quibus per lineas rectas conjunc&is fpatium

tum a

clauditur, per menfulam aut goniometricum determinando, qux me-

Chart

thodusetiam in montibus , qui vallibus anguftis a fe diftinguuntur,

ebfervari poteft

angul

perpe

PROALEMZA IV

turam

Cin ]

175. Determinare latitudinem fluvii (Pzg. 85.)

Vs QX

RESOLUTIO

bilits

107 q

Eligatur in litore oppofito frutex, arbor, fexum,

las qu

aut quodcunque objcctum ,. & fimul menfüretur bafis 4B

i Ti

In utroque puncto goniometrico accipiatur angulus, &

t of

refolvatur triangulum .A4BC

Si in Z fuerit re&us, in-

$1

Q'qui

noteícit perpendiculum Z

ierit obliquus, pona-

ur ex C demi(la perpendiculapis ad bafin CE, inveni-

tur

NI

APPLICAT. PRINCIP. AD PART. &c. 119

futt

dut

tur ex noto 44, latere 4C, & re&o cathetus CE, a

1

, fe-

qua diflantia bafis a littore Le fübtrahenda eft, ut ha-

beatur vera Ce.

In menfula in 4 & B collocata ab.

lonen

fcinditur pun&um C, unde demiífa perpendicularis in fca-

qoam.

aC die

13 dat diftantiamv CE, & dein fübtra&o eE, vera-m Ge.

"Ollli-

PAOR OOSB LOE M As-V,

Tg-

1 fem-

176. Dimetiri & defignare curíum fluminis.

1 gra

(Fig. 121.)

diop*

R-E SOLTU TÁTI!IoO.

ie (i«

bores

TTendatut catena fecundum curfum fluvii v.g. 4B,

& obfervetur angulus, quem acus magnetica cum di-

re&ione linez 4B facit, reli&ta catena in' terra, menfu-

rentur perpendicula verfus fluvium ia iis locis, ubi cur-

yt quie

vatute fünt, & in chartam trausferantur, 'extreimitates

( alia

perpendicularium curvaturam exhibebunt, abgulus vero

trane

tcurte

dire&ionem linee 45, qui angulus du&a verticali, fi-

gatium

tum acus magnetice reprefentante, transportatorio in

je ne-

charta efformatur. Catena ex B ver(us C tenfa obfervetur

mntüty

angulus, quem acus cum BC facit; menfurate ut prius

perpendiculares, & in chartam trauslate dant curva-

turam, fluvii, angulus vero dire&ionem yerfus C; ex

Cin D, ex D ulterius procedendum , donec totus flu-

94J

vius ex una parte abfolvatur. Si latitudo ubique :xqua-

biliter crefcat, oppofitum litus feparata dimeufione

non eget, fed fufficit, poft viginti, triginta aut cente-

xufil

nas quasvis hexapedas latitudinem metiri ($. ?re.) & eam

: ABv

in mappa defignare; at fi latitudo femper varlet, & li-

J

tus oppofitum a. precedenti multum diíferat , eadeim

m

quoque dimenüo(ex altera parte repetenda erit.

ona*

relil*

uf

H 4

€ Q-

1230

CAPUT-TX.

d

CORBRBOGLLEARIUM.IL

n$

qu.

I77. Menfula quoque hzc dimenfio peragi poteft, applicana

Qul

do ea, quse íuperius dedimus principià.

Ele&a in .4 (tatione

pofitisque circa litus, ubi curvatur? notabiliores funt baculis,

t&í

ducantur ope dioptrarum direciones verfus quemvis baculum,

s

diftantig ab .4 menfurate ex fcala in menfulam transferantur,

T

Mi

punda hec rectis conjuncta curfum usque ad ultimum baculum

defignabunt.

Menfuretur dein diftantia AB ; du&a in menfula

Prog

linea ab ex fcala determinetur, transferatur menfula ad B , ex

tis ]

»

quo punco rurius coliimationes verfus baculos novos fiant, &

plic

meníurate diftantie ex fcala abfcindantur, habebitur continuas

Và.N

tio fluminis; Inde ad C & hinc ufque ad finem procedendum ,

todi

notando ea, quz fuperius de adverfo litore diximus. ,

üc

C-ORUDLLARIYUS 1.

i

178. Hac quoque ratione limitum agrorum , fi valde irre-

at

gulares funt, figüura conftrui & mappa confici

debet, fi fcilicet

tic

accurata dimenfio ceconomica requiratur, methodus ab ea non

ebí

differt, quam fuperius $. I74. tradidíimus, hinc ea hic rurfus

zllon répetimus,

turl

COROLLARIU fHM.

T.

fü$,

k

179. Dum rivi permeabilis curfus depingendus eft, ftatio-

nes hinc inde accipi , & lineis rectis totus quafi claudi poteft ,

unde demiíffa ut prius perpendicula ipfum alveum dabunt; vel

eredi in cuvaturis notabilioribus baculi per dioptras interfecari

fur.

& media rivi linea determinari poterit, cum plerumque latitu&

Jit

dines rivorum per incon&agtem aure temperiem quotidie mu-

t£entur,

line

1

PROBLEMA(VL

180. Determinare fitum & figuram vie inter

agros, aut etiam in civitate, in fylva &c. (Pig.

lem

Iþata,

xzz))

if

RESCOTLUTIO.

:&,

hts

Commodiffime id ruifus przeftatur acu. magnetica &

4

x

|

catena menforia. Sociorum unus cum extremitate cate-

l

nsae

d

g

2

—

»

APPLICAT. PRINCIP. AD PART. &c. tat

na fecundum viam progreditur, quamdiu linea recta

plieat«

durat, v. g. ufque in B ; alter in dire&ione catens cir-

latione

cuium , quo acus clauditur,, eidem ita applicat , ut diop-

&culis,

tre,feu pun&a O & iC in eadem re&a cum catena fint,

eulum,

ànfot ,

acus gradum aliquem in circulo abícindet, qui in ma»

nuali libello fimul cum longitudine catense notetur,

culum

tenfüla

Progredienie primo verfus C, alter ftationem in prio-

j

tis loco 5 fumere debet: acu directioni catenze ap-

nt, &

plicata, gradus iterum, qui diverfus a priore cum cur-

tnuas

va via effe ponitur, ab illa oflendetur, qui cum longi-

dum ,

tudine catene in manuali notetut, ita in C, D, E,

&c operatio fiat. Mappam conüru&urus, in 4 duc re-

&am verticalem, qua fitum acus deliyznet , & conitru-

jttt«

atut angulus obfíervatus, cujus unum latus eft ipfa ver-

cilicet

ticalis, alterum prima vie directio, in quam longitudo

j non

"rfb

ebfervata transferatur, habebitur pun&um B; du&a

rurfus in £ verticali, conítruaturangulus hic obferva-

tus, & in AC notetur ex fcala,menfurata longitudo

BC, hag methodo D, EÉ, &c.poterunt determinari.

fatis

CIOROLLARIUM

I.

