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 Cl. Georg. Ignat. De Metzburg, In Universitate Viennensi AA. LL. Et Philosophiae Doctoris, Matheseos Professoris Caes. Regii Publici, Et Ordinarii Institutiones Mathematicae In Usum Tironum Conscriptae Metzburg, Georg Ignaz von Em - d Á d a od rl us * at x —— IAMO : E n EuE s. s o d * LA n PA T UMA OMA 9 [] — L V a'.. oism s gt g x P — d - ,&Wuää — a. —€— — —— E a Iu r m - — al j B t m E s A * u ln m : m E 36 E d AE s. iÉ N- — ** TOevEE: i E NV — s " L a N Wi d E^ NEDORC r s STKAS N 2 "ER E N ii W d u — * — d —— J ^ ewiubeis 17 um M ( — z — — C Em Il [ i l p | CN N [ Í ] nta$X— | ss N — a | || I H i l 1 l ] l i r wO ,rb 1 NI u j p ww B Tübingen w 3 s T d Es — CL. GEORG. IGNAT. p& METZBURG., IN UNIVERSITATE VIENNENSI AA. LL. ET PHILLOSOPHUE DOCTORIS, MATHESEOS PROFESSO- RIS CAS$, REGII PUBLICI, ET ORDINARIEI INSTITUTIONES MATIIDLNLATICI IN USUM TIRONUM CONSCRIPTJE. TOMULUS Ill m d NI E . N -—,,J/ 4&,/*17 Mfb As CNSS. t BD cocmu ^A p .;ä -W'Iïü AR | A : | l ï SSSOENS UA i | N «/'»z-.g- P t f ! ! | | PUTEE j XX In. ll u L A. Tymu l i ii A NS N z - MÉS d ETE L]ïäRL L. ET ï FAVO AV v —— m o VIENN.JE, Trris JOAN. 'THOMHE woz. nD£. TRATTNERN, C4ERS.REG. AULZE TYPOGR,. ET BIBLIOEOQL.Z4E. ;MDC(;LXXVIL GN2 ( ( C - c c t m 2$ æxï&Qk&jþgkaU£XJ€mkw4&&*3 Qæ&ä&ä&m f2 TGTE L-JDHEQDEFE DEISIRGI gc-m rmï! G d s iDEEEJDUDEEGF E&JWbümääuDäEErhmm & WWT*W**P&W wrwwamwä - li 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* $64 149 JOTIQ) eMiS iv 5l ff ? e—-fn E B- d * w eA» I Trzg : 2 E ia [——M Q u » — [l l 'Lþd/ e du P A 633 PA v I PAU NI PVAVA Z & D n ut, &ïrdm Il tom ^L ngulony *n eleme H [OLU D IT l x OC ONES ]a. ) äooQä E CA ,ng/ ,a&; JS 0 ?RR ':-i; 73 r r (5 et ] & ux " 35€ [6] f8lfs)(a]s] P g g'ï " 1 1 r UD EJE]'Z];(?M 35 *Iän YT t 22 U:FDQU x* uääü P 3 l f'gä ! S LL G Y J ICUCIR EHBE 5?"*3*0:1& E]EI'S; M ä | DA (ä Nvþ(. , e2c ^ at '*-J)M ) —— :szz::gg:zw& :&QW:&:ä f *s 9j 1d f II BM EIN L A A 107 ä-1 m AAIGONONMIINIAS II AON.US l æ-—ä SPTIUNSORWESRUN — , l$ COAOU T E m , 14j 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 403 4 ,45 4 80 40 p5 41 Ü 6535 1 YÍs) 50 [5 b 80353 ] ^ 00 ! M li & 16l Jle ouec y r QU P 2j iL 1 | j 19 3j 13 OT C u 90 IPBO) 22 00 |4 10 4 53 I^ 9 Pi 7070 4l 312 46 Q00 46 ] 5I j 252 E Re- B ; Redu&io angulorum ad centrum. 