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4 -A f i 


cy 



77ze Dibner Library 
of the History of 
Science and Technology 

SMITHSONIAN INSTITUTION LIBRARIES 



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c!> 2 

omni Logiftica Mathematica, 
Amplifsi mi ^ Facillimi, & 
expedittfimi explicatio. 

Authore ac Inventore, 

IO ANN E NE PER O, 
Barone Mercbifton ii, 
cyc. Scoto . 





ETAFFfiBV 1{Cj I , 

Ex officina ANDRE* H ART 

'Bibliopola, c I :>♦ d c* xiv, 


























































































































































































































































f’e. flluBrifiimo, & optima fiei 

PRINCIPI CAROLO, 

Potentifsimi, & Inviftifsimi, 

IACOBI D G 


magna? Britannia?, Francia?, <$c 

Hibernia Ppgi*, filio unico,IValliaPrin 

dpi, Duci Eboraci, & Rothefaice, magno Scotice 
Senefcallo ? ac Infularum Domino ? 
r D. D. r D a 

c 

Vum nullum fit fiudium , veldottrina 
genus (illusirifime Princeps) quod ge¬ 
nero fa ac heroica ingenia , adprae lar a 
quaque & fui limia magis acuat 5 con- 
trdque tarda & infulfa pellor a magis 
obtundat , quam mathefis: non miran- 
\ dum ett eruditos & magnanimos prin¬ 
cipes eam magnopere praterit is omni¬ 
busfeculis in deliciis habuiffe , imperitos 
vero & ignavos homines eandem velut ignorantia fua & ignavia 
hoHem y femper odio acerrimo profequutos e ffe. Cur non igitur no- 
vitium hoc no firum inventum, cum obtufa ingenia & humi re- 
pentia refugiat ,ad fublime Celfitudinis tua ingenium&patrocini¬ 
um confugiet & tranfvolabitf Prafertim cum nova hacL&gartth - 

A % morum 




morum methodus, omnem illam pristina Mathefeos in calculo dif¬ 
ficultatem ( qua ahoqui gener ofam tuam indolem offendere poffet) 
pemlm e medio tollat : ad fuhlevandam memoria imbecillita „ 
tem ita fe accommodet , uttllius adminiculo facile fu, plures qua- 
fiones mathematica* unius hora /patio } quam frislind (f commu¬ 
niter recepta formafimum, tangentium, f fecmtium, 

abfolvere-j . ideo que tua Celfitudini tanto gratius hoc in¬ 
ventum for e Jperamus, quanto facilem magis & expeditam reddit 
LogiHicam * quid enim jucundius,& in omni difciplinarum gene¬ 
re pr aft antius e (fe pofit, quam praclara quaque & fublimia , exa- 
Be, ex tempore facili negotio, nulloque vel temporis vel laboris di- 
Jpendio expedirem? Rogamus igitur (ferenifime Princeps)ut mu- 
nufculum hoc ficet exiguum, & longe infia meritorum & dignita - 
tis tuafaftigium,certifimi tamen ohfequiipignus &fymbolum,pro 
humanitate tua boni confulas. Quod fi te feci/fe intellexero,vel hac 
fola ratione animos mihi jam morbis pene confeBo addideris , ad 
alia propediem, his fortafje majora cr tanto principe mage digna 
moliendum. Jnterim illullrifimos tuos parentes magna magna 
Brita nni M' luminar ia, t e quepraciarum tampraclarafir ~ 
pis ramum, & futura notfra tranquUitatis fpem, diu nobis incolu¬ 
mes fervet & protegat fummus ille Rex Regum, & D ominus do- 
minantium , cui omnis honor & gloria in aternum tribuatur. 

Serenifsimse tu# Celfitudinis obfe. 
quio addi&iffimus 


lo annis Neperts, 


f z!M I\ I F I CVM 



‘ Trafatio. 


Vum nihil fit(chariffimi mathematum cultores) 
mathematica? praxi tam moleft^n 3 quodq- Lo- 
giftas magis remoretur,ac retardet, quam mag¬ 
norum numerotu multiplicationes,partitiones, 
quadratpque ac cubicpextra&iones, qua? prp» 
ter prolixitatis tpdium, lubricis etiam erroribus plurimum 
funt obnoxip: Coepi igituranimo revolvere, qua arte certa 
& expedita, poffem di&a impedimenta amoliri. Multis fub- 
inde in hunc finem perpenfis,nonnulla tandem inveni prpcla- 
ra compendia, alibi fortaffe tra&anda: verum inter omnia 
nullum hoc utilius, quod una cum multiplicationibus, par¬ 
titionibus,& radicum extra&ionibus arduis & prolixis,ipfos 
etiam numeros multiplicandos , dividendos, & in radices re- 
folvendos ab opere rejicit, & eorum loco alios fubftituit nu¬ 
meros, qui illorum munere fungantur perfolas additiones,, 
fubftra<5liones,bipartitiones,& tripartitiones. Quod quidem 
arcanum, cum (ut cptera bona) fit, quo communius, eo me¬ 
lius: in publicum mathematicorum ufum propalare libuit. 
Eo itaque libere fruamini fmathefeosftudiofi ) & qua a me 
profedum eft benevolentia, accipite* Valete. 


A 3 Ad 




Ad Ledtore ni Trigonometme 



Q Vi Coeli, atque foli, & finuofos aequoris arcus 
. Metiri, &curvos vis numerare gradus • 

Quique per expanfum tenfas cognofcere redas, 

Men furat radio quas Geod^ta fuo : 

Madeanimo,inftaoperi ,Lo gar ithmis utere,quoshic 
Rara Caledonii, dat Baro, gemma foli. 

Fruftra erit hinc multis tabulas extendere chartis: 

Fruftra erit & calamo crebra litura tuo. 

Qua? nifi multiplici nunquam potuere priores, 

Adu uno hic facili tu numerare queas. 

AdS ophiam quid ferto novi, quod priftina vincat, 
Quifquis ab ingenio nomen habere cupis. 

Patricius Sand^us. 


In Logarithmos D. I. Neperi. 

D Egiomontanvs fertur per aphprefin,atque 
Profthefin ignotos elicuiifefinus: 

Atque aliquos aliquot finuum qua?fita Logiftas 
Confimili legimus notificaffe modo : 

Nemo tamen eundos poterat fic folvere nodos, 

Aut certa redam lege docere viam . 

Mufarum Nep e r v s honos, & gloria gentis 
Scotigena?, parvo proflat utrumque libro. 

Nominefic N E P A R, Par i l i fit & omine Non Par, 
Quumnonhac habeat NEPAR in arte Parem* 

- Qsr 




Aliud • 


DV c h an an e tibi Nipervm adfcifce fodalem, 

^ Floreat & noftris Scotia noftra viris : 

Nam velut ad fummum culmen perdu&aPoefis 
In te ftat,necqud progrediatur habet: 

Sic etiam ad fummum eft culmen perdu&a Mathefis, 
Inque hoc ftat, nec quo progrediatur habet. 


Ad LeBorem. 

<■ r r*. < , I t 

U Ic liber eft minimus, fi fpedes verba, fedufum 
^ Si fpe&es, Lc&or, maximus hic liber eft- 
Difce> fcies parvo tantum debere libello 
Te, quantum magnis mille voluminibus. 


Andreas Ivnivs 

f'hihfojfhirt ProfeJJor m 
Academia Edinburgcna. 








IN LOGARITHMOS. 


Q Va tibi cunque finus , tangentes atque fecantes 
• Prolixo frse jiant 5 atque labore gravi: 

t^dbfque labore gravi , & fubito tibi 7 Candide LeUor % 
Hac Logarithmorumparva tabella dabit . 



,*v. 



MIRIFICI 

Logarithmorum canonis defcri- 

ptio, eiufque ufus in utraque Urigonome - 

tria, ut etiam in omni Logiftka mathematica, 
amplifsimi , facillimi 3 & exfe* 
ditifstmi explicatio, 

LIBER l* ' 





C A P V t 1. 

De Definitionibus. 


In ea aequaliter trefeen dicitur 3 quum funitus 
eam deferibens , * qualibus momentis per aqualia fn~ 
ter valla progreditur» 


€$mio t 


7 


f 


y 

o 


|T 5T i jg 10 ta 

s £ siri 


t* 

*i 9 ib 


Sit pun&usA, a quo ducenda fit linea fluxa alterius 
pun&iqui fit Biluat ergo primo momento B ab A in C. 

1 Sccub- 








fc LIBER I. CAP. I. 

Secundo momento a C in D» Tertio momento a D in E* 
atque ita deinceps in infinitum dtfcribendo lineam A 
CDBF, &c. intervallis A C,C D, D E, E F & ceteris 
deinceps sequalibus, & momentis aequalibus defcriptis*. 
dicetur hsec linea per definitionem fuperius traditam 
«qualiter crefcere. 

Co- 'Unde hoc incremento quantitates a qui^-different es temporibus aqui* 
rol - differentibus produci efb neciffe* 

lari- Vt in fuperiori fchemate unico momento B ab A iti 

um. C, & tribus momentis ab A in E progreflum efh Sic fex 
momentis ab A in H, & odo momentis ab A in K. Sunt 
autem illorum momentorum unius & trium, & horum 
fex & odo differentia? aequales , fcilicet duorum. Sic 
etiam erunt quantitatum illarum A C, & A E a & harum 
AH, & A Kdifterentiae CE, &H K, squales» «qui-dif- 
ferentes ergo, ut fupra. 

^♦Def. Linea proportwnaliter in breviorem decrcfcere dicitur , quumpnn 
Bns eam tranfeurrens aqualibus momentis , fegmenta abfeindit ejufdtm 
continuo rationis ad lineas d quibus abfeinduntur . 


0 0 (2 & & & (2 00000 


cC y 

dMmefoi 


S' 

X 


-i —f-t- 




S K 


* J y t CC &c. 

3 4 / &7 $j)icu.n. 

Exempli gratia. Sit linea finus totius *. « proportio» 
naliter minuenda. Sit pundus traufcurfu fuo eam mi¬ 
nuens 0. Sit denique ratio Tegmentorum Ungulorum ad 
lineas a quibus abfeinduntur, utq.radq.s. Qua ergo 
ratione fecatur q.s* in r, eadem ratione (per r o^TEucl.J 
fecetur et, a in y. atque fic C. tranfeurrens ab a, in y pri¬ 
mo inorner to ab et. a abfeindat y, relida linea feu fi- 
nu y, Ab hac autem y .» procedens 0 fecundo mo- 
mento abfeindat fimile fegmentum quale eft q. r ad q e 
s,quod fit y.^i relido finu A*y. A quo proinde tertio 
momento abfeindat 0 fimili ratione fegmentum As, re¬ 
lido finu t.a A quo fimiliter quarto momento abfein- 
datur (fluxu €) fegmentum s £, relido finu £. «. Ab hoc 
Q, *>, quinto momento abfeindat £ eadem ratione fe- 

gmen- 





IIB. I. CAP, L S 

gmentum £n,relido finu ».<»♦ & ita deinceps in infinitum. 
Dico itaque hic finus totius lineam *♦ » (ex prsemifla,de¬ 
finitione ) proportionaliter decrefcere in finum«.». aue 
in alium quemvis ultimum in quofiftit £.& fic in aliis. 

Cor. Vnde hoc aqualibus momentis decremento 3 ejufidem etiam rationis 
proportionales limas relinqui est necejfie. 

Qt^e enim fuperius eft continua proportio finuuna 
minuendorum ,a a> y dc k q > 3 &c. 

atque fegmentorum ab eis abfciflbrum * y, y A ^ e, 
» 1» 1 3 8 c k k Eadem erit neceffario'etiam 
finuum relidorum proportio,{cilicet,^ *, f «>> e a, £ *>, 

tc0,}ico 3 &t Aa f utex ip.prop.5.& iiiprop.y.Eucl.patet* 

3 -Def. Quantitates fiurda y fieu numero inexplicabiles y mtmeri$ quam proxi¬ 
me definiri dicuntur , numeris majufculu , qui a veris /ardarum 

valonbm unitate non differant , definiuntur* 

Vt fit femi-diameter feu finus totus rationalis nume¬ 
rus 10000000* erit finus 45. graduum radix quadrati 
200000000000000. quas furda feu irrationalis & nume¬ 
ro inexplicabilis eft, atque inter terminos 7071067. mi¬ 
norem, & 7071068, majorem includitur. Ab horum 
itaque utrovis non differt unitate. Surdus igitur finus 
ille 4/. graduum quam proxime dicitur definiri & expli¬ 
cari quum per numeros integros 7071067 vel 7071068. 
negledis fractionibus definitur. In magnis etenim nu¬ 
meris ex fragmentis unitatis fpretis nullus error fenfi- 
bilis emergit. 

4*def. S ynchroni motus funt y qui fimul & eodem tempore fiunt* 

Vt in fuperioribus efto quod B. moveatur ab A in 
C.eodem tempore quo C movetur ab a in y.dicenturre- 
&tx A C, & a y fynchrono motu deferibi* 

5 Def. Quum quolibet motu & tardior & velocior dari pofiit , fequetur nt~ 

6 po- ceffano cuique motui aquivelocem {quem nec tardiorem, nec velociorem 

ftul. definimus) daripojfe . 

6.def. Logarithmus ergo cujufque finus , eft numerus quam proxime defi~ 
niens lineam y qua aqualiter crevit interea dum finus totius linea pro - 
portionahtef in fimum illum decrevit, exiflente utroque motu fynchrono $ 
'atque initio aquiveloce, 

B 2 Exem- 


LIBER I. CAP. I. 


4 


JHoni» 4, 


f 


4 

X 


G 


f 7 

h ± 


8 o 

l£— 


10 11 
K 'N 




£ £ £ £ Te T~T ~7 


Q~&/c. 


ci jr 


g 0 fi $ e g 0 em 

1 


s< 

!• 


= 4 “ 

e 


if ^ ° 0 * 

% $ & 7 8 j?.iq;xliu 


10 


S ^ 

&—&—= 


JU 


Exempli gratia Repetantur ambo fuperiora fchemata 
& moveatur B, femper & ubique eadem feu aquali velo¬ 
citate qua coepit moveri C initio quum eft in a* deinde pri¬ 
mo momento procedat B. ab A in C, & eodem momento 
procedat £ ab a in y* proportionaliter: erit numerus de¬ 
finiens A C logarithmus lineae, feu finus y ». Tum fecun¬ 
do mometo promoveatur B a C in D, & eodem momen¬ 
to promoveatur proportionaliter C a y in /, erit numerus 
definiens AD logarithmus finus *T«. Sic tertio momento 
procedat aequaliter B a D in E , & eodem momento pro¬ 
moveatur proportionaliter C a £ in e, erit numerus defi¬ 
niens A £ logarithmus ipfius finus * &>. Item quarto mo¬ 
mento procedat B in F, & j3 in £> erit numerus A F Loga¬ 
rithmus finus ( «. Atque eodem continuo fervato ordine 
erit (ex difinitione fuperius tradita ) numerus A G loga- 
rithmusifinus »®,AH logarithmus finus / «. A I log. finus 
K«. AKlog, finus\& jtain infinitum. 

Cor. Zlndf finus totius tooooooo* nullum feu o e fi logarithmus :■ <jr per 
eonfcefuensy numerorum majorum finu toto logarithmi funt nihilo mi¬ 
nores* 

Quum enim ex definitione pateat,quod a finu toto de- 
crefcentibus finibus, a nihilo accrefcant logarithmi, ideb 
contra crefcentibus numeris (quos adhuc finus vocamus) 
in finum totum, fcilicetin iooooooq, decrefcant, in o, ftu 
nihilum logarithmi eft necefle. Et per confequens nume¬ 
rorum crefcentium ultra finum totum iooooooo. ( quos 
iecantes aut tangentes, & non amplius finus vocamus) Io- 
ga^thmi erunt minores nihilo# 










LIB. L CAF. IL _ # 

Itaque logmihmos (mmm, qm fem fer majores nihilo funt , ahun~ 
dantes vocamus 3 & hoc figno -j«-, aut nullo praenotamus. Logarithmi 
autem minores nihilo dcfe&ivos vocamus 3 praenotantes cis hecftgm**-** 

<l Admonitio . 

E Rat quidem initio liberum cuilibet fmui , aut quanti- 
tati nullum feu o ? pro logarithmo attribuiffe; fed pro¬ 
flat id prae caeteris finui tori accommodaffe: ne unquam in 
pofterum vel minimam moleftiam parturiret nobis ad¬ 
ditio & fubftra&io ejus logarithmi in omni calculo fre- 
quentiffimi. Cacterum etiam quia finuum & numerorum 
fi-nu toto minorum frequentior eftufus: eorum igitur lo- 
garithmos abundantes ponimus: aliorum vero defe&ivos, 
ctii contra feciffe initio liberum erat. 


C AP* II. 

De Logarithm. propofitionibus* 

P roportionalium numerorum 3 aut quantitatum, aqui-dijfer entes furti 
Logarithmi* 

Vt proportionalium finutrm, fcilicet y ®, qui fe habet 
ad g », ut * to, ad \ Logarithmi refpedive funt nume¬ 
ri definientes AC, AE, AH, & AK, (ut per def. 6 , 
patet.) differunt autem AC, & AE differentia C Ei 
atque AH, & AK differentia H K. Sunt autem ex 
i. def. & fuo corollario C E, & HK aequales:sequi-dif- 
ferentes igitur funt Logarithmi praefatorum finuum pro¬ 
portionalium. Et ita in omnibus proportionalibus. 

Nam quas affectiones & fymptomata Logarithmi ab 
ortu&^enefifuaacquifiverint, eas inpcfterum retineant, 
efi: necefle. Ab ortu autem & in genefi fua imbuun¬ 
tur hac affe&ionc, &feaec lex illis praeferibitur, ut fint 

B 3 sequi- 


£ o o LIB L GAP. II. 

aequi-differcntes, quum eorum finus feu quantitates fint 
proportionales (prputexdcf. Logarithmi 4 & utriufque 
motus patet, & in cbnftruftione logarithmorum amplius 
aliquando patebit,) proportionalium ergo quantitatum 
sequi-differentes funt Logarithmi. 
p ro - Ex trium proportionalium Logarithmis, dapium fecundi feu medii 
poC 2 » tninutum primo,aquatur tertio. 

Quum> per prop. i. differentia logarithmorum primi 
& fecundi squetur differentiae logarithmorum fecundi 3 c 
tertii,id eft,fecudus minutus primo aequetur tertio minus 
fecundo: Ideo addito fecundo ad utrumque aequationis 
latus proveniet bis fecundus feu duplum fecundi minu¬ 
tum primo aequale tertio, quod erat probandum. 

J>r*. Ex trium proportionalium logarithmis , duplum fecundi feu medii 
pof 5« a $Mdtur aggregato extremorum ♦ 

Ex praecedente prop.2. duplum fecundi minutum pri¬ 
mo sequatur tertio. Vtrique aequalium laterum adde pri- 
mum 3 & exurget duplu fecundi aequale primo & tertio, id 
eft,aggregato ex tremorum,quod erat demonftrandum. 
Pro- Ex quatuor proportionalium logarithmis^aggregatum fecundi & tertii 

po£ 4* minutum primo equatur quarto. 

Quum per i .prop. ex quatuor proportionalium loga- 
rithmis, fecundus minutus primo, sequetur quarto minus 
tertio, utrique aequalitatis lateri, adde tertium , & fient 
fecundus & tertius minuti primo ^quales quarto, quod 
erat propofitum. 

Pro- Ex quatuor proportionalium logarithmi< aggregatum mediorum ( fe- 

poC y. eundi, fcilicet,& tertii) aquatur aggregato extremorum, primi iciddi- 
cet } & quarti* 

Per prop* 4. praecedentem, fecundus & tertius minuti 
primo erant aequales quarto: utrique squalitatis lateri 
adde primum, & fiet fecundus» plus tertio aequalis quar¬ 
to, plus primo, quod demonftrandum erat. 

Pro- Ex quatuor continue proportionalium logarithmis triplum alterutrius 

pcf.6» mediorum aquatur aggregato extremi remoti^ dupli vicini. 

Per fecundam prop. duplum fecundi feu medii minu¬ 
tum primo eft aequale tertio: & per tertiam prop.duplum 

hujus, 


1 


LXB. I. C A'P. IXI. # # % 

Ilujus, quod eft, quadruplum fecundi minutum duplo 
primi, aquabitur aggregato fuorum extremorum,videli- 
cet quarto plus fecundo. Xam fi ab utroque sequalitatis 
latere fubduxeris fecundum , fiet triplum iecundi mimi- 
tum duplo primi aquale quarto, hujus rurfus squalita¬ 
tis lateribus adde duplum primi > & exurget triplum fe¬ 
cundi aquale quarto plus primi duplo : qtrod proban¬ 
dum fufcepitnus. 

dmonitio* 

H Vc ufque Iogarithmorum genefm & fymptomata 
explicavimus:quo vero calculo,quave logiftica me¬ 
thodo habeantur, hoc loco explicandum foret. Sed quia 
ipfum canonem integrum , ejufque logarithmos omnes 
cum fuis finibus ad fingulas quadrantis minutias primas 
exhibemus, ideo in tempus magis idoneum do&rinam 
conftru&ionis logarithmorum tranfilientes, ad eorum* 
ufum properamus, ut praelibatis prius ufu,& rei utilitate, 
cetera aut magis placeant pofthac edenda, aut minus 
faltem difpiiceant filentio fepulta- Praeolor enim erudi¬ 
torum de his judicium & cenfuiam, priufquam caetera 
in lucem temere prolata lividorum detrectationi expo¬ 
nantur. 

C A P, I II. 

Defcriptionem compledtens tabula logarithmo- 
rum, & feptem ejus columnarum. 

Se- ITyRima columna eft expreffe arcuum ab o. in 4/. Gra.crefctntium: 
Ctio.l.X & fubmtelligkur ejfte etiam fuorum ad femicir culum reliquorum* 
Se- Septima amem columna eft arcuum a quadrante in 4S* gradum de- 
dio.2. crefcentium : & fuhwtelhgitUY ej]e etiam fuorum ad fcmicircnlum re- 
liquorum . 

S e- ZJnde alterius columna arcus, funt arcuum alterius e regione rejpon- 

£tio*3 . dentium complementa . 

B 4 tAtquc 


UB I. C AP. IIT* 

4 * tsftqtie in prima exprimitur omms trianguli nttilimi reffian» 
gttli angulus acutus minor . 

/* In feprima autem ei e regione co /locatur ejufdem reBanguli an« 

gulas acutus major. 

£■ Infecunda columna funt (Inus aycmmprima columna, 

7. Sunt que hi crus mimis fiub tendens minorem angulum reUan* 

guU, cujus bafis, fen bypotenufa e [i jinus totus. 
t* In fexta columna jum Jinus arcuum fiptima columna, 

f» Sunt que hi cras majus fubtendens majorem angulum ejufdem 
reclanguli, cujus f dhctt hypotenuja e/iJinus totus. 
i 0« Unde omni triangulo reBitineo reBangulo Jit aquiangulum & 
fimde exfim toto y & Jinu fecunda columna, & fim Jexta ei e re¬ 
gione rejpondente* 

st. Tertia columna continet Logmthmos arcuum 4 & finmtm fi* 
nifirorum . 

12* Qui funt Logarithmt proportionis er uris minoris reclanguli ad 
ejufdem hypotcmfmt* 

8S* Itemque hi funt aicuum, & (Inuum dextrorum Logarithmi 
eompU mentorum , quos antilognithmos appellamus* 

14* Quinta columna continet Logarithmos arcuum , & fimum de - 
xtrorum . 

sj* Qui funt Logarithmi proportionis cruris majoris reBanguli ad 
ejufdem bypotenufam . 

16, Itemque hi funt arcuum & JinUttm finifirorum antilogarith* 
mijfeu Logarithmi complementorum* 

#7. Quarta denique fu media columna continet differentias inter 
loganthmos tertia columna , & quinta* Unde duplex ef hac co* 
lamna, abundans & defecHva. 
iS* tAbundames 3 funi differ em ip , qua oriuntur exJub/lraBiont 
logarithmorum quintp a logarithmis tertia . 

/ 7 , TdtfeBiva Vero , funt differentia orta ex fubduBione loga~ 
rithmorum tertig k logarithmis quinta : qua ideo funt mineres 
nihilo* 

20 1 D jfemtia abundantes dicuntur dijferentiaks mmtri } ar€mm 

finifirorum , 

2/. Suntque logmthmi proportionis minoris cruris reBanguli ad 
mfdcm erus maius * 

Itmqm 


LlB. I. CAP, IIL p 

Item que funt Logaritbmi facundorum, (ive tangentium arcuum 
finifirorum. 

Differentia autem defe&ivp dicuntur numeri differentiales ar • 
cuum dextrorum» 

%+. Sunt que Logarithmi proportionis majoris cruris rc&anguli ad 
cjufdem crus minus ♦ 

£/. Itemque funt Logarithmi facundorum, five tangentium arcuum 
dextrorum » 

x6* Omnis etiam arcus (inifi er, ejufjue ad Jemicircutum reliquus 9 
dicitur arcus complementi arcuum , fimum , & Logarithmorum 
dextrorum , atque differentidtum di fcftivmum. 

St contra , omnis arcus dexter , ejufjue ad f mi circulum reli¬ 
quus » dicitur arcus complementi arcuum , fimum , cr Logarith - 
morum finifirorum, atque different talium abundantium» 

- ~ • - ■■, • . . ■■• ■ • r - 

<» Admonitiones. 

M . TT/r notandum esi, fi Logarithmos tertia coftmn* 

JlJL defe&ivcs feceris ( prapojito fcihcet figno ,) fient 
Logarithmi hypotenufarum, jht fec antium arcuum dextrorum fe- 
pirnp columna ♦ 

sp. Et hi etiam fient Logarithmi proportionis hypotemfa reBanguU 
ad ejufdem crus minus. 

go. Et fi Logarithmos quinta columna defebliuos feceris, fient Loga- 

rithmi hypotenufarttm , five feeuntium arcuum finifirorum prima 
columna. 

Fient etiam hi Logarithmi proportionis hypotennfie reUanguli ad 
ejufdem crus ma jus . forum quia ad reBtltneorum fidentiam compa¬ 
randam, fili (inus, torum que arcus, & logarithmi cum dtfferentia- 
hbusv adSpharieorum autem, jpretis fimbus,joli arcus, & eorum 
logarithmi,&differentiales fufficiunt: ideo hypotemfas & facundos 
tabula exclufimus: ficui i & (mus ipfos in Jp hanc is negligi volumus* 
Offendemus tamen obiter te poffe (fi libuerit) as omnibus fatu expe- 
,dite m retlilmeis uti, in Jp hanc is vero minime. 


C 


CAP, 


m 


LIBER I. C A P. IV. 


CAR IV. 

De ufu tabula, & numerorum eius* 

tangentium, £r fecantium pracifein tabulis fu is repertorum f 
d:io I l ^jLogaritbmos non minus praetfe dare. 

Per fe£t i i. & 14. cap. 3« reperto finii dato in fecun¬ 
da, aut feptima columna noftr^ tabulse, reperietur ejus 
Logarithmus in ejufdem lineae tertia'vel quinta columna. 
Habentur igitur ficexa&e finuutn tabulatorum logarith- 
mi* Tangentium autem & fecantium numeris in fuis ta¬ 
bulis repertis habentur arcus. Ex arcubus vero cognitis 
noftra tabula exhibet tangentium logarithmosE u diffe- 
rentiales cum fignis fuis in media columna per fe&. 2 2, & 
25. Et fecantium logarithmos reciproce in tertia & quin¬ 
ta columna, prtepofito tamen his-figno per 2 8, 

& 30. Habentur igitur finuum J tangentium,& fecantium 
tabulatorum logarithmi* 

Exempla finuum, 

S inus 6946584. Logarithmum qusero. Sinum illum 
praecife reperio in fepunda columna refpondentem 
arcui 44. Gr. o. m. & in eadem linea terti# columnas ad~ 
flat illi 3643349. fuus logarithmus, quem quadivi. Item 
finus 7213574, queratur logarithmus, Sinus hic inve¬ 
nietur refpondens'arcui 46, Gr. 10. m, & ei vicinus 
3 266204. logarithmus e/usqu^fitus. 

Exempla, tangentium. 

Q Vseratur tangentis 2186448. logarithmus. Huic 
jtangenti in luatabula refpondetarcus i2 Gr.20.rn. 
& huic arcui in mediacolumnatabul# noftr# refoondet 
logarithmus,feudifferentialisabundans 152030/4^11^- 
litus* Item fi tangentis 4573 6291, logarithmum quadi-* 
veris, offendes in tabula tangentium e;us arcum 77. Gr. 
40. m. hujufque arcus in tabula noftra differentialem ean¬ 
dem, dcfe< 9 :ivam tamen, fcilicet — 1 5 2 03 064. 