Jleü,

;Td

r81. Menfula ih 44 collocata , collimetur verfus B , & men-

fecarl

furata B ex ícala in menfula ahícindatur , translata meníula in

atitu4

Bita collocata, ut ad 4 coliimanti regula fuper AB jaceat, col.

m

limetur verfus C, & menfurata DC ex fcala notetur, & fic ulte-

rius ut jam repetitis vicibns monftravimus.

COROLLARIUM

IT.

IS2. Patet,fi viaacclivis aut declivisfuerit, in mappis aü-

xt,ef

tem tantum horizontzles linez reprafententur, lengitudines menfu-

i.

ratas effe hypothenuías, que'a catheto feu horizontali linea

differunt,

Hinc angulus elevationis, aut depreffüonis fumendus

€ft, ut ex hoc noto angulo v. g. 4 (Fig 123.) cum re&o D &

latere AC innotefcat 4D vera diftantia. Si enim CD quadratum ab

-AC quadrato fubtrahatur, differentia erit .4D quadratum , & ra-

»a&

dix erit 4D ; ($. 45. Trigon.) Pra&ice id ita infütui poteft

ate

(Fig. 124.) fiat «b equalis longitudini 4C', biffeda «b ducatur

H 5

femi-

A?

CAPUT X

I22

'Qute

femicireulus, ex diametri pun&o a applicetur ad peripheriam al-

fecuri

titudo €D

vut fit CD z ad

erit db horizontalis quaxfita

ER

enim adb in femicirculo angulus re&us, proin ad & cd cafheti,

trae

a&c vero hypothenufa

Si dioptre femicirculo inftru&ae fint,

]ic in

angulus in ipfa collimatione obtinetur

Si acclivitas magna non

nfac

fit, poterit in ipfo 4 libella (de qua fequenti caplte) aut alia

f05 ml

ratione altitudo. AB sxqualis altitudini DC obfervari , quod fa-

eile obtinetur faclendo BC horizontalem,

fcii &

lortot

COROLLARIUM

III.

diren

183. Subterranex vix eodem modo determinari poffunt, ve-

rum hoc in geometria fubterraneafeparatim & propriisad id pra-

paratis inftrumentis docetur,

*f

PROBLEMA VII

Suenay

&i$x

€onyu

184. /Edificium dimetiri & illius ichnographi-

teifec

am conficere,

d

tenm

und

RbESUL,UIIO

Nta,

Dropor

Si muri & partes precipue angulis re&is conjun-

öenMï

guntur, tantum opus erit, longitudines metiri, atque ex

v x

fcala in mappam transferre; fi anguli obliqui fint, eo-

rum quoque ratio habenda erit, menfurantur autem , ut

in geometria diximus, produ&o uno latere, ut habea-

fur complementum , quod fubtractum a 180 dat angu-

lum folidum quafitam ; aut produ&o utroque latere ha-

Vifa

betur verticalis , zequalis interiori quzfito. (Fig. 125.)

Si 4 & ac xquales aífumantur, obfervato b, cum c in

triangulo ifofceli illi equalis fit, innotefcit 4, & proiu-

de A quxfitus. Habitis angulis & longitudinibus latitudo

d

quoque murorum determinari debet, qui ad feneftras ,

haC

aut accuratius adhuc & fecurius ad januas menfurantur,

ex (

haru

quoque ,

& feneftratum diverfe latitudi-

fu;

nes & intervalla catena, aut pertica menforia acci-

Qx!

plantur; & in manuale transferantur; idem de fcalis,

P

(

APPLICAT, PRINCIP. AD PART. &c. 123

"

m al

puteis, furnis

aliis partibus notandum ,. quz» omnia

, Ht

fecundum dimeuficnes accepte cx fcala . in. cliartam

let])

transferantur

& in.fua loca cifponantur

Diagonales

fint,

hinc inde ab argulis ad oppofitos angulos duc&tz labo-

i fon

t alia

rem facilitant, & ad commilios errores minores emendan-

d &

dos multum. juvart

Finita bafi aut infimo plano zdi-

ficii ad contignationes fuperiores procedendum,

quin

hortorum etiam adíitorum fitus & figura eadem methodoe

T£*

dimenfione capi & in mappa ædxhuo adjungi poffunt,

pta«

COROLLARIUM

185. Dum zdificia turribus cin&a funt, vuti in antiquis

edificiis & arcibus (Fig. 126.) contingit, centrum turris inveni-

niendum eít, quod methodo ordinaria determinatis tribus pun-

&is v. g. 4, b , c, in ipfa turris circumferentia per lineas rectas

hi

conjun&is, in quarum biffe&ione

perpendiculares. fefe in ein-

teifecantes eriguntur, preftatur

invento hac rat)one centro des

terminatur circulus per tria puncdta a, b, c, tranfiens , qui ro-

tunditaten turris dabit

du&a diaméter & juxta ífcalam determi-

nata, in biffe&ionis pun&o dabit centrum, ex quo circulus in

proportionata menfíura defcribi poffit; ac in f prolongetur, ex d

du

demiffa perpendzcularls, determinabit punca m

& din charta,

ex

ubj müri laterales cum turri conjungendi fint

30«

ut

PROBLEMZA. VII

lac

u-

186f Ichnographiam oppidi, przfidii, aut Ci-

"

vitatis wnücere

J

RESOLUTIO

1-

Eligatur pun&um in foro quodam majore, atque

inde ad omnes irtroitus plotearum , fixis ibidem prius

baculis collimatio fiat

diflantie menfuratze ad chartam

ex fcala transferantur ,

fi

1ili

"7

vl etiam fiat coliimatio ver-

fus ftatuas,

fontes, turres ? qua ex ea ftatione confpi-

ci poüunt, quod pro direc

C&ione & confirmatione ope-

ratio-

[

124

GCAPUT Ix.

enQ

rationis fervit , dum lines ex ftationibus in eodem puna

€to objecti femel determinati coincidunt.

Hec jam

jiqu

nl

pun&a, baculorum fixorum in charta notata, in pla-

tea etiam fignafa relinquantur, ut in reditu iterum in-

Yeniri & dimenfiones comprobari queant; poífunt prz-

terea collimationes ad angulos domorum fieri, que fo-

p

rum fimul & plateam fpe&ant, per quod latitudo pla-

ia en

occut

tearum haberetur, qus dimenfione dein reali verifica-

tur.