1ta di Latus 1OO. hexapedarum. 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I2 02 -R UUX C CA ï ïä 2522 6 ä 9 9833 936 50 20 15,24 1812 : ä 32 2836 2840 31 rP Vr m vt [910 Lal MO Q5 n O7057àÀ'0 O0 2I 57|26 20|3 ä 44 35 739 3 43 54 23 98|28. 9132 51|37 33|42 15|46 56, [o)f O& Otu1 OCY OC OU OO IO 4 O2 m 7 10 O 0ONDO O O WO MX 00 00^4 OWU OO lD m M 24 49|29 46|34 44|39 42|44 40|49 37) OA -U d 2 ä l2fe r2 25 58/8I 9/36 21|4t 33|46 45|51 56 310 4 C300 G Qo ^4 OV iO -- -- C 00SO O 55[32 18|37 41;43 448 27|53 50 QIIE-4 4033 12/38 4444 17|49 49|55 21 ä 9!II 17 1333 81|39 30|45 950 48|56 2 6* C o2 Oouvt f 3334 15|39 5845 40|51 2357 5 43|11 95 ï lOæQXOXKOUlQJUI hMO b2 b2 hO NO bO — b M 0R k M r tO d0 12 RO 0 O Q GO^4 CV 4 00 E O^A OSIO — OxO-DOONOxH CNO210 4 :O CO O OO O ää 4034 O4|40 8]45 SO|5I 34[57. I8. gä ïä 44TTT 28 (;ïä Ununoo O.b5 n £atus 20O. hexapedarum, )ped. o Q2 4 5 6 7 Io 15 30 1 X I 30 I 45 IS 2 30 IO go.|1 o 2 29 2 59 3 29 29 4 58. 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I0| ] io 90 : 2O0|I * 2 19 00 2 $9 9 19 15 o 592 29 3 28 Q O3 r2 4 27 4 57 E 20 3911 3713 16 4 24 3 58 6 32 p i 48|t1 144 5 39 M Z 19 8.4 3 r/ ao 57|i 5213 494 46 6 41 Q3 IQ 8 323 9 33 30 35 6j2 17|4 23|5 29 dERÁHui 7 40 CO P. OW/AQO I2 m EN [* $9] 9 52 10 57 95 s 40 14!2 414 8 36 949|II 3 I2. 17| M0 45 212 3l5 45 9 27|10 48.12 9 13 30 &45 sss 5o 2812 &X x tO FUA UO i C 10 24l5 19 IO IS5|II 42|13 IO 14. 38 80 j 95 2 4|3 4216 49 IO. 57|12 31 I4 5 60 ä 39|3 5716 16 CA O Cn Oo 7 O I9 II 84|13 I4|14 53 16 33 I5 39; 6601 65 R Qo I [9) [ 20 23 12 7|13 51 I5 35 17 19 6c 1 ANS S SKONDCOOON! OVUx 4 Q2 P2 rl 46 I2 34,14 22,16 1O QicoO0o0O0O0 OVtOoc U cowD OO 59 iT I7 57, 101 15 ä 14 II 12 55 | i I4 46|16 37 Ig 27, 151 80 z Iií 17 I3 IO IS 3|I6 56 I8 48: 8ol1 85|1 Ó O05 OV E J* 0502 t9 t9NO NCO 25 I3 19/15 I3 IAZ I9 2 9O|I Cnoxcn n b CA J Q3 m OO Q3 Q32 Q9 G2 G2 dx . 9 CA Un U u p - p .. 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Q2 r 036X H 05 SobRRAGUOUOo D AO CO^À OVU - O2 10 M U l9 OV^3 00 7 7X IO 58 391 n dem: EC 23|10 33|X1 43 : $, 60|I 55 I2 BN o SS| II 9j12 24 651 €An J O2 20 DO 93|Ii 4O|I2 58 Gc| j 7Oo|1 43 46,19 7|12 28 /O 78|1 4|12 907|13 SO [5 80|1 ä 17/12 41]14 6 P 95|1 51 Cenun - t 25|12 50|14 I6 FO PO HO PO BO k o t QNUA C QO F4 OCC v ON E 0 O O2 LO BO BO bOORO RO BO LO PBO BO Bob J . 