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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 382r, page id 00068196:0759
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X =
X = lam XF = l + dm X = (+ X ellm. X = l + 4 ln + Ellmm
Xx = i + um
X = V + 3Um 
= 1 R alma cum = I1+ 2tm + mm
X = i +alem +6Ulmm
X = I5 + 5(a m + 10 l mm
X = I5 + 5 Vm + 15 (emm + 20 l n 
X = 17 + 7 lom + 2t mm F 35 l’ nd
= B3 + 3Ulm + 3lmm + m
= V + aBm + 6Ulmm + 4ln + m
= V + 5Im + 1ol hum + ollm + 5 bm 4 m
= l6 + 6( + 15(2on 120 lint slbuislm + m
=I7 + 7 (6m + 21 ( mon +35 ( + 35 (3, 2 llm +D In L m
Regula ad namaros coefficientes.  sit T forma n, quaeritur coefficiens, dibuntur exponentes 413,
 is erit numerus hnionum, rerum 4+3, seu 4+3, 413-1, 4+ 4 1, :, 1. 2. 3
 X = 13 + 3Um jam Um = Im X, fit X3 = 1 + 3lmX seu 13 = X - 5lmx
nta
(om = lm il seo 11 = XX - 2lm fit l’ m =lu XX = aUmm et l = X +- 4Bm -6mm = p - auX a binx + Gllmm
= X
II = XX - 2lm
13 = X - 3 lmX
it = x- 4 lmxx + allmim
V = X - 5 linX3 + 5UlmmX
 m = Im l = lm X - 3lhamx  l’um = UmmX
fiet l.= 3 - SImX + rellmmX seu ( = X - 5 lmX + 5 UmmX
16 = X - 6 lmx +9 lmmXX 4 - 2 l m
Em = lm  = lm x - allum xX + 2(3m lum = Umm l = Umm XX - 2 m
1 = p - 7 lmx + 14Umm X - 7 Pm X 
1 = X6 - o lmX + 24 llm X - 12 (3m = X - 6lm X + gUmmax + 18 l’m3
18 = X - 8Im  + 20 UmmX _ 16PmAX + 2l m
=15ILmmXx + 30 (3M3
lm = lm 15 = lm X3 = Gllmm X B + 5B mX Cmm = Umm l = Umma 23 ( ex
V m = VmX et fit l = x - 7 lm X + GClmmX - 35 (m X = X - 7lm X + 14 lbum - 7 ln X
12 = X) - 9 lmX + 27 UmmX - 3d(m X + 9lm X
10 = X - 10 Im p8 + 35 Umm X - 50 ln X + 25 (tn XX - 2 l ’m
Iu = X - 1 lmX) +44 UlmnX - 77CnX + 55 lin X - 11 L iX
ju - X = clmu +sq llmmX = 112 ln X +105 (i X - 36 l ’inxx +. 2 lm
l = X 3 - 13lm X + 65llmmd - 156(n X + 192( .  - 91 la X + 13 Cm. X
Et ita porro, columnae sequentes oriuntur additione numerorum
columnae praecedentis, sius prime autem columnae in qua extet lon prosigitum 2)
2i UmmX +63(2m57
35 (2m 3X
sit jam aequatio cubica data X + pX _ Y9 = 0 fiet pe lm = -p. 3 et P+ m = - q. et M = - p. 3l) nnich. 3
et 13 - p (. 27(3) e = - q et fiet 1 4+ 9 ( = p. 27 et (3 = 1 = 49 +  p) - L. 4
vel X - 3p X = q sit p tm sit X - Uem fiet p = lm et 9 = (+ m3 et  m - p: l et M = p. 1
sit aequatio surdesolida X f pX +) PFX X - 5 pX + 5ppX = q
et 13 + p C.(3) = 9 et 16  913 = - pd firt 1 = 2 4 + N(4+ 99 - p3) et m = 29 - s (199 - p3)
_ sit X - SpX + spp X = 9 fint ln = p et ( + n = q ergo (3 + p (. 1) = q seu 10 - 915 = -p seu
(10 - q 13 + 199 = a 49 - p et 13 = = q + Y (499 - p5) et m = L9 - 1( 19 - p
Et generaliter si numerus par n sit impar, posita aequatione
X - npX .= + (  (a - 1. n. 2 - 1), X   (u -2. n. 3. n. 4 - n. 4)
(n - 3. n- 4. n - 5. n-6
1. 2. 3. 4
M. S. n- 6 ) X n - B
n - 4. n - 5. n - 6. n -7. n - 8
1. 2
n- 6. n - 7. n - 8) 
1. 2. 3
nug0
(n - 5. n - 6. n -7. n - 8. n - 9. 1 -10 n. 7. n . 8. M - 9. M - 10)  6.  Ma 12
1. 2. 3. 4. 5
1. 2. 3, 4.5. 6
1. 2.3. 4
etc  = 9. suet et sit X = l + m
in
2
et lm = p et M - C p: L
1. l. 3. 9. 5
et In + pC. () = q et Jin = q1 = - p
seu jin - q 1 +  49 = 4 99 - p et l = + q + Y (299 - p n)
et l = 2) (29 + Y (3 49 = p : )) et m = 3 (.= q - 1 (+ 99 - pr7)
_ et X =  Y(+ 9 + Y C99 - p.)) + YC2 4 - ( 449 - p 2 1)
( qu i C794 - p 2)
vel p = n (= 4 + Y ( 94 - p:)) +
Hic calculus paribus imparibusque gradus est communis. Et credo etiam ad fractos
si P sit radius et X sit subteus
arcus circuli erit  subtensa arius circuli nun qui sit ad priorem, ut n ad 1. Hinc cum p sit major quam 9
et irrationales exponenles poterit applicari.
un qumine unitate erit Y( 44 p) seu et q= 1) quantitas imagina
sit ergo vadiqu 1 ares subtesa arcus  saberr erit n cubcensa arcus qui sit ad qum englabtusa qut t ad A erit
a  e   + fC 1  23)
31 = 4. 6. 3
Bisa
ot
entus
3
5 6
2 2n L 13
n+3 - 25
ergo 
Ergo 9
 reu
aTan + gR + h) =
25n + es=
isras
et proiit
13:
25 -1 d f = L
y
3 = 25 +95
nn +
182
Ego  nuran -2
n+ 1, n- 3.
B. p -, nt rontis - 9
4 20
5. 1. 3 - 10
1. 5. