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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 94r, page id 00068201:0187
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Dc Aequatioisds per
Emos snum,
Hac Schedula sub finem confinetia Resolutio
si sit Z n +n.
omnium aequationum a per logarithmos
et inventiu geomehica logarithim inomitst
n m +n 162mn 27
datis loganthm nominum; et quantitatibus;, et appropo
lognrthis ac nomi quantitatibus nominum coced
alterius bimmii unum cum proposito nomen
commun
ghei min, + 3mn mon i
 mn + amn Emancts nempe   + 4nim + 6mn + 4mu + n
amu mobmn tn
 Cami nr in , A n 4 n5, + smu, mon, 13, sei
debet esse cubus, conferem b m +n + m 7 4 fere
p3+2q prus, + P y  n + 2mn 7 c
ZBH m + n +6 mn Z6 fuc
et qnn see itrerermu
ergo B2c+C A) et C 1 + Unde hoc tandem colligo:
Erpo plae ZB n +  + smn quZ ema r 2 amu L
gepimea ne+ amue.
73 nd +r + 3mn, man
n +ima +omn +omn +5mu in
5mn m venin HOmn + 13
me6nin +5 mu  onn mn omni +n
6mn m na +20 nn + 5an + n+
  492 + nZ H - 9Z
ent mer Ze et n A   ert
Z4m ru + amau in trmatu - ema,  vel Zt e a  amnEmen , +  m- n B)
n
s
 
76 mtn +6mn 2mon (, (2 En -  mir, ( vel Z  6mn monta, + m = n F - 6m n 1. m - n, 1s
ri in in 17 mumen B
hive Z6 H 6mn man(B -6m 
3mp. ph +snpE
dissecariposset hoc propotitio in duos al multiplum ipsius mtntP. a
scilicet excerpemus: nmmnm, +Xm p (Em nB + Imp, et restat:
m f, 2mn tem a
Ergo p8. ein Z
conferatur aequatio ii:
videamus jum an quadrati ab m+ntp multiptum invenire possimus.
excerpendo: n m+ 2mn +P+ mn +2mnp + mp
restabit: mn + 2mnp + mp + n = harp +3 ap +p. sive estabit n+ p, (, Dm. et n +B, B)
in Z + in  nrp, Z
quod etsi nihil
et nep 1os mndo. nocen ud aea nip B 
eZ   qucZ sit jam aequata.   at  + al 70. pro 3 pono 2tC
+ 3eZ +c
 ati. +ahc
 erit m 3C. et Cn + cp, 352 + ahi. sine n +   3C + at
iie denique 2 4 6c.h + a h + 137C reisan C4 9 ali c + ati
fier: 53C X + 44 ah C  a l C + a h C  a t3 ppo quae reapse cubria est.
dederit quod desideravimus, haec tamen binomiorum expressio memurabile exhibet
Purcuio hmi quaedam memorabilia duci circu logaridimos.
Log  a b est liquationes om reducius ad aequationes interlogarithmos binomiorum
log b y + f. log y + 9 B log y + h. + log y + 1.
qem a s
diti numderi constantis, qui etiam est cognitus, ut  10 - 100 + 1000 = 10000 etc. es ad
b - 12 + 17 =  esc. seu log. br 7 - b
u
omnia poterunt divid: per   100 + 1000 erc. ideoque
ac si dica, log b. f V - b + b - b etc.
17b eideoque in aequatunt
10- - 100 4 1000 - 10000 ex.
per inde erit evanescinti illo multiplicato  dvisore perpetur
Notuadum est tamen, ipsum b. esse numerum minorem unitate seu
fractionem. Jam pe Ergo si sit b vel Z p a m+n. binomium tunc erit loganthimus a im + n
aequali) a. men mtnt id estjam
ZCm gemmo,
+2 () m 13 + 3in d man
2  1 7 + 2no uu B enin
m m  m. es logaritimus a m; et n - n + 1= n et logarithmus a n.
et quonian seris infinita -Emne +3ma men = ann  m n 1. e venit
feliciter ut qui termini multiplicantur per, 2 vel 3. etc iidem per eosdem numeros dividuntur
hinc fit u divisa tota serie per mn. it series fiat progressionis glometricae itque
atque ideo omisso primo termino-nn 1. reliqui summa erit. mn
Supersunt permini qui supra in C, qui non sunt nisi binomia facta ex 1 +mAn
nin, vel mn, Cx. ita tamen ut scrupus solida sibi respondentia quagantur in terminos
uniqus scriei descendentis. Prima senes descendens est OD. ubi primus temminus mn
quin nihil in covariabile, secundus min. et mn. certius na +mu et. Ubi rursus hoc com
modum habemus quod numeri divosores, progressionis qui sunt progressinis authmeticae
evanescurr ob respondentes eosdem multopticatores ex natura binomii. Mraeterca etram
totum O.  dibidi potest per mn. restat ergo ut illiquarum deatum senerum in infinito
procedentium, quam prima D in cipit ab mn. seconda e incipit ab mn. reliqua i sequens
md nt ab dtis maiperd ser in quam quae libiet sesoli polesi pa sin euas propestiones
inen imn in