050500020Q09 ) ho N M E bed obepboy OO COV4» MO 10 - OO 9 7 R2X CO NO NSO OO QNUA 09 QX u1 u u u i. . i . 0309 N) TO i R A. 0505 !9 G 109 4» ROMWO rO O2 r2 CO b2 QO OO r ^3 t2 M^ Oi 0 OU u i -P.»g35Co 1o 60 qQ0 00 00 004 L^ OvOWU - 4 60 I0 19 H Q5 Q3 t3 m CA CO 00 4 00 OO2 A & 0 QOVOONO ^X d& O t v O*O O VO 9N 00 0074 J OUU O IQ m lO MO N o nmUXUX OO2 -t "Ev BR 1 o 0O ONO NO 0000 QVUA.R O2 I9 i M 901 ss 28x12 54[14——20— : ï : r Xj j| FEEE!!!E!SE!!-E R edu&io angulorumad Centrum. Latus SOO. hexapedarum Yo pPed..o 6 o [] UIU 44 4 p 4 44 [ 4 41,.4 4( 44 4 I I2 ] ) IO I9 I8 24 3o OjI 36 42 48 54 à W 24 36 48|i 12/1 g I 48 451 I5 I8 36 54 I Il|i 291 472 n i2 40 63 20 28 47|t IO|I 34]t 5712 21|2 ä 8 3 3i $4 P 25 29 581 27|1 5612 252 5413 ] 52 4 21 93 30 S|I O9iI 442 18/2 53|3 2614 35 1O E omUXOCMO O 108 35 4O0|I 59/2 38,3 1813 574 [* 5 56 m r Mi 40 44|t 122 578 41|4 259 54 € 38 I2 Q5 -b &n G on GX 9930 45 48|t 25/3 14)4 2/4 5215 4 QO ^ m O0 7 17 yiT "l 43 5o 93 I 38:3 31|4 2415 16]6 7. 94 EN 19 2 571 503 45/4 42|5 886 31 8 25 M2 0500 3X OWU -- O0 pQ R [00] 16 32 6 CUnun -b O2 I 5913 594 585 57/6 56 ] 8 56 QuOoo7- OVro -b 65 II 147 I9 9 21 g 17 19 70 1314 7|4 d 297 37| 9 41 E u 5 75 20|4 387 I|.9 58 IÍ.4 !9 27 Q rgo 234 93 H2 m e€ CYOOND à 4617 2|IO IO II !7 Ti 04 ä ä 8 IO I6 P 85 äï ä M od Md 54 J& G Q GA CX G 517 &nQt.. 05 i G o2 IQNO - GXO00 OV^4 C O 11 25 ig9 9 90 ä CO 74 Ui 02 O.U XO O2 01 ^4 OOO Go QO O2 Q3 Q2 OO I9 BO D9 L2 P9 f 27|4 & 4416 5318 MO N2 NO O0 CO gO 04 QU dU - Go IO|YO I9lIE 28 Latus 60O. hexapedarum. TO Iped 3 4 6 Yo lO I5 20 ao 35 40 5o |1D IO IO 20 30 40 501 O|I IO|I 20|I 39 ] 2795 I5 I5 ao 59|I 141 29|t 44|t 5912 28 1342 20 20 39 9|I I8jI 38|1 58/2 18j2 3712 16 49 25 24 48j1 I2|I 712 2512 4913 1413 63 30 29 26|I 552 9 QO 523 203 4914 46 19 38j1 r IIj2 173 5014 29 pn 40 37|t I4|I u o2 273- 414 234 QO t m OS CX m N d ;912 45 4I|I 21j2 423 2 3j4 Do 2416 o 1 441X 28,2 45 I9 10 $8 35 47|i 5643 23/5 $1j6 Go n 4 342 83 : 187 49 16! 0 Sot 3912 184 ORMND 377 ix,, 58 65 $21 4442. 974 55/7 4 39 7O0 A 4317 35|4 I1:8 58 113 ')8 75 55|1 51l2 41|4 o3l8 I3 TE go 56|1 5312 QO OVH C QO m t2 & O OO 4644 3118 24 n 95 571 5412 484 tONO b2 0O r2 QO t2 3748 aI u1 571 952 Cn un P - 4 05 N I f I2 r Q3 Go Q3 Q2 Q9 O0 Q2 Q9 b F2 l2 P2 494 qxoOS IQ QNVUOQ OQWO No OOA MO 19 4 4 U Qni uU Ui -4 L.