Yus

Inde jam menífula ad aliam ftationem per priores

menfuras determinatam transferatur, & ita dirigatur, ut

pun&um in menfula verticaliter pun&o in terra corre-

fpondeat, regula vero cum dioptris ad lineam jam de-

32

terminatam applicata , baculum priorem oftendat; fi

t

Q

regula in menfula aliis punétis prius per menfuram de-

terminatis applicetur , accurate baculi in his locis

qu

mu

erecti confpicientur.

Collimetur rurífus verfus novum

gu:

quoddam pun&um, & diftantia menfuretur, quae novum

ib

quoque in menfüla pun&um correfpondens determina-

bit, & ita confequenter per lineas rectas menfuratas &

ope dioptrarum & fcala in menfulsm translatas proce-

lo

dendum, donec ad quoddam determinatum pun&um per

lt

plures plateas redeatur, & pars, & cum pluribus par-

p

tibus totum claudatur.

ü

COROTLLARIUM TI.

n

bi

187. Ubi latitudines platearum variant, & a linea recta

ti

deviant, fpecialiter id annotandum eft, quod per plures ftatio-

nes praeftatur ;íi neque domus juxta du&um linez re&e zdificate

fü

ft

fint, cutvature earum quoque in confiderationem venire debent,

idem de ftatuis, puteis , aliisque occurrentibus rebus intelligen-

dum.

COROLLARIUM

II.

Y88. Ex hac methodo, que a prioribus traditis principiis

nil differt , deducitur, initium dimenflonis non neceffario in fo-

zo Drimario capiendum eífe, fed ubi commodum fuerit, inquo-

' cuns

APPLICAT. PRINCIP. AD PART. &c. 125

tl p'tma

[

éunque civitatis angulo prima ftatio eligi poterit & inde ad re-

& m

liquas ftationes dimeníio produci; cafus, circumftantie extraordi-

in pb

narig, majus commodum, autíimile quidpiam hec determinant.

um it«

COROLLARIU MM —IIH,

X pte-

189. Nec mirum debet videri cuipiam , acum magneticam ad

Ug f

procurandum menfule debitum fitum . hic non adhiberi, ferramen-

b pli-

ta enim, qus in feneftris, januis, receptaculis &c.

hinc inde

itiica-

occurrunt, facile mutationem in illa inducent, quod in fylvis,

viis, agris &e. timendum non eft.

[!

Jriotej

tur, ut

COROLLARIUM /IV,

Cotté

190, Sifcala major accipiatur , ita ut cujusvis etiam domus

i de

magnitudo, aule interiores, anguli, anfra&us viarum , differen-

ht; fl

ti» latitudinum exigue diftingui queant, nihil horum megligen-

m de«

dum, fed accurata dimenfione in mappam redigendum , nonita

quidem ut fuperius de ichnographia domus diximus, fingulas

locis

murorum latitudines annotando , fed ita , utdomus a domo diftin-

tV Uffl

guatur, & àule interiloris magnitudo appareat, uti conípiceretur

[rnii

gb eo, qui fupra civitatem directe volitaret,

uinae

BCHOLION,

us &

?E'OCG'

I91I. Egregie id preftitit Mathematicus Cafareus, & philo-

fophice direcor facultatis Cl, Nageliug, qui urbis Viennenfis,

m pet

una cum fuburbiis, ichnographiam elaboravit, & cupro incifam

j P&f'

publici ufus fecit; non angulus in platea, non areola exigua im

domo eft, quam non accurate exprefhífet ;

atque in tam vafta

dimenfione, ubi tot platearum varietates & ambag£es moleftias

novas adferunt, mirifice tamen' coherent omnia, & veram ur-

bis nofire inaginem reprzfentant.

Pretium operis is tantum

rea

wfimare poteít, qui ipfe difficultates in dimenfionibus cum men-

(atio"

fula elaborandis expertus eft,

Partem hujus mappa ? in minori

firate

fcala Fig, 127. exhibet.

;ent ,

ligene

r m

-ipïis

nfv

quo

CA-

fl

T5

6

CALIPUT X

fete ul

um

que ti/

ead c

ljlit?l1

g d

CATLTUT X

[fig, X

kiem à

: DE LIBELLATIONE,

b 41

E

inftruan

DEFINITIO

I

tübulo a

tum it

I92. Libellare eft differentias altitudinis objefftd-

tür (Fiz

tum, aut ftationum duatum a centro telluris & aber-

efte fup;

6

tür ver

rationes a linea horizontali determinare.

-

miento .

C O-R'OS L.ARJU.Mi

dini oc

titudin

193. Cum he obje&lnrum diftantie majores, vel minores

effe poffint, atque hocipfo vatii altitudinum anguli diverfis cor-

transfer

difpofit;

re&dionibus fubje&i oriantur, patet libellationum quoque metho-

dum diverfam effe, illam ubi diftantia major J4Majorem ; altéram

(rrigen,

Minorem appellant, in qua objecorum diftantia 300 pedes nort

ilt!inque

excedat.

Cum illa nonnifi in o^fervationibus aftronomicis, dum

n (,

v. £. £radus meridiani obfervatur, occurrat, .hrec autem quotidias

fanti ,

ni ut ita dicam, ufus fit, dum aquse vel adlocum in ufus civi-

hotiey

les aut militares: vela loco ad pericula & inundationes evitans

das, derivande funtj de hac tantum minore agemus, in qua

mnec refraá&io aut alia corre&io obfervationes angulorum ingre-

ditur,

I

Qit, c

lunadg

DEFINITIO I.

0 inff

Üiettem

194. .Libella eft inftrumentum , quo differentias al-

diriti,

titudinum examinamus, id eft, quo ad libellandum

utimur.,

patiar !

facile ;

-SCHOLION »

&in 1

Mnnihi

[95. Vario modo libelle conftruunturs fimplicifima eft,

Wi ap

æ*s bul

dum tubo orichaleino .4B ( Fig. 128,) in extremitate utraque

4 & B ad angulos re&os inclinato tu^i vitrej cd , af inferuntu£,

u

"J;þati

& glutine,firmamtur, neaqua in'ufa effluat, hxc aqua ex legibus

hydroftatice, licet inftrumentum fitum verticalem non habeat,

T

fefe

—

DE LIBELLATIONE

127

fefe ad libellam componet, ita ut altitudo aque femper in utro-

que tubo zqualiter a centro telluris diftet, qu: igitur objecta in

eadem cum fuperficie aque hinc & inde conípiciuntur, eandem

habent altitudinem

cum autem collimatio per fuperficiem aque-

am diífcilior

fit

tubulo vitreo

dioptra in annulo firmate

(Fig. 129-) adjungi poffunt, qua furfum & deorfum ad fuper-

ficiem aqueam usque promoveantur, una pars prominens foramel-

fo5^a4

d, altera filis b, e, ad angulos rectos fefe interfecantibus

inflruatur ita, ut per foramellum afpiciens in dioptra, oppofito

tubulo applicata, fila ex eadem parte videat.