E%m- 



LIB. J. GAP* IV* 


u 


BxemfU fecantium» 

S Ecanti 18118009* refpondet in tabula fecantium ar¬ 
tus 5 6 . gr. 3 o. m. & huic arcui in tabula noftra con¬ 
venit reciproce defe&ivus .— 5943212*logarithmus fe- 
cantis 18118009. fuprafcripti* Sic fecantis 131183 37- in¬ 
venies logarithmum-- 2714255. A'fecantis 13960592. 

offendes logarithmum •— 333 6533. 


2. 


Numerorum datorum , & in tabulis firmum 3 tangentium , c tr fe¬ 
cantium non repertorum, logarithmos aflimare. 

Numerum dato fimillimum, five is fuerit dati decu¬ 
plus , centuplus, millecuplus, ioooo pIUS ’ iooooo l ’ IUS) aut 
1 oooooo p!us ’ quaere in fecunda, aut fexta columna tabulae 
noftrae ? aut{imavis,intabulistangentium,aut fecantium: 
& hujus arcum nota, ejus enim logarithmus e tabula no- 
ftraelicitus, eil quem quseris. mentetamen rcfervando, 
aut memoriae gratia notis exprimendo numerum loco¬ 
rum, feu figurarum multiplicitatis. Vt fi quaeratur loga¬ 
rithmus numeri 137.in tabulis non reperti.- reperies in¬ 
terfiniis numeros 14544* 136714. & 1371564. & inter 
tangentes 13705046. inter fecantes vero numerum 
13703048 , quieft omnium datofimillimus, dummodo 
ejus ultima? vel dextimae quinque figurae deleri fubintelli- 
gantur* hujus ergo fecantis 13703048, & fili arcus 43«gr. 
8. m. logarithmus (per pr^ced, aut per fefih 28. & 30. 
cap.3 )quasratur,& invenietur.—31503 3 2.qui pro loga- 
rithmo dati numeri 137. etiam habetur: recordando ta¬ 
men ultimas quinque figuras abfeindendas efle, aut me¬ 
moris gratia exprefle hoc modo fignadas-3150332. 

--00000. Similiter fi per tangentem 13705046. fupe- 

rius expreffum quaefiveris logarithmum numeri t 37. ex 
tangentis illius arcu '5 3-gr. 53.m.invenietur (per fed.2 5 ) 
in media columna-3151790* logarithmus illius tan¬ 

ge- tis 13705046. qui quia excedit 137. datum qnirqi c 

locis feu figuris, ideo--3157790.»— 00000. erit lo- 

garithmnsnumeridati 137. Tanto tamen minus exactus 
e.i hic logarithmus , quanto magis 13705046* eft difii-- 

C 2 mi! is 


\ 







LIBER I. CAP. IV. 
milis numero 13700000 Jlu ceties millecuplo dati.fed hic 
error partes unitatis no exuperat. Si tandeperfinum 
fuprafcriptum 1371 j^.qufffiveris logarithmu dati 137. 
is (per hanc, & n.fedt cap. 3. ) deprehendetur efle 

19866327-oooo* Nec fecus operandum erit figno + 

quando numerus figurarum datae quantitatis excedit nu¬ 
merum figurarum finus ei fimillimi, quod raro contingit* 
ut, fi quaeratur numeri feu diferetse quantitatis 232702. 
logarithmus,invenies in tabula linum 2 3271, ei omnium 
(imillimumded unica deeft huic figura. Hujus ergo loga- 
rithmo tabulato ( per fedh 1 1. cap. 3.) reperto , qui eft 
60631284. adjiciatur unicacyphra figno -}- interpofito, 
&fiet 60631284+0, pro logarithmo numeri 232702. 
quslito* Sed modus logarithmos scidimandi omnium 
optimus ell, quo primo creati funt: de quo alibi. 

3 % ZJtidc, ut [aperior e prima fe Shone logarttbmifimplices,&puri exhi¬ 

bentur: ita hac procedente appofitis cyphris impuri emergunt . 

4» Similium fignorum logarithmos addere , ef aggregatum utrwfquc 
eum figno communi exhibere* 

Vtcx additione —56312, ad— 73493- provenient 
-129807. Itemque addito 4216. ad -f 1392 . produ¬ 
cuntur p6o8. Sic 3219- qo* ad 4360-ooo, faciunt 

757 p -00000. 

J* *Di fi milium fignorum logarithmos addere} efl differentiam eorum 

cum figno majoris numeri exhibere. 

Vt ex additione-210, ad 3 32.producitur ^ 122. 

Item ex additione —■ 210, ad 192. producitur-18., 

Sic — 210, -j-000. ad 332—oo* funt 122^-0* Item 
— 210—000,ad 192-j-00. funt—18—o. 

6* Duorum logayithmorum , hic illius hcfeBuvus jlle autem hujus abun¬ 
dans proprii dicitur: cum & numerum , & cyphras communes fm eof 
dem: figna vero omnia -f- & ■- penitus contraria habeant ♦ 

Vt abundantis 56312. defe&ivus efl-56312,Item 

abundantis 56312*—00. defe&ivus efl-56312-^00* 

Sic abundantis 56312-^00. defe&ivus efl -—56312 





LIB. X. CAR IV 'tjf 

t^b undant em fubftrahere, efl ejm defcBivum addere. 

Vt fubftrahereabundantem 563ia.ex — 73495.16013 
< erit, quod addere illi lis tiefe&ivuni , qui ( per 6 . ) eft 

—- 5^312. ad eundem - 73495. nentque (per 4. 

praemiflam)-12£807* Sic fubftrahere 563 12 -j- 00.ex 

•—73495—ooo.tft idem quod addere—563 i2-|-oo;. 
ad -—7349 5 —000. fiuntque (per 4.& 5. procedentes } 

—— 129807--OOOOO, 

g, Defectivam fubpraherc esi ejus abundantem addere* 

Vt fubftrahere defedtivum-4216 ex + 5392. 0(1 

idem quod addere 4 116. ad 5392* & (per 4-) produ¬ 
cere 9608. Sic idem eft fubftrahere-42164-00. ex 

539 2 + °* quod addere 4216*—00, ad 53924-0* &: 
producere 9608-—o. 

^ Logantbmum numero-tenus augere vel minuere ftdvo valor e fri&i- 

no y eft ad illum addere , aut ab eo fubftrahere quemvis ex logarithmisJe~ 
qiientibtiSyfcilicct 29029842-^0, vel46091684-^-00* ve 1 69077927 
•JjfOOQtVcl 92109969-^0000,. vel 1191192100000, nihil pror- 
fas ftgniftcantibus. 

Vt fit Logarithmus 39 r 5 6- —o. cui fi addideris illo¬ 
rum quemvis , ut exempli gratia, 230258424-0.-fiet 
inde 23064998. major numero, valoie autem prorfus 
idem qui 3915 6 -o. Namque hujus 39156-o. lo¬ 

garithmi, quantitas feuvalor numeralis (per 12, & 13. 
fedt.feq* hujus ) eft 996 0920. a quibus deme uni¬ 
cam figuram ultimam,prout o.notat,& fiet996092. 

Illius autem Logarithmi 23064998. va’or numeralis 
(per feq feft* 12. & 13. hujus ) eft etiam 996092. idem, 
qui prius, 

Exemptum minutionis . 

S it Logarithmus 25.451769. minuendus, i quo fi 

fubduxeris 230258424-o.relinquitur 2425927_o. 

CjUfiem valoriSjCujus prior hic 2 5451769.^01 fimplicis 
'& puri Logarithmi 2425927. valor eft decuplus valoris 

C 3 utriuL 












IIB L CAP, IV* 

utriufvis eorum. Sunt ergo eorum valores invicem ^qua¬ 
les. Nihil enim aliud fignificat additio Logarithmi 
2.3025842 + o, quam quod valor numeri cui additur, 
(ic decupartiendus, & huic decima? parti cyphra unica (it 
adjicienda: fubftractio vero e/ufdena fignificat valorem ' 
logarithmi a quo fubftrahitur decuplari, & ab hoc decu¬ 
plo cyphram unicam abjici, remanet itaque in utraque 
priftinus valor. Sic 46051684+ °°* additus fignificat a J 
centefimam partem valoris duas cyphras adjici: & fub- 
ftradhis, quod a centuplo dua? cyphra? rejiciantur; & fic 
de reliquis fupra exprefiis. 

Si itaque ad logarithmum minutum aliquot cypbrisaddideris, aut k 
hgarttbmo autto cyphris [ubfflraxcris ait quem ex logarithmu fuprafcri- 
ptu totidem cyphrarum, producetur ex impuro iogarithmnspurus ejuf- 
dem valoris « 

Vt in fuperfore primo exemplo fit logarithmus impu¬ 
rus 391 56 o*purgandus a cyphra fua& figno,ad¬ 
de ergo illi 23025842-1-0, fiet inde, ut fupra 23064998. 
logarithmus purus priftini valoris. Sic a logarithmo 
63 584468 + 00 impuro fi fubftraxeris 46051684+00, 
totidem fcilicet cyphrarum , relinquetur logarithmus 
17532784. purus 3 & ejufdem valoris, cujus prior ille 
impurus. 

Si ad logaritbmnm numero defeblivum addideris y aliquem ex ftpra- 
dtBis logarii hmis mna feHionis numero majorem , proveniet logaritb - 
mus ejufdem valoris numero abundans • 

Vt ad logarithmum —28595270 — 0000. adde 
quemvis ex numeris nona? fedfionis numero majorem. 
v« g. 46051684+00. & fiet inde 17456414—00. ejuf¬ 
dem valoris, & numero abundans. 

Looarithmorum in tabula noHra mmero-tenus inventorum finus 
tangentes, Jecantcs feti numerales valores quofcunque exhibere poteris, 
percap , 3. feti, it. 14*22* 2j. 28 . jo*fiye fnt puri , five impuri* 

Vt logarithmo 36. graduum &40, minutorum , qtii 
efl 5155724. in tertia columna, refpondet fuusfinus 
5971586. infecunda: &ejus defe&ivo-5155724. re¬ 

fpondet in tabula fecantium 16745970. fecans 53.gr, 



LIB. I. CAP. IV i$ 

20. m. Item logarithmo difterentiali 29507^4410quar¬ 
ta columna, refpondet tangens (in fua tabula) 7444724, 
& ejus defedtivo -— 2950794. refpondet tangens 
13432331. graduum fcilicet 53* & 20, minut. Si'c loga¬ 
rithmi 2204930. in quinta columna numeralis valor eft 
in fexta columna 802123 2. finus fcilicet gr.53.dd 20. m. 
&ejufdem defeftivi fcilicet— 2204930. numeralis va¬ 
lor eft fecans 124 66913* conveniens gradibus 36. & 
40. min. 

Exemplum impurorum, 

S it logarithmi impuri. 97796—o, inquirendus valor. 
huic numero-tenus refpondet in tabula noftra finus 

9902681, a quo aufer dextimam figuram ( prout-o. 

indicat) & fient 990268.valor logarithmi 97796—— o* 
qu aditus. Sic logarithmi 25451769-f.oo, valor eft 
78459100, quia logarithmo 25451769. puro refpondet 

in tabula noftra finus 784591. Item logarithmi - 

3 13^-00. in quarta colum na, apud gradum 46, re¬ 

perti, valor erit 103553, quia tangens 46. graduum eft 
103 5 5302.Sic logarithmi —63 50305—oo.in tertia co¬ 
lumna apud gradum 32.reperti, valor eft 188708, quia 
fecans complementi 32. gtaduum , fcilicet 58. graduum, 
efi 18870800. cujus duseultima;& dextima; figuras 00. 

delenda; funt propter-00 annexa logarithmo. 

Zogari timorum datorum , in tabula noftra non refertorum numera¬ 
les valor es aftimare. 

Ad vulgares Geodefias fufficit plerumque,logarithmi 
tabulati propinquioris dato,numeralem valorem pro da¬ 
ti accipere, verum fi propius ad metam accedere defide- 
ras, logarithmum datum per nonam hujus numero-te¬ 
nus auge, vel minuefalvovalore priftino, donecaut in 
t ibula ieperiatur 3 aut alicui tabulato fatis fimi lis devene¬ 
rit, & hu/us logarithmi valcr perpijcmifiam inventus,eft 
quem quxris. ut exempli gratia , queratur valor hujus 
logarithmi 2314972 i-j-o. cui in tabula non reperitur fi- 
milis vel fatis propinquus, verum fi ab illo fubduxe- 






U LIB I, C A P« V* 

ds 23025842-1-0. relinquetur 123879. cuifub Si.gradii 
reperietur fatis propinquus & fimilis 123881. cujus finus 
9876883. per praemiftam inventus, eft valor oblati ioga- 
rithmi 23 14972 1 o.qusefitus. 


s 1A dmonitio. 

P Ro hac (e&ione, & fecunda hujus monitum volumus,, 
numerorum datorum logarithmos, & contra loga- 
rithmorum datorum numerales valores (ubi non repe* 
riuntur in tabula) omnium accuratifRme exhiberi per 
modum ipfumquo creantur, aut refolvunturlogarithmi, 
qui eft, ut a finu dato per media Geometrice proportio¬ 
nalia defcendas, donec in proxime minorem finum tabu¬ 
latum perveneris: fimiliterab hujus logarithmo tabula¬ 
to dcfiendas etiam per totidem media Arithmetica con¬ 
gruat horum ultimus erit illorum primi logarithmus: & 
contra per refolutionem, ut a logarithmo dato per media 
Arithmetica in Logarithmum tabulatum proxime mino¬ 
rem defcendas, & ab hujus valore tabulato fimiliter etiam 
defcendas per totidem media Geometrica & congrua 1 Sc 
horurn ulcimus erit numeralis valor illorum Loganth- 
morum primi. Verum qua: asqui-differentia Arithmetica 
cuique continuata: proportioni Geometricae conveniat & 
fit congrua, exquirere non eft mediocris ingenii. Quare 
de his ( De > afpirante) ubi de Logarithmis condendis & 
creandis agetur, amplius aliquando differemus. 

C A P. V. 

De amplifsimo Logarithmorum ufu^ & expe¬ 
dita per eos praxi. 

Pro* trium proportionalium Logarithmis , dato logarithmo 
bl _• medio & altero extremo,reliquum extremum,ejufveproportionalem ? 

ma 1 ♦ vel arcum , per unicam duplationem & ftbfiraUiomm dare* 

Quum 


LlB. I. CAP. V. ^ rf 

Quum j 5 er ^(^nd^ |r 6 p«cap^* 3 iip 1 utit niifcdii (fcilicet 
logarithmi) minutum 'altero ex tremoru aequetur reliquo* 
ideo a duplo medii logarithmi dati aufer logarithmum 
extremi datum, & relinquetur Toganithmus extremi qua> 
fiti: cui in tertia, quarta, aut quinta columna tabute in- 
ventae refpondet arcus in prima & feptima: finus autem 
in fecunda, aut fexta: & fui fecanres aut tangentes in ta¬ 
bulis fuis per cap. 3, fed. 1.2 .6. n. 14. 22. 25. 28,30. 

pro extremo quafito habentur. 



Exemplum* 


D Entar looooooo.primu proportionale^ 7071068* 
fecundum, queratur tertium. Id vulgo exquiritur 
medium in fe quadrate multiplicando, & hoc quadratum 
per primum dividendo. Sed nos facilius medii logarith¬ 
mum 346573?* duplando , & ab hoc duplo, quod eft 
6931470. logarithmum primi (qui eft o ) auferendo : & 
ita reflat 6 p3 i470.1ogarithmus quaefitus:cu/us arcum 30. 
graduum^ linum 5 000000. (fcilicet proportionale qua> 
fitum) juxta eum invenies. Sunt ergo 10000000. 
7071068.5000000.tria proportionalia>quorum ultimum 
fola duplatione, & fubhraciione acquilivimus, quod pol¬ 
liciti furnus, Item duo proportionalia 10562556. pri¬ 
mum, & 7660445. fecundum, aut fakem eorum logarith¬ 
mi -— 547302, & 2665142. dentur* Tertium fic habe¬ 
bis. Ab hujus duplo 5330258, aufer -— -547302 , & 
( per 8. fed» cap. 4.) producitur logarithmus 5877600. 
33. graduum,& 45, min. cujus finus 55 55702, eil tertium 
proportionale quafitum. 

Frob. Sx trium proportionalium logarithmi*.dat is logarithmis extremis 3 me~ 
2. dium , ejufcjue proportionale , & arcum per urucam additionem & hi - 
partitionem dare , 

Quum per fe< 5 t. 3. cap..2, duplum Logarithmi medii 
aequetur aggregato extremorum , ideo extremorum Lo~ 
garithmos adde; producum bipartire, & emergt tLoca- 
rithmus medii: atque inde medium & medii arcus inno. 

D tcfdt 


tl LIRER h CAP. V* 

tfifctt in columnis, & per fe&iones, ut fiipra* 

Extmph gratia* 

Dentur ex trema iocooooo. & ^oooooo.qysrittir 
4iutn. Id vulgo acquiritur multiplicando data illa irm« 
cem, <& produci radicem quadratam extrahendo. Ve* 
rum nos fic facilius. Datos extremorum Logarifchmos* 
o primi, & 5531470* ultimi addimus, & aggregatum 
553147°. bipartimur, fictque 3465735. optatus medii 
Logarithmus. Vnde ipfum medium 7071068,& ejus 
arcus 45.gr* rationefupradi&ahabentur. Jtem finfcex~ 
trema data 10562556. & 5555702* eorum Logarithnif 
■— 547302. & 5877600, Horum additorum futnma cil 
5 3 3 0258;. per fed. 5 . cap*4, quam bipartimur, Sc&g 
2 665145. Logarithmus, & e/nsarcus 50 graduum: Et fi* 
nus, fcu medium proportionale quaefitum eft 7660445» 
fola additione, & bipartitione inventum*, 
frob* Ex qudtuor proportionalium Logarithmis, datis tribus 9 eorum ve ar* 
cubus , invenire quartum Logarithmum, ejufjue Jinum> <& arcum per 
unicam additionem , & fubJbatHonem, 

Iu boc problemate, quseficum femper pro quarto fta* 
tujmusata.ut datorum primum fe habeat ad fecundum, 
ut tertium ad quaditum. Quumque ita conftitutorum 
aggregatum ex Logarithmis fecundi & tertii mi nutu Lo- 
gorithmo primi, aequetur quarti Logarithmo per 4, fe5fc, 
cap 2*. Ideo Logarithmos fecundi & tertii adde.& liinc 
aufer Logarithmu primi j & pro veniet Logarithmus quar¬ 
ti qu^fiti: & inde ipfum quartum, & ejus arcus*, 

SxejnpE gratia*. 

Sit ut 7660445. ad 5848078* ita5000000. ad quan¬ 
tum, quod qu$rimu$ # . Hoc vulgus acquirit.ducendo (e® 
eundum in tertium, & dividendo per primum; Tu au¬ 
tem fic facilius Logarithmos fecundi 153088; .& tertii 
6531465* addes, fiet.70845 57: a quo auferes Logarith- 
mum primi,quiefi 2665 149 ,& relinquetur441 94 o 8 ,Lo- 
garithrn.us quarti: cujus finus 6427876* eft ipfum quar¬ 
tum defideratum , & ejus arcus efl4p; graduum*, Idem 
grovenket fi (fpretis finibus ) fidium darentur tres fui ar« 

' CUSfO, 


LIB. f. CAF. v; xr 

80.gr. & 30. gr. NatnqttecxLogarithmis 
arcuum 80. gr. & 30. gr. ablato Logarithmd 50. gr. r&- 
manebit Loganthmus 40. gr. Et ita ipfe arcus 40. gn 
in nctcfcetabfque finibus, eorum ve multiplicatione aut 
divifionc» prout initio polliciti fumus» 

%^AUud exemplum . 

S it ut tangens feu foecundus numerus 43. gr. ad fi- 
ruam 57- gr* j ita foecundus feu tangens 3 5, gr ad fi- 
*ium quartum tacitum, cujus arcum negledis & fpreris 
tam finibus quam tangentibus, fic inveniemus. Loga- 
rithmumdifterentialem 35. gr. fcilicet 3563784.inmc* 
dia columna inventum ad Logorithmum 57.gr. videli¬ 
cet 1759372. in quinta columna locatum addimus: i 
produdo videlicet 5323156. diflerentialem 43. gr* qui 
eft 6p86p8, fubducimus,& relinquitur 4624458» Lo- 
garithmus quarti (finus fcilicet,) quo in tertia colu¬ 
mna per nfed, cap. 3. reperto, reperies juxta eum 
In prima columna 3 p.grad. 2. minur. fere qui efl ar¬ 
cus quaefitus quarti proportionalis feu finus fpreti* 

Hac ratione proportionalium arcus, abfque eorum 
Anibus, tangentibus, fecantibus, aut proportionali» 
bus quibufeunque acquiruntur. 

Qup J certe compendium ad triangulorum planorum 
angulos dimetiendos ,& ad univerfam fphasricorum Tri» 
gonometriam conducit plurimum ; ut fuo loco pate- 
bir. 

Frob* continue proportionalium datis extremis eorumve arcu* 

. * bm f mal orum quodvis , eorumve arcuum^ quemvis invenire 9 in* 

Jutta fmplict tripartitione pro ardua cubica radicis extraBione 4 

Quum in horum Logarithmis, triplum cujufque 
medii aequetur aggregato extremi remoti & dupli vici¬ 
ni , per prop. 6. cap. 2. Ideo duplum Logarithmi ex¬ 
tremi alterutrius ad Logarithmum extremi reliqui ad- 
de,& produd»m tripartire, •& proveniet Logarithmus 

P 3 medii 


LIBER I. CAP, V. 
medii priori extremo proximi, & eodem modo alterum 
medium. Vt exempli gratia : Sint extrema, primum 
4019246. ultimum vero 305615 56. Quaeruntur media, 
quaeabfque extra&ione radicis cubica: dic invenies. Dato¬ 
rum Logarithmi funt 9090051. & -—547302 ad illius 
duplum 18180102. adde hunc fiet 17632800. qui tri¬ 
partitus producit 5877600,Logarithmum , cujus finus 
5 5 5 57 ° 2 > eft prius medium quaditum. Item fimili modo 

ad hujus-547302, duplum, quodeft. - 1094604. 

adde illum 9090051. & producetur 7995447, qui tripar¬ 
titus producit 2665149 , Logarithmum , cujus finus 
7660445. eft pofterius medium etiam quaduum. Qna- 
tuor itaque proportionalia continua fuat 4 o 2 9 2 4 6. 
S 5 557P2.7660445. & 105625 5 6. 

tAliutU exemplum* 

S int extrema data 14142135 & joooooo.iiljus in tabu¬ 
la fecantium inventi legarithmtis in tabula noftra ed 
——34.6573 hujus vero 5000000. Iogar thmus efl: 

6931470. cujus duplo 13862940, adde- 3465735. fiet 

2 ? 3 P 7 20 5 ‘quemtripartive,&: fit-^346573 5 . 1 ogarithmus 
medii proportionalis minori extremo 5oooooo.proximi, 

quod eft 7071068. Sic duplo —346573 5. quod cft_ 

693 1470. adde +69 31470, & fiet inde o.feu nihil, quod 
tripartitu etiam reddit acujus finus & valor eft 10000000. 
pro reliquo & majore medio. Quatuor itaq. hxc continue 
proportionalia funt 14142135, 10000000. 7073068. 
5000000. 

CONCLVSIO. 


E X his pr abbatis judicent eruditi quantum emolumenti adferent illis 
logarithmi: quandoquidem per eorum additionem multiplicatio , per 
fubftratttonem divifio, per bipartioneextraBio quadrata, per tripartitio 
nem Cubica y gr per alios faciles profapharejes omnia praviora calculi ope* 
ra evitantwr:cu)usrci jpecimcn generale hoc priore libro exhibuimus. Se¬ 
quente autem de eorundem proprie & particulari ufu in nobili illa Gcerne- 
irU jpecfe 7 quo Trigonometrio dicitur , traUotitri fumus . 


Finis prioris libri 




LIBER SECVNDVS. 

De canonis mirifici Logarith- 

morum praclaro ufu in 

Trigonometria* 

CAP* I. 

Vum Geometria fit ars bend metiendi, 
Dimenfio fit magnitudinu propofitarum, 
niagnitudi.nes figuram ( potentia faltem) 
confiituant, figura fittriangulu, aut trian¬ 
gularum: Triangularum vero compofitum 
fit ex triangulis , quibus fuifque partibus menfuratis, 
menfurabitur& illud , illiufque partes omnes. Certum 
igitur eft ex triangulorum dodrina omnis Geometrica; 
quadlionis Solutionem Logifticam pendere. 

Triangulum aut re&ilineum eft, aut Sphericum. 

De retflilineis, prop. i. 

EcUhnei tres anguli .aquantur duobus retiis. 

Vnde duobus datis, aufer eorum aggregatum ex i8o, 
gradibus,&proveniet tertius,Iicm unico ex i8o.gradibus 
ablato, reftat reliquorum duorum aggregatum. 

T> 3 Redi- 





a-i 


LIB. II, C AP. I, 

Re&ilineum aut reftangulu eft, aut obliquangulum. 
Ia re&angulis crura vocamus, qua: re&uni angulum 
ambiunt: hypotenufam, qnx fubtendit, 

PrOp» In rettangulo hogarithmut cruris aquatur aggregato ex t>»gambmo 

2* duguli ei oppojiti , & Logarithmo hypotenufit. 

Quum ex Trigonometrix principiis pateat,alterutrum» 
vis crusie habere ad finii anguli ei oppofituit hypotenula 
ad finum totum :&(per prop.y.cap.2»lib.i#)horu quatuor 
propqrtionaliu logarithmi fecundi & tertii,aequetur loga- 
rithmis primi & qt#rti:quarti aute Logarithmus fit o,feu 
nihil(per collarium 6dcf.capi.lib. i. JIdeo(ut fiiprajLo- 
garithmus cruris aequatur aggregato ex Logarithmo an* 
guli quem fubtendit, & Logarithmoliypotemife* 

COroI« XJnde hypot enufe pruris, gr anguli quem fhb tendit,duobus qusbwf° 

cunque datu, tertum , atque inde reliqua omms 
nt iangult paries innosejccnt , 


w *f 


Qiiia enim haec tria cum finu tote 
conftituunt quatuor proportionalia t 
certum eft eorum quodvis quarto loco 
pofle confli tui, & per 3.probi* cap, 
lib. i. acquiri. 

Vt trianguli oblati A*B. C. in A re- 
dlanguli,detur hypotenufa ;B. C. f§8?s 
cum crure A B P384. Quaeruntur an* 
guli obliqui C. & B. Ex Logarithmo 
igitur A. B.635870—000, aufer Lo« 
garithmum B C 634799 —000. Su* 
perfunr 7071 Logarithmus anguli C s 
cui m tabula refpondent 8p g. $| pro 
ingulo C* & ex adverfoo g 56? pro 
: tjm complemento, angulo fcilicetB. 





IlBv It CAP, i. «£ 

Vice verfa Udetur angulus C, cum crure re- 

anguli A. B., & queratur hypetenufa B. C. 

Ex Loga ithnrto A, b, 6 3 5 8 7 o _— 000 aufer Lo* 
gar ithmum anguli C. 1 o 7 1, & provenient 6 3 4 7 5? p 
—000 Logarithmus B,C, p 3 8 5; hypotenufie quae- 
fitae. 

Tertio fi datis B. C, & angulo C. qu^ratinr 

A B,: adde Logari thmum B. C 6 3 4 7 9 9 -o o o, ad 

1071 Logarithmum anguli C, & producentur 635870 
«— 000 Logarithmus numeri 9384 cruri A. B. quaeli- 
to relpondentis♦ Necfecusjpfum crus reliquum A. C 
ex, angulo B. ( qui eft complementum anguli C.) ;am 
cognito habetur. Atque ita-omnes hujus redanguli par- 
tesinnotefcunt,. 

frop; reciangulo Logarithmus cujufvis cruris, ett *qud- 

r Iu aggregato ex dffertmtdt oppofiti anguli , Logmthmo reliqui 
cruris*. 

Quum ex vulgari do&rina triangulorum con¬ 
flet; quod alterutrum crus fe.habeatad tangentem fjbi 
oppofiti; anguli , ut reliquum crus ad linum totum: & 
quum f per prop. 5. cap. a.lib, 1. ) ex his quatuor pro¬ 
portionali busLogarithmi,mediorum (id cft, diflferentia- 
Ifs anguli, & Logarithmus cruris eum ambientis) aequen¬ 
tur Losarithmis cruris eundem, fubtendentis, & finus 
totius (qui eft nihil , feii o.) ideo Logarithmus cruris 
eft aequalis aggregato, &c, ut fupra, 

CoroL Vnde ex cruribus refti, & ungulo alteri eorum oppofito * 
duobus qwbufcunque datis , tertium ( per hanc ,) atque proinde cp* 
ierp omnes re&angdi partes (per praced. ) innotefcent* 

Quandoquidem ha?c tria cumfinu toto confit 
tuant quatuor proportionalia , certum ell eorum quod- 
vis quarto loco poffe collocari ■, & per 3 ♦ probi, cap, 5* . 
Iib« a . acquiri* 

P 4 


Vfc 


tL.CAP.il* 

-£f r Vt praecedentis trianguli A B C, in A re- 

danguli datis cruribus A B, 93 84* & A C, 13 7. 
Quaeratur angulus B. Ex Logariihmo A C* 
42924534—000. aufer 635870—000, Lo~ 
garithmum A B. & provenient 42288664, dif- 
ferentialis anguli B, o g 56. ri, qu&fiti._Verum 
(i dentur crus A C, 137-& angulus B.og. 56.11, 
habebitnr crus A B auferendo 42 2 88 664* diffe¬ 
rentia lan anguli B. a Logarithmo A C. quieft 

4.2 9 2 4 5 3 4-o o o. Inde enim proveniens 

i63 5870-ooo.eft Logarithmus numeri9384« 

qui crus eft A B-quadicum. Tertio datis crure 
A 8,9384. & angulo B, o g, 56. ii: ut habeatur 

crus A Cadde 635870--ooo,Logarithnumi 

cruris A B» ad 42288664,. differentialem anguli 
B. & provenient 42924534—000icgarjth- 
mus 137, cruris AC,quadIti. Hypotenufa 
autem B C per praced. prop. habetur. An¬ 
gulus etiam C, patet, quum fit complemen¬ 
tum anguli B, jam cogniti. £t ita per hanc, 
& praemiifam,ex latere quovis,& parte alia qua- 
^ vis redanguli datis reliqua; omnes ejus partes 
innotefeent* 

Completam ergo habes re&angulorum redilineorum 
fdentiam: fequitur obliquangulorum. 


De triangulis redilineis prariertim obliquangulis* 

GAP, II. 

I 2\( omni triangulo, Aggregatum ex L&garithmis anguli 
cu]ufvis i & lauris eum ambientis aquatur aggregato ex Logarith* 
tnis lateris, &anguli eis oppofit orum* 

: Quia omnium laterum ad oppofitbmm angulorum (I- 
nus t adem eft ratio: & Ita fedum ex anguli cujufvis fina 
redo, & latere quovis eum ambiente, aequatur fado ex la¬ 
tere fubtendente priorem angulum, & ftnu anguli fubten- 

1 ii i 









LIB. II. CAP. II. sj 

fi a priore latere. Ideo (perprop. 5. cap.3.lib.i.)aggre¬ 
gatum ex Logarithmis &c. aquatur, ut fupra. 


Corol, Vnde ex duobus angulis quibufcunque data Jpeciei, & fuisfubten- 
dmibus,fi tria dantur , quartum q nodatu que > atque inde eat era omnes 
trianguli partes irmotefcent* 

Horum enim quatuor proportionalium quodvis quaefi- 
tum poteft quarto loco conditui,per 5. probh cap, 5< 
lib« 1 ♦ inveniri. 

Vt obliquangnli A. B, C 
detur AB. 26302,& B C* 
57955 4 & angulus C. 26 
graduum •. Quanraturque 
angulus A, qui fic habe¬ 
tur* Adde 5454707.;—00 
C Logarithmum B C* ad 
8246889 Logarithmum 
feilicet C 26 graduum, &: fient 13701596-,—00. Hinc 
aufer Logarithmum AB, qui eft 13354921-00, re¬ 

flant 346675 Logarithmus 7 5 graduum, & paulb pluris, 
anguli feilicet, A quaditi, fi A prodicatur acutus : alioquH 
1 o 5 g (per 1. & 1. fedt. cap* 3 ♦ lib* 1 ♦) fi pronuncietur ob- 
tufus* 

Vice verfa fi detur angulus A jam 7 5 graduum, atque 
angulus C, & latus B C - ut fupra / & quaeratur A B. adde 
5454707—00 Logarithmum B C.ad 8246889 Logarith¬ 
mum anguli C, fient, ut fupra, 13701596—00, a quibus 
aufer 346675 logarithmu anguli A.provenient 13 3 5492i 
—00 Logarithmus lateris A B, & numeri ejus 26302 
quasfiti. Habitis jam angulis A> 7 5 g. & C. 26 g, erit 
a ngulus B. 79 gr. per 1* hujus. Ex quo jam habito, non 
fecus acquiritur latus ei oppofitum A C. 58892, quam 
nuperrime ex angulo C* innotuit latus ei oppofitum 
A B* Itaque jam patent omnes hujus obliquanguli par¬ 
tes. 



Jn obliquangulu crura vocamus , qua angulum quemvis ambiunt : 
bafim qua fitbtendit , 

E 'f n obii - 



** ElB II CAP II. ^ 

prop*, h obliquangulis , Loganthmus aggregati crurum fnbdutltis dfim « 
5 mu fatta ex jLogarithmd differentia crurum , & dffetentialt fcmi-ag~ 
gregati fuorum oppofitoruw angulorum , relinquit deferentialem fena* 
Afferentia eorundem . 

Quia, ut aggregatura crurum ad differentiam crurum* 
ita tangens femi-aggregati fuorum oppofitorum angu¬ 
lorum, (e h&bet ad tangentem {emi-differentiae eorun¬ 
dem, Vnde analoga funt, & (perprop. i. cap. 2.1ib. i.) 
eorundem differentiae {eu exceffus funtaquales. Ne- 
cefTario igitur (per prop. 4. cap. 2». lib. i.) concludimus 
ut Cupra. 

Coroj, ZJnde ex dmbtls cruribus , & angulo comprahenfo , innotefeunt (per 
hanc ) anguli reliqui oppofiti : atque inde ( perpram ffam ) reliquum 
latus. 

Nam fubdu&o Logarithnioaggregati crurum, a fum- 
ma fadte ex logarithmo differentiae crurum,& differentia- 
Ii femi-aggregati oppofitotum angulorum additis , pro- 
Teniet differentialis femi-differentiae eorundem angulo¬ 
rum : qua femi-differentia addita ad femi-aggregatum 
di^um* proveniet angulus major, & fuhftradta minor. 

Vtrepetrti fuperioris obliqnanguli A B C. dentur 
crura, A B- 26302, & B C- 57P55 5 & angulus comprae- 
henfus B. 79 graduum. Quaerantur autem reliqui an¬ 
guli A,& C. Aggregatum crurum AB.&BC eft 84257, 

ejufqe Logarithmus eft 
247 3 8 81 p_^o. differentia 
autem eorundem A B, & 
BC eft 31653, ejufque 
Logarithm. eft 34529210 
— o. Quumque B angu¬ 
lus detur 79 g. erit (per 1. 
hujus) aggregatum angu¬ 
lorum A & C, graduum lor, femi-aggregatum vero 50 
g. 36 , cujus differentialis eft —— 1 9 3 17 6 6, quo ad 
3452921 o.—-o addito, fient 3 2 597444— o.. hinc abla¬ 
tis 24738819—-o, provenient -j- 785^625 differentia- 
lis graduum 36,.qui funt femi-differentia angulorum, 

A te 


3 





II CAP II %y 

A & C quaditorum. Hanc ergo femi differentiam 24* 36 
addeadfemi-aggregatum 50.36,fient gradus, pro 
angulo A quajfitorum majore, & fubftrahe eofdem 244 
gradus ab eildem $0 i gradibus, & relinquentur 26 gra¬ 
dus pro angulo B quotorum minore. 

De0- J n obUqumgulis vera bajis ffmper efl vel aggregatum cafuum : & 
mtio tunc differentia cafuum bajis alterna vocatur 1 vel vera bajis efl dffe- 
.r enti a cafuum: & tum aggregatum cafuum vocamus alternam ♦ 

Vt trianguli A B C. cafus 
B minor eftAD: cafus ma¬ 

jor eft D C. Cafuum ag¬ 
gregatum A C efl: bafis ve¬ 
ra. Et in hoc triangulo 
aufer cafum minorem A D, 
feu ei aequalem D E a cafu 
majore DC, relinquetur 
differentia cafuum E C, quam bafim alternam vocamus. 
Contra vero in triangulo BBC cafus minor eft D E (cui 
aquatur D A) Cafus major eftD C , & cafuum diffe¬ 
rentia B C eft bafis vera-* Cafuum autem aggregatum, 
fcilicet AC, bafim alternam vocamus, 

Prop. In obhquanguiis fttmrna Logarithmorum aggregati & differentia 
6 crurum , est aqualisfumma Logaritbmomm baftuw, vera , & alterna. 
Quia bafis vera fe habet ad aggregatum cru: um, ut 
different : acrurum ad bafim alternam: ideo (per prop. 5. 
cap. 2. lib. i .) neceffario concludimus,bafium Lcg irith- 
mos aequari Logarithmis aggregati & differentia crurum* 
P . t ut fupra. 

® ZJnde ex obliquangulo datorum laterum, funt duo reBangula nota - 
rym bypotenufarum cum altero atjttfjue crure, ejua (per 2,hujus) rcli~ 
cptas etiam omnes obliquanguli partes notas reddunt. 

Nam addito Logarithmo aggregati crurum ad Lcga- 
rithmum differentia? crurum , & hinc ablato Logarithmo 
bafis vera?', proveniet Logarithtrus bafis alterna?, per 
prop. 4. cap. 2. & probi. 3. cap. 5. lib. 1. Ha um ita¬ 
que bafium femi-aggregatum eft: cafus majordemi-diffe- 
rentia vero cafus rnmor* Vt fttperioris trianguli ABC 

E 2. dentur 





LIBER II. GAP. II. 
denturlatera,videlicet crus A B 26302,&crus B C 5795: 5, 
& bafis A C 5 S 89 1, & quasiantur extera# Aggregatum 
crurum eft 84257 , ejufque Logarithmus eft 247388 rp 
—o. Differentia crurum eft 316 5 3,ejufque Logarithmus 
eft 345 29210 — o. Hos Lcgarithmos adde, fient inde 
59268029—00,4 quibus aufer 5293461—ooLogarith- 
mum bafis AC, reflant 53974568 Logirithmus numeri 
45286 bafis alternx t quam ad veram adde , fient inde 
104178, quorum dimidium eft 52089, D C, cafus major. 
Eandem ab eadem aufer, fient inde 13 606, quorum dimi¬ 
dium eft: 6803, A D cafus minor. 

Rectanguli itaque A D B. habitis jam , hypotenufa A 
B,& crure altero A D; atque reiianguli B D C habitis,hy¬ 
potenufa B C,, & crure D C, innotefeunt ( per 2. hujus) 
anguli rectangulorum apud A & B & C,& per.confequens 
omnes etiam obliquanguli oblati partes ex prxmiftis pro¬ 
palantur. 

Nec feclis agendum foret fi darentur latera trianguli 
E B C, & exterae partes quxrantur. Ex cruri bus enim 
bafi vera E C , innett fcit bafis alterna A C. atque ex his 
uterque cafus, & extera, ut fupra. 

CONCLVSlO. 

P \Erfe$dm igitur & completam jam habes omnium trian¬ 
gulorum refltlineorum do firmam , qua fi aliquantulum operofain 
Logarithmis retiarum variabilium inveniendis videatur: jn motibus 
tamen planetarum computandis ( in quibus fcihcet eccentricitates or¬ 
bium , elongationes lAugium & apogaomm , epicydorum diametri, o* 
alia refla , eadem & invariabilts permanent) eorum logarithmi exafle 
femel notati, femper in poflcrum,fine ulla mutatione fubfervient , miran¬ 
da certe facilitate 3 & certitudine . 

Sequuntur jam Spharica triangula , omnium difficillima , ut vulgo ab 
aliis traduntur , per Logarithmos tamen nojlros , omnium facillima» 

De triaiv* 


LIB. II. CAP; III. 2 9 

De Triangulis Sphericis, 

CAP. III. 

T T r ungulis Sphericis ungulus omnium quadranti quan~ 

t it at e proximus , £7 latus eum flubtendem dubia fiunt, &An ejufdem 9 
an diverfa fint flfltciei i mfi id aut computas, aut hypothefis prodat . 

Duorum vero obliquiorum angulorum quilibet eft cjufdem fpeciei, 
cujus eft latus eum fubtendens . Vnde alterius data , reliqui patet Jpe- 
cies* 

Si trianguli angulus aliquis propinquior fit quadrarrti , quam latus 
eum fubtendens 3 erunt duo ejus Utera ejufidem fpeciei , gr tertium qua¬ 
drante minus . 

Si vei o trianguli latus aliquod propinquius fit quadranti } quam eo 
fubtenfus angulus: erunt duo ejus anguli ejufdem flpeciei, & tertius 
quadrante major* 

Triangulum Sphaericum aut efl quadrantale , aut non* 

Quadrantale efl cujus aut latus, aut angulus aquatur quadranti. 

Vnde, non re&anguli quadrantalis (ciendam seque fa¬ 
cili, ac re&anguli comparari poiTe, docemus. 

Quadrantale triangulum aut eft multiplex* autfimplex. 
gMultiplex quadrantale aut eft trireblangulum 5 aut bireUangn- 
Ium. 

T vir eFk angulum eft cujus fingula partes quadranti aquantur» 

Vnde omne triangulum ? cujus trium panium non oppofit arum finguU 
quadranti aquam ur,TrireFlangulum eft. 

Bircflangulum eft, cujus duo tantum anguli, & fuaJubtendentia la¬ 
tera (igillatim quadranti aquantur* 

ln omni bireU angulo angulus obliquus aquatur fluo fubt en dent i la¬ 
teri. 

Omne Triangulum cujus pars aliqua aquatur quadranti , & angulus 
aliquis obliquus aquaturfino flubtcn denti } VnreFl angulum eft* 

Omne Triangulum habens duas quaficunque partes figillatim qua¬ 
dranti aquales, & tertiam inaqualemgBircUangulum eft* 

Caler a quadrantalia fimpheia dicuntur • 


E 3 


De fim- 


lib. ii; cap iiil 

De fimplicibus Quadrantalibus» 

CAP IV. 

Q Vidrantale fimplex ett, cuius unica tantum pars quadranti aqua* 
tur , c at era autem quinque partes funt non quadrantes. 

Harum quinque partium non quadrantium , Tres qua d re&o angu¬ 
lo , feti quadrante latere , fitu remotiores funt 3 infua complementa con¬ 
vertimus , & retento priftino ordine omnes quinque in circularem , feu 
pentagonalem fitum fiatmmus 3 & circulares vocamus* 

Sit primo triangulum B P S in B re&an- 
gulum. Ejus quinque partes obliqua;, 
feu non quadrantes, funthar. B P latus 
ambiens re&um. P angulus obliquus al¬ 
ter. P S latus fubtendens redum, S an¬ 
gulus reliquus obliquus. S B reliquum 
latus ambiens re&mPro quibus nos facili* 
a oris calculi gratia affurnimus latus, BPip- 

fum.-complementum anguli P; Comple- 
BS s mentum lateris PS a complementum an r 
guli S, atque ipfum latus S B, &fervatoJnaturali fitu 
has quinque partes .ordine {latuimus, ut a margine, & 
circulares vocamus. 

Similiter fit fecundo triangulum quadrantale fimplex, non 
re&angulum (ex centris folis orientis, poli, &zenith 
fadum ) S P 2 , in datere 2 S quadran¬ 
tale. Ejus quinque partes non quadran¬ 
tes priflinte funt. 2 angulus alter am¬ 
bitus a latere quadrante. Latus P Z di* 
flantia poli azenith. P angulus fubtenfus 
a quadranteXatus P S diftantia poli a So¬ 
le, & angulus denique S alter angulorum 
quos quadrans arnbit. Pro quibus nos 
ad faciliorem computum noftrum affumi- 
musipfum angulum Zjfeu P 2 S, quiefl 
arcus plagae Solis a feptentrione. Comple¬ 
mentum P 2 , quod efl ipfa elevatio poli: 
Complementum anguli p, feu anguli 

ZPS 










■ af-LP 
cmfe/F^P 
mpC P. 


IIB IL C AP* IV. , ; ¥ 

EPS quod eft differentia afcenfionalis 9 idefl, diffe¬ 
rentia temporis ortus vel occafus Solis ab hora fexta* 
Complementu lateris P S quod eft Solis declinatio:&an¬ 
gelum ipfum S feu P S Z 3 quem angulum politionis Solis 
(refpe&u fcilicet poli & zenith) vocamus. Has quinque 
partes etiam circulari vel pentagono fitu ftatuimus, ut a 
margine, & circulares vocamus* Nec alise fient circula¬ 
res partes fuperioris 
tjianguli re&anguliB 
P S, fi P polum, Sfo- 
lem, & B cardinem 
borealem feu fept^i- 
trionalem pofueris* 
Fient enim latus BP 
elevatio poli, com¬ 
plementum P diffe¬ 
rentia afcenfionalis. 
Complementum p S 
declinatio folis, cbiru 
plementum S angu¬ 
lus pofitionis folis; ac 
denique B S plagae 
felis. Quse funqese¬ 
dem prorfus circulares partes, qua; fupra, & eodem fitu 
levorfum quo ille dextrorlum dilpofitse.Et itain omnibus^ 
quadrantalibus tam redangt Iis,quam non reftangulis. 
Hinc fit quod plurima fint triangula tn partibus fiuis naturalibus haud 
conformia,qua in fartibus his circularibus prorfus conveniunt^ hac 
nohra circularium met hodo rejolvuntur. 

Vt fatis lucide apparet in duobus fuperioribus triangulis B 
P S, & P Z S conjun&is* In quibus omnes naturalis 
partes ( pr$ter P S & B S hujus , & P S & P Z S illius ) 
prorfus differunt: circu’ares vero partes omnes (ut fupra 
didumefl) conveniunt* 

Hac circularium partium uniformitas manifeflifiime patet in rcdkin- 
gulis faElis in fuperficte globi ex quinque circulis magnis , quorum pri¬ 
mus ftcet fecundum , fecundus tertium , tertius quartum ^quartus 

E 4, quin— 



Solis 


s/ITr 1 - 

LPoJvtio tus 
JolLs £r’ contvL 

BSVafUan. 




LIB It CAP, IV* 

quintum,quintus dmiq. primum ad recloi angulos .-relique vero fetliones 
tmnes ad angulos obliqms fient., 

Exempli gratia ; Meridianus regionis D B , fecat hori¬ 
zontum B E inpundoB. Horizon B E fecat circulum E 
C,qui folem ambit (id eft, qui circa folem tanquam polum 
ducitur) in pundd E* Circulus E C, qui folem ambit,fecat 

meridianum folis C F in 
pundo C. Meridianus folis 
C F aquatorem F D in pun¬ 
do F: & tandem oquator 
F D fecat meridianum re- 
E gionisD B inpundoD. Et 
omnes ho quinque fedio- 
nes in pundis. B. E. C* F* 
D orthogonaiiter & ad re- 
dos angulos fiunt: fadis 
coteris fldionibus in pun¬ 
dis Z.p,S* 0 *Qjid angulos obliquos* Fientqueexhis fe- 
dionibus redangulaquinq.p B S,S F 0,0 E Q^QP Z,& 
Z Cp, quorum quamvis partes naturales differat,& in lin¬ 
gulis triangulis varientur,circulares tarne quinque partes 
easdem funt, quas fupra, abfque ullo diferimine, 

Eadem circularium partiti uniformitas^patet etiam in quadrantalibus 
non r eB angulis ,facl i sin fuperfeie globi ex quinq^punUis } quoru primus 
diftet a fecundo y fecundus k tertio, tertius d quarto , quartus d quinto,& 
quintus a primo diftantiis & arcubus aqualibus quadranti 3 alia veta 
punllorum di flantia inaquahs fnt quadranti. 

Vt in eodem procedente fchemate puda,P a Q/Qjib 
S, S ab Z, Z ab 0 ,atque O a p, diftant fpatiis quadranti 
ssqualib 9 : at vero P ab Z,Z a QsQab 0,0 ab S,& S a P, 
diftant ab invicem arcubus non quadrantibus. Et fient ex 
his diftantiis quadratalia non rcdangulaquinquejp Z 
ZQs 9 >OP s 5 osp j&Sp Z:quoru quamvis naturales 
partes differant: partes tamen circulares eodem & im¬ 
mutabiles hic permanent, quo fupra, Scilicet, elevatio 
poli, differentia afcenfionalis , declinatio folis, angulus 
pofttionis folis, & plaga folis: quo omnibus fuperioribus 

trian- 






LIB II CAP II. ^ ^ 33 

triangulis ex sequo conveniunt, nec his duntaxat folis,ve- 
rumetiam omnibus triangulis qua? oriuntur ex interfedio- 
nibus ca?tcris horum decem arcuum ad integros circulos 
produdorum: qua?,quia plurima & confufa funt, milfa hic 
facimu$#Hac epitome fatis eft monuilfe omnent confufi- 
onem naturalium partium 3 & fuarum regularum» his pau¬ 
cis circularibus partibus &fua regula unica evitari,ac tolli» 
Quinque circularium partium , tres femper in quajh&nem cadunt, 
quarum dup dantur , iertta queritur. 

^Atque harum trium una eft intermedia , & duo. funt extrema , qua 
fcilicet intermedia aut circumponuntur, aut opponuntur. 

Verbi gratia , Sint partes tres in qua?ftione propofita: 
ha?, plaga folis, elevatio poli, & differentia afcenfionalis: 
quarum, elevatio poli pars intermedia dicitur, & reliquae 
duae extrema? ei vicina?,aut circumpofita? vocantur verum 
fi tres partes in quseftionem cadetes foren;t,declinatio fo- 
lis 3 elevatio poli,& angulus politionis folis, vocabitur (ut 
prius) elevatio poli intermedia:fed declinatio folis,&an* 
gulus politionis folis^extremsea media remotae,feu ei op' 
pofica? dicentur, par ratioeft in rchquis quinque* 


Logmthmus inter medi a aquatur dijferentiahbus circumfb- 
fitaru extremarttyfeu antilogarithmis cpfofitaru extremaru. 

Hoc theorema probatur indudione omnium trium 
partium feu triplicitatum , qua? ex quinque circularibus 
partibus quadrantalis prioris B P S redanguli, conditui 
poffunt,&in quseftionem cadere* pofterioris autem non 
xedanguft P Z 5 triplicitates omittimus,quia ejus omnos 
pa’tescirculares(ex i8,& ip,& 2o.pra?miffis)eaede pror- 
fus funt quantitate,qua? prioris«Qipnqueerg6 partiu cir¬ 
cularium redanguli BP S, (qua? funt B S ,feu plaga folis 
orientisrcomplementum B S P. feu angulus politionis lo- 
lisicomplementu S P,feu declinatio folis.-compJementum 
S P B,fiu differentia afcenfionalis :& P BJeu elevatio poli) 
tres illae quae in qurftionem extremaru circumpofitarum 
ca luntffunt aut primo B S,complem*B S p,& compl.S p: 
aut fecundo compl.B S P,compl. S P,& compl.S P B: aut 

F tertio 


14 


LIBER II. CA P. IV. 
tertio comp.S p cempLS p B,& p Braut quarto compl. 
S P B , P B, & B S: aut quinto funt p B,B S, & cora» 
plera. B S p. 

Verum quia in omni- us bis triplicitatibus, Tangens 
alterius extremse eft ad finu n rcditai intermediae,ut finus 
totus ad tangentem reliqua extremas ( pro ut ex vulgari¬ 
bus demonftrationibusTrigonometrias pater.jldeo^per 
noftras demonftrationes prop.5 cap. 2 . 1 ib.i.)Logarithmi 
mediaru(qui funt Logarithmus folius intermediae per co- 
rol.d.def.cap 1 iib. 1 .)asquantui logarithmis tangentium 
utriufque extremas. Sed Logarithmi tangentium harum 
extremarum funt differentiales earundem (ex fed. 22. & 
25. cap. 3. l;b.T.) Logiritlimu,igitur folius intermedias 
asquatur differentialibus circumpoficarum extremarum, 
ut priore parte Th.orematis aiLruimus. Sequitur po- 
fterioris partis confirm ;tio. 

Earundem ergo quinque partium circularium, tres il¬ 
lae quasin qusefiiuncm txtremavum intermedix oppofiti- 
rum cadunt/unt aut primo p B,comp.B S P,& compj.S 
p Braut fecundo B S,comp,S p>& P Braut tertio compl. 
B Sp,comp.SP B 4 &B S.aut quarto cop S p,p B,&.cop. 
B S praut quinto denique comp.S ? B,B Scomp.S p. 

Sed in omnibus his triplicitatibus feu quinque cafibus, 
finus redus complementi alterius extremas fe haber ad fi- 
num redum intermedias, ut finus totus ad fimum re dum 
complementi rdiquas extremas(quod fufius a. Regiomon- 
tano^CopernicOjLansbergiOiPitifco, & aliis demoftr; tur, 
quam ut brevi hac epitome repetendum fit )ide 6 per no- 
fhas demonftrationes (prop.5, cap. 2 . 1 ib.i ) Logarithmi 
complementorum harum extremam aquantur Loganth- 
mis mediarum,id eft, (utdidu eft)Logarithmo folius in- 
termedias.At Logarithmi complementorum harum extre¬ 
marum oppofitarum funt earundem ipfaru partium anti- 
logarithmifexdef, fed.i? & i6.cap.3.1ib.i.)fequitur ergo 
in his cafibus.quod log rrthmus folius intermediae aeque¬ 
tur antilogarithmis fuarum extremarum oppofitarum, ut 
afierit pollerior theorematis pars. Tctum itaque theore¬ 
ma con- 


/ 







LIB. II. C AP« I V. & 

ma conflat, P-rseter hanc probationem per indu&ionem 
omnium cafuurmqui occurrere poffuntjpoteft idem theo¬ 
rema lucide perfpief ex ip* & 20 a - praecedetibusjnquo- 
rum fchemate,homologa circularium partium coftitutio 
earundem analogiae flmilitudinem arguit: ita ut quod de 
una intermedia & fuis extremis circumpofitis, aut oppo- 
fitis vere ^nuntiatur. de caeteris quatuor intermediis & 
fuis extremis refpe&ive cirmcumpofitis» aut oppofitis 
negari non poffit» 

Portfmd generale* 

H inc [equitur in quadrantalibus fimplic ibus , qued ex duabus parti* 
bus quibufeunque datis tertia quavis wnotefcet* Semper enim aut 
intermedia quaritur, & ejus logaritbmus habetur addedo differentiales 
circumpofitarum extremarum datarum :aut altera extraparu quaritur» 
$ ejus differentiatis emergit ex fubfraBione dfferentia/is reliqua ex* 
trema data k Legar ithmo intermedia nota:ut in quinque prioribus tripli - 
citatibus re&arguliprae e dentis theorematistotidem non reB anguli: 
aut intermedia quatit ur 3 & ejus Logaritbmus provenit addendo antilo* 
garithmos ofpojitarum extremarum datarum : aut denique altera extre¬ 
marum oppofit arum quaritur:& ejus antilogantkmus exfubduBione an* 
tiiogarithmi reliqua extrema oppofita data ex Logarithmo inter media 
nota habetur ♦ Xdt in quinque pcfterioribus cafibus reti anguli prae e den¬ 
tis theorematis ffr totidem non re&anguli • florum autem Logarithmo- 
rum^antilogar it Imorum, & dfferentialium jam inventorum cuilibet re- 
fpondent duo arcus diverfayum fpecierunn Ex Jpecie igitur quafiti arcus 
per fecundam 3 tertiam 3 quartam hujus, aut per hypothefim notajpfe arcus 
verus tnnotefcet . 

Vt in priore exemplo raptim*, Tres quaeftionis partes 
circulares funt, plaga folis, elevatio poli,& differentia af- 
cenfionalis,id eft,In reclangulo B P S, partes B S, p B, & 
complem.S p Btvel in non re&angulo quadrantali p Z $, 
partes pZS> compl P Z,& compl S P Z: quarum trium 
dentur extremae circumpofitasXcilicet plaga folis orientis 
B S, vel p 1 S,70 gr: & differentia afcenfionalis compl S 

p B,vel compl* SPZ, i^gr. * 4 * 27 *• &inter - 

F % media 


LIB II CAP IJII. 

media pars P. B, vel compl. P Z, qu$ eft elevatio poli. 

Addatur ergo differenti alis 70* gr., viz.--10106827 

ad differentialem 1 6 gr.24.27. videlicet ad 12226180. 
& provenient 2119353, Lcgarithmus 54 graduumpro 
elevatione poli quadita, 

oAdmonitio. 

P Rapter elevationem poli hoc modo inventam ,habetur 
etia fecundo eadem praxi plaga folis ex elevatione po¬ 
li^ angulo pofitionis folis.Item tertio angulus politionis 
folis ex plaga folis,& ejuftlem declinatione datis. Quarto 
declinatio folis ex angulo pofitionis folis, & differentia. 
afccnfionahV Quinto differentia afcenfionalis ex decli¬ 
natione folis, & elevatione poli,; 

Secundum exemplum, 

D Etur plaga folisorientis B S,feu P. Z; S^o.graduui: 

& elevatio poli 54, gradumquse eft P. B, aut compl, 
P. Z, Quaeratur autem differentia afcenfionalis^ fcilicet 
comph S. P. B , vel compl. S. P.Z, Et, quia hiefimiliter 
extrema partes circumponuntur intermedia?, ergo aufer 
differentia'em plagae folis feu 70. graduum , qui eft. 
—10106827. ex Logarithmo elevationis poli ^.gradu¬ 
um- fcilicetex 2119353 . & provenient inde 1,2226180,. 
differentialis graduum 1 < 5 , 24 27 fere., arcus differentias.; 
afcenfionalis qusefita?i 

Admonitio. 

A D hujus exempli imitatione habetur fecundo de^- 
clinatio folis ex differentia afcenfionaIi,& elevatione 
poli datis. Item tertio angulus pofitionis folis ex decli¬ 
natione folis, & differentia afcenfionali.. Quarto pla¬ 
ga folis; ex angulo; pofitionis, folis,, & declinatione 

^ e/ufdem. 


LIB II. CAP. IV. 


37 


cjufdem. Quinto elevatio poli habetur ex plaga folis, & 
angulo pofitic>nis folis* Item contra habetur.Sexto dif¬ 
ferentia afcenlionalis ex declinatione folis, & angulo po¬ 
litionis folis datis. Septimo declinatio folis ex angulo 
politionis folis, & plaga ejus.* O&avb angulus politio¬ 
nis lolis habetur ex plaga folis & elevatione poli datis. 
Nono plaga folis ex elevatione poli , & differentia afeen- 
fionali. Decimo tandem elevatio poli habetur cxdiffe¬ 
rentia afcenlionali, & declinatione folis datis.. 


T ertium exemplum• 


I N pofteriore exemplo e/ufdem feptimse tresqusflionis 
partes circulares proponuntur htE, declinatio folis, 
elevatio poli , & angulus politionis folis. Eaefuntin 
redangulo B. P, S. compl* P S. B P- & compl. B S P, 
& in non re&angulo quadrantali P. Z.S, ex iunt, compl. 
P.S.compl. Z.P.& Z.S.P. Quarum trium dentur ex¬ 
tremas oppofiras/cilicetdeclinatio folis, qua? ell compl. 
P. S. it. gr* 3 i*\& angulus politionis folis ,qui eft 

compl. B. S. P, fiu Z. S. P. 34 gr. ip' 21" fere.Et quadra¬ 
tur intermedia pars B. P, f*u compl. Z p, quae cft elevatio 
poli. Addatur ergoantilogarithmus ii.gr 3 5 '.Ji", qui 
eft 206271 ad antilogarithmum 34.gr.19'. 2 l "> eft 
1 pt3082,provenient 2iip35 3.Loganthmus 54. gradu^ 
um pro elevatione poli quadita.. 


dmonitio. 


P Rseter elevatione pol i hoc jam modo inventa 5 poteris 
fecundo per eandem praxim habere plaga felis ex ejus¬ 
dem declinatione, & differentia afcenlionali datis. Tertio 
angulum politionis folis ex differentia afcenlionali & ele¬ 
vatione poli. Quarto declinationem folis ex elevatione 
poli. & plaga folis. Et quinto invenies differentiam af- 
cenfional:m ex plaga folis &angiTo politionis folis da- 





1 % LIB. II. CAP. IV. 

■ - 4 ; t ■ - • - " - - • - . f> 

££udrtnm exemplum* 

D Emr declinatio folis cOmpl. S. P. ii.' gr.3f.51", 6c 
elevatio poli B, p, feu compl. p. Z graduum ^4. 
Quaeratur autem angulus politionis folis compi.B.S.P* 
feup,S,Z : &, quia hicfimiliter exirem^ partes interme¬ 
dia opponuntur, igitur auferendus erit antilogarithmus 
it* graduum 3f. 5^1 , qui eft 206271. exlogarithmo 54. 
graduum, qui eft % 119353. &fupererunt i9i3o82.antilo« 
garithmus 34.graduum ty.ijffere, qui funt angulus po« 
fttionis folis quaelkus® 

/•■v . * ^ r i . 

Admonitio. 

P Rarterangulu politionis folis hac prima praxi acquifi- 
tumjhabetur fecundo eadem praxi declinatio folis ex 
datis differentia afcenliona!i&plag*i folis. Tertio habetur 
differentia afcenfionalisex datis elevatione poli & angulo 
politionis folis. Quarto elevatio poli invenitur ex plaga 
folis & ejufdem declinatione datis. Quinto plaga folis ac¬ 
quiritur ex angulo politionis folis &differecia afcenliona- 
li.Sexto(cotrario ordine)angulus politionis folis invenitur 
ex plaga folis & differentia afcelionali datis.Septimb decli¬ 
natio iolishabetur ex angulo politionis folis,&elevatione 
poli datis.Odavo differentia afcenfionalis ex declinatione 
folis^ ejufde plaga invenitur. Nono elevatio poli habetur 
exdatadifferentia afcenlionali, & angulo politionis folis. 
Decimo tandem acquiritur plaga fclLgex elevatione poli, 
& declinatione folis datis. 

Atque ita ad imitationem horum quatuor exemplo¬ 
rum , triginta varia? folvumur quaeftiones circularium 
partium, hoc eft, triginta varia? qua?ftiones in quadrantali 
re<ftangulo, & totidem in non re&angulofolvunturhoc 
porifmate , beneficio unius tantummodo additionis 


LIB. II. CAP. V. . o $9 

^el fubftra&ionis. Csterum ad huelligentiam pofterio- 
ris parcis hujus porifmatis,de arcuum fpeciebus, vide ex¬ 
empla, (tertium, quartum, quinuim, & fextum)fequcntia, 

• ' v i v, .■ ' v ' ' ■' '• r --* • ■ v ■■ - ■* ;'vU - .i. '> 

C,- 

De non quadrantalibus mixtis* 

-it: "V - : :CAP. V. 

H AFlcnus qiiadrant alium } fequitur triangulorum Sp haricorum non 
jundrant alium do Urina» 

Non quadrantale eH triangulam Spharicam, cajas nec latus , nec 

angulus quadrans e Fle 

Non quadrantale reducitur ad bina quadrantalia , Si d vertice ad 
ejusbafun (prout opus fuerit extenfam ) dimittatur.perpendicularis,, 
nui quadrans arcus* 

'Pirpendiculam cadit intra triangulam ,ft anguli apud bajim fmt 
ejufdem (pecie/: extravero fi divcrft '. & contra ♦ 

Ouadrans arctes cadit extra triangulum^fi crura fmt ejufdem 
ciet: Intra vero fi divo fau & contra. 

Ex non quadrantalis fex partibus, tres data f lum fafficiunt ad 
reliquarum [dentiam comparandam : mfi forfan trium datarum , 
quarum una- aiteri opponatur , tertia fu propinquior quadranti , quam 
ahera e jufdem generis data . In hoc enim cafu requiritur etiam dari 
m/cies partis qua tertia opponitur, ut reliqua fetantur . 

L Hrnus:ca{us exempla funt quartum & fextum exem- 

plmn fequendum* 

<partes tres datp aut m fcellanea funt 3 aut pura» 

tJMtfcellinex f mt , quarum una. eH chverfi generis k reliquis 

duabus, ' 

y t cum dantur duo latera, & a- gulus aliquis: aut 
duo argi.lt cum latere aliquo. 

Iri partibus mijceilaneis datis , fi ab tUo ttym no lateris dati 
in culpes reliquo termino fit angulus datus, cadat ad baftnperpendicu¬ 
laris aut quadrans d,aum illum angulum fubtendens s reduietur non qua¬ 
drantale ad bina quadrantalia f per nonam fttl. cap. 4. hitcus) fc tbilia». 

Vn de gr non quadrant alis partes ( quia cum horum qua - 
irantaim 'm tibiis 3 aut parttum reliquus communes funi) 

F 4 j fate 


4 «» 



LXB. II. GAP IUI* 

facile initotefcunt , cognitu tamenprius per 2 , $* & 4*fc&, cap, j f bu« 
jiu > 4M ex hjpothefipartium fpcctebus. 

Exemplum duorum Uterum , & anguli interp 0 - 

fiti datorum, 

V T fit fufas & exercitii gratia) Triangulum Spheri¬ 
cum non quadrantale in fuperficie primi mobilis 
deferiptum P* Z. S. polum,Zentth, & folem referensrcu- 
jusfexpartes funt,latus P.Z , quod eftinterftitium poli 

& Zenith, feu complementum ele¬ 
vationis poli. Latus Z. S , interfti- 
tium Zenith & folis, feu comple¬ 
mentum altitudinis folis* Latus P. 
S. interftitium poli Sc folis, feu com~ 
plementum declinationis folis ab 
aquatore. Angulus Z, P. S. hora 
diei,feu tempora horaria sequatoris. 
Angulus P* Z* S/quse plaga 
eft, feu azimuth folis a fe- 
ptentrione. Angulus P. S. 
Z, qui angulus efHitus & 
politionis folis ad polum 
& zenith. Harum fex par¬ 
tium dentur tres quacun¬ 
que miftellanese. Verbi 
gratia, angulus horarjus Z.P.S.42. 2$. 5^.fquihoram 
fecundam 4$. 59« 5<f. pomeridianam notat) & latus 
P* Z. 3 4. complementum elevationis poli. a tque lacus «P 
S* 69- complementum declinationis folis. Ex quibus,ut 
acquirantur tres reliquse partes, ab. Z. termino lateris. P 
Z* dati,dimittatur perpendicularis Z. M , aut, fi mavis, 
quadrans Z, H. angulum datum Z. P. S. fubtendens, re- 
ducenfque non quadrantale oblatum P. Z. S. ad duo 
triangula in angulo M* quadrantalia, qusefuntP.M. Z, 
&Z. M. S, ut in primo fchemate: vel(fi varietate delefte- 
risj ad duo triangula in latere Z. H. quadrantalia , qme 
funt Z» H t P, & Z. H*S*ut in fecundo fchemate: Quorum 

" qua- 



I 



LIB II. C AP. V. o 4f 

quadrantalium omnes parces per p.fc&cap 4.bujus acqui- 
res.Nam ex dati s P Z 3 4.aeque Z p M/eu Z p $ 41*9' 
5^'.invenies perpendicularem Z M 22.if 4 T & egulam 
P Z M. 52.4^. 3I & latus P M 2^.26 2? quo ablato i 
P S 69, reftat M 842,3 3.3 *quo & perpendiculari Z M -jam 
cognicis,inveniesperp*fe< 9 :cap.4-hujusangulu MSZfe» 
qusefitu p S Z,31 4 f • & latus qusefitu S Z 47. atque angu - 
lum M Z S 67 .38« ii quo ad P Z M 5 2.4 £ 38 .addito, fit an¬ 
gulus reliquus qu^fitus p ZS 120.24.49", Tres itaque ha¬ 
bes partes quaditas officio perpendicularis Z M, primi 
fchematis. Eafdem quoque officio quadrantis Z H 
fecundi fchematis venari poteris. Ex datis enim , i*t 
fupra, PZ 34. &Z P S fcu Z p H 42 29.5?»^ 
venies per eandem p arn c ap.4- hujus, angulum ZH P. 

1 f v 47,&angulum P Z H 142. 4$, 3$,fr iatus PHn6. 

2p.ex quo aufer P S 69 *reflat SH 47-26'* 29, quo 8c 
angulo apud H 22. rL^*.jam habitis,invenies per 9 cap. 
4. hujus, angulum HS Z 148. 53,5sbejufqueadfemicir- 
culumreliquumfciltcet31.5. $ .angulum P SZ qusefitumt 
atque latus qusfitum S. Z 47.Denique angulum HZ S 

2 2.21.4.p quoexH Z P t42.46'.38.ablato, reftat anguitts 
reliquus qusefitusP Z S120.2 4'. 49'‘.prorfus iit fupra. 

Admonitio. 

* s, V- f 4 • y* - ■ 

H V/us exempli, imitatione novem variae fblvuntur 
hujus fr cujfifquetrianguliquiteftiones.Ex datis enim 
elevatione poli, hora diei, & declinatione folis illius dici, 
habetur (uffupra) primo azimuth feu plaga folis, fecun¬ 
do altitudo folis * Tertio angulus pofitionis folis. Item 
datis declinatione folis, angulo pofitionis folis, & alti¬ 
tudine folis, habetur quarto plaga folis,quinto elevatio 
poli, fexto hora feu arcus horarius* Item datis altitu¬ 
dine folis, plaga folis, & elevatione poli, habetur fepti- 
rao hora diei, o&avo declinatio folis, nond denique an¬ 
gulus pofitionis folis. 

G S mnium 


I l B II; C A P 



d ecuncium exemptum anorum angu* 

Urum ^ & Uteris inter datorum^ 

P Recedentium fchematum datis angulis , ho¬ 
rario fcilicet Z. P* S* 42. 25/. 5p,° & plaga: folis 
P« Z. S; 120, 24',4p-^um complemento elevationis 
poli, latere fcilicet interpofito p. Z. 34. Tres exte¬ 
ra? partes exquiruntur* Nam habitis primo (utfupra ) 
Z.M.22. iC* 47*, &p.M« 26. 2<5V2p*.&angulop.Z ?M. 52, 
46'* 38". quo exp* Z* S. 120. 24'. 4p v '* ablato relido- 
que angulo M.Z.S. 67.38'. 1 ?♦ ex hoc atque Z. M jam 
notis,invenientur tandem latus quxfitum Z Si 47^ Z 
S* M. live angulus quxlitus Z- S. p, 31. 6 '♦ f. atqueATS. 
42« 3 3 # 3T» quo ad p. M* addito, fit reliquum latus qux¬ 
fitum p* S, dp* Hafque beneficio perpendicularis pri- 
mifchematis habes, necfecus eafdem ofBcioquadrantis 
fecundi fchematis acquirere poteris: acquiruntur enim 
(per nonam quarti hu/us ) ex datis, anguli P. H Z. & 
P* Z* H,& ex hoc fubdudo P* Z. S. dato, refiat S, Z, H* 
quo, & angulo P, H.Z.jam notis propalantur exterx om¬ 
nes partes e 


Admonitio, 

• " .■ nin2 dc 

H Vjus exempli imitjti^rW novem varix foivuntur 
hujus, & cujufque trianguli quxftiones. Ex 
datis enim (ut fupra) hora diei, elevatione poli, & 
plaga folis, habetur primo declinatio folis, Secundo 
angulus politionis folis,tertio altitudo folis. Item datis 
hora diei, declinatione folis, & angulo politionis fo!is,ha- 
betur quarto altitudo folis, quinto plaga folis, fexto eleva¬ 
tio poli Jtem datis angulo politionis lolis^ltitudine folis, 
& plaga folis acquiritur feptimo elevatio poli o&avo hora 
diei, nono declinatio folis? 


Tertium / 


tIB II 


V* 


Tertium exemplum duorum datorum laterum ^quorum 
quadranti propinquius fubtendit angulum datum . 

P Rxcedentiurn fche matum dentur latera P. Z. 34. & 
eo quadranti propinquius Z.S. 47, cum eo quem hoc 
fubtendit angulo Z. P. S.42. 29.59. acquirantur perp. 
jfe&.cap4-hujus Z.M.22.11.47. &P-. 2 » M. 52.45.38» 
& P. M. 26*20 29.& fimili modo habebis Z.S.M.fcuquadi¬ 
tum angulu Z.$.Pxertiffime enim fcitur hic per 2, fent,cap a 
g*hujus, minor quadrante, fcilicet effe 31« & non 

effe 148,53.5?. Habebis etiam angulum M. Z.S.67» 
38* i?-quoad P. Z. M. 52. 4S. 3 8* addito , fit reliquus 
qnxfitus angulus P. Z. S. 120, 24.49. Habebis denique 
& M* 8,42.33*3tiquoad P.M.26.26.29 addito.fit quxfi- 
tum latus P,S, 69, Nec fecus eadem acquirere poteris 
(fi libet) officio quadrantis Z. H.fecundi fchematis. 

Quartum exemplum duorum datorum laterum , quorum 
quadranti minus propinquum fubtendit angulum 
datum: magis autem propinquum fubten¬ 
dit angulum data tantum fpeciei, 

D Hntur latera Z S,47 ,& eo quadranti minus propin¬ 
quum P Z 3 4,, cum eo quem hoc fubtendit angulo 
Z S P 31 oT' deturque quod quem Z S fubtendit (an- 
giiltis fcilicet S P Z.) fit fpecie minor quadrante r Di- 
miffo itaque ab Z, ad bafim P« S. perpendiculari Z.M. (ut 

prius ) aut quadrante Z 
I. ( ut hic ) fubtendente 
angulum datum Z# S, P. 
Acquirantur per p. fe< 5 L 
4. hujus extera: partes, 
ut ( exercitii.& varietatis 
gratia) ex hujusfehema- 
tis quadrante Z. I acqui¬ 
res anguium ZI.S. 22.11*47., &I. Z, S. 157.3$. u.^&S. 
132. 33. 3i.&fimilimodo habebis angulum I.P.Z,& 

G 2 per 




44 


LJR II CAP V. 

per confequens angulum quasfitum S« P. 2 * 42* 2J: 59. 
Quia ex hypothtfi exprefle quadrante minor declaratur, 
alioquin nili ejus daretur fpecies, foret (ex prima cap, 3« 

& quinta fedb hu;us cap. ) incertus: potuit enim aliter 
fuiflfe 13 7,36.1". Habebis etiam fic angulum I, Z.P. 37* 
13. 2'Vquo ex I.Z. S. 157« 38 - 11« ablato, relinquitur re¬ 
liquus quaefitus angulus P.Z.S. 120, 24 4^. Habebis de- 
nique & I. P. 63 . 33* 3'f. quo ex I S. 132» 3 33 % abla? 
to, remanetqusefimm latus P. S. dp. 

Eafdem etiam metas attinges (i beneficio perpendi¬ 
cularis Z, M. primi fchematis > partium logifiicen qusdi« 
veris. 

aA dmonitio. 

\ 

P Rsccedentis tertii & hujus quarti exemplorum imi¬ 
tatione,ododecim vari* folvuntur hujus & cujufque 
trianguli quseftiones.Ex datis cnim(ut in tertio exemplo) 
elevatione poli, altitudine folis, & hora diei, habetur pri¬ 
mo plagafolis, fecundo angulus pofitionis folis, tertio 
declinatio folis. Item datis (ut in hoc quarto exemplo ) 
elevatione poli, altitudine folis, & angulo pofitionis (olis, 
habetur quarto plaga folis, quinto hora diei, fexto decli¬ 
natio folis. Item dacisaltitudinefolis,declinatione folis, 
& hora diei, habetur feptimb angulus pofitionis folis, 
ocfavo plaga folis, nono elevatio poli. Item datis altitu¬ 
dine folis, declinatione folis plaga folis, habetur deci-~ 
mo angulus pofitionis folis, undecimo hora diei, duode¬ 
cimo elevatio poli. Item datis declinatione felis,elevatio¬ 
ne poli, & angulo pofitionis /olis, habetur decimotertio 
plaga folis, decimoquarto altitudo folis, decimoquintb. 
hora diei. Item datis declinatione folis, eletatione poli, 
& plaga folis, habetur cecimofexto hora dici, decimofe- 
ptimp angulus pofitionis folis, & decimo-o£tavd altitudo 
feclis. 


LIB II. CAP. V. 
gmntum exemplum duorum datorum angulorum ^ puerum 
quadranti propinquiorem latus datum fubtendit. 

T Rianguli P. Z. S. primi fchematis } dentur anguli P*$« 
Z. 31. %.", & co quadranti propinquior S. P.Z. 42* 
29 ♦ 59. cum latere Z. S. 47. hunc fubtendente. Ex quibus 
p. S. Z. &S#Z. habetur (per nonam quarti hujusjperpen-* 
dicularis Z. M. 22, 11,47. & caterae partes quadrantalis 
S* Z. M.fcilicet M. Z. S» 67. 3§.«. & M S 42.3 3.3'f. Sicut. 
& ex perpendiculari hoc cum dato Z.P. S. fcu Z.P. M. an- 
gulo,habentur partes omnes quadrantalis Z, M* P. Scili¬ 
cet primo latus quofituin P. Z; certifirme enim fcitur hoc 
( per 2.fentent.cap. 3. hujus) minus quadrante, videlicet 
ede 34. non autem e(fe 146, Deinde habetur p. Z» M. 52. 
4$, 3tfVquo ad S. Z. M. 6j. 3 n. addito, fit quxfitus an- 
giilus pZ. S 120,24.49.. Vltimo habetur p. M. 26. 2$.. 
2% quo ad M. S. 42* 33. addito t fit reliquum latus 
quofitum p. S. 69. Has etiam ipfas partes aliter (fi ma^ 
vis) ex duobus proxime procedentis Schematis quadran¬ 
talibus Z. LS. & Z I.p. acquirere poteris* 

Sextum exemplum duorum datorum angulorum. quorum 
quadranti minus propinquum fubtendit latm 
datum , magis autem propinquum 
) f fubtendit la tus data tan*. 
tumfpccki*. 

T Rianguli P. Z.S. primi Schematis dentur an¬ 
guli Z.p. S. 42. 29. 5^.& eo quadranti minus propin» 
quus Z. S.p.3 1 * 3 « ^ .cumeum fubtendente laterep.Z.34, 
Deturque quod angulum Z.p.S fubtendens, (fcilicct la¬ 
tus Z. S.) fit fpecie minus quadi ante. Ex his datis quaesa¬ 
tur perpendicularis ZM.22.11.47. & ca tero quadrantalis 

P,Z.M.psrtes 3 fcilicttP.ZM.52.4^38 & P.M.2 6.2 J.Zp.Si- 

cut.&cxperpedicularihoccudatoZ.SMfc uZ.S P.31. £.| 
quaeratur partes omnes quadrantalis Z M.S.fcilicet primo 
latus optatu2,5,47,quia ex hypotl efi exprefie quadrante 

G 3 minus 


S 


IIB. ^1^ 0%P-V» 

minii $ declaratur, alioquin potuit fuiffe 15 3>Nam (per x * 
cap, 3. & quintam hujus ) incertum eR nifi ejus exprefTe 
detur fpecies» Deinde angulus M.Z.S. 6 f. g.g it. quo ad 
M« 2. P. 5 2* 4^. 3 addito, fit optatus angulus p. Z. 5. 
1 20, 2 4,45* Denique habetur S.M. 31 • Quo ad 

P. M. 2 b. 26. 29 addito, fit optata bafls P. S.‘ 69 . Eafdem 
retiam partes ex duobus, quadrantalibus P. H.Z. & S.H» 
,Z. fecundifcHematis, qu^m facillime acquirere potens, 

^Admonitio. 

• '4 . j «* *■* • V ' / •' 

• - _ Dii ‘ ' * Si , 

F R^cedentis quinti & hujus fexti exemploru imitatio* 
ne,o<ftodecim varise folvuntur hujus,& cu/ufque trian¬ 
guli quaeftiones.Ex datis enim (ut in quinto exemplojan- 
gulo politionis folis,hora diei,& altitudine folis, habetur 
primo elevatio poli, fecundo plaga folis, tertio declinatio 
falis. Item datis fut in hoc fexto exemplo) hora diei, an- 
gulo politionis folis,& elevatione poli,habetur quarto al¬ 
titudo folis, quinto plaga folis, fexto declinatio folis. 
Item datis hora diebplaga folis, & altitudine folis, habe¬ 
tur feptimo declinatio folis, o&avo angulus politionis fo- 
lismono elevatio poli .Ite datis hora diefplaga folis,& de¬ 
clinatione folis, habetur decimo altitudo folis,undecimo 
angulus politionis folis, duodecimo elevatio poli. Item 
datis plaga folis , angulo politionis folis, & declinatione 
folis, habetur decimotertio blevatio poli, decimoquarto 
hora diei, decimoquinto altitudo folis. Item datis plaga 
folis, angulo politionis folis, & elevatione poli, habetur 
decimofexto declinatio folis, decimofeptimb hora diei, 
decimo-o&avo altitudo folis. Atque ita hujus folius ca¬ 
nonis methodo, quinquaginta quatuor varis folvuntur 
quseftiones ejufdem trianguli non quadrantalis» Cetera; 
inferius folventur. 

j Ex his itaque patet quod duorum angulorum & fuorum 

fnb tendentium Uterum tribas dat is quarti[dum Logmthmns innet ep 

cet 


LXB n. C A P. VI. 


cet } tacita etiam quadrantalium defecriptione, tAb Aggregato enim edi 
hogarithmk angulidr laterisfehi adjacentis datorum? aufer Logarith- 
mum tertii datu>& proveniet inde Logarithmus quarti quafti 0 ip(umque 
quartum^nji fet, incerta Jpecui, innotefcet ♦ 

Vt ex iuperioribus tertio, quarto, quinto, & fexto ex¬ 
emplis percipi poteft. Angulorum enim bafis Z. P. S, & 
>u Z. S* P. & fuorum fubtendenttum crurum Z.S. & Z, P. den¬ 
tur tria, qu^ (verbi gratia) finr crura Z. S. 47 , ejufquc 
Logarithmus 31.28580, & Z. P. 34, ejufque Logarithmus 
5812606, cum huic adjacente angulo Z. P. S. 42,2$, 59. 
cujus Logarithmum. 3921720 adde ad 5812605, fit 
97343 2 6 (Logarithmus fdlicet taciti & fuppreffi perpen¬ 
dicularis Z. M. vel anguli Z.H. S.feu Z. I. P. ) a quo au¬ 
fer, 3128 580 remanet 6605746 Logarithmus quarti Z.S, 
P. qusefiti, Ipfum itaque quartum Z. S. P, erit 31. ' 6 . ^, 
Quouiaai per 2. cap. 3* minus quadrante arguitur» 
Contra autem datis Z.P. 34.ejufq*Logarithmo 5812606* 
& Z. S. 47, ejufque Logarithmo 3128580 , cum huic ad¬ 
jacente angulo Z, S P. 31. 6 > 5* ad cujus Logarithmum 
660 5746,adde 312 8 58o,fitaggregatu(ut fupra)97343 2 6, 
i quo aufer 5812606, provenient 3 921720 Logarithmus 
quarti quaditi, fcilicct Z. P. S,cujus arcus per i.fed.cap^, 
incertus eft an fit 4*. 29. 59. an 137.36. j. nifi declaret, 
hypothefis ina]6rne, an minor fit quadrante. 


De non Quadrantalibus puris* 

C A P. V X. 


H Adienus de partibus mifcellaneis datis: Sequum 

tnr purse. 

Purd fiunt tres partes e) ufdem generis datre* Sunt que aut 
tria latera data y & quaeruntur anguli; aut tres anguli dati 3 & quarm* 
tur Utera* 

' Admonitio- 


P Vr& quamvis (tmp licitate priores , ob difficultatem tamen, 
earmdm merito hic pojleriorem fortimtuy locum . 

A .. - G 4 


In irum- 


LIB. XI, CAP V. 

1« triangulis Spharick primo fumma ex Lgmthmh crurum fuh- 
diiU t a fumma ex Ligarithmis Aggregati & differentig fcmbajis & 
femidiffereniU crurum, relinquit duplum Logarithmi dimidii anguli 
verticalis. . , 

Quia docent Regiomontanus libro 5. cap. 2 , de trian¬ 
gulis^ alii, ut redangulum comprehenfum fub finibus 
rectis crurum, fe habec ad quadratum finus totius : Ita 
differentiam finuum verforum bafis & differenti# crurum 
fc habere ad finum verfum anguli verticalis t quum au¬ 
tem ut illa differentia ad hun c finum verfum, ita redan- 
gulum fadum cx finibus redis aggregati & differenti^ fe- 
mibafis & femidifferentiae crurum, fe habet ad quadra¬ 
tum finus redi dimidii anguli yerticalis ( eft enim 
noviffimum hoc redang ulum ad illam differentiam finu¬ 
um verforum , & hoc ultimum quadratum ad illum 
finum verfum in ratione 5000000. cupla, exiftentefinu 
toto 10000000.) Ideo fiquetur quod, ut redangulum 
fub finibus redis crurum fe habet ad quadratum finus to¬ 
tius , ita redangulum fadum ex finibus redis aggre¬ 
gati & differentias femibafis & femidifferentiae crurum* 
dehabebit ad quadratum finus redi dimidii anguli verti¬ 
calis; & pereonfequcns (cx corol* def,d.cap.i«&prop«4. 
cap. 2. & probi, 3. cap.j.lib.i.) Summa ex Logarithmis 
crurum,fubduda ex Logarithmis aggregati & differentiae 
femibafis & femidifferentiae crurum, relinquit duplum 
Logarithmi dimidii anguli ver ticalis, ut fupra. 

Secundo , Summa ex Logarithmis crurum fubduCki d fumma ex Lo- 
garithmis aggregati & differentia femibafis & femiaggregati crurum, 
relinquit duplum antilogarithmi dimidii anguli verticalis . 

Non enim aliter fe habet fumma ex Logarithmis ag¬ 
gregati & differentia: femibafis & femiaggregati crurum 
hujus propofitionis, ad fummam ex Logarithmis aggre¬ 
gati & differentiae femibafis & femidifferentiae crurum 
praecedentis propofitionis, quam'duplum antilogarith- 
mi dimidii anguli verticalis hic, ad duplum Logarithmi 
ejufdem dimidii anguli verticalis fuperius? quod alterius 
loci cft demonftrare. 


&€dmmti* 


LIB. II. CAP.V I. 


■i 


49 

<*A dmonitio. 

f. T N Spkxricis etiam, bafes veram & alternam, eodem fenjh capimus, 
.L (juo m reUilineis, nimirum alteram pro aggregato, alteram p ro diffe¬ 
rentia cafmm» 

{£. T trito different talis fcmibafis vera data , fubdu&m ex fiwtna diffe¬ 

rentiati um fcmidggrcgato & femidiffer entia crurum, relinquit differ en- 
tialem jemibafis alterna. 

Hujus ratio fundamentalis eft, quod ut tangens femi¬ 
bafis verte fe habet ad tangentem femiaggregati crurum, 
ita tangens femidifirentiae crurum fe habeat adtangen- 
temfemibaQs alterna?. Tangentium enim Logarithmi 
funtfuorum arcuum differentiales per led,22,& 25,cap-3, 
lib.-i. Vade hanc tangentium analogiam fequetur illa 
Tuorum Iog«rithinoium,feu differ e ntialium aequalitas per 
prop. 4xap.2n1b. 1. Verum quia hujus analogia? tan¬ 
gentium fundamentalis , hadeniis ignotae, demonftratio* 
nema me forte requirent Ledores, eam ideo, quantum 
hujus compendii brevitas patitur, hicexplicabimus. 

Sph^ra itaque A F P G incumbat plano H IK Q,,ut fc 
invicem tangant in communi pundo A, a quo per Sphae¬ 
ra? centru o erigatur reda A o P fecans fupremum Sphae¬ 
ra: Hemifpha?rium in pundo P.eritque ita AoEperpen- 
petidicularis plano H I K deinde angulo A defcriba-. 
tur in Sphterte fuperficie triangulum A *7 in ^acutum, 
aut A 'a ]s in 9 > obtufum, & protradis femicirculis A* P,& 
A yPifeu A £ P, polo*, intervallo Ky , feu ei aequali *£ 
ducatur circulus tAfry, fecans * P in * & * A in & A £ y 
in pundis# & y. Ex pundo* in arcum A @y dimittatur 
perpendicularis arcus a y.. Erunt itaque hic A Acrus ma¬ 
jus, a y vel a /3 crus minus , A y & A £ bafes, alte¬ 
ra vera , reliqua alterna, A ^ differentia crurum, & A 5 
aggregatum crurum , quia * *, & a <A ex conflrudionc 
funt aqualia minori cruri a y feu* His peradis, & 
fuppofito P vicem gerere oculi aut lucidi cujufpiam , ab 
eodem P in fubjedum planum HIK Qdimittantu£,radius 

H P y 


fQ 


LIBER II. GAP. VI. 

p 



Qj 

P y fecans planum in c, & radius P /2 fecans planum in b: & 
quia y /3 A tti eodem plano feu criculo funt cum lucido P a 
erunt fuas umbrae c b A in eadem reda. Similiter ab eodem 
pundo P* in idem planum dimittantur radius P « fecans pla- 
numdn e, & radius P «T fecans planum in d. & qtaia s in 
eodem funt plano & circulo cum lucido Pt ideo luas um¬ 
brae e d A erunt in eadem reda. Pmerea quia P o A, 
















LIB, IX. CAP*YL "$i 

eft plano orthogonalis feu redhngula,ideo triangulaP.A. 
d, & P .A. e : atquc P A b,& P A c.funt in A redtangula,at- 
que ideo etiam A d efttangens anguli A.P <X, feu A P.d, 
&Ae eft: tangens anguli A P «vel A.P e fic etiam Ab eft 
tangens anguli A P£ vel A P b 5 & A c eft tangens anguli 
A P y vel A P c,pofito gnomone feu firiu toto P A,& quia 
Ad eft ta.ngens anguli A P <A& A P eh eft dimidium angu¬ 
li A q S' per 20.prop.3 .Eucl.(qu'od hic fit in centro, ille 
in circuferentiajideo A d eft tangens dimidii anguli AoA 
feu (quod ide eft) dimidii arcus A J\quod eft femidifferen- 
tia cruru.Similiter quia A e ed tangens anguli A P qangu- 
lus autem A in circuferentia fit dimidiuanguli A O sin 

centro, ideo A e ed tangens dimidii A O e,-feu dimidii ar¬ 
cus, A s,quod ed fena i aggrega tum crurum» Simili modo 
inbaftbus vera & alterna erit Ab tangens anguli A P & 
feu dimidii anguli A feu dimidii arcus A & quod ed 
altera femibafis: atque A c erit tangens anguli A P y y 
feu dimidii anguli A Oy t feu dimidiiarcus A y, quod ed 
rdiqua femibafis. Quamque jam odenfum fit quod A b 
fit tangens alterius femibafis, & A c tangens rtliqu^ femi¬ 
bafis, atque A d fit tangens femidifferentise crurum, & A 
e tangens d mi aggregat i crurum. Dico quod ut A b tan¬ 
gens femibafis verae fe habet ad A e tangentem femiaggre- 
gati crurum, ita A d tangens femidifferentiae crurum ad 
A c tangentem femibafis alternae: vel contra ex alterna 
veram faciendo, ut A c tangens femibafis verse fe habea t 
ai A e* jitem femieggregati crurum: Ita A d tan¬ 
gens femidifftrentise crurum ad Ab tangentem femibafis 
aIcernar. Quod fic probo. §\ pun&a b c d c fint in eodem 
circulo,erit (per 3(5prop.3-& i6prop.6», Eticlid.)ut A b 
ad A cjta A d ad A c. & contra,&c.ut jam diximus. Ve- 
runi ourufta b c d e cadunt in eodem circulo: Omnis enim 
circuli, in fuperficie Sphaera? deferipti umbra a lucido in 
v eadem fuperficic, quod pon eft in circuli peripheria pro¬ 
cedens cirpulum facitperfe&e rotundum in plano ortho- 
gonoad r- diam, qua? a lucido per ^centrum Sphaera? pro¬ 
greditur, ut ex Opticis, & aftroiabii fabrica patet. At hic 

H 2 circiu 




LIBER II. CA P. VI» 
circulas /'£ y un fphaeras fuperficic defcribitur, & lticidup, 
e£l extra circuli peripheriam,quasq. ab eo procedit re&aper. 
centrum(videlicet P o A)eft ad planum orthogon&Nccef- 
farid ergo ejus circuli umbra, quas in punda d»b.c,e. incidit 3 
circularis eft.& perfede rotuda«Ergo ut fehabent A.b.ad A. 
e. Ita A.d.ad A,c.& cotra/id eft,ut tangens femibafis verse ad 
tangentem femiaggregau cruru, ita tangens femidifierentia? 
crurum ad tangentem femibafis alterna ♦ & per confequens, 
differentialis femibafis verae,fubdu&us ex fummadifferentia 
liumEamiaggregatf& femidifterentise crurum aquaturtdiftc- 
remiali femibafis alternas,quas demonftrandafufcepimus. 
XJnde trianguli Spharici datis tribus lateribus,habetur triplici medo 
Angulorum quivis . 

Brimui modus eft, ut latus quodvis ( pracipne quadranti proximum) 
pro bafifiatuas. Inde [emidifferentiam crurum,&ad femibajim addas, 
& a femibafi fubfirabas: produBi & refidui Logarithmes addas, hinc 
auferas aggregatum ex Logarithmis crurum,reliqui bipartiti Logarith- 
mi arcum duplices, & proveniet angulus verticalis, at que ita eat eri. 

Vt trianguli P Z S repetiti, dentur latera P Z 34 gr. & Z S 
47 gr* & $ P dp gr, Quaerantur anguli,primoque quadranti 

proximus P Z S angulus, quem S P 
69 (latus fcilicet quadranti proxi¬ 
mum) fubtendit. Hoc itaque S P 69 
pro bafifiatuatur, Inde femi-diffe¬ 
rentiam crurum p Z, & Z S, videlicet 
6 . 36. Et adde ad femibafim 34.36. 
fientque4i aggregatum: & fubftrahe 
ab ea, fientque iS» refiduum. Loga- 
ritbmos graduum 41,fcilicet4215044,& graduum 28, lci- 
licet 7561472 adde, fient 11776516. Similiter crurum F«Z- 
34 , & Z S. 47. Logarithmos 58x26o6,&3i2858oadde> 
fientq*8p4T 186.quibus ex 11776516 ablatis,fient 2 83 5 3 3 o: 
cujus dimidio Logar. 1417665. refpondentem arcum, vide¬ 
licet 60.12. 24- duplica, provenient 120» 24. 4y. angulus 
verticalis P.Z. S. quafitiis. Nec fecus angulos reliquos, fi 
libet, invenire poteris*, facilius tamen per 9. cap. 5. hujus in» 
&ptcfcent> quia per 2. fentent» cap. 3. funt certas fpecici*. 

Secundus 




LIB. IL CAP, VI. ^ 53 . 

Secundus modtts eft } ut laure quovis (prdcipue quadranti proximeJ 
pro bafi Jiat uto, Semi-bajtm & adfcntii aggregant» erarum addas , & 
ab eodem fubff rabas ; froduBi & rcfidut Lcgaritkmos addas 3 & bine 
auferas aggregatum tx logarithmis erutu , reliqui bipartiti aut; legant lo¬ 
ra i arcum duplices .& proveniet inde angulus verticalis: atque ita eat eri . 

Vt cjufdem trianguli P,Z.S,(corsftituti utinpramifTa) 
femibafim 34.36. & ad feroiaggiegatura crurum 40. 36. 
adde, fient que 75. & ab eodem fubttrahe, fie n tqu e d^uo- 
rum 75.& 6 graduum logarithmos 346683, & 22 582951 
adde, fientque 22929634. Hinc aufer aggregatum cxLo- 
garithmis crurmn, quod (ut fupra) eft 8941186,fientque 
13988448. Qua* bipartire, fient inde 6994224 antiloga* 
rithmus conveniens-arcui 60, iz* 2^~. cujus duplum 120. 
24» 49. eft(ut fupra) quaefitus angulus P; Z 5 veruculis. Cae- 
tcros licet etiam hoc modo , facilius tamen.per 9. cap. 5; 
hujus invenies angulos. Sunt enim per 2., fentent. cap. 3.. 
notse fpeciei. 

Tertius modus efb\ut latere quovis pro bafi pojito, differentialcmfe- 
mi-aggregati crurumad dijfcrerittalcm femi-differ enti a crurum addas, 
& a produth auferas dfferentialem femi-bafis vere , & proveniet inde: 
dff&entialis fcrni-bafu alterna: quarum (cmt-bafium frmma cH cafm 
major, & differentia cafm minor, duodijtinguentes n [langula , qu&g? 
fuas 3 &tpfiits oblati trianguli partes dmnes (per nonam cap, 4. & olla- 
vam cap. S, hujus) notas reddunt. 

Vt propofiti trianguli P. Z. S< datis lateribus ut fupra, 
quaerantur anguli apud bafim ZP.y.&Z SP. Semi-aggre- 

gatiim crurnm P Z,& Z S. eft 40.36. 
SemidifFerentia crurum eft 6.36. 13 - 
11us differentialiseft 15772 96, hu¬ 
jus vero eft 217 21209. Quos adde, 
fient 23298505. Hinc aufer femi- 
bafis verse 34. 3 6. differentiakra 
3750122* remanent 19548383 , dif¬ 
ferentiatis 8.3.31, pro fem i bafi alter¬ 
na, addeergo femibsfes 34.30.& 8.3,3'{.fient inde 42.3 3.3'f, 
pro tnajore cafu M S.&fubfirahe S.^T.a 34.3 6.relinquen¬ 
tur 26.25,2^ pro minore cafu P M. horu itaque cafuum offi- 

H 3 cio ha- 





' LIB.IL CAPLVL 
cio habes duo jamredangula in M. Lcilicet p MZ, & S 
M Z: qbx &pefpendicularemZ M, & angulos verticales 
p Z Mj& SZ Mjaut, li libet, ipfttm p Z S , patefaciunt: 
(per nonam cap^/S^odfcavam cap.5 hujus.«)$ed hisomif- 
flsad quxfitos bafis angulos 2 p S , & Z S p, redeamus. 
Cafus P M , 26, ia, 29. jam acquifiti difFerentialem 
69 85518 (per 9, cap.4,paddead differentiarem comple¬ 
menti p Z, fcilicet ad dilferentidlem 56 ± c{ui eft —- 
3937709 , provenient -f- 3047809 Logarithmus com¬ 
plementi anguli Z p S , quod complementum eft 47.36« 
similiter calus S M, 42,3 3', etiam acquifiti dif- 

ferentialem 85.3239 (per eandem nonam feci.) adde ad 
differentialem complementi p ZXcilicet addifterentialem 
43 gr.qui eft d98698jprovenient -p 15 51937 Logarith- 
mus complementi anguli Z S pjquod complementum eft 
58.5 3 . 5 ?. Memor autem hic fis non ipfas partes p Z* 
34, &Z p S, autpz 47, & ZSp,fedfuacomplementa, 
viz. 56 gr.&47. 36. r. & 43 gr* & 58, 53.5f* circulares 
partes hic dici per fecundam cap. 4. hujus. Verus itaque 
angulus quarfitiis Z p S eft 42,29* 52* & Z S p eft 31. 
ut etiam ex fed. odava/cap. 5 . hujus patet* 


o /liud ejufdem trianguli exemplum. 



E Odem triangulo p. Z. S ♦ aliofitu conftitntoj fit.?. Z. 

bafis 5 & datis lateribus ut fupra * quaeratur angulus 
P. Z. ?. Crurum itaque S. P. 69- & P &• 34 fimi-aggrega- 
tum eft 51.36^ ejufque difterentialis — 2288650: femi- 

differentia vero eft 17. 30, 
ejufque differenbalis eft 
+ 1 1542 341. Quos dif- 
ferentiales adde erit fum» 
ma -p 92 5 3 691. a qua au¬ 
fer differentialem dimidii 
bafis S Z videlicet dif¬ 
ferentialem 23*36. qui eft 
8328403,remanebit925 2 88 difLerentialis arcus 42.21 Au 
profemi-bafi alterna. Adde ergo femi-bafes 42, 21,11-& 


23 ♦ 3 *o • 




' ZTB. II. .'CAP. VI. ff 

'23. ?o.prouenient 65. $L u; pro majorecafu ST, & tum 
fubftrahe23. 56. a 42, 21. if v remanent 18. 51.11. pro 
minore cafu T X , vd T Z. Hujus ergo diflferentialem 4 * 
10749201, adde ad difFerentialem complem, Z P, fcilicet 
ad difFerentialem grad, 5<5,qui eft — 39-37709,& prove¬ 
nient inde 6807492 Logarithmus complementi an¬ 
guli P ZT. Arcus autem in tabula refpondens huicLo^ 
garrthmo 6807492 ex advei fo , efi graduum 59. 39. ri* 
pro angulo p Z T , cujus anguli p Z T } quum angulus 
quasfitns P Z S 5 fitad femi-circulum reliquus (quod Tcm- 
per occurrit quum bafis alterna eft major vera ) erit ne- 
ceffe pZ S.cfie graduum 120.24, min.49.fed. alioquin fi 
bafis vera alternam fuperayerit, coincident anguli p Z T, 
& P Z S, & sequales erunt. 

Admonitio. 


T Res jam'habes veros modos inveniendi angulos cx 
datis lateribus , quorum unoquoque tres varia fol- 
vuntur hujus, & cujufque trianguli quasftiones, £x datis 
enim elevatione poli, altitudine Tolis’, & declinatione To¬ 
lis, dubitantibus fatisfit ad quaftionem qua vel plaga To¬ 
lis, vd fecundo angulus fitus & politionis Tolis, vd tertio 
hora diei queritur. 

Huc ufque ex lateribus invenimus angulos . SupereB ex 
angulis invenire latera ♦ 


In omni triangulo jpberico mutari poffunt later a in angulos . cS- angu¬ 
li in latera: ajj . rupis tamen pius pro mico quovis angulo , & fuo jub- 
tendente Utere (uu ad (emicirculum reliquis . 

Exempli gratia . 

Efto triangulum QJ}. T, cujus fint an¬ 
guli 0^47- R iii. Sc T 34. Sumamus 
primo pro angulo quovis 5 videlicet pro R 
s 11 futim adfcmieirculum reliquum,quod 
efl 69. grad. Hko hos angulos 47. 69. & 
74..mutari pofie in latera, & fiet fuperius 

H 4 trian- 




f$ LTB. IL C A P» VI* 

triangulum p Z S. In quo p Z» eft 34 grad, 2 S, eft 47 
grad-& p S eft 69 grad. ut edam ex illius trianguli, p Z 
S,angulis,fient hujus mutuo latera. Nam Z S P angulus 
grad, 31. a/f illius, eft latus Qjt hujus: &angulus Zp 
S grad. 41* 29. 59. illius,eft latusR T hu/us: & tertii an¬ 
guli illius, qui eft grad. 120,24.49. reliquum adTemi-cir¬ 
culum, quod eft 59.3^. 11. eft latus Q^T hujus. Gu/us 
rei de monft ratio nem exhibent Barthojomsus PitifcUs, 
Adrianus Metius, & alii. Eam igitur hac epitome mini¬ 
me repetendam cenfeo. 

i tz* TJride trianguli Spherici datis tnbtu angulis y facili cenverfione ae*° 
qmmntUY Utera . 

Vt praecedentis trianguli QJR T dentur anguli 0^47. 
R ni.&T 34 Querantur autem latera, Pro angulo quo¬ 
vis unico , verbi gratia ( utfupra ) proR m. fumatur 
ifuum ad femicirculumreliquum 69. grad. Inde pofitis 
47, dp, & 34. pro lateribus, ut in triangulo fuperiore p 
Z S fa&um eft, per quemvis ex tr ibus modis fuprafcriptis, 
quaere illius anguk> 3 ,& inveniescontra latus 47.angulum 
42,29. 59 contra latus 34, angulum 31,^. & contra 

;Iatus 69 (quod pro 111 pofuimus)reperies angulum 120. 
2.4,4^. Ideo in triangulo oblato QJiT, pro latere R T. 
Tubtendente angulum Q^ 47 > pone 42,29.59, Et pro la¬ 
tere QjT.fubtendente angulum T, 34, pone 31.4.5, ve¬ 
rum pro latere QJT ftibtendente angulum R m , pone 
59. 35.1l.q1ve funt reliquum graduum 120. 24. 49 . 3d 
Tem i-circulum : quia prius pro 111 Tumpferas fuum ad 
mi-circulum reliqunm, fcilicet 69 . Et ita ex angulis per 
converfionem acquires latera. 

Admonitio . 

E X hac laterum per angulos datos inventione 
tres vari# Iblvuntur hujus, & cujufcunque trianguli 
qmeftioncs. Vt in triangulo P Z S, Ex datis, hora diei, 
plaga folis, & angulo ficus vel politionis folis, h#c praece¬ 
dens fatisfacit quseftioni, qua vel primo elevatio poli, vel 

fecunda 


LIBER II. CAP. VI. 57 

fecundo altitudo folis, vel tertio declinatio felis quera¬ 
tur. Exodtava itaque fe£Lprocedentis cap. 5. &fepti- 
ma ac duodecima hujus, {exagitata habes variarum quo„ 
ftionum folutiones, que in quodque triangulum cadunt, 
Nechisplures ex multiplici trium quarumlibet partium 
compofitione oriri poflunt variationes. Perferam igi¬ 
tur habeS;& abfolutam triangulorum tam Sphericorum, 
quam planorum dodtrinam* 

qoNavsia 

Aiis ergo jam oslenfium efiquodfint, quidfint. (fi cuius 
^i| ufus fint Logarithmi: Eorum enim beneficio abfique 
multiplicationis 5 divifionis > aut radicum extra&ionis 
mole Hi a , omnis Geometrica quaslionis ficlutionemlogifii- 
cam promptifii me exhiberiJum apodeiclice demonfiravimus , 
tum exemplis utriufique Trigonometria docuimus , Promi fi • 
fium itaque mirificum Logarithmorum canonem habetis , 
eiufique amplifsinmm ufium : qua fi vobis eruditioribus gra . 
ta fore ex referiptis veflris intellexero , animus mihi adde¬ 
ntur , ad tabida condenda methodum in lucem etiam proferen¬ 
dam. Intcrim hoc brevi opuficulo jruamini , D coque opifici 
fiummo, omniumque operum bonorum opitulatori laudem 
fiummam (fi gloriam tribuite. 

; ■ v • > • r ( - * «- 

; ; .. 

.... 4 • ^ ; 

Errata ante le&ionem emendanda. 

I yJg.42.Hnea 10,pro 13 lege iuTag.43JM-22.pto % kge^Jbtda 
lm.33.pro cE S .lege Et SJ.Pag.4Sjw-16.pro 5000C00.cupla lege 
50oooco. cupla , 0"intellige quinquies miRies millecupla* Tag49‘ 
iin. 27 prox p infige x Pin ifiao.s0Jin.2pro criculo lege circulo 
Tag.s3Jtn.12.p10 Sadi.kgeStc.Tag.s6Jin.3pro angulis hen tjege. 
angulis, dent.ibidem Un.28.pro mi-circulum lege {emi-circu?lum* 
Cetera prudens LeUar facile emendabit 4 

Sequitur Tabula feu canon Logarithmorum. 


I 



mm\ 

' Sinus, | ] Lojrarithmi j 

Differentia j 

UsArithmi J 

Sinus ^ 


© 

o 

infinitum 

infinitum 

0 


looooooo 

60 

i 

2909 

81425681 

81425680 

1 


iooooqoo 

59 

2 

5818 

74494213 

74494211 

2 


/999999« 

58 

3 

8727 

70439564 

70439560 

4 


9999996 

57 

4 

11636 

67562746 

67562739 

7 


999999 3 

56 

5 

H544 

653 31315 

65331304 

11 


9999989 

55 

6 

17453 

63508099 

63508083 

16 

[ 999.9986 

54 

7 

20362 

6 1966595 

6x966573 

22 

i 9999980 

53 

8 

23 271 

60631284 

60631256 

28 

! 9999974 

5i 

9 

26180 

59453453 

594534*8 

33 


9999967 

51 

IO ' 

2 9088 

58399857 

58399814 

43 


9999959 

5o 

11 

31997 

57446759 

57446707 

3i , 


9999950 

49 

12 

34906 

56576646 

56576584 

62 


9999940 

48 

13 

3781? 

55776222 

55776149 

73 


9999928 

47 

M 

40724 

55035148 

55035064 

84 


9999917 

46 

I? 

43 6 3 1 

54345H5 

54345129 

96 


9999905 

45 

16 

46341 

53699843 

53699734 

109 


| 0999892 

44 

17 

49450 

53093600 

53093577 

123 


9999878 

43 

18 

51359 

52522019 

52521881 

138 


9999863 1 

42 

19 

55268 

51981356 

5 1981202 

134 


9999847 

41 

20 

58177 

51468431 

51468361 

170 


9999831 

40 

2 l 

6 1086 

50980537 

50980450 

187 


9999813 

39 

22 

63995 

50515341 

50515137 

205 


9999795 

38 

13 

66904 

50070827 

50070603 

224 


9999776 

J7 

24 

I 69813 

49645239 

49644995 

244 


9999756 

36 


, 727 21 

49237030 

49236765 

265 


9999736 

35 

26 

| 75630 

48844826 

48844539 

287 


99997*4 

34 

27 

78539 

4846743I 

48467122 

309 


9999692 

33 

28 

81448 

48103763 

48103431 

331 


9999668 

3i 

29 

843*7 

47752859 

47752503 

356 


9999644 

3. 1 

30 

87^65f 

474!3851 

47413471 

381 


9999619 

30 













































































































Gr. o 


mtn [ Sinus 

j Logdrithmi | Differentia | iAgdYithmi 

i j Sftws 


3° 

3 i 

3 z 

87265- 

90174 

93083 

47413852 

47085961 

14676848 3 

47413471 

47085554 

46768049 

381 

407 

434 

9999619 

9999593 

9999566 

30 

29 

28 

33 ' 

34 

35 

95992 

98901 

101809 

46460773 
46162254 
45872392 

46460312 

46161765 

45871874 

461 

489 

518 

9999539 
99995 11 

9999482 

27 

26 

25 


548 

579 

6 11 


36 

37 

38 

104718 

107627 

110536 

45590688 

45 3 1 67 1 4 
45050041 

45590140 

453 i 6 i 35 

45049430 

9999452 

9999421 

9999389 

24 

23 

iz 

39 

40 

4 5 

H 3445 

116353 

119262 

44790296 
44537332 
44290216 

44789652 

44536455 

44289505 

644 

677 

7 f ! 

. 9999357 

9999323 

• 9999289 

2 I 

20 

19 

42 

43 

44 

45 

4 6 

47 

48 

49 
5 ° 

51 

53 

122171 

125079 

127988 

44049255 

43813959 

43584078 

44048509 

45813177 

435832?? 

746 

782 

819 

9999254 
9999218 
999918I 

l8 

I? 

16 

130896 

133805 

336714, 

433593 < 5 o 

43139582 

42924534 

43358503 

43138686 

42923599 

857 ’ 

896 

935 

9999143 

9999105 

9999065 

15 

H 

j 3 

139622 

34M31 

145439 , 

42714014 
42507833 
42305826 

42713O39 

42506817 

42304768 

975 

10 1 6 

1058 

9999025 

9998984 

9998942 

12 

11 

10 

148348 

151*57 

154165 

42107812 
41913644 
41723175 

42IO67II 

41912499 

4I72I986 

IlOI 

1145 

1189 

9998900 
9998856 
99988 ]' I 

9 

8 

7 

54 

5? 

56 

I 57°74 

359982 

162891 

41536271 
41352795 
41172626 

41535037 

4 J 35 15 15 
41171299 

1234 

1280 

1327 

9998766 

9998720 

9998672 

6 

5 

4 

58 

59 

! '' 165799 

168708 
! 171616 

41006643 

40821746 

40650816 

41005268 

40820322 

| 40649343 

J 375 
1424 
1473. 

9998025 

9 P 98577 

9998527 

3 

2 

1 

60 

1 1745*4 ; 

40482764 

i . 

40481241 

- 

1523 

8 

9978477 

9 

0 

nnn 
Cjr , 
89 





















































































































Gr. 

I 

min | 


I 


+ 


Sinus [ 1 Lo^drithmi [ DrfferentU [ iogurhhm j | Sinus | 


o 

I 

z . 

.74524 

177433 

Ih’0341 

V 40482764 

1 4031748 3 

j 40154899 

40481241 

40315909 

.40153273 

1523 

1574 

1626 


9998477 

9998426 

9998374 

60 

59 

58 

3 | 

183250' 

39994918 

I 39993239 

1679 


9998321 

57 

4 i; 

186158 

■ 39837448 

39835715 

1733 


■. 9998267 

56 

5 

( 189066 

3968242 r< 

39680633 

1788 


9998212 

5 5 

6 

I9I975 

39529765 

; 39527922 

'«43 


9998157 

54 

7 

i 194883 

393794 0 7 

39377508 

Io99 


9998101 

53 

8 

197792 

39231274 

39229318 ( 

1956 


9998044 

52 

9 

200700 

.39085307 

39083293 

2014 


9997986 

5* 

xo 

203608 

38941441 

38939368 

2O73 


9997927 

50 

n 

206517- 

38799612 

38797479 

2I33 


9997867 

49 

S2 

209425 

38659767 

38657573 

2194 


9997806 

48^ 

J X 

212333 

38521858 

38519603 

2255 


9997745 

47 

5 4 

2I524I 

38385824 

3-8383507 

2317 


9997683 

46 

15 

2l8l49 1 

38251613 

38249233 

2380 


9997620 

45 

36 , 

22IO57 I 

38119183 

38116739 

2444 


9997556 

44 

17' 

223965 | 

37988481 

37985972 

2509 


999749 i 

43 

7* 

226873 

37859471 

37856890 

2575 


99974 2 5 

42 

19 

229781 

37732105 

37729464 

2641 


9997359 

41 

2p 

232689 

37606339 

37603631 

2708 

1 

9997292 

40 

21 

235S97| 

37482135 

137479359 

2776 


9997224 

39 

22 

238505 

373594 1 ) 8 

373*6613 

2845 


9997155 

38 

23 

241413' 

37238269 

i 37235354 

2915 


9997085 

37 

2-4' 

244321 

37U8532 J 37115546 

! 2986 


9997014 

36 

25 

247229 

37000208 

36997150 

3058 


9996943 

35 

- 2,6 

25OI37 

36883272 

36880142 

3130 


9996871 

ii. 

' 

27 

25 3°4* 

3676769O 

36764487 

3203 


9996798 

33 

28 

255953 

36653428 

36650151 

3277 


9996724 

32 

2-9 

258861 

36540448 

36*3739«1 

3352 


9996649 

.ii 

30 

I 261769 

i 1 

l364*8748 

36425320 1 

i 1 1 

3428 

. 


9996573 

30 


















































































































Gr. i 


min | Sinti.t | [ Logdrithm | Differenti * ] logdYtthm i j 1 


3 0 

3 1 

3 z 

261769 

264677 

267585 

36428748 

36318272 

36209009 

36425320 

363 14768 
36205427 

3428' 

3504 

3582 


9996573 

9996496 

99964.19 

1 3 ° 

2 9 

j 28 

33 

34 

3 < 

270493 

273401 

276308 

36IOO924 

34994OOO 

35888207 

36097264 
3599026 I 
35884388 

3660 

3739 

3819 


9996341 

9996262 

9996182 

27 

26 

. zt > 

36 

37 

38 

279216 

282124 

285032 

35:783520 

35679917 

3557738 0 

35779620 

35^75935 

35573316 

3900 

3982 

4064 


9996101 
9996019 
9995937 

24 

23 

2i 

19 ; 

40 

41 

28794° 

290847 

293755 

35475892 

35375415 

35275935 

35471745 

35371184 

35271619 

4147 

4231 

4316 


9995854 

9995770 

9995685 

21 

20 

19 

4*2 

43 

44 

296663 

29957 ° 

302478 

35177444 

35079909 

34983320 

35173042 

3507542 0 

34978743 

4402 

4489 

4577 

1 

9995 5/99 

9995512 

9995424 

18 

'7 

43 : 

4 6 

47 

30558? 

308293 

3 11200 

34887<$52 

34792895 

34699029 

34882987 

34788141 

34694185 

4665 

4754 

4844 


9995336 ’ 

9995247 

9995 H 7 

15 

14 

1? 

48 

49 

50 

314108 
317015 
319922 

34606036 
34513899 
34422 606 

34601101 
34508872 
34417486 

4935 

5027 

5120 


9995066 

9994974 

9994881 

12. 

11 

10 

9 

8 

7 

5 i 

3 3 

322830 

' 325737 
328645 

3433214 ° 

34242484 

34153629 

34326926 

34237176 

34148226 

5214 

5308 

5403 

1 

j 

9994787 

9994693 

9994598 

54 

5 5 

56 

33*552; 

334459 

337367 

34076549 
33978246 
3389 170 1 . 

34071050' 

33972650 

33886007 

5499 
5596 

5694 

9994502 

9994405 

9994307 

1? 

4 

57 

-58 

-59 

3.40274 ; 

343184 

346088 

33805893 

3 3720820 
33636464 

33800100 

33714927 

33630471 

5793 

5893 

5993 

f 

9994208 

9994109 

9994009 

3 

2 

1 

60 

3 + 8»?5 | 

*i 

1 3 ^?? 2 iii 7 

. 

33546723 

. 

. 6094 

j 

31 

9993908 

3 

0 

HMW 

Gr. 

88 




a z 




















































































































G r. x 


2 Hh •<— 

min | Sinus { i Lo^drithmi J Differentia [ logdrkhmi j | Sinus 


© 

1 

2 

348995 
35 1902 
354809 


33552817 

3 3469860 
33387588 

335467*3 

33463664 

3 3381289 

6094 

6 196 
6299 


9993908 

9993806 

9993703 

60 

59 

58 

** 

5 

3577 16 


33305993 

33299590 

6403 


9993599 

57 

4 

360623 


3 3225056 

33218549 

6507 


9993495 

56 

5 

363530 


33144770 

33138158 

66 12 


9993390 

11 

6 

366437 


33065128 

3 3058410 

6718 


9993284 

54 

7 

369344 


32986107 

32979282 

6825 


9993177 

53 

8 

372251 


3290771* 

32900779 

6933 


9993069 


i) 

375158: 


3z8*99*3 

32822881 

7042 


9992960 

51 

io 

378064 


3275274° 

32745588 

7152 


9992850 

50 

11 

380971; 

, 

32676149 

32668887 

7262 


9992740 

49 

IX 

383878 


32600139 

32592866 

7373 


9992629 

48 

n 

386785 


32524706 

32517221 

748 5 


9992517 

47 

14 

389692 


32449837 

3 2442 239 

7598 


9992404 

46 

15 

39*598 


3*375 ? 16 

32367814 

7712 


9992290 

45 

16 

395505 


32301761 

32293934 

7827 


999*175 

44 

17 

398412 


32228539 

32220596 

7943' 


9992060 

43 

lX \ 

401318 


$215585* 

32147793 

8059 


9991944 

4* 

19 

404225 


32083692 

32075516 

8176 


9991827 

41 

20 

407131 


320I2045 

3200375 r 

8294 


999 1709 

4° 

2 1 

410038 


31940909 

31932496 

«413 


9991590 

39 

2 2 

412944 


31870276 

31861743 

. X533 


9991470 

38 

x3 

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H 7935 ? 3 

14527^2 

266401 


9737116 

50 

11 

2280676 


14781128 

14514046 

267082 


97364*3 

49 

12 

2283508 


14768719 

1450095? 

267764 


973575*9 

48 

I 2 

2286340 


147*6325 

14487878 

268447 


973 * 12 4 

47 

M 

2289172 


14743947 

14474817 

269130 


9734457 

46' 

ic 

2292004 


14731585 

14461771 

2698 14 


9733793 

4 * 

1 6 

2294835 


14719239 

14448740 

270499 


9733 * 2<5 

44 

17 

2297666 

• 

14706909 

14435724 

271185 


97324*8 

143 

i8l 

2300497 


1469459? 

14422723 

271872 


973*789 

42 

19 

2303328 


14682297 

14409737 

27256O 


973 * I 2 Q 

41 

zo 

2306159 


14670015 

14396766 

-273249 


9730450 

4 ° 

21 

2308949 


14657749 

14383810 

273939 


9729779 

39 

22 

2311819 


14645498 

14370868 

2746 30 


9729107 

38 

23 

2314649 


14633263 

14357941 

275322 


9728434 

37 

24 

2317479--. 


14621044 

14345029 

276015 


9727760 

36 

2 5 

2.320309 


14608841 

143 

276709 


972708? 

3 * 

26 

1313158 


14596654 

I 43 I 92?0 

277404 


9726409 

34 

27 

2325967 

J 

14584483 

I4306384 

278099 


972*733 

33 

28 

2328799 

I 

14572318 

14 * 93*33 

278735 


972*056 

32 

29 

2331625 

j 

14560189 

I428o697 

279492 


9724378 

3 i 

-ja»— fcT.fr 

3° 

1 1354454 


14548066 

I4267876 

280li>0 


9723099 j 

30 



















































































































Gr. *3 


min J Simis I | Logarithmi } Differentia 1 hg&ithml | j Sttws 


30 i 

31 j 
32 1 

2.3 34454 
2337182 
2340110 

»4548066 

14535958 

14523866 

14267876 

14155069 

14242177 

280190 

280889 

281589 


9723699 

9723020 

9722340 

30 

29 

28 

33 

34 
.3? 

36 

37 
3^ 

2342938 

2345766' 

2348594 

14511789 

I44997 2 7 

14487680 

14229500 

I4II6737 

I4203988 

282289 
28 2990 
283692 

. 1 - 


97*1659 

9720977 

9720294 

27 

26 

25 

2.35 142.1 
2354148 
2357075 

14475648 

14463632 

14451631 

14191253 

14178533 

I4I65827 

384395 

28509? 

285804 


9719610 

9718926 
9718241 

2 4 

23 

2-2 

39* 

40 

4 1 

4 2 

43 

44 

2359902 

2362729 

2365555 

14439645 

14427674 

14415718 

14153!35 

14140457 

14127793 

286510 

287217 

287925 


97*7555 

9716868 

9716180 

21 

20 

19 

2368381 

2371207 

2374033 

I44 0 3777 

1439185) 

14379941 

I4H5H3 

i4id , 25Q7 

14089887 

288634 

289344 

290054 


9715491 

9714802 

9714112 

18 

17 

16 

46 

47 

2376859 

2379684 

2382509 

14368046 

14356166 

14344301 

14077281 

14064689 
140521 ii 

290765 

291477 

292190 


9713421 

9712729 

9712036 

15 

14 

13 

48 

49 

50 

1385334 

2388159 

1390983. 

14332451 
14320616 
14308796 

14039547 
14026997 
14©14461 

292904 

293619 

294335 


9711343 

9710649 

9709954 

12 

II 

10 

5 8 

52 

53 

2393808 

2396632 

2399456 

14296991 

1428 5200 
14273424 

14001939 

13989430 

13976935 

295052 

29577° 

296489 


9709258 

9708561 

9707863 

0 9 

8 

7 

54' 

55 

5 6 

2402280 

2405 104 
2407927 

14261662 

14249915 
I423818 2 

13964453 

13951986 

I393953 2 

297209 

297929 

298650 


9707165 

9706466 

9705766 

6 

5 

4 

57 

58 

59 

2410750 

2413573 

2416396 

I4226464 

I4214761 

14203072 

13927092 
13914666, 
13902253 

299372 
300095 
300819 


9705065 

9704363 

9703660 

3 

z 

1 

60 

| 2419219 


| 13889854 

301544 

7 


9702957 

> 

0 

win 
Cj r 
76 














































































































14 ^ “ 4 “ ' 

min f Sirus | | Lofaritbmi j Dtjferenti& j logdrithmi [ j Sinus f 


o 

I 

% 

2419219 
2422041 
2424863 


14191398 

14179738 

14168092 

13889854 

13877468 

138650'5 

301544 

302270 

302997. 


9702957 

9702253 

9701548 

60 

59 

58 

-> 

3 

2427055 


14156461 

13852737 

303724 


■9700842 

57 

4 

2430507 


14144844 

13840392 

304452 


9700135 

56 

f 

2433329 


I 4 I 33 2 4 2 

1.3828061 

305181 

i . 

9699428 

c 5 

6 

2436150 


14121654 

13815743 

305911 


9098720 ? - 4 

7 

2438971 


14110081 

1380343» 

300642 


969801i 

53 

8 

244179 2 


14098522 

13791148 

307374 


?; 697301 

52 

9 

2444613 

/ 

14080977 

13770870 

308107 


9696590 

5 i 

io 

2447434 


1 4 ° 75*447 

13766606 

308841 


9695:879 

5 o 

11 

2450254 


14063927 

13754555 

309576 


9695167 

49 

V 

12 

2453074 


1405 242v 

i 374 21 17 

310312 


9694454 

48 

13 

2455894 


I4040940 

13729891 

3 11049 


969374 0 

47 

14 

2458714 


14029465 

1 3717679 

311786 


9692025 

46 

*5 

2461533 


14018004 

13705480 

312524 


9692309 

45 

16 

2464352 


14006557 

13693294 

313163 


969H93 

44 

^7 

2467171 


I3995124 

I 3 6 8 I I 2 I 

314001 


9690876 

43 

18 

2469990 


1 3983705 

13608961 

314744 


9090150 

42 

19 

2472809 


13972300 

13656814 

315486 


9689439 

41 

20 

2475628 


13960909 

13644680 

3 16229 


9688719 

40 

21 

i 2478446 


13949532' 

13632559 

316973 


9607998- 

39 

22 

2481264 


13938168 

13620450 

317718 


9687277 

38 

23 

2484082 

i 

13 9 2 6 818 

I3608354 

318464 


9686555 j 

37 

24 

2486900 


13915482 

13596272 

319210 


9685832 

36 

23 

2489717 


13904159. 

13584202 

319957 


9685108 

35 

26 

2492534 


13892850 

13572145 

320705 


9684382 

2 4 

27 

249535 ' 


13881554 

13560100 

3 2 1 454 

> 

9683057 

33 

28 

2498l68 


13870272 

13548068 

322204? 


9682931 

32 

%9 

2500984 


13859004 

135 36049 

322955 


9682204 

3 i 

3 ° 

1 2503800 


13847749 

13524042 

323707 


9681476 

30 





















































































































Gr. 


14 Hr 

min | Sinus, [ ] L/Ogayithmi j Differeritiit j lotJitYithwt j I Sittttf 


3 o 

3 i 

3 2 

2503800 

2506626 

2509431 

13£47749 
13836508 
13825280 

13524042 
13512048 

5 3 500066 

323707 

324460 

325214 

9681476 

9680747 

9680017 

30 

29 

28 

33 

34 

11 

26 

37 

3K 

2512248 

25 15064 

2 517879 

13814066 

13802865 

13791678 

I3488097 

I3476I4I 
I3464I98 

3M969 

326724 

3 27480 

9679287 
967855 6 
9677824 

17 

26 

25 

2520694 

2523509 

2526324 

13780504 

13769343 
j 1375819 5 

13452267 

I3440348 
13428441 

328237 
328995 
329754 , 

967709I 

96763571 

9675623 

24 

*3 

22 

39 

40 

41 

25291 38 
253 1952 
25247 66 

^ 13747001 
13735940 
.1372483.3 

13416547 

I34O4665 

I 4392796 

330514 

.331275 

322037 

9674888 

9674152 

9673415 

21 

20 

19 

4- 

43 

44 

45~ 

46 

47 

25'37)8 o 

2540393 

2543206 

13713739 

13702648 

13691590 

13380939 

I3369O94 

13357*62, 

5 iSOO 

3 35f<>4 
354318 

9672677 

9671938 
9671 199 

18 

17 

16 

2546019 

2548832 

2551845 

13680535 

13669493 

. 136584554 

13345442 

1 3 333 6 34 
13321838 

3 35°9 3 : i 
335859 

336626 

9670459 
9669718 
9668976 

15 

14 

J 3 

48 

49 

50 • 

2554458 

255727 3 
2560082. 

13647448 

13636445 

13628454 

13310054 

13298282 

13 28 6 5 21 

3 37394 

3 38 i 6 3 

3 389 33 

■ 9668233 
9667490 
96 66746 

12 

11 

IO 

51 

52 

53 

2562894 
2565706 
25685 17 

«3614476 
i 3603 5 11 
13592559 

13274772 

I3263035 

13 2 5 1 3 10 

13239597 

13227896 

13216108 

339704 ; 

340476 

341249 

96 66001 

9665255 

9664508 

9 

54 

5 5 
% 6 

2571328 

2574539 

2576950 

13581620 
13570694 
: 13 5 5 9 78 i 

3 42023 
342798 

' 343573 . 

9663761 
966301 3 
9662264 

U 

f? 

58 

5 9 

2579760 
258 2570 
2585380 

: 13548880 

13537992 

14527117 

13204531 

1319^866 
13181213 

344349 

345126 

345904 

9661514 

9660763 
9660011 

965 9258 

3 

2 

I 

0 

60 

* 2588190 

13516255 

13165572 

346683 


rmn 



i 



75 




































































































































Cir. 


i? 


? 5 Hh 

| j lo^drithmi | Differenti# j lovdriihmi j [ Sfow 


© 

I 

2- 

2588190 

2591000 

2593809 

13516255 

13505406 

13494570 

i3i«5?7i 
i3«57943 
i 3146 3 26 

346683 

347463 

348244 


9659268 

9658505 

9657751 

60 

59 

58 

3 : 

2596(518 

13483746 

13134720 

349026 


9646996 

57 

4 

2*99427 

13472934 

13123126 

349808 


9656240 

56 

5 

2602236 

13462135 

13 1U544 

350591 


9655484 


6 

2605045 

«34?1348 

13099973 

351375 


9654727 

54 

7 

2607853 

13440573 

13088413 

35* 160 


9653969 

53 

8 

2610661 

I34298iO 

13076864 

35 2 946 


0653210 

52 

9 

2613469 

I34I9060' 

13065327 

35 373 3 


0652450 

51 

IO 

2616277 

13408322 

13053801 

354521 


96 51689 

50 

11 

2619084 

I3397596 

13042286 

355 310 


9650927 

49 

12 

2621891 

13386883 

«30307X3 

356100 


9950165 

48 

13 

2624698 

13376182 

130ip29 I 

356891 


9649402 

47 

14 

2627505 

13365493 

13007810 

357683 


9648638 

46 

15 

2630312 

l«33HX«7| 

12996341 

358476 


9647873 

45 

l6 

2633118 

i3344i?3 1 

12984883 

359270 


9647108 

44 

17 

2635924 

13333*02 

12973438 ; 

360064 


9646342 

43 

s8 

2638730 

13322863 

12962004 

360859 


9645575 

42 

19 

2641536 

I33I2237 

12950582 

361655 


9644807 

41 

20 

2644342 

13301625 

12939171 

362452 


9644038 

40 

21 

2647147 

13291022 

12927772 

363250 


9643268 


22 

2649952 . 

I328O432 

12916383 

1 364049 


9642498 

38 

*3 

2652757; 

I3269854 

12905005 

364849 


9641727 

37 

24 

2655562 

13259288 

12893638 

365650 


9640955 

36 

2$ 

2658366 

13248734 

12882282 

366452 


0640182 

35 

26 

2661170 

I323819I 

12870936 

367255 


963 9408 

34 

27 

2663974 

13227660 

12859601 

368059 


,9638633 

33 

28 

2666777 

132I7I4I 

I2848278 

368863 


’9637858 

32 


2669580 

13206633 

12836965 

369668 


9637082 

31 

30 

267238 3 

I3I96I37 

12825663 

370474 


9636305 

3 ® 


74 










































































































Gr. 15 


15 

min I Sinus» 

| Jj^arithm 

1 1 - 

! Differenti# 

| logdrithmi [ | 

Sinus j 


3 o 

2672383 

13 19^137 

12 8 2 5 6 6 3 

370474 


9636305 

30 

3 ^ 

267*186 

13185553 

12814372 

371281 


96355*7 

29 

fi 

3 2- 

2677989 

13175181 

12803092 

372089 


9634748 


33 

26^0792 

13164721 

12791823 

372898 


9633969 

27 

'34 

2683595 

13154*73 

12780565 

3737°8 


9633189 

26 

33 

2686397 

13143837 

12769318 

374519 

! 

9632408 

25 

36 

2689199 

13 i 334 I 3 

12758082 

375331 

i 

9931626 

24 

37 

26920OI 

13123000 

12746856 

376144 


9630843 

*3 

38 

2694802 

13112599 

12735641 

376958 


9630059 

22 

39 

2697603 

13102210 

12724438 

37777 *i 


9629275 

21 

40 

2700404 

13091833 

12713244 

378587 ! 


9628490 

2.0 

4 1 

2703205 : 

13081468 

1270206^ 

379403; 


9627704 

19 

42 

2706005 

13071114 

12690894 

380220 


9626917 

18 

43 

2708805 

13060771 

12679733 

381038 


9626129 

17 

44 

2711605 

13050440 

12668583 

381857 


962534» 

16 

45 

2714405 

13040120 

12657443 

382677 


96 * 455 * 

1? 

46 

2717204 

13029812 

12646314 

383498; 


9623762 

i 4 

47 

2720003; 

13019515 

1263.5195 

384320; 


962,2971 

i 3 

48 

2722802 

13009229 

12624086 

3X5143 


9622179- 

12 

49 

2725601 

12998955 

12612988 

385967 


9621387 

11 

50 

27284 0 0 

12988692 

12601901 

386791 


9620594 

IO 

51 

273 1198 

12978441 

1 12590825 

3876164 

\ 

9619800 

s> 

32 

-733994; 

12968201 

125779759 

38844*; 


9619005 

8 

53 

2736794 

i 2957972 

• ». 

J 12568703 

389*69 

i 

9618209 

* < ~ 

7 

54 

2739592 

12947755 

12557658 

390097 

! 

9617413 

6 

55 

2742389 

11737549 

12546623 

390926 


9616616 

5 

5 6 

2745186 

12927354 

12535598 

„391755 


9615818 


57 

2747983 

12937171 

12524584 

392587 


9615019 

3 

58 

! 2750780 , 

. - 12906999 

1251358$ 

393419 


9614219 

2 

59 

275.3577 

12896838 

12502586 

f • 

394252 


9613418 

1 

45 O 

17563731 

12686689 

12491603 

395086 


9612617 

0 


min 










































































































16 + 

win\ Sinus [ | logarithmi | Differenti* \tog<ttrkhfai | f Sirns | 


o 

I 

i ‘ 

275:6373 

2739169 

2761963 


12886689 

12876331 
12866423 

12491603 

12480630 

12469667 

395086 

3959 U 

3 ? 6 75..6 

9612617 

9611815 

9 61.IQ 12 

60 

59 

58 . 

■3 

276476l 


12836306 

12458714' 

' 397592 

96 10208 

57 * 

4 

2767336 


12846200 

12447771 

398429 

9609403 

46/ 

5 

277 ^ 31 ?J 

- 

12836103 

12436838 

39926:7 

9608598 

5 5 

6 

f 2773146 


12826021 

12425915 

4OO106 

9607792 

54 

i 

2776941 


12813948 

12415002 

4OO946 

4OI787 

9606985 

53 

8 J 

1778755 


12803886 

12404099 

9606 17*7 

57 

9 

2781329 


12793833 

12393206 

4O2629 

9605368 

5 i 

IO 

1784313. 


12783793 

12382323 

403472 

9604559 

50 

11 

2787117' 


12773766 

12371450 

404316 

9603749 

49 

; 

12 

2789911 


12763748 

12360587 

405 161 

9602938 

48 

13 

2791704 


127 5574 1 

1^349734 

4O6O07 

9602126 

47 

14 

2793497 


12743743 

12338891 

4O6854 

9601313 

46 

15 

2798290 


[1 273576 P 

1.2328059 

4O77OI 

9600499 

45 

16 

1 '2801082. 

• 2803874 

| 

12723785 

12317236 

408549 

9599685 

44 

17 

l 

1 

1271582 i 

123.06423 

409398 

9598870 

>3 

18 

2806666 

{ 

i 

12705868 

12295620 

410248 

9598054 

42 

19 

2809458 


12695926 

j2284827 

41 1099 

9597^37 

4 i 

20 

28 12230 : 

t 

12685995 

12274044 

411951 

9596419 

14 o 

2 1 

. 26 15043 

\ 

i 

\ 

12676075 

12263271 

... 4I2804 

9595600 

39 
: 38 

2.2 

; ,281783:2 

\ 

1 

i i 6.6 6 166 

1225 25.08 

. 413658’ 

9594781 

23 

. 28-20.62 3 • 


r 2656.267 

. 12.24 J 7.54 

..41491 V 

959396S 

37 

24 

282 34.14 

• ' 

* - 

12646379 

6223 '10.10 

415369 

9593 HO 

36 

i? 

2 6 

2826244 

t 2828994 


12636501 
12626633 

12220175 

12.209550 

416226 
417083 

9592318 

9591495 

35 

34 

27 

2831784 

i 

I2'6i 6776 

: 12197836 

417941! 

i 9590072 

! 33 

28 

2834374' 
•• 28-37 3-^4 


1.260692.9’ 

. 12188129 

418800 

419660 

; 9589848 

1 V* 

29 

1 

125 97 ° 9.3 

1 2r 17-7 4 3 3 

9589023 

! 3 :* 

30 

,»2840153 I 

l 12587267 

12166746 

420521 

9586197 

3 ° 























































































































Gr. 


w in\ Sintis | | ' Lyrdrithm ] Differentia j hgtftihm ' | [ Smus 


3 ° 

3 i 

3 Z 

33 

34 

ii 

36 

37 

38 

39' 

40 

41 

2840153 

2842,942. 

2845731 

2848520 

^ 2851308 
2854096 

I2587267' 

12577452 

12567647 

12557853 

I2548069 

12538296 

12166746 I 

12156069 
12I454OI ] 

42052I 

42I383 

^422246 


1 9588 197 1 

9587371 

9586544 

30 

29 

28 

27 

26 

25 

24 

23 

22 

’ 12334743 
12124094 

121134^5 

423I-IO 

4^3975 

42484I 

9585716 

9584887 

9584057 

2856884 
1859672 
2862459 , 

* 12528533 
12518780 
12509038 

12102825 
12092204 
12081593 

415708 

426576 

427445 

\ 

9583226 

9582395 

9581563 

2865246 

2868034 

2870^19 

I-2499306 

; 12489585 

I 247 9874 

12070992 

12060401 

I2049819 

4283,14 
429 184 
430055 

; 

958073O 

9579896 
9579O61 

s : 2 I 
20 

*9 

l8 

17 

16 

4 1 

43 

44 - 

45 

46 

47 

28736OS 
2876391 
2879177 , 

12470174 

11460484 
12450804 

120 39247 
I2O28684 

I20l8l30 

430927 
431800 
432674 

■ 

j 

9578225 

9577389 

9576552 

2881963 

2884748 

2887533 

12441134 

12431474 

12421824 

I2O07585 

I I997049 

I I986522 

433549 

434425 

4353°2 


9575714 

9574875 

9574036 

15 

34 

13 ^ 

48 

49 
fO 

28903 18 
2893103 
2895888 

12412184 

12402554 

. 12392934 

II976OO4 

H965495 

11954996 

436180 

437059. 

437938 


9573196 

9572355 

9571513 

12 

1 I 

IO 

?7" 

5^ 

53 

Z898672 j 
; 2901455 j 
29O424O 

' 12383324 
: 12373724 
12364134 

II944506 

1 I 934 P 25 

11923553 

438818! 
439699I 
440581 ! 

i 

95 7 o6 7 o 
9569826 
9568982 

3 

i ; 6 

r 7 

k -6 

5 

4 

54 

55 

56 

2907023 ; 
29O9806 

2912589 

j 12354554 

j 12344984 
] 123354-5 

H91309O 

11902636 

I 1892192 

441464 

442348 

443233 

9568137 i 
9567291 I 
9366444j 

5 7 
38 

59 

29 15'37 1 

2918153 

2920935 

12325876 

12316337 

12306808 

U881757 

11871330 

I 1860912 

444119 
448007; 
4458961 

j 

i 

9765596 ' 

1 9564747 

9 5 678 98 

; 3 
< 2 

X 

60 

2.923717] 

1 

.12297289 

j I 18505O3 ’ 

446786 

*> 

; 

72 

9563O48 

0 

^7 * * 

7 ^ 

/ .7 


e 



























































































































Gr. 17 

17 4“ _ 

min j Sinus [ f Loourithnii j Differenti a [ logAnthmi f f Sinus ! 


0 

I 

% 

2923717 

2926499 

2929280 

12297289 

12287780 

12278280 

11850503 

11840104 

11829713 

446786 

447676 

448567 


9563048 
9562 197 
9561345 

60 

59 

58 

3 

4 

5 

2932061 

2934842 

2937623 

12268790 
12259310 | 

; 12249840 i 

11819331 

11808958 

M798594 

449459 

4 ?° 35 * 

451246 


9 560492 
9559639 

9 558785 

r ?7 

56 

55 

<5 

2940403' 

12240379 

11788239 

452140 

• 

9557930 

54 

7 

2943183 

12230928 

11777893 

4 ? 3 ° 3 ? 


9557 0 74 

53 

8 

294^963 

1 12221487 

117675 56 

453931 


95562 37 

52 

9 

' 2948743 

12212056 

11757228 

454828 


9555360 

51 

10 

29? i 4*3 

12202634 

11746908 

4 ?? 7 2 6 


9554 ?° 2 

50 

11 

29543 ° 2 ! 

12193222 

11736597 

456625 


9553643 

49 

12 

2957081 

12183820 

11726295 

4 ? 75 2 5 


955 2 ?83 

48 

*3 

29 5 " 9 8 60 

12174427 

11716001 

458426 


9551922 

47 

14 

2962630 

12165044 

11705716 

459328 


955106 1 

46 

iG 

2965416 

12155671 

1169544° 

460231 


9550199 

45. 

16 

2968 194 

12146308 

11685173 

461135 


9549336 

44 

17 

297 ° 97 2 ' 

12136954 

i 11674914 

462040 


9548472 

43 

18 

29737 ?° 

12127616 

11664665 

462945 


9547607 

42 

39 

2976527 

12118276 

11654425 

463851 


9546742 

41 

20 

29793 0 4 

12108952- 

11644194 

464758 


9545876 

40 

21 

2982081 

12099637 

11633971 

465666 


9545009 

39 

22 

2984857 

1 12090332 

11623757 

466575 


9544 I 4 1 

38 

2 3 

2987633 

12081036 

11613551 

467485 


9543 2 7 2 

37 

24 

i 2990409 ’ 

12071749 

11603353 

468396 


9542403 

36 

2% 

: 2993^5 : 

12062472 

11593164 

469308 


9541533 

35 

26 

2995960 

12053204 

11582983 

470221 


9540662 

34 

27 

2998735' 

1204394? 

| 1 1572810 

471135 


9539790 

33 

28 

3001510 

12034696 

11562646 

472050 


9538917 

32 

29 

3004284 

12025456 

11552490 

472966 


9538043 

31 

3 ° 

3007058 

12016225 

; 11542341 

473884 


9537169 

3 <? 


• • t » > * i 


7 2 


















































































































*7 




r/1 ? n | Simis | j Lo^driihmi J Differentia j logdYithwti j j_ Sinas 


30 

31 

3 2 

300705« 

3009832 

3012606 

12010225 

12007004 

11997792 

II54234I 

11532202 

II522071 

473884 

474802 

475721 

9537169 

9536294 

75354*8 

30 

*s> 

28 

33 

3015380 

11988589 

II5II948 

476641 

953454* 

z7 

/Z 

34 

3018153 

119773 96 

11501835 

477561 

9533664 

i G 

33 

3020926 

I1970212 

II49373O 

478482 

953 2786 

*5 

36 

3023699 

1196 1037 

11481633 

479404 

9531907 

24 

37 

3026472 

1195187* 

II47I545 

480327 

9531027 

*3 

38 

3029244 ; 

I19427I6 

II461465 

481231 

9530146 22 

39 

3032016 

11933567 

11451373 

482176 

9529264 

21 

40 

5054788 

11924431 

II44I329 

483102 

9528382 

20 

41 

3037559 

II9153 0 3 

I1431^74 

484029 

9527499 

19 

42 

3040330 

11906 184 

1I42I227 

484757 

9526615 


43 

3043101 

I1897074 

II4IIl88 

485886 

7525730 

*7 

44 

3045872 

11887773 

11401157 

486816 

95*4844 

16 

43 

3048643 

11878881 

11391134 

487747 

95*3758 

15 

46 

3051413 

11869778 

II38 iII 9 

488679 

95*3071 

*4 

47 

3054183 

11860724 

II37IH3 

489611 

9522183 

liL 

48 

3056953 

l1851659 

I I36I I 15 

490544 

9321*94 

12 

49 

30597*3 

11842603 

113511*5 

471478 

9520404 

11 

50 

3062492 

11833557 

I 1341144 

492413 

95195 14 

10 


3065261 

11824520 

11 3 3U71 

473347 

95186*3 

9 

5 2 

3068030 ! 

Il8l549 2 

I 1321206 

494286 

7517731 

8 

33 

3070798' 

11806473 

IT311240 

495224 

9516838 

7 

54 

307 3 5 66 

11777463 

I I3OI3OO 

496163 

7515744' 

0' 

55 

3076334 

II788461 

II291358 

477103 

9515050 

5 

56 

307910* 

11777468 

II 28I424 

498044 

75 H*55 

4 

57 

3081869 

11770484 

II27I498 

49*8986 

7513257 

? 

<8 

3084636 

I I761509 

I U6I580 

4797*9 

7512362 

2 

55 

; 3087403 

II752543 

U 2 51670 

500873 

9511464 

-J * « 

1 


60 


9510565 
















































































































Gr, 18 


18 _. 

min | Simis ( j Logurithm 1 Differentia j logdrithrni ( | Sinus 


o 

1 

2 

3090170 

3092936 

3 ° 95?02 


<~> l 

II <7435 86 

■11734638 

1 1725699 

11241768 

11231874 

11221988 

O 1 

50l8l8 

502764 

5037X1 

9510565 
9509666 
j 9508766 

60 

59 

58 

3 

4 

5 

3098468 

3^01244 

3103999 


I 1716768 

1 1707846 
11698933 

I 12 12 109 

I 1202239 

11 192377 

504659 

505607 

506556' 

9507605 

9 506963 

9 506061 

57 

56 

55 

54 

5 3 

52 

51 

50 

49 

6 

7 

cS 

3106764 
3109329 

3112294 

— 

J1690029 

1168 1133 

I1672246 

I llh'2523 

I I172676 
11162837 

507506 

508457 

509409 

9505158 

9504254 

9503349 

V 

10 

11 

3I 

3117822 

3120586 

11663 368 
11654499 
11645638 

UI53OO6 

III43183 
11133367 

510362 

5II316 

5 12 27I 

9502443 

9501536 

9500629 

12 

13 

14 

3 H 3349 

3126112 

3128875 


116 36786 

I1627943 

11619109 

11123559 
» II 3 3759 

1 I 103967 

513227 

5 14184 
515142 

9499721 

9498812 

9497902 

48 

47 

46 

if 

16 
17 

3131638 ' 

3134400 

3137162; 

: 

1161028 3 

11601466 

11592658 

IIO94182 

II0844O3 

IIO74837 

51610j 

517061 
518021 

9496991 

9496080 

9495168 

45 

44 

43 

18 

19 

20 

3139924 

3142686 
3 H ?448 


11583858 

11575067 

11566285 

IIO64876 
11055123 

11045378 

518982 

519944 

520907 , 

9494255 

9493341 

9492427 

42 

41 

,40 

21 

22 

2 3 . 

24 

2 5 

2 6 

3148207 
3150970' 

3 1 5 " 3 73 1 


31457511 

11548746 

115 39989 

IIO3564O 

11-02 5910 

HOI6178 

521871 

522836 

523802 

9491512 
9490596 
9489679 

39 

38 

57 

3156491 

3159251 

3162011 


115 31240 
11522500 
11513768 

IIOO647I 

10996763 

10987062 

524769 

525737 

526706 

9486761 

9487842 

9486923 

36 

35 

34 

2 7 

28 

29 

30 

3164770 
3167529 

3170288 


11505045 

11496330 

11487624 

IO97736.9 

IO967683 

I0958004 

527676 

528647 

529620 

9486003 

9485082 

9484160 

-> ^ 

3 2 

ti? 

3173047 


11478926 

IO948332 

f 

530594 

2483237 

rl 

30 






























































































































18 -f 


tnin 

j Sinus» 

I 

j Logrttthmi 

| Differ enti# 

| looMii hmi | | Sin0 s i 

3° 

3 173047 


11478926 

10948332 

530594 


9483237 

30 

3i 

3175809 


11470237 

10938669 

531568 


9482314 

2 9 

31 

3178563 


11461556 

<0929013 

53*543 


9481390 

28 

3 3 

3 181321 


11452883 

10919364 

533519 


9480465 

27 

24 

31S4079 


11444219 

1090972-3 

534496 


9479539 

26 

33 

3 186837 


11435963 

10900090 

535473 


9478612 

25 

36 

3 189594 


11426919 

10890464 

536451 


9477685 

24 

37 

3192331 


11418279 

1 10880845 

537430 


9476757 

23 

38 

3195108 


11409644 

; I087I2 34 

538410 


9475828 

' 22 

~ 

3197864 


11401021 

1086 163Q 

5 39 39 1 


947489S 

21 

40 

3 200620 


11392406 

I0852033 

540373 


9473967 

20 

41 

3203375 


1 j 383800 

10842444 

541356 


9473035 

19 

4 1 

3x06130 


11379202 

IO832862 

542340 


9472103 

l8 

43 

3208883 


11366612 

10823287 

5433*5 


9471170 

I? 

44 

3211640 


11398030 

10813719 

5443 1 1 


9470236 

l 6 

44 

3214395 


11349456 

I0804I58 

545298 


9469301 

15 

46 

3217130 


11340891 

IO7946O5 

546286 


946836 6 

14 

47 

3219904 

, 

11332334 

10785059 

547*75 


9467430 

13 

48 

3222638 


11323789 

10 775520 

548265 


9466493 

1 2 

49 

3223412 


11315 244 

IO763988 

549256 


9465555 

1 I 

50 

. 3228163 


11306711 

IO756462 . 

550249 


9464616 

ro 

5i 

3230918 

• 

11298186 

IO746944 

551242 


9463677 

y 

fi 

3233671 


11289670 

10737434 

552236 


9462737 

8 

53 

3236423 


11281162 

IO727931 

553231 


9461796 

7 

54 

3239173, 


11272662 

10718436 

554226 


9460854 

6 

55 

3241927 


11264170 

IO708948 

55 5222 


9459911 

5 

56 

2 24.4679 


11^55686 

IO699467 

556219 


9458968 

4 

5 7 

3247430 


11247210• 

10689993 

557217 


9458024 

3 

58 

3230181 


11238742 

1068052.6 

558216 


9457079 

.2 

59 

3292932 


11230282 

10671066 

559216 


9456133 

1 

60 

3253662 


11221830 

1 066161 3 

560217 


9455186 

0 































































































































19 


“T _ , 

j i lofTd^ithmi | Dijfcrentix | logArahwi f j Sn ms ) 


0 

1 

2 

325 5682 
3258432 
3261182 


* .> » 

112218301 
I 1213386 

I1204950 

1066 16 13 
10652167 
10642728 

560217 
561219 
562222 


94551,86 

9454239 

9453291 

60 

59 

58 

3'"" 

32631:31 


I I 196522 

10633296 

563226 


9452342 

5^ 

4 

3266681 


I I l88 102 

10623871 

564231 


945*392 

56 

5 

326943° 

1 

I I 179690 

10614453 

565237 


9450441 

c5 

6 

3272179 


Il»7!286 

10605042 

566244 


9449490 

54- 

7 

3274927 


I 1 I62889 

10595637 

567*5 2 


944^538 

3? 

8 

3277675 


11154500 

10586239 

5 682.61 


9447585 

52 

9 

3280423 

ii 

1II46I19 

10576849 

569270 


9446631 

5 1 

io 

3283171 

. 

IU 37746 

10567466 

570280 


9445676 

5o 

11 

3285918 


11129381 

10558090 

57**91 


9444720 

49 

12 

3288665 


II121024 

10548721 

572303 


9443764 

48 

*3 

32914*2 


11112675 

10539359 

5733*6 


9442807 

47 

14 

3294*?9 


II104334 

10530004 

574330 


9441849 

46 

15 

3296906 


I1096000 

1052065^ 

575345 


9440690 

45 

16 

3299652 


11087674 

10511313 

576361 


9439931 

44 

17 

3302398 


11079356 

1050 1977 

577379 


9438971 

43 


18 ! 

19 

20 

3305144 

3307889 

3310634 


11071046 
11062744 

11054449 

10492648 

10483326 
10474010 

57>>398 

5794*8 

580439 

9436010 

9437 0 48 

9436085 

21 

22 

2 i 

3313379 

3316123 
3318867 ; 


1104616 2 

11037883 

110296 12 

10464702 
10455401 
10446107 

58 1460 

582482 

583505 

9435 * 2 * 

9434158 

9433*93 

24 

*5 

26 

3321611 

33*4355 
3127098 


11021348 

11013092 

11004843 

104368 19 

10427538 

10418263 

584529 !. 

585554 

586580 

9432227 

9431260 

9430*93 

27 

28 

29 

5 3 * 964 * 
333*785 
: 33353*4 


1099660 2 
10988368 
10980142 

10408995 

io 399733 

10390479 

5876O7 

588635 

589663 

94 * 93*5 

94*8356 

9427386 


42 

41 

40 


30 j 33 3*069 l 


39 

3 ^ 

3 J 

26 

3 ? 

34 

"33 

32 

35 


1097192.3 J 10381231 




590692 


9426415 


30 









































































































Gr. 

* 9 

min ( Sinus » 

19 

4 

[ j L$jgarithm f Diffe 

• 

rentia j lo a Arithmi | | Sinus j 

3 ° 

3 1 

3 2 

3338069 

33408 il 

3343333 


10971923 

10963712 

10 9 3 3 3 0 9 

10381231 

10371990 

10362756 

590692 
591722 
, 392753 

- 

9426415 

5425444 

9424472 

30 

29 

28 

3 ? 

34 

33 

3346294 

3349033 

3331776 


10947353 

10939123 

10930944 

10353528 

10344307 

10335092 

3937^3 

594818 

395852 

9423499 
9422525 
94 21 33 ° 

2 7 

26 

23 

36 

37 

E- 

36 

40 

41 

3334316 

3337236 

3339996 


10922771 

10914606 

10906448 

10325884 
103\6682 
1030748 6 

596887 

597924 

598962 


9420575 

94S9599 

9418622 

24 

2 3 
22 

3 3 6 27 3 6 

3363473 

3368214 

10898298 
10890136 
1088 2021 

10298297 

10289113 

10279940 

600001 
601041 
602081 


9417644 

9416665 

9415685 

21 

20 

19 

• 4 - 4 ^ 4 *, 

4^ kw ijj 1 

3370933 
337369 ! 
3376429 

1 

10873894 

10863774 

10837661 

10270772 

10261610 

10232454 

603!22 
604164 
605207 


9414705 

94 1 37 2 4 
9412742 

18 

I? 

l6 

43 

46 

47 

3379167 

3381903 

3384 'H 2 

■ 

1-0849333 

10841437 
1083 3366 

10243304 
10234161 
1022 5024 

606251 

607296 

608342 


9411760 

9410777 

9409793 

13 

14 
13 

48 , 

49 

3 o 

3387379 

4390516 

3392832 


10825282 

10817106 
jo 8 o 9 137 

10215893 

j0206770 

10197653 

609389 

6IO436 

61I484 


9408808 

9407822 

9406836 

1 2 

I I 

10 

3 1 

3 2 

33 

3393388 

3398324 

3401060 


10801075 

10793021 

10784974 

10188542 

10179438 
10170340 

011533 

«i 3 !k ’3 

614634 

1 

/9403849 

9404861 

9403^72 

9 

8 

7 

34 

33 

56 

37 

38 

39 

3403793 

3406330 

3409263 


107769 34 

10768902 

10760877 

10161248 

10152162 

10143082 

6J5686 

6I674O 

617795 

— 

9402882 

9401891 

9400900 

6 

3 

4 

3411999 

34^4733 

3417467 


10752860 

10744850 

10736847 

10134009 
10124942 
10114881 

6 18851 
619908 
620966 

9399908 

9398913 

939792 1 

3 

2 

1 

0 

*«/» 

7 o' 

60 

3420201 

10728852 

I 0 IO( 

5827 ' 

^622025 

> 

7 

9396926 

0 




























































































































20 


min\ 

Slms | ! lo^drithnii } D‘fjcrenti.e [ 

lowiJimi j [ 

8 7/1/ 


o 

3410201 

10728852 

IO 106827 

622025 

9396926 

60 

i 

3422934 

10720865 

1005778 1 

623084 

9395931 

59 

z 

3425667 

10712885 

10088741 

6 24144 

9394935 

5$ 

3 

^428400 

10704912 

IOO79707 

625205 

9393938 

57 

4 

3431133 

10696943 

IOO70678 

626267 

t 

939294° 

56 

5 , 

3433855 

10688984 

IOO61654 

6273 3° 

9391941 

<< 

6 

3436397 

10681030 

IOO52636 

628394 

9390942 

54 

7 

34393*9 

10673085 

IOO43626 

629459 

938994* 

53 

8 

3442060 

10665147 

IOO34622 

6 3052.5 

9388941 

52 

9 

34.44791 

10657216 

IOO25624 

631592 

9387939 

5 i 

IO 

3447522 

10649292 

IOOI6632 

632660 

9386937 

50 

11 

345°*53 

10641373 

IOO07646 

633729 

9385934 

47 

IZ 

343*983 

10633465 

9998666 

634799 

9384930 

48 

I? 

3433713 

10625562 

9989692 

63 5870 

9383925 

.47 

J 

14 

3458442 

10617667 

9980725 

6 36 942 

9382919 

46 

13 

3461171 

10609779 

9971764 

638015 

9361913 

45 

16 

3463900 

10601898 

99628 IO 

639088 

9380906 

44 

17 

3466629 

10594024 

9953862 

64016 2 

9379898 


18 

3469337 

10586157 

9944920 

641237 

9378^69 

42 

19 

3472085 

10578297 

9935984 

642313 

9377880 

41 

20 

3474813 

10570444 

9927054 

643390 

9376870 

40 

2 l” 

3477540 

10502596' 

9916130 

0444 v - > 6 

9375659 

.39 

22 

3480267 

10554760 

9909213 

645:547 

9374847 

38 

23 

3482994 

10546929 

9900 20?, 

646627 

9373834 

37 

24 

34«57*i 

~r 

O 

C\ 

O 

9891390 

047700" 

9372820 

36 

25 

3488447 

IO531280 

9882496 

648790 

9371806 

35 

26 

3491173 

IO523474 

98736OI 

649873 

9370791 

34 

27 

3493899 

[ 10515669 

98647U 

650958 

9369775 

33 

28 

3496624 

i 10507871 

9855827 

652044 

9368758 

32 

29 

3499343 

j IO5OO080 

9846949 

653131 

9367740 

31 

3 ° 

I 3502075 

IO492295 

1 9838076 

654219 

9366722 

30 



























































































































ni in | Sinus | | Logarithm J Differentia j lotrdyithmi j f Sintii 


30 

31 

32 

3502075 
3504799 
3507323 , 


10492295 
10484516 
10476745 

9838076 

9829209 

9820349 

6 54219 
655307 
656396 


9366722 

9365703 

9364683 

35 

29 

28 

33 

34 
3 ? 

3510247 
34 3 ^971 

3 4 14^94 


1046898I 
1046I225 
10453476 

9811495* 

9802648 

9793807 

657486 

658577 

659669 


9 363662 
9362640 
9361618 

27 

26 

2 5 

3 6 

37 

38 

3518417 
3521140 
3523862 


10445734 

10437999 

10430271 

978497 2 

9776143 
9767320 

660762 

661856 

6 6 2,9 5 * 

1 

9360595 
9359571 
93 54 6 

24 

23 

22 

39 

40 

43 

3526584 
3529306 
35 32027 


10422550 
i 0414836 
10407 12 9 

9758503 ’ 

9749693 

9740889 

664047 
665 143 
666240 


9357521 

9356495 

9355468 

21 

20 

19 

42 

43 

44 

3534748 

3437469 

1 „37740 190 | 


10399429 

10391735 

10384047 

973 2091 
9723298 

97 * 45*0 

667338 

668437 

669537 


9354440 

93534 11 

9352382 

18 

17 

16 

43 

4 6 

47 

3 542910 
3545630 

3548350 


10376366 

10368692 

10361024 

9705728 

9696951 

9688 179 

670638 

671741 

672845 


9 35 1352 
9350321 

9349289 

15 

*4 

*3 

48 

49 

50 i 

355 1070 
3553789 

3556508 


10353362 

10345706 

io 338057 

9679412 
9670650 
9661894 

673950 
67505 6 
676163 


9348257 

9347224 

9346190 

12 

11 

10 

51 

5 3 _ 

3559227 

356 [945 
3564663 


io 330415 

IO322780 
10315152 

9653144 

9644400 

9635662 

677271 

678380 

679490 


9345*55 

9344*19 

9343082 

9 

8 

7 

34 

r* ^ 

•) 

5 'J 

3567380 

3570097 

2 572814 


1030753* 

10299916 
10292308 

9626930 

9618204 

9609484 

680601 

681712 

682824 


9342045 

9341007 

9339968 

6 

5 

4 

57 

58 

5 _P_ 

3575 5 3 i 
3578247 
358^963 

"1 

1-0284707 
10277113. 
10269526 

9600770 j 
9592062 
. 9583 360j 

683937 

685051 

6 8 6 16 6 


9338928 j 
9337887 
9336846 j 

:> 

2 

1 

60 

| 3583679 

1 


10261946 

1 9574664 

, 1 

f 687282 


! 9335S04 

0 

miu 

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21 

min 


It 


15 

16 

I 7 _ 

18 

19 

20 

21 
22 

24 
2 5 
26 


+ 


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0 

1 ' 

z 

3583679 

3586395 

3589110 


10261946 

10254372 

10246804 

9574 66 4 

9565973 

9557^87 

687282 

688399 

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9335804 

933476I 

93337I7 

60 

59 

J>8 

-7 

3591825 ' 


10239243 : 

9548607 

690636 


5332673 

57 

4 

3594540 


10231688 

9539932 

,691756 


9331628 

56 

5 

3597254 


10 224140 

95 31263 

692877 

1 

9330582 

55 

6 

3599968 


10216598 

9522599 

693999 


9329535 

54 

7 

360 2682 


10209063 

9513941 

695122 

{ 

9328488 

53 

8 

3605 3 95 ' 

1: 

io 20153.4 

9505288 

696246 


932744O 

, fa 

.9 

3608108 

} 

110194012 

9496642 

697370 

j 9326391 

51 

IO 

3610821 

;/ 10186496 

9488001 

698495 

9325341 

50 

I 1 

3613533 ) i 10178987 

9479366 

699621 

f o2 221200 

4.0 

12 

• 3616245 


10171484 

9470736 

700746 


9 323236 

40 

13 

3618957 


10163988 

946 2111 

701877 


9322186 

47 

14 

3621669 


10156498 

9453491 

703007 


9321133 

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362438Q 
3627091 
3629802 


36325 12 
3635222 
36379 32 

3640642 

3 * 4 $ 35 f; 
3646060 


1014901f 
10141538 
10134067 


94448 77 
9436268 
9427664 




3648768 

3651476 
3634184 


10126603 
101I9145 
101 i 1694 

10104249 
10096811 
10089379 


9419066 

9410473 

9401886 


704138 
70 5 270 
706403 

767337 

708672 

709808 


9320079 
93 19024 
9317969 


9316913 
9315856 
9314798 


45 

44 

43 

42 

41 

40 


9393305 

9384730 
9376I60 


27 

28 

29 


36568 02 

36 


595 99 


3662306 


1008 1953 
10074533 
10067120 

10059713 

100523 12 
10044918 


9367595 
9359035 
93504^1 
5 > 34 1 r> 3 I 
9333386 

9424847 


7IO944 

71208I 

7 


13219 


9313739 

9 312680 
9311620 


20 


7 H 358 
7*5.498 
716 6 3 9 

7177821 

718926 
720071 j 


9310559 

9309497 

9308434 

9307371 

9306307 

9305242 


39 

38 

37 

36 

35 

44 

3 3 

3 z 
31 


| 3665012 


10037530 

93 i 6313 j 721217 


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9304176 j , 

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Cjr. 2i 

21 + 


miri | 'Sitim f | Lsffar/thmi ] ‘Differentia ] logdiithmi ( f S inu$ J 


30 

3 z 

3665OH 

3667718 

3670424 


1003753° 

10030148 
10022773 

9316313 

9307784 

929926 1 

721217' 

722364 

723512 


9304176 
9303109 
9302042 

3 0 

29 

28 

3.3 

34 

33 

3673130 

3675835 

3678541 


joo15404 
10008041 
10000685 

9290744 

9282232 

9273726 

724660 

725809 

726959 


9300974 

9299905 

9298836 

27 

26 

29 

36 

37 

38 

368 i 146 

368 395 1 
3686655 


9993335 

9985991 

9978653 

9265215 
, 9256729 
9248238 

728110 

729262 

730455 


9297766 

9296695 

9295623 

24 

23 

22 

3~ 

4° 

41 

3689359 

3 691062 

3 694765 


9971322 
9963997 
9956678 

6£29753 

923127 3 
9222798 

73*569 

732724 

73388o 

L 

9294550 

9293476 

9292401' 

21 

20 

19 

41 

43 

44 

3697466' 

37OO37O 

-3702872 


9949366 

9942060 

9934760 

9214326 

9205865 

9197406 

735037 

736195 

737354 


9291326 

9290250 

9289173 

18 

17 

16 

m 

45 

46 

47 

3705574 

3708276 

37 10977 


9927466 
9920178 

9912896 

9188952 

9180503 

9172059 

738514 

739675 

740837 


9288096 

9287018 

9285939 

i5 

14 

48 

4 9 

5o 

3713678 

3716379 

3719080 


9905620 
9898350 
9S9 1086 

9163620 
9144186 
9146757 

742000 

743164 

744329 


9284859 

9283778 

9282697 

te. 

12 

11 

IQ 

fi 

f 2 

53 

3721780 
3714480 
3727 i79 


9883828 

9876577 

98693 32 

9138333 

91x991? 

9121502 

745495 

746662 

747830 

« 

928 1615 
9280532 
9279448 

9 

8 

7 

54 

55 

56 

3729676 

3732577 

3735275 


98.62093 

9854860 

9847633 

9113094 

9104691 

9096293 

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750169 
751340 

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9278363 
9277278 
9276 192 

6 

5 

4 

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58 

9 

3737973 

3 740 6 71 

3743360 


9840411 

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9087900 

9079512 

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9275105 
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| | Logarithtr.i j 

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loedrithmi i j 

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9266 38 i 

5 5 

6 

3762243 

9775 696 

9012595 

763IOI 

9265267 

5 4 

7 

3764938 

9768535 

9004252 

7642.83 

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53 

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9761380 

8995914 

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9 26 3096 

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51 

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Z I 

3802635 

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i 8887977 

780931 

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2 Z 

3805325 

96618 36 

8879709 

782127 

9247676 

7 V 

27 

3808015 

9654770 

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28 

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J 9241020 

j y 

7 

29 

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J 92 3 9908 

31 

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38268 34 

9605468 

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Gr. 22 


22 •+ i -- 

mm | Sinus t [ ) Dolavit hmt | Differentia | looxYithmi [ f Sinus I 


3° 

3i 

3 2 

d 5 
34 
33 

3826834 

3 8 2 9 3 21 
3832208 


9605468 

9598448 
95 9H34 

881373 2 

8805506 

8797285 

79 1736 
792942 
794149 


9238795 
92 3768 2 

9 2 3 6 5 6 8 

30 

29 

28 

27 

26 

i 5 

3834895' 
3 837581 
3840267 

9584426 

95774*4 

9570427 

8789069 

8780859 

8772653 

795357 

796565 

797774 


9235453 

9234337 

9233220 

36 

3842953 


9563436 

87644^2 

798984 


9232103 

' 24 

37 

38456.38 


9556451 

875 6256 

800195 


9230085 

23 

:S ! 