- OU209 h r gþwwvwmh QYOYO OO OA ut 4 O2 x M OC ^3^405^4 0074 05 CO ^ ^4 74 747 QOv ORUA Ut 3818 G2 Q2 t2 H C - 00.» ^ ONIO UA r QV OO ^ QO KO CL EE N2 NO SO NO0 CO QO CO ^ ^4 O OX - 4 OO I9 h42ï - ;SEBINGELE RIX 11 Reduäio angulorum ad centrum. Latus 7OO. hexapedarum. i? d. $ io o* [4] 4 14 44 41 41 44 44 4 LE l 4 I3 17 AT 26 30 38 42 1O 17 26 34 43 SI|I OT «S8|I 17|I 25 10 I5 I3 25 38 S81|1 4|1 16|I 39 42|1 59|2 j igo 17 33 So|I 711 24lX 41|I 5812 14|2 al 48 20 25 p2 42|I 3a|t 93|1 44 5192 262 46/3 713 28 25 130 25 491 38|2 3|2 2742 52/3 16/3 41|4 30 35 28 56|1 53|2 21|2 4913 1738 454 13j4 42 33 40 g1|I Q35 HO m 373 913 4044 13|4 44|5 I6 40 45 35]1 54/3 28/4 38/5 135 47 4 50 38|1 8|3 P 1|5 3916 16 95 40|I 21|4 225 2/6 42; 2 30 I6o 42|1 4016 22/7 60| $6 ?65 44|1 CnUx J 92 r 566 407 25 TO 461 916 55/7 42 M u7S 48|1 ä 207 87 54 TO. 80 48 I ää ä 277 15 49|1 43/6 317 : $! 985 49|t O5 Q5 Q3 Q3 QD H2 FO LO 6O 6O X U HO VO CX O G o 65 F2 hO bO b9 bO bO bQ bQ B k k uB sO-oUGREBTER tO RO PO PO 4 Qo Qo Go Q9 O2 t9 I2 r2 BO bO bO R o [ l Qu o OuYoo0r OW O Tede d& Q2 Q5 02 02 Q3 Q2 I9 19 ää CACAUX J O2 10 2 Cb OCU H-ä Ggi Ui un u o u g -R 446 3317 ä ä li d5 9o 90l Latus 8ooc. bexapedarum. 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Io 19095 8 3 io t d t 4 dd /i 4l 44 44 44 41 44 44 4 j q IO 16 20 Ij2 IO I3 23 27 30 33 B Io I3 19 26 33 40 46 53 59|i ) I5 92o 3o 40 50 591 9II I9!t 29|t 39 u d I3 26 39 g2!1 SiI I8|I 21I 451 58|2 II 1à 18 25 16 32 48|t 4|1 20|I 371 53|2 912 25 41 ( 6 30 I9 38 971 I6|I 35|t 352 14|2 3|2 5213 II 164 4'21 35 29 44|X 28/1 502 IIj2 33j2 I7|3 89 45 16 40 25 4911 13|I 382 272 9113 A I25 4 45 27 $4l Ol|Y 482 4213 30 96 16 50 29 g M2 281X 572 56|3 253 Q2 O2 m 53 26 4) 95 E 842 I3 p 60 33 3912 u o2 3l 40 93 65 35 44 £JA - 05 O M O2 r NO 46 70 36 48|2 C CO OO CO $r $r $ 37 5I ä ää 39 80 37 212 303 &A -R O2 I9 CVUO IGX 16 1548 4 38 t9 QO t2 OO r CO m 20 Q 9 85 BM u rj m R m o rmÁ m W A o o OV ONAA - E2 O ONOS iM bed gqeb g o M oA Ro 5412 3213 p d M Q c6CA O O0 OV OVUX 19 LO RO hO Mb OVROMOQUSE MU O.O 2CG AO0202 M Ul OO o oU GgUAiU! 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