Examen dioptra-

rum ita inftitüitur? in extremitate linee meníurate 4 & B fivane

ieftoe

ur (Fiz. 129.)baculi, in medio vero C,quod accurate menfuratum

eífe fupponitur, ftatuatur inftrrumentum, Collineetur verfus baculum

; aDet

.4, & notetur punctum vifum A4M; inverfo inftrumento collinee-

tur verfus baculum B , & notetur pun&um N,

'Translato: inftru-

mento in .4, notetur baculo pun&um P

quod eft equale altitu-

dini oculi, feu foramello in dioptrà, examinetur differentia al-

titudinum .MA, & PA —— MP, & hec differentia ex N ad F

ftiores

$ cot«

transferatur ut fit MP — Np; debebit per dioptras, fi fila rite

metho»

difpofita fint, pun&um p videri, fecus fila in utraque dioptra

corrigenda erunt, ut pun&um p videatur; ob angulos enim ino

[téram

utrinque zquales, & latera MM & :o zqualia, cum eadem altitu-

lég not

dine Co, erit pun&orum JM & m eadem a horizontali linea di-

; di

oodias

ftantia, proin Mm honzontahs Hnea, & illi parallela Pp etiam

horizontalis.

us civie

evitané

SCHOLION

II

un-qua

| g^

196. Ufitatior libella eft, que aqu» bullam aeteatm fubfti-

tuit, cujusjam fuperius $. I2. defcriptionem d

dxmus cui hoc fo-

lum addendum eft, quod pro libellatione (Fiz. 130.) dloptms & ful-

cro inftru&a effe debeat; unius faltem pedis fit, bulla aeris me-

dietatem femper tubi fuperet, latitudo vero interior dimidii fit

ns il

digiti.

Interne bene polita fit

ut bulla in motu affri&um non

ndum

patiatur

quod ex motu zquabili bulle tubulo nonnihil elevato

facile advertitur ; tubulus vitreus annulis aurichalcinis includitur,

& in regula firmatur, ita tamen , ut dum rectificanda eft libella,

nonnihil attolli, vel deprimi poffüit, quod cochlea uni extremi-

tati applicata facile preftatur

Ex legibus aerzometrie certum

j Cü)

ef, bullam aeream femper fummitatem petere, & liquorem ut-

pote graviorem aere ad infima defcendere

traque

linc in zqualibus

gntuf ,

inclinationibus bulla zequalem locum in tubulo occupare, in hori-

eülbus

zontali vero fitu in medio hgrere debet,

$i igitur planum , cui

abeaf n

im-

[dí

aj

128

CAPUT X.

imponitur, nonnihil fit inclinatum bulla verfus partem aitiorem

quibu

funt; J

afcendet, ibique fubfiftet, inverfa libella ita, ut pars inferior

jam locum fuperiorem obtineat, rurfus in eadem a puné&o me-

fin

(V $

h3

dio diftantia, fi libella jufta fuerit, herebit. Examen ab eo

diir f

Pro libellatione

quod precedenti fcholio dedimus, non differt,

(i.13

libella longiori regulae aut orichalcine aut lignez aut ferrez in

ttpoftt,

extremitate utraque dioptris iníftru&w (Frg. I31.) imponitur,

qieat, |

& tota machina in fulcro ita firmatur, ut ope cochlee & rotze

fflat, 1

dentatm horizontaliter pofitz horizonta'iter quoque circumagi,

liter o

& ope alterius cochlez verticaliter erectw fua cochlea inftructe

verticalÓter quoque in exigua licet diftantia moveri queat, Si

ftationes longiores eligrantur, regule loco tubus, cui libella im-

ponitur, adhiberi poteft, (Fig. 132.) ubi tamen collimationes

l

xeciproce moleftiores funt, cum fempertubus inverti & fitus ho-

£a fur

rizontalis rurfus eidem procurari debeat, ut hinc & inde objes

10 Ctm

étum capi poffit.

Remedium huic attulit laudatus jam .fepiu$

2ontalit

Liesganigius, dum tubum quoque reciprocum , in utraque extre-

Qojeli :

qinitate obje&ivo inftru&um adhibuit, cui tubulus vitrum oculare

continens ex utraque parte citra motum tubi inferi poteft 5. ve-

ruüm hec ad libellationem majorem pertinent, quam aftronomis re-

linquimus,quzeque in Cl. Scherfferi curfu mathematico in Parte II,

geometrie defcripta & demonítratalegi poterunt, fi quis accura-

2

tione fumma, ut fecundorum etiam errorem emendet, opus ha.

Qiffete

beat.

SCHOLION Il

197. Goniometricum etiam in libellatione fervire poteft ,

cum enim horizontalis cum verticali, ad centrum telluris ducta,

|

femper angulum rectum facere debeat, íi goniometrico vertica-

bell

liter ere&o perpendiculum in centro fufpenfum accurate nonage-

tione

fimum eradum abfcindat, diameter & proinde dioptre ibidem

lendo,

afixe fitum horizontalem habebunt.

At cavendum ef, ne ip-

T n

fum filum inftrumentum tangat, neque etiam multum ab eo di-

ftet, iUud enim liberum ejus motum impediret, aliud facilem

puu&

errorem obfervationi gradus nonnazefimi induceret, hinc radi

&ltitüd

tantum de^et à filo, ut ofcillatio adhuc libera maneat.