3848323 


9549472 

8748065 

801407 


92 29866 

zz 

39 

3X51008 


9542498 

8739878 

802620 


9228746 

21 

40 

3853692 


953553° 

8731696 

803834 


9227625 

20 

41 

3856376 


; 9528567 

8723518 

805049 

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9226504 

I9 

42 

3859060 


95216 10 

8715345 

806265 


9225382 

18 

43 

3861743 


9514659 

8707177 

807482 


9224259 

i7 

44 

3864426 


9507713 

8699013 j 

808700 


£2231,35 

16 

* 

45' 

3867109 


9500773 

8690854 

809919 


9222010 

15 

46 1 

3869791 


9493839 

8682700 

811139 


9220884 

14 

47 | 

3872473 


94869 * I 

,8674551 

812360 


9219758 

13 

48 

3*875 *55 


9479988 

8666405 

813583 


921863I 

12 

49 

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94-7307I 

i 8658264 

814807 


9217504 

11 

50 

3880518 


9466160 

[ 8650128 

816032 


9216376 

10 

5 l 

3 8 8 3 1 9 9 


9459254 

8641996 

817258 


9215247 

9 

$2 

3885880 


945 2 3 54 

863 3.870 

818484 

i 

9214117 

8 

53 

3888560 


9445460 

8625749 

81971l 


9212906 

7 

54 

3891240 


9438571 

8617632 

820939 


9 21 1 8 5 5 

6 

55 

38939i9 


9431688 

8609520 

822168 


9210723 

5 

56 

3896598 


, 9424810 

8601412 

823398 


9209590 

4 

57 

3899277 


9417938 

8593309 

824629 

/ 

9208456 

3 

58 

3901955 


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9 2073 2 

2 

59 

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9206185 

1 

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I 39073 1} 


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23 


/ 



23 


mln \ 

Sinm 

| Lo?<trithmi | Different i <& 

logdrhhmi j | 

SinUl | 


Q 

3907311 


9397354 

8569026 

828328 


9205 049 

6Q 

1 

3909989 


9390504 

8560941 

829563 


9203912 

59 

2 

3 9 12666 


9383660 

8552861 

830799 


9202774 

58 

3 

3915343 

s 

9376821 

8544785 

832036 


9201635 

57 

4 

3918020 


9369988 

8536714 

833274 


9200496 

56 

* 

3920696 


9363160 

8528647 

834513 


9 199 356 

^5 

6 

3923372 


9356337 

8520584 

835753 


9i98 2'i5 


7 

3926048 


9349520 

85 12525 

836995 


9197075 

53 

8 

3928723 


9342708 

8504470 

838238 


9I9593I 

52 

9 

393i39& 


9335902 

8496420 

8 39482 


9194780 

5i 

Io 

3934072 


9329101 

8488374 

840727 


9193644 

50 

11 

3936746 


9322306 

8480333 

841973 


9 !92499 

49 

*2 

3939420 


9315516 

8472296 

843220 


9191353 

48 

J 3 

3942093 


9308731 

8464263 

844468 


91.90207 

47 

3 4 

3944766 


9301952 

8456236 

845716 


9189060 

46 

3 5 

3947439 


9295178 

8448213 

846965 


9187912 

4* 

16 

3950112 


9288410 

8440195 

848215 


9186763 

4.4 

*7 

3952784 


9281647 

8432181 

849466 


9185614 

43 

X8 

3955456 


9274890 

8424172 

850718 


9184464 

42 

3 9 

3958128 


9268138 

8416167 

85 197 1 


9183313 

41 

20 

3960799 


9261392 

840816-7 

853225 


9182161 

140 

21 

3963470 


9254651 

8400171 

854480 


918I009 

39 

22 

'3966 140 


9247915 

8392179 

855736 


9179856 

38 

22 

3968*810 


9241185 

8384192 

856993 


9178702 

37 

2 4 

5 971460 


9234460 

8376209 

858251 


9177547 

36 

25 

3974149 


9227741 

8368 231 

859510 


6 376391 

3* 

2 6 

3976818 


9221027 

8360257 

860770 


9175235 

34 

27 

3979487 


9214319 

8352288 

862031 


9174078 

33 

28 

3982155 


9207616 

8344322 

863294 


9172920 

32 

29 

3984823 


9200918 

8336360 

864558 


917176 i 

31 


39874^1 


9194226 

8328403 

865623 


9170601 

3° 


66 

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pzin 1 Sinus | ] Lffgarttbtnt [ D ffnrrtm (lopjrithfni f I Sinus f 

„ - - -... ---- - i i i m u !< « ' « »'—.»0"i n i r« 


30 

31 

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3990139 

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9194226 
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867089 

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9 170601 
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3 ° 

29 

28 

33 ! 

34 

35 

3995493 [ 
3998359 

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9174181 

9167510 
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8304558 

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9I67H7 

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27 

26 

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24 
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37 

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4003491 
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9 16 1297 

39 ' 

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42 

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4019478 i 

4022141 

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9156627 

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18 

17 

16 

45 

46 

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48 

49 

50 

4026467 

4030130 

4032792 

9094473 
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9081262 

8209568 

8201679 

8193794 ; 

8385913 

8178036 

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88.6186 

887468 


9153115 

9 i 51943 
9150770 

9 H 9597 

91484^3 

9147248* 

15 

14 

13 

I 2 

I I 

io 

40 3 5454 
4038115 

4040776 

9074664 

9068071 

9061483 

888751 

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891320 

51 1 
52 
52 

40 43437 
4046097 

4048757 


9054901 

9048324 

9041752 

8162295 
81 5443 1 

8146571 

892606 

893893 

895 18 I 


9146072 

9144895 

9143718 

9 

8 

7 

6 

5 

4 

54 

55 

56 

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24 


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r/iln f Simis 


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3° 

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nnn | Sintis j J Logdrhhmi | Differentia j Lgdrithrm J ] $ttsm f 


0 

1 

2 

42 26183 
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mih\ Sinus» f | Lo^msthmi j Differenti^ j logdritbmi] I 


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Gr. 16 

2 6 4 * 


min 

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1 

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0 

43&37 1 * 


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30 

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Gr. 27 

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fHtn\ Sima | | hw Arithmi ] Differentia |loaArithmi f [ Sintis j 

0 

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Gr, 

27 

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ij 


4 - 


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30 

3 * 

32 

46 s 7486 
4620066 
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7727344 

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28 

33 

4625225 


7710596 

6507067 

1203529 


8866076 

27 

34 

4627804 


7795022 

6499975 

120 5 047 


8864730 

26 

35 

4630382 


769945- 

6492886 

1206566 


8863383 

25 

36 

4632960 


7693886 

6485800 

J208086 


8862035 

24 

37 

4635538 


76583,24 

6478717 

I209607 


8860687 

23 

38 

4638115 


7682766 

6471637 

I2 TII 29 


8859338 

22 

39 

4640692 


7677212 

64645 60 

12 1265 2 


8857959 

21 

40 

4643268 


7671662 

6457485 

1214177 


8856639 

20 

41 

4645 844 


7666I16 

6450413 

1215703 


8855288 

19 

42 

4648420 


7660574 

6 443 344 

I217230 


8852930 

18 ’ 

43 

4650995 


76.55035 

6436277 

I218758 


8852583 

17 

44 

4653570 


7649500 

6429213 

1220287 


8851230 

16 

45 

^4656 145 


7643969 

6422152 

I22i8'i7 


8849876 

15 

46 

465871*9 


7633441 

6415094 

I223348 


884852I 

14 

47 

46 612 93 

i 

763-919 

640803 9 

I224880 


8847I65 

13 

A o 
A . 0 

4663866 


7627400 

6400987 

I2264I3 


8845809 

12 

49 

46 6 643 9 


7621885 

6393938 

1227947 


8844452 

1 I 

* 

5 ° 

4669012 

. 

7616374 

638,6893 

1229481 


8843095 

IO 


52 

53 

54 
5 5 

56 

57 

59 

60 


4671584 
4674 1 5 6 
4676727 


76 10807 

7605363 

7599863 

6379850. 

6372809. 

6365771 , 

4679298 
4681869 
4684439 


7594367 

7588875 

7583387 

6358735- 

635 1702 
6344672 

4687009 

4689578 

4.69214/7 


1 75 77 903. 

7572422 
j 7566945 

6337645 

6 330620 
6322598 

1 4694716 


: 7561472 

6316578 


1231017 
123 2554 
1234092 

7 l 3032 ~ 

1237173 
1238715 


1240258 j 
1241802, 1 
1243347 


g§ 4 l7 '37 

8540378 

8839018 

8837657 
8836295 
34932 


9 

8 

7 


8833569 

883220.5 

88308417 


1244894 


8829476 


4 

5 
4 

3 

2 

1 

a 
























































































































Gr. ' 28 


28 “ 

min f Sinus [ ] lo^irithmi [ Differentis j logarithwi [ [ Sitius j 


0 

1 

2 

4694716 

4697i84 

46998^2 


75^1472 

7556003 

7550538 

6316578 
630956 1 

6302547 

1244894 

1246442 

1247991 


8829476 

8828110 

8826743 

60 

59 

|j8 

r“~ 

4 

5 

4702419 

4704986 

4707553 


7545076 

7539<$l8 

7534^4 

6295535 

6288526 
628 1520 

1249541 
1251091 
1252644 


8825375 

8824007 

8822638 

57 

56 

55 

6 

7 

8 

4710119 

471268? 

47*5*50! 


7 >2.8714 

7 / 523268 
75178^6 

6274517 
6267517 
6260521 

I254197 
1255751 
1257305 

1 

r8821268 
8819898 
8818527 

54 

53 

<2 

9 

10 

11 

471781? 

4720380 

4722944 


7512388 
7506954 
7501524 

6253528 

6246538 

6239550 

1258860 
1260416 
I26i974 


8817155 

8815783 

8814408 

51 

50 

49 

ii 

13 

*4 ( 

4725508 

4728071 

4730^34 

7496097 

7490674 

7485255 

6232564 
6225 581 
621860 1 

1263533 

1265093 

1266654 


8813034 

8811659 
8810285 

48 

47 

46 

I? 

I* 

^7 

47 , i33»?7 
4735759 1 
4738321 ' 


7479840 
747442 8 
7469020 

6211624 

6204649 

6 197676 

1268116 

1269779 

1271344 


8808907 

88075 30 
8806152 

45 

44 

43 

18 1 

19 

20 

4740882 

4743443 

4746004 


7463616 

7458216 

7452819 

6190706 

6183739 

6176774 

1272910 
1274477 
1276045 


8804773 

8803394 

8802014 

42 

41 

40 

21 

2.2 

13. 

24 

*5 

26 

2f 

28 

29 

4748564 
475 H24 

4753 68 3 


7447426 

7442037 

7436651 

6169811 

6162853 

6155896 

”148941 
6141991' 

. 6135063 

1277614 

1279184 

1280755 

1282327 

1283900 

1285474 


8800633 
8799251 
8797869 

8796486 

8795102 

8793717 

39 

3$ 

'37 

3f 

35 

34 

37*6242 

4758801 

47^1359 

7431269 

7425891 
7420517 

4763917 

4766474 

476903,1 


7415146 

7409779 

7404416 

6128096 

6121152 

6114211 

1287050 

1288627 

1290205 


8792332 

8790946 

8789559 

33 

32 

3i 

j® 

47.715 88 


7399057 

6107273 

1291784 


8788171 

3o 


6 1 
















































































































Gr. 28 


2S _ Hr 

mim | . Sinm j ] Lifj.rithmi j Differentia | lotdYhhnn | j Shus j 


31 

32 

477x588- 

4774144 

4776700 


7399057 

7393701 

7388349 

6107273 

6100337' 

6093404 

1291784 
1293364 

1294945 


8788171 

8786782 

8785393 

30 

29 

28 

33 

34 
33 

4779 2.55 
4781.810 
4784365 


7383001 

7377657 

7372316 

608 6474 

6079547 

6072622 j 

1296527 
12981 xo 
1299694 


8784003 

8782613 

8781222 

27 

26 

25 

3 6 

37 

38 

35 

40 

415 

47-86919 

4789473 

4792026 


7366979 
7361646 
7356316 

6065700 

6058781 

6051863 

1301279- 

1302865 

1304453 


8779830 

8778437 

8777044 

24 

23 

22 

4794579 

4797x32 

4799684 

— 

7350990 
7345 668 
7340349 

6044948 i 

6O38036 
603I126 

1306042 

1307632 

1309223 


8775650 

8774 2 55 

8772859 

21 

20 

19 

42. 

43 

44 

4 T 

46 

47 

4802 236 
4804787 
48073?? 

7335034 

73297^3 

7324415 

6024219 

6017315 
6010413 

13 10815 

131240 8 
1314002 


8771462 

8770065 

8768667 

«8 

X7 

16 

4809888 
4812438 
4814988 


7319 x 1 x 
7313811 

7308514 

6003 514 
5996618 

598972-3 

I3X5597 

1317193 
1318791 


876 7267 
8765868 
8764468 

15 

14 

13 

48 

49 

50 

4817537 

4820086- 

482263? 


73032 21 

729793 1 

7292645 

5982831 

5975941 

5969054, 

1320390 
. 1321990 
x 3*359i 


8763068 

8761665 

8760263 

12 

11 

10 

51 

52 

53 

4225x83 

4827731 

4830278 


7287363 

7282084 

7276809 

5962170 

5 954288 
5948409 

1325193 
1326796 
1328400 


8158860 

8757456 

8756051 

9 

8 

7 

54 

55 

56 

4832825 

4835371 

4837917 


7271538 ’ 
7266270 
7261006 

594X533 

5934 6 59 

5927787 

1330005 

1331611 

*333 2I 9 


8754646 

8753 2 4o 

8751833 

6 

5 

W| * 

4 

57 

58 

59 

4840462 
4843007 
, 4845552- 


7255746 

7250489 

7245236 

5920918 

591405 x 

5907187 

1334828 

1336438 

1338049 


8750425, 

8749016 

874x607 

•» 

2 

2 

1 

60 

4848096 

, 


7239987 

5900326 

1339661 


8746197 

0 

miti 

gr a 


h 

























































































































29 
min | 

Sinus [ 

j Logarithmi ] 'DijferentU 

) logdiithmi ( 

[- Sinus ■ 

1 

o 

4848096 

7239987 ’ 

5900326 

1339661 

8746197 

6©- 

i 

48 50640 

7234742 

5893468 

i 34ii?4 

8744787 

59 

z 

481 3184 

722 9 590. 

5886612 

1342888 

8743376 

58! 

3 

4855 727 

7224262 1 

58 7975 9 

I344593 

8741964 

57 

4 

485 8270 

7219327 

5872906 

134<5l 19 

8740551 

56 

5 

4860812 

7213795 

5866059 

1347736 

8739137 

55 

6 

4863354 | 

72085 67 

5859213 

13493*4 

[8737722 

54 

7 

486589^ 

7293342 

58 523 68 

I359974 

8736307 

53 

8 

4868436 

719812! 

5845526 

1352595 

8734891 

52 

9 

4870977 

7192903 

^5838686 

I3542I7 

8733475 

5i 

IO 

4^73*17 

7187689 

5831849 

I35584O 

8732058 

50 

11 

4876057 

7182478 

5825014 

1357464 

8730640 

49 

12 

48785 96 

7177271 

. 5818182 

1359089 

8729221 

48 

13 

4881135 

7172068 

5811353 

I36O7I5 

8727801 

47 

14 

4883674 

7166868 

5804526 

I3623 42 

8726381 

46 

15 

4886212 

7161672 

5797791 

1363971 

8724960 

45 

l6 

4888750 

7156480 

5790879 

1365601 

8723538? 

44 

17 

4891287 

7151291 

5784059 

I 36723 2 

8722116, 

43 

18 1 

4893824 

7146106 

5777242 

1368864 

872069 3 

42 

19 

4896361 

7140924 

5770427 

I-37O497 

8719269 

41 

20 

4898897 

7135746 

5763615 

1372131 

8717844 

40 

21 

490143 3 

7130572 

5756806 

1373766 

8716418 

39 

ZZ 

| 4903968 

7125401 

5749999 

1375402 

8714992 

38 

2 3 

[ 4906503 

7120 234 

5743195 

1377039 

8713565 

37 

24 

, 4909 037 

7115070 

5736392 

1378678 

8712138 

36 

25 

4911571 

7109909 

5729591 

I3803 l8 

8710710 

s -7 

26 

4914105 

7104752 

5722793, 

1381959 

870928 I 

34 

27 

4916638 

7999598 

5715997 

I5836OI 

8707851 

33 

28 

4919171 

7094448 

5709204 

I385244 

8706420 

32 

29 

4921703 

7089301 

■ 

570 2413 

1386888 

8704989 

31 

39 

4924235 ( 

7084158 

5695625 

I38853 3 

8703557 

■ 


















































































































r. 29 


29 

min 

Simis 

| | Log Arithmi 

“ 4 * — " 

j 'Differenti* 

| logArithmi J 

Sinus 

f 

3° 

4924235 


7084138 

3693623 

138853? 


8703357 

30 

31 

4926767 


7079018 

5688839 

1390179 


8702124 

29 

32 

4929298 


7073882 

5682036 

1391826 


8700691 

28 


4931829 


7068749 

5675275 

1393474 


8699257 

27 

-/ -7 

34 

4934359 


7063620 

5668496 

1395124 


8697822 

2(5 

151 

4936889 


7038494 

56617^9 

1396773 


8696386 

25 

36 

4939418 


j 7033372 

5654945 

1398427 


8694949 

24 

37 

4941947 


7048233 

5648173 

1400080 


8693512 

23 

38 

4944478' 

■ 

S 7043138 

5641404 

1401734 


8692074 

22 

39 

4947004 

1. 

7038026 

5634637 

1403389 


8690636 

21 

40 

494953* 

, 

7032918 

5627873 

1495043 


8689197 

20 

41 

4952039 


7027814 

362 1 ri1 

1406703 


8687757 

19 

42 

4954586 

1 

7022713 1 

3614351 

1408362 


8686316 

l 8 

43 

4957H3 


7017613 1 

5607593 

1410022 


8684873 

17 

44 

4939639 

i 

7012321 1 

5600838 

1411683 


8683431 


43 

4962163 


7007430 

5594083 

1413345 


8681988 

15 

45 

4964699 


7002342 

5587334 

1413008 


8680544 

14 

47 

4967215 


6997238 

5380386 

1416672 


8679100 

I? 

48 

4969740 


6992177 

5573840 

1418337 


8677653 

I* 

49 

4972264 


6987099 

5567095 

1420004 


8676209 

I I 

30 

4974788 


6982023 

5560353 

:X 

1421672 


8674762 

IO 

51 

4977311 


6376934 

5553 <>I 3 

1423341 


8673314 

9 

52 

4979834 


6971886 

5546875 

1423011 


8671866 

8 

53 

4982336 


6966822 

5540140 

142668 2 


8670417 

7 

54 

4984878 


6961761 

55334°7 

1428334 


8668908 

6 

55 

4987399 


6936704 

5526677 

1430027 


8667518 

5 

56 

4989920 


6931630 

5519949 

1431701 


8666067 

_ 4 

5 ^ 

4992441 


6946600 

5513224 

143337 6 


8664613 

3 

58 

499496 1 


69 4 1 533 

3306300 

1433033 


8663162 

2 

59 

4997481 


6936309 

3499778 

1436731 


8661708 

1 


f 5000000 


',6931469! 5493P59 

1438410 

; 

8660254 j O 

1 


i 



J 

i 


1 | 



\ g ra > 


6 o 

li 1 



















































































































tmn 

I Swu> | 

| Lo/rarithwi j Different i<2 | 

toprtrithw? f 

i 

j 

Sinus 

) 

° | 

5000000 

6931469 

5493059 

1438410 


8660254- 

60 

1 

5002519 

69 2 643 2 

5486342 

1440090 


8658799 

59 

2 j 

5005038 

6921399 

5479628 

1441771 


8657344 

58 

3 ' 

5007556 

6916369 

5472916 

»44345 3 

j 8655888 

f 

5y 

4 

5010074 

69I1342 

5466206 

1445136 

8654431 

56 

5 

5012591 

6906 3 Ip 

5459498 

1446821 

j 8652973 

55 

6 

5015108 j 

6901299 

545 2792 

1448507 ' 


8651514 

54 

7 

5017624 

6896282 

5446088 

1450194 


8650055 

53 

8 

'5020140 

689 I26'9 

54393^7 

1.3,5 1882 

[ 8648595 

52 

p i 

5022656 

6886259 

5432688 

1453571 


8647134 

5i 

30 

5025171 

6881253 

5425992 

1455261 


8645673 

.50 

II 

5027686 

6876250 

_ ‘ t 

54!929p 

1456952 


8644211 

49 

12 

5030200 

6871250’ 

5412605 

1458645 


8642748 

.48 

13 

5032714 

6866254 

5405915 

1460339 


3641284 

47 

14 

5035227 

6*8 61126 1 

5399227 

1462034 


8639820 

46 

IS 

5037740 

6856271 

5392541 

146 3730 


86383.55 

45 

l6 

5040253 

6851285 

5385858 

1465427 


8636889 

44 

17 

5042765 

6846302 

5379177 

1467125 


8635423 

43 

18 

5045277 

6841323 

5372499 

1468824 


8633956 

42 

19 

5047788 

6836347 

5365822 

1470525 


8632488 

41 

20 

5050299 

6831374 

53<9I47 

1472227 


86310I9 

40 

21 

5052809 

6826405 

535 2 475 

1473930 


8629549 

39 

22 

5055319 

6821439 

5345805 

1475634 


8628079 

V8 

2 3 

5057829 

6816476 

5339137 

H77339 


8626608 

37 

24 

5060338' 

68115 16 

5332471 

1479045 


8625137 

-> 6 
2 

2 S 

5062847 

6806560 

5325808 

1480752 


8623665 

35 

2 6 

5065355 

6801607 

5319147 

1482460 


8 6 2 2 19 2 

34 

27 

5067863 

6796657 

5312488 

1484!69 


|8620718 

33 

28 

5070370 

6791710 

5305831 

1485879 


j 8619243 

32 

29 

5072877, 

6786767 

5299177 

1487590 


| 8617768 

i 3i 

30 

5075384’ 

6781S27 

5292525 

1489302 


> 8616292 

30 
























































































































30 

min 


T. 

T _ 

Sinus j [ Lcgdruhmi ] ‘Differenti & | hmriihmi [ | Sinus | 


30 

31 

32 

5075384 

507785)0 

5080396 


"6781827 

677689O 

6771956 

5292525 

5285874 

5279225 

148=9302. 
1491016 
1492731 


8616292 

8614815 

^613338 

3° 

29 

28 

33 

34 

2 > 

5082901 

50854 06 

5087911 


6767O26 

6762099 

6‘757 1 7 < ? 

5272579 

5265934 
5259291, 

1494447 

1496165 

1497884 

— 

8611860 

8610381 

8608901 

27 

26 

25 

24 

23 

22 

36 

37 

38 

5090415 

5092919 

5095422 

—' 

6752255 

6747338 

6742424 

5 2 5 2 6 5 1 
5246013 
5239377 

1499604 

15 013 2 5 
1503047 

8607420 

8605939 

8604457 

39 

40 

41 

5097925 

5 100427 

5102929 

6737513 

6732606 

6727702 

5232743 
5226 112 
5219482 

1504770 ’ 

1506494 

1508220 


8602975 

8601492 

8600008 

21 

20 

39 

4 2 

43 

44 

5105430 
5107932 

5110431 


6722802 
6717905 
6713011 

5212855 

5206230 

5199607 

1509947 
1511675 
1513404 

— 

8598523 

8597 0 37 

8595551 

18 

17 

16 

45 

46 

47 

48 

49 
30 

5112931 

5115431 

5117930 

5120429 

5122927 

5125425 

i 

6708120 
6703232 

6 6 9 8 3 48 

5 192986 

5 186367 
5179751 

1515134 
1516865 
1518597 

8594064 

8592577 

8591089 

15 

H 

33 

6693467 

6688589 

6683714 

5173137 

5 166525 
5I599H 

1520330 

1522064 
1523800 


#589600 

8588110 

8586619 

1 2 
il 

IO 

51 

32 

53 

5127922 

313 0419 

513 29 1 6 

6678842 

6673974 

6669 109 

5153305 

5146699 

5140095 

1525537 

1527275 

1529014 

1 8585327 

8583635 

[8582142 

9 

8 

*7 

~6 

5 

4 

54 

55 

56 

51,35412 

5137908 

5140403 

6664247 

6659388 

6654532 

5133493 

5126892 

5 120293 

1530754 

1532496 

1534239 

«1 

8580649 

8579155 

8577660 

57 

58, 

59 

5142898 

5143393 

5147887 


6649680 
664483 i 
6639985 

5113697 

5 107103 

5 100511 

153^983 

I5377 2 8 

,1539474 

8576164 
8574668 
8573 371 

3 

- 1 

i 

60 t 5150381 

t 

i 

; 

6635142 ! 

509392-1 

11541221 

■ 1 


857167} 

! 

0 

mm 


59 


59 ‘ 




































































































































31 # ^ 

min | Sinus f | logarithm \ Differentia [ logarithmi f | Si rms } 


o 

I 

r i 

z 

3 

4 

5 

51503&I 
^ I42874 

4I55367 

663 5142 

• 6630302 
6625465 

6620631 
6615801 
6610974 

5093921 

*o87332 

5080745 

I54122I 
1542970 
1544720 


857i 6 73 

8570175 

8568676 

60 

59 

58 

5157859 

5160351 

5162843 

5074160 

5067578 

5060998 

1546471 

1548223 

1549976 


8567176 

856*675 

8564173 

'57 

56 

*5 

6 

516*334 

6606150 

5054420 

1551730 


8562671 

54 

7 

5167825 

6601329 

5047844 

1553485 


8561168 

53 

8 

5I703I5 

6596512 

5041271 

1555241 


8559664 

52 

9 s 

5172805 

6591698 

5034700 

1556998 


8558160 

5i 

IO I 

5175294 

<5386887 

5028130 

I5<8757 


8556655 

50 

ii f 

5177783 

6582079 

5021562 

1560517 


8555 H9 

49 

12 

5180271 

<^77*7* 

5014997 

1562278 


8553643 

48 

13 

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31 

min < [ Sinm | J lo^arithmi 

+ l 

Differentia 

| logarithmi [ J Sinus j 

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o 

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o 

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Simis i 1 logdnthmi \-Di fferenti* \logdrithnn [ [ Sinus j 


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Sinus } 1 lo?arithmi I fflfj&entid \logdrithmi\ ( Sinus f 


30 

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27 

J J 

6234248 


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J T 

35 

6 23 6522 


4721624 

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25 

3 6 

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4717979 

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246 51.3 7 


7815205 


2 '4 

3 7 

6241069 


4714336 

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2 3 

j8 

6243342 


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2469784 

’ 

75 11574/ 

22 

39 j 

6245614 


4707056 

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7809758 

21 

40 

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4703419 

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20 

4 1 ( 

6250 156 


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54 

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5 3 

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39 » 

| $inm | f lorarithmi j Differentia j logarithmi [ j Shiuf | 


n 

• 6293204 

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6297724 

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59 

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55 

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r 7760465 
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52 

9 

10 

1 T 

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4398902 

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50 

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41 

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; 6340360 
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38 

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32 

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24 

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tnm I Sinm [ ) Logarithmi ] Differenti* f logdrithmi f j Sinus \ 


0 

1 

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26 67590 
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39 

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440 9019 
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6 

7 

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■ 6441236 i 
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6445685 j 

4398648 

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2 679819 
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7645466 

34 

33 

32 

6447909 

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49 

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13 

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4377933 

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42 % 
41 

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Gr. 40 

40 ~ 1 l 

m in f Sinus j j lo&rkbwl Dffiwti* 

1 \wArithwi | 

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f 

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31 

3 2 

6494480' 
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30 

29 

28 

33 

34 

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36 

37 

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650 H 14 
6*03324 
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4306116 

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1**9629 

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27 

26 

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6*07742 
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429*925; 

42924 3% 
428914^1 

1*41964 

1536077 

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2753961 

2736433 

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24 

23 

22 

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40 

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6^1436^ 
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4285752 

4282365 

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1518420 

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2761447 

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7587031 
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43 

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6*20984 
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4 2 733 97 
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43 

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2776459 
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15 

14 

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48 

49 

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7366874 

27 

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34 

6764477 

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850416 

3058585 


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35 

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6768760 

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6770901 

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3896348 

827063 

3069285 


7357031 

22 

39 

6775181 

3893189 

821225 

3071964 


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21 

40 

i 6777320 

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815388 

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6779459 

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: 6790144 

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30 

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29 ' 

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33 " 

34 

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: 3249479 

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r 7 

46 

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47 

48 

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3682636 

42 48* I 

3*578,35 

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13 

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3263413 

7217601 

12 


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3676390 

413177 

7213588 

II 

30 

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31 

6927728 

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5 2 

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33 

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34 

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57 

58 

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3 66602 

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44 

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4-4 

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Sfam J | Livdrithm j Differentia \hgmithm ] | Sinus f 




o~~ 

Tt 

6946584 


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3294213 


7193398 

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1 

6948676 

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6 

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5.4 

7 

6961216 

5 36223O7 

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33*3917 


7179238 

53 

8 

6963304 

3619308 

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33*6738 


7 *772I3 

52 

9 

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296750 

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48 

13 

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7 167078 

47 

14 

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46 

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16 

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6982071 • 

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25OI9O 

3 542204 


7158958 

43 

18 

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19 

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238550 

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7154895 

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3350723 


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40 

21 

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22 

6992476 

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3356411 


7148796 

38 

2 3 

6994333 

3374531 

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3359157 


7146762 

37 

24 

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3571560 

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3 362105 


7144727 

36 


6008712 

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3364955 


7142691 

35 

26 

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7140655 

34 

27 

7002866 

3562656 

19*997 

3370659 


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33 

28 

7004942 

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l86178 

3373513 


7136581 

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2-9 

7007018 

3556728 

180359 

3376369 


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3° 

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174541 

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44 






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m>n | Sinus j f Loasrithmi j Differentia | lo^rithmi { f * Sintis f 


3° 

3i 

3* 

7009093 
7011147 
7013 241 


3553767 

3550808 

3547851 

f 174541 

168723 

162905 

3379**0 

3382085 

3384946 

7132504 

7130465 

7128425 

36 

29 

28 

33 

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3544895 

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27 

34 

7017387 


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26 

35 

5»™, 

7019459 


3538989 

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3393538, 

7122303 

25 

36 

1 7021530 


3536038 

139672 

3396406 

7120261 

24 

37 

7023601 


3533089 

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3399275 

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38 

7025671 


3530142 

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f 3402146 

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22 

39 ! 

7027741 


35*7197 

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3405019 

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4° 

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7112086 

20 

41 

7031 <879 


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3410770 

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7°?3947 


3518371 

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3515432 

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16 

45 

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7042213 


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7046342 


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