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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 14
# Folio 18v, page id 00068032:0036
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Aequatio X  ab X + ac. po. Unde Analogra:
Analogia
X3
a B aX +C unde Loga
 B
Uende equatio Logarlhmica: (83 L oyX  log l X + C p ) Log a
Videamus an ne dliceat ad Logarithmos logarithmicrum:  1p - 2 L0g + Eog axEC
ergo L0g. -Loga + L0g. B X FC p L0g. 3togX sed nihil hinc lucis
L0gX
Omnis formula ad quosdam factores reduci potest, in quibus nem nisi X. simplex
si
cum quadam cognita.
sit lga X +C. semper potest reduci ad L0g X+  ponendo enim B ep c.
sive epi ac. fiet Log x +C , + L09 6 - Log a _ + L09 X + e
ex eadem aequatione Analytica plures elici possunt Loganthinicae.
ut Aequatio proposita etiam En hanc Analogiam resolvi potest:
 X + ab na. porendo ab ess quantitatem negatroam p =af. fiei X tabp X- af
ac 1 X jam X af B X+ a) x saf. unde fiet equation o ganthimica
L0g X+ saf + Logen quaf - log. a = 109C  Leg - a = L0g X explicando  per + et f per
et transponendo aequationem habebimus: +L0g X - saf  +2 Loga A
oatur notabilis, minirum hos tres nmeros +LogX
tres numeros quo incogni Logarithmo effectos  Log X + taf
+R Cog. C
rebi jam res
inrognitam includentes esse progressionis Arithmeticae, snde sequitur Logarithmos eotum propessio
nis esse logarithmicae Et him mechodum haberi inveniendi summam trium numero
rum progrestio nis cogarithmicae. Sed haec nihil ad rem nostram
Nec dum video quid magnopere ex aequa tionibus logarithmicis ducid possit.
Addenda sunt Mengoli inventa.
An forte aliquando effici potest, ut ipsae quaedam quantitates logaricunicae, p
iroguit fiant incognitae a quarendae, vig. exponente incognitarum ordinariam de
X+ bX CX+ d, X + 6+
quo eralibir
CogX + Cog. X + b p Log a + LogE. it est: x + bX - ac 1 0.
viy
videndum an caliquid ghribuere possit Hyperbolae Ditura ut aequatio est
=3 Cog X L 209 X + e B cognitae quantitati ad
quam aequationem quaeibet aequatio cubica reduci potest
Ergo sit Hyperbola cus aentu A. vertex E. la AB BE
ABE rect. ACAX. CFp C. cognita ordinate FG, Cd, BE
debet spatium a EBCDE bis ablatum a sputio DCFGD semel
relinquaere quantita tem cognitae equalem, quaentur punctum
de. 
Nde erder quid
idendum ut forimulae omnes aequationum in cpesitas 
denta incognita in compositas exhis: X. x+b. X+C. X +d. resolvi possint,
XCX +b X+C Y at e. fiei enim: X + bX + db X = a  0.
videamus an conferi possit date:  X + lX + amX  - a e p.
d
fiei b+d∞ 1 seu d l - b
adeoque ab⊓ 1b = 6 am, unde fit aequatio b2 - 1b - am 10. uno gradu inferior, ideoque solubilis
et semper hoc modo quaelibet fierula redui aequatio comparari dt aequationi fact cujus mcdgnita
partes hoc modo productae, qu cujus rei ratio est, quia idem est ac si omista are, aequaetionem
X3 + lX +am X ad est X + IX + am, factae exradicibus cquarates ubi semper eadem
ejusdem gradus redit aequatio. Alius via semper efficere poteris, ud aequatio cubica resolvatur in
sominos factorum simplicium ape selius aequationis simplicis, ut si ponas: X 1 IX -am X + a e.
ubi lartum formuta - lx uam X + a c. esolvendi, in factores quod ope aequationis rationalis
seu simplicis, seu uno gradu ea formuta inferions fiript. Eodem modo quadrati quadratica
per quadraticam resolvetur in factores smplicres, imo forte adhuc per simplicem, methodo
Slusiana unalogiarum. Posito Jam hoc modo propositam aequationem in factores simplices
resolutam, nihibalunt producetur lius X. + Eog X + R (+) cog X + 9 erc it
e omnibus cognitis ad una commune mensuram redaoris tentando facilius inveniretur
sed quiniam lentare incommodum, ideo arte quadam motus adhibita effici poterit, ut seriper
summae omnium logaithmorum, recta ejusmodi, quocunque modo sumatur X haberi possit aequalis recta
da et apparebit, quando ea accedat ad datum; ibi vero postea invento senet puncto circiter
exactissime posten id in quadam posiore cuodo descripta Cogarthinica, mox ex actrsseme vivenietur
in Tublicissime descrisata, as rebiquim tentando, momento ab solvendo
ses firmatives par
seut ramene, moneatut les memeront une regle, les negalives rameneront un poit,ut la tegle