Ut mo-

iledioy

qus perpendiculi facile fiftatur, vafi aqua repleto immitti poteft,

e!fe P

S'UHUOTLION

IV,

m E

398. Pro ipía libellatione adeffe debent duz perticz , figu-

BFyt

rP quadrangularis, YO aut I2 pedes longe, uni lateri divifio-

num partes, (Fig. 134) pedes & digiti fcilicet, inciduntur, in

qui*

Mg

DE LIBELLATIONE

129

altiorem

quibus afferculi rectangulare$ fürfüm & deorfum promoveri pofe

inferios

n&9 m

funt; in afferculis crux aut circellus (Fig. 135) efformatur, di«

ftn&o ab eo , quem aífferculus habet, colore; ut motus ifte fa-

n ab eg

ellatione

cilius fiat, pertice a tergo ita imittuntur , ut inter ipfas crenas

(Fiz. 136) aliud Hgnum , afferc ulum gerens, afcendere & defcende-

erreg in

ponitur ,

re pofit, quod manubrio armatur, ut ad debitam altitudinem attolli

queat, fimul autem cochleam gerit, ut inomni locofirmiter pere

& tota

cumagi,

fftat. 'Infma parspertice ferro munitur, üt in terram verticas

liter ufque ad divifionis initium infgi pofüt

wirudte

eat, Si

SCHOLION V.

yella ime

matioref

fitüs ho

199, Poteft etiam in nonnullis cafibus ea libella applicari, des

qua fuperius $. 13 mentionem fecimus , que extriangulo equilate-

de 0bje*

| fapiuf

.fo cum perpendiculo conftat, cujus ope pertica tertia (Fig.gg.) horis

zontaliter poni, & differentia vel utriusque pertice vel alterius

p extteé-

obje&i immediate examinari poterit,

. geulare

, ve

omis re*

PROBLEMA L

parte IT,

accun.

200. Libellate terminorum 44 & F^ altitudiüts

opur Iie

differentiam fupra horizontem. (Fig. 147.)

RESOLUTIO,

: potet,

s dida,

Perticis in 44 & FF verticaliter ere&is ponatur li-

bella in medio c, & verfus utramque partem collima-

vertici

nonage"

tione infüituta a fociis tabellze cum fignis ita vel attol.

ibidem

lendo, vel deprimendo difponantur, utaccurate in filo-

, i

; e di*

rum interfectione appareant, annotetur, in quo divifionis

pundo E & D fint; ab 4E fubtrahatur DF, erit AG

facílent

altitudinum differentia quafita, cum enim ob ditantiam

Ur mo*

mediocrem ED, & .4B, horizontales & parallelo

Poteüo

elle ponantur, erit £4 — DB, & G4A— FB & DF

— EG , proin 4E— EG, feu DB — BFze G feu

B F altitudinis differentia,

z ügu"

divifo*

ts j

JMetzb, Math, P Iii.

k

& 0,

qui

:O ALE X.

130

eofer

COROLLARIUM.

ag !

vedilnis

$ol. Si diü"antia longior fit, facile patet, eandem opera-

Bonem íepius inftituendam effe, & differentias altitudinum inter

D abl

diverfas perticasita combinandas effe, ut ad extremas duas,qua-

qum differentia indagatur, referantur, Sit altitudinum differentia

$

inter ftationes 4 & B, (Fig. 138.) feu differentia inter B & M qux-

[2

It

yenda.

Pofita libella in I & ere&is perticis in.4 & C quzratur

?

per libellam differentia inter .4D & CE — 44; igitur in .4 &

C eft

DA — EC hoc e& DA — Da z 4a creícens.

'Fransla-

Y

?

ta pertica ex A in F & libella in 2, quzratur differentia inter

CG & FH — xF-cyzzab fed decrefcens, cum profundius fit R

2!

quam C, igitur relate ad priorem eft 4a — ab —— Ab, "Tranf-

làta pertica ex F'in Z, &libella ex 2 in 3, queratur differen-

tia inter KF & LI-—U&-—18-bc,

fed creícens, proin ad

priorem .4b addenda, & habetur 4c differentia altitudinum, Tan-

dem pertica Z ad M & libella ad 4 translata , quaratur inter NI,

& Mo diferentia, habebitur ea NI — Mo--NI— Nrzcrl&

rI] — MQ — dc crefcens , igitur fi ad priorem, .4c addatur cd

232:

erit Ad vera altitudinum differentia quaefita, eft enim ob paral-

lelas Ad — M B.

$i altitudines crefcentes per -]-, decrefcentes

per—— exprimantur, poterit obfervationibus pera&is reducio in

fne fieri, & vera differentia detegi,

P

Sit 4D obfervatum

IO,

EC 9. erit a

84

p-—-

m

3——-

Vs ;

^ $Sit GC .

6, FH z 9, erit ab

&i d

Sic KF

4s; LIcs $, ep b mX

det

Sit NZ

nuut

8, Moc- , eriü cd — 6 el-

pro Q

16— 3 * 13

diftar

eroim differentia altitudinum inter 4 & M — 13 pedibus.

Xime

portio

ferenti

Ka

COROLLARIUM.

diftant

209. Si libellatio accuratior petatur; ita ut linearum etiam

vetam

partes in confiderationem venire deberent, correciones, de

quibus fuperius mentionem fecimus, adhibendz effent, vel libel-

4

la ita femper difponenda, utaccurate in medio inter utramque

10j

perticam ($. 197.)ftatuatur, tunc enim nullum nec ab inítru-

Vto !

mento nec a refra&ione nec aliunde erroris periculum erit, ex

fere n

utraque enim parte, íi quis error fuerit, equaliter obfervatio-

hr

pnem afficiet, proinde altitudo zqualiter ex utraque parte corri-

£enda foret, vera igitur differentia hac ratione etiam erronea

ob-

DE LIBELLATIONE.

13*

ebfervatione obtinetur.

$i in medio ftatio accipi non poffet

utraque feperatim accurate menfuranda foret, & ex tabula cor-

] Operte

ve&ionis altitudo . competens diftantie fubtrahendaz partem hu»

m inter

jus tabule ex Cl.

Scherffero hic exhibemus.

45, qu-

fetentia

B-

[v

D]

:

$9,94. hexap.

ctaeecus

! M Qg«

124,

Weratür

152,

n4&

176,

Tosslae

197;

ja inter

216,

us t R

230,

Tranf-

249,

lifferene

264,

1in ed

279,

? Jan-

292,

iter M,

895,

digit O lin.

—u&

318,

ar d

336

3415

N OR "- ^4 Q O ^4 Q 09 H U0 ^A Co. 60

I14E1111122 3111

LIT11112T111TET341

EEITITIITS ETE TL E]

3 &c &c

) paral«

Centes

idig 1n

SCHOLIOSN.

203. Ufus Tabule fequens eft: longitudo feu diftantia li«

L

belle ab obje&o in prima columna ponitur, cui competens corre-

;

&io ab altitudine obfervata fubtrahenda inaltera columna refpon-

-

det.

Si pro intermediis diftantiis corre&ionem habhere velis v. g.

du

pro diftantia 325 hexap. cum elevationes crefcant ut quadrata&

E

difauntiarum, fiat inter differentias quadratorum diftantie proe

xime majoris, & minoris, & inter differentias elevationis proe

portio, fcilicet ut differentia quadratorum diftantiarum , ad dif-

ferentiam elevationis, ita differentia quadratorum minoris & mez

difantiz ad elevationem correfpondentem , que addita priori dat

vetam correctionem , ut fequens exemplum docet,

etiam

o2

.

