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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 137r, page id 00068195:0273
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simplice
coefficientes terminorum in potentia polynomii sic definiuntur: sit
polygonium X = a + b + C + drete Erit X = 4. + 4ab +6 a b + 12 a bc +24 abcd. in
hac potentia, quae est quadru biquadratus ipsius X, sit terminus aebc, quaeritur ejus coefficiens 12
Is est numerus omnium en tranum tabitatum traospositionum diseentium ipsius abe, velut it alac
abc. a d. bac cab. bca. cba. abac. acab. abcadacba, baca. caba. Cum autem hic
inoliri alibi docuimus, ut pro abe sit I2 t, = 12. pro a BC sit 1.2. 3. 45. 6. 2 - 5. 6. 7 = 210
1.2.
1.2. 3. 1.231.2
loefficientes terminorum in potentia combinaoria bolynomii etiam definti possunt ex prioribus,
modo sequenti. Voco autem poientiam combinutoriam velynomi cum aebibe xep 
non solum adhibentur X pro potistomio Va +b+C ret C. sed et y pro binionibis omnibus ab tac +bet et
et Z pro ternionibus omnibus vabet abda acd + bed + ek. e BB pro sxternionibus omnibus, et ita
porro.  Eiit  atb+c+det vel a = X  et qu ∞XX- 2y  vel xx = aa + 29, et
a = X - 3Xy + 3Z. Et ita porri.  ute ritu numerus coefficiens ipsius XyZ, (ex. gr.)
in ab vel X.. Fiet ut 3 numeus dimensionum apparentum ipsus X9Z a 6 nundrum occultarum
q(nim x habet 1, y habet 2, Zhabot 3, quia Y intet ab inti vabet abC) ut 6 mimeris
permatabilitatum apparentium ipsius XyZ, (seu coefficiens ipsius abc in potentia polymmmii
X9Z est 1. 23 6 = 1
simplice) at quaesitum 12. e Itaque coeficient iimar eb pus dimenti a
signum in potentia smiplice est semper + 1. in potentia combinatoria e positu simpade t ter pus dimendi daa 
XK = aa F2ab X = a + 3a b +6abc X= a +44b +6ab + 12 abc + 24 abcd 
ains diberum. Prenius Gnia p a edi dinaeniim t
X= a +5ab+10ab+  +30 a bc +6 abcd + 120 abcde
X = a +64b+15 ab130ab ab,60abc +120abcd + 9 abC,
1c
10a
Brevint qad figuum forma qua abique d di  tstimersidou apparex ts  diff 
a = X
 p 91s 6250b 12 2000
=X- 25973Z. 1.
X=X- EKyt AX Z+ X(- 5XB. 5 y215)
asqrent
=X 4Xy+ 4X22y 242
-X - 6Xy + B XZ +9X ( - 6XB, 6 X - 27 4 6y B13
si a, b,c, et suit 1, fit, X = 12 3 456 y = D, 1, 3, 619,15 20, 0, 1, 4, 10 209 ( = 0,0pe, 1, 5,15 P= 0,0, 0, 0 110 ad vi
si ergo rudices aequationis ponantur esse Data invatioore somme curu omres
+qu YX.3 + etc. Ponendo radices es , suEt 
esse a; vel o pont C, etc. prit a = B et g a ab =  et abC = Y; L habust
ex
 X +eXC
et erit aa = lp - 29 et  = p - 3p9+ 3Y et a = p - 4 ppq. + 4pY + 29)-
porri a = p3 - Sp9+SppYF5p99- Spes - 5qY + 5F. Et ita porro
jerine
, sed ita et ponafattentum adesses 
10
a, to e di
noinvestidaletiat
ntis jam et
Assuno jam instar hypotheseos, quor raclix quaesita aequationis X, sit pelyno-
1s
m P1 4, C, ∞5, B
mium, seu X = le ex quae Lm, etc sunt quantitates itrationales, posito
oeus cuique est aepartu
 sen reliagtnone sato
porro Turpono easdem vel earum potentias conitinationessue quaslunque non dare ali
quid rationale, em, lina, Um, et nisi forte eo cisu quo omnes simul ducuntur in
se invicem. Si itaque nulla talis forma ex ipsis  (exc epta forte illa unica) dat aliqid rationale.
aqurgens
lrcditd
elic _ 129.3
1.2. 351
120. 6 - 0
rimm (parium) habit X = a
idem signum cum
38a
3125
sese
q
67
6250
ane
68)
t
628
3125
D
unt