M

$, de

339»

*

— 8318, 1^

p 3252 ERN 318,;-2 :x

libe"

10903284 — 10118761: I0"'2—10552$—10118761: 7'^ fere

umque

lin. 7d ees

eadem

m

ergo 325 dant corre&ionem fubtrahendam g96e 1

fere methodo, qua in Logarithmis partes proportionales quafivi»

[, 0X

sfaïiO'

TOUS,

corrie

toneà

—— Ó

"

[x]

SCHO-

v

132

CAPAT X

$CHOLION m

204. Generaliter notari debent in libellatione fequentia,

Cum tantum differentie altitudinum accipiantur,

non erit opus

accuratam pundi aut crucis in afferculo efforma

tz altitudi

nem

habere,fed fi utrumque equaliter elaboratum fuerit,

proinde di.

)

ftantie punci medii a limbo inferiori in utroque eadem fit, fuf-

ficiet eam divifionis partem notare , quam limbus inferior indicat,

& differentia vera habebitur,

Pertics verticaliter erigantur , im

quo .commiffus error facile detegitur, fi mota nonnihil pertica

fignum fupra filum dioptre, ubi obfervator conftitutus eft » afcen-

dat, $Si enim verticalis eft pertica, fignum refpe&u fili diops

&C

11

trarum maximam quam habere poteft, altitudinem obtinet, pro«

in filum dioptre ad circulum, qui per motum pertice efficere-

p

q

tur, tangens eft, igitur nunquam fupra illud afcendere poterit,

Dum fiuuvii declivitas libella examinatur, nonnunquam neceífa-

rium eít, ut rips quoque altitudo varia h'abeatur, hzc igitur fe-

paratim menfuranda, & differentie in utraque fluvii parte anno-

tande funt.

Dum unius tantum obje&i differentia a loco ftatio-

nis queritur, idlibella ordinaria preftatur, quod fequens proble-

P

ma refolvit,

li

^

PROBLEMA I.

.205. Determinare differentias altitudinis libel-

la ordinaria cum perpendiculo.

RESOLUTIO

^

P

Pofita (Fig. 1 39.) in 4 pertica verticaliter, ap-

plicetur ad illam altera ex B horizontaliter , quod ope

iel

libelle examinatur, erit A4E altitudo petita, ob hori.

zontales enim parallelas 4D & EB et AE BD , fed

AE eft altitudo quxfita, ergo etiam BD; divifiones

M

——

in pertica AE notatz in hexapedis, pedibus & digitis di-

menfionem dabunt.

22b

Si

DE LIBELLA TIONE.

133

Si declivitas longior fit, quam ut una pertica ab 4ad B

fequentia,

pertingat, fepius eadem dimenfio repeti poteft, & in-

L erit. oput

vente altitudines partiales in unam fummam colligen-

dümdinem

de, ut Fir. I4O exhibet. Hzc methodus caltellatio dici

ptoitde die

N

folet.

em fit, fut-

t indicat,

Wwanttr , fr

SCHOLION.,

h] perüm

eft , afcen«

206. Pote& etiam libella ad inveniendas altitudines feü

accefübiles, feu illas, que inacceffbiles funt, adeoque repeti-

1 &i diope

Xinet, Dr0«

tis ftationibus locum non prebent, exhiberi, que ex principiis

proportionum refolvuntur ut fequentia problemata oftendunt , un-

t& efücetee

de applicatio ad alios cafus petenda,

ote poterit,

un neceffas

ecigitur fe-

PR;OBLEM6SA. —liI

parte anpo-

looo ftatio«

207. Invenire altitudinem acceffibilem v. g.

jens ptobles

muri cujusdam, zdificii &c ope libellae (Fig. r41.)

RESOLüTIO.

Pofita libella in 4 fecundz dioptrz applicetur. ver-

inis [ibel-

ticaliter baculus, & notetur pun&um e, per quod radi-

us vifualis, applicato oculo prime dioptre in 5, ver-

fus altitudinem / tranfit.

Menfuretur ex b demiffo per-

pendiculo ad 4 diftantia «B, qua eft sequalis ?d, erunt

riangula- )ce, bdf fimilia, ergo erit )c : ce z Ld: df,

litet, ap*

fed bc eft longitudo libelle, ce altitudo in baculo ob-

quüd ope

fervata, bd diítantia menfurata, eryo invenitur df quar^

ob hori*

tus terminus, ad quem additur altitudo perpendiculi

BD, fti

ha z dB,ut habeatur totaf B. Sit longitudo libelle — 2,

qivifiones

altitudo ce z— 5, dillantia a& z 12,perpendiculum a —

4, erit 2: 32 12 : I ergo tota CB — 19 -- 4

'—220

Si

T3

PRO-

134

CNPUT-'X

fsin

PROBLEMA IV.

le d

dinem

H

208. Invenire altitudinem inadcefübüem C5

:EF:

ope libellze, (Fig. 142.)

N

RESOZLUTIU

Pofita libella in 4 collimetur perg àad C, & note-

ipitüt

tur altitudo gf/, & pun&um .4 correfpondens pun&o D,

o0^

translata libella in eadem recà ad A, collimetur per k

273

ad C, notetur &j & pun&um / in terrà correfpondens

puncto A; menfuretur 4[.

Erit triangulum Dfr fimi-

2

le triangulo DEC, & triangulum. At

fimile triangulo

&bferva

heC ,: quee duas fequentes dant analogias

ia

$ adve

D. f6

DE: EC

damus

hi

.

ik

d

hE: EC

1atis

dmaxim

etlam.

Sit longitudo libelle Df & i za; fG eb,.Krize;

Dh que eft qualis 4 menfurate zz d, AE —y, DE

& py

æd-i—'y, EC:: X.

I

"enta : PDzdd-y:«

[

b

d*h3/1 E

x

g

[

a

c —

—

J

x feuc?a4

x:y

C :b

r

qnmcmcem

led uä

erit ex vquo perturbato

dely : 5,

À d

fa&a zquatione erit cy — ib

dum |

bd

ia i d

zi

debet.

(-(:--b)y.—

que in

bd

triatoy

) —

DZ Y

c— b

€9 foj

€ei

eri

Invento y habetur x ex fecunda analogia

Enm(

[

CL

Mu

,bd

-

xX

e

;

e—b

fer"m

Tgitur multiplicetur diftantia menfurata per altitudinem

W 2

b ilq

minoremin baculo acceptam, & dividatur per differen-

a

tiam altitudinum in baculis obfervatarum , habetur ba-

kh()—

fis

RS.

DE LIBELLATIONE

135

fis incognita y, fiat dein proportio, ut longitudo libel-

le ad altitudinem majorem ita bafis inventa y ad altitu-

1 CB

dinem D E, ad quam addita altitudo perpcndh.t,larm DA

zz Eb dat totam BC quafitam.

Sit 822,.b

in.tu4. d.....SO, DA4zz35,

X 50

2

erit y

s T5

4,...... o

| Wt

& D.

igitur :

: 4-150: 300 z EC, ergo CB 400 -- 5,

393

per k

»ndens

COROLLARIUM I

7 fini

209, Rationem quandam effe inter altitudines in baculig

angu

ebfervatas & inter diftantias libelle ab objecto facile detegitur,

ft advertamus, quod,

quo

pPropius ad locum cum libella acce-

damus, eo aititudines in baculo obfervande fiant majores, di-

flantie vero minores, itaut fiad E perveniretur, altitudines fint

maxime & diftantie nulle, hine facile concludendum, diftantias

v;-c

etiam intermedias fuis relatw;s baculorum altitudinibus recipto-

ee proportionales eife,

DE

COROLLARIUM II

210. Dum in eodem horizonte libelle poni non poffunt ,

fed una ftatio altior altera fit, (Fiz,143.) ea que de pruno triangulo

Àhki diximus, hic quoque notanda funt; prius fcilicet y invenien-

H

dum 8

ut €o noto inveniatur x, ad quod EB vel /2A addi

debet, ut tota CB habeatur, Ponatur linea ?h ad DC produci us-

que in p, & ex p transferatur longitudo libelle usque in z erit

triangulum pmnzquale triangulo Dgq, ob parallelas zquales pmn &

D4 ex conftru&ione,

& re&os in g &n, ergo erit a priore cafu in

eo folum dlf*erentla, quod ad du*annavn priorem d feu ph ac-

ceíferit Dt ,

& ad altitudinem x feu CF accefferit FE — f, (EB

enim eít vquale altitudini hbel'æ) igitur & D£ & f querendum

eft, ut ex priore formula y inveniatur,

Ef autem f — pt , dif-

ferentia altltudmum, quae ope obfervationis, ut in libellando

(s.

202,) monuimus, habetur, unde ob triangula Dgg, dtp fi-

r

milia eft

um

——

——

dinem

gq: gD um pt : D£,

fferet

.

f

& gg

»qDzcadayptz-f,erit

L ]

b

.

« mf

Ut D

14

Mens

fis

CAPUT.X.

156

Menfuretur difantia ftationum Do, & fubtrahatur prius inventum

ä

——

?

us

G.

Unde in triangulo DEC erit bafis

Dt,. habetur £o —— p

DE :::Df—ä-—fo—-i—-oEm'g— -- d--y & ECzzx--f & erit

*

.

analogia; a

.

.

xX - f.

b:-gf—æ«d—l-'y

$ubfituendo loco.x valoremin priore problemate lnäentum?

& determinatum

——

Á

Cy

[

Lj

erit a: b ——

-- ïg— -&—-f Fa&a æquationa: erit

b

aey

|

—

jf

-.oaf

;- - bd-b5y; reciu&ïs fra&ionibus eft

j

y

-- af

af -- bd -4- by

1

—M

af Iaf -- bd

y

—

—

N

e —b

E

Idem omnino quod prius invenimus. Inde jam *þatet, qu2curn-

Z

que difantia affumatur, y femper per — —— exprimi, In praxi

igitur menfurata ftationum diflantia, & altitudinum differentia

queratur Df, inferendo, ut altitudo baculi primi ad longitudi-

nem libelle ita differentia altitudinum inter flationes ad termi-

Zodie.

2um quartum D£; hoc Dt fubtra&um a diftantia menfurata Do dat

«uantitatem d in formula prius determinata, hinc y erit —— di«

ftantie d du* in altitudinem baculi remotioris b, divifum petr

differentiam altitudinum in baculis obfervatorum.

Exemp, Sit altitudo c obfervata — 7, libella —— 2 , altitu-

dob —4, differentia altitudinum inter flationes — f —— IO, di-

-

305

.

ergo Dt — 50

ftantia fiationum, — 50, crit 4 : 2 — 10 1 ;

f

4

r

Q

X 4

r8o

we b r-45x-4, ptolny —

—

——

:;* GO,

erze

7-—4

3

—

eum z —— 60, & f — IO, erit x --f feu Cg

G

70.

bi

^A

«

;

ä

DE

DE LIBELLATIONE.

157

iüventüm

DE GjOEODESÍTA.

erit bafit

j & erit

DEFINITIOL

210. Geodefía äf,t äifæributio agrorum in partes pe-

titas.

dyenturtt ,

CORdLLARIUM.

oni erit

2II. Neceffe igitur eft, ut tota prius dimenfio , & delineatio

a£ri habeatur, ut quantitas ejus determinari & per d

lvifionem.

inibus d

indagari poffit, quantum cuivis parti obveniat,

hzc dein divifio

juxta regulas in geometria q. 138. defcriptas i

n charta defigna-

rl & rurfus ex charta in 3grum jipfum transfer

ri debet,

Quan-

titates autem apud nos per jugera defignantur,

4 u

DEFINITIO li.

. [n praxi

212. Jugerum eft portio agri. 1600 hexápedarum

differentia

quadraticarum , dupla ipitur eft jugerti Romani, quod

w,»,gïmdi-

$CO folum hexapedas habebat, & erat fpatium , quod

; ad termi-

a duobus bobus per diem arari poterat,

n Do t

t z

jivifum pet

SCHOLION L

:1, altiti

IT3. In vineis jugerum triplum eft jugeri Romani, igitur

2400 hexapedas habet

f;IO,di

; Cujus partis quartz feu 600, paxs decima

pz9*

feu.60 hexapede Lib

ra VOcari aífolet.

f0 e

SCHOLION IL

214. Dum agros aut fp

atia quecunque intert plutes diü;-i-

buimus,

follicite cavendum eft, ne angulofa nimis & in arc&um

€oeuntia fpatia poffefforibus afgnemus, fed quantum fieri poteft,

re&angularia aut faltem trapezia fint, ut commode arari aut co-

l pofünt.

Dum puteus adeft , autpratum, aut aliud quidpiam,

quod communi ufui eífe debeat, ita divifio inftituenda eft, ut

ad illud 'ücuivis acceflus pateat , proinde unum fingularum par-

aum

pb

155

CAPUT X

b

tium latus ibidem deünat, ne peralienos fundos viam fibi facee

ge cogantur,

gni

la

t

SCHOLIO(N n

qit

AC

quan

er$, Dum colles occurrunt, non fpatium obliquum incll-

fatum , fed horizontale eidem refpondens in quantitate. divifio-

iliqu

nis exprimendum eft, non enim plus capitacclivitas collis, quas

menf

doon

fhypothenufam reprxfentat , quam fpatium horizontale catheti,

fi'obje&a omnia perpendiculariter ad centrum tellurisfeu tan-

betu

euam radii eidem infiftere ponantur, ut fit in ewdificiis, fegeti-

gio,

bus, arboribus, hominibus &c. poterint enim res viciniffimz perpen-

dahi

diculariter in bafi erecte furfum ufque continuari ut Fig. 144. ex-

hibet, & apparebit, nec una quantitate plus fuperius quam inferia

us poni poífe, proin hypothenufa ad cathetum ex angulo, elevae

tionis obfervato, & ex hypothenufa menfurata ut $. I8O. dixiw

fint

1»us, reducenda erit.

Id tamen verum eít, preferendam eífe

Acclivitatem bafi horizontali, dum de arboribus aut aliis, qus

zem

multum excrefcunt, fermo eít, facilius enim ibidem fefe rami

fit,

explicabunt, cum quivis fupra »slium -emineat, proin plus etiam

chart

fruduum ex iisdem fperari, aut fpatium etiam, quod in infima

dti n

obje&orum parte intermedium majus relinquitur,'in alios u«

lia te

fus applicari poterit; ni tamen 'ventus, aut alia incommoda,

patar

quibus loca elevatiora magisexpofita fünt, rurfus damnum mi-

Qut Q

Bentur,

Hinc plures quoque homines inacclivitate poni non pof-

Ybus

funt, quam inferius in plano horizontali fubüfterent , (Fig. 145-)

Tunt

licet ibi facilius fefe «explicarent, ut de arboribus

diximus ;

abl

dum autem de quantitate plani, aut rebus in ea jacentibus,fer-

[A]j

ino eft, negári non poteft, acclivitatem amplius fpatium quam

vefpondentem horizontem dare, longiorem enim . hypothenufam

effe quam Cathetum nemo eft, qui ienoret, (F'g. 146.) Hinc

plures jacere poterunt in acclivitate, plures currusdifponi, quame

in fubje&o horizonte, & fic de aliis.

iui;-

^

l

0pt (

SCHOLION IV.

toto j

*Xàm

$16, Dum fine calculo quantitatem quandam, cujus jam

8quy

fieuram , &perimetrum dimenfam habemus, examinare volumus,

v

Sd pra&ice ita inftitui poteft,

Formetur in eadem charta , in

«qua ichnographia eft, fcala quadratica, aliquot hexapedarum

ejus magnitudinis, ut exciffa fenübile pondus faciat, excindatut

d ean

dein tota figura, reje&is fcilicet omnibus partibu;, quea

non

i

^5

9

DE LIBELLATIONE.

Ó

ibl at«

&on pertinent; in bilance valde àccurata, in qua etiam unius

grani differentia fenfibilis eft, fruftum charte, in qua fcala: &

alia major, in qua tota figura depi&a eíft, ponderetur; infera-

tur, ut pondus primum ad ífecundum, ita quantitas hexapedarum

quadraticarum fcale ad quantitatem hexapedarum totius figure,

Hac ratione totius etiam regionis in mappa geographica depictae

m ll«

quantitas haberi poteft; efformetur fcildicet ex eadem charta ,

e divillo.

aliquot milliarum quadraticarum fcala (quze in omni mappa in

li, qua

menfura longitudinisfemper invenitur, fi huic igitur longitudini,

? Caneti,

duorum, trium aut unius etiam milliaris latitudo adjungatur, ha-

vfeu tat«

betur quantitas quadratica) & ponderetur; exícindatur dein re-

s, fegeti-

£io, cujus quantitas queritur, faca ut prius proportie quafíita

V perpen

dabit milliaria,

ET.

»m infetiw

SCHOLION iv

lg. elevae

217, Cum rarius accidat, ut chartz folia ejusdem ubique

(fo, dixiv

ndam effe

fint craffitiel, adeoque frufta diverfa veram ponderis proportio-

aïïiS; qué

nem non fervent, neque tam accurata bilanx femper ad manum

fefe rami

fit, que exiguas, de quibus hic agitur, differentias indicet,

ylus etiam

chartz plumbea folia fubitituuntur, qus ope machinz, in qua cylin-

dri metallici fuper lamina metallica moti frufta plumbi in fo-

ín. infna

, alios us

lia tenuia & ejusdem ubique craílitiei, & ponderis extendunt,

:ommdda,

parantur,

Facilius enim in iftis exigui etiam ponderis exceffus

aut defe&us advertitur.

Ichnographia tota uni aut pluribus ta-

um n

libus foliis connexis imponitur , & ope ítyli limites in iis effor-

j n pof*

mÍuantur, ut partes fuperflue, & exteriores hac ratione notatoe

Fir. 5

diximus s

&bfcindi poíffint; idem de fcala notandum, reliqua ut prius di-

ibus fere

é&um eít, infüituuntur.

ium quam

SCHOLION VI.

I

ottenufam

73) fiv

9I19, Poterit etiam hac ratione divifio fpatii dimenfi infti-

)ïx-h güam

tui; habito enim toto pondere , cuivis parti competens pondus

ope diviüonis arithmetice determinatur, & abícila interea ex

toto, judicio oculi, parte petita v, g, tertia, quarta &c pondus

examinetur , quod abfcims, aut adje&is partibus ponderi petito

zquale reddi poteít, que dein pars cum (fuis frufillis ichnogra-

cujus j&m

phie impoíita petitam partem defignabit.

6 VOlumIEsi

gt , !

apelarim

gxcin53

tul

g

us

$CHO-

£0*

*

140

CAP. X. DE LIBELLAT.

SCHOLION VIL

219. Si ichnographia tota craticulis quadraticis nt ita di-

cam, claudatur, facilius adhuc ejus quantitas determinabitur ;

Ined

cum ipfa fpatia quadratica ibidem efformata cernantur , quae in

unam fummam colligi poffunt,

ut Flg. 147. exhibet; .conftrue

"m

untur àutem. vel ab hexapeda ad hexapedam, aut a quarta patte

0

fsillaris ad quartam, aut 3 milliari ad milliare pro ratione

|

lre

01 0

iI51T 20

fcilicet fcale & accurationis, que exigitur. Quadratula, que in

201 88

limitibus occurrunt, fubdividenda funt , Ut partium etlam ratio

N

haberi poffit,

u

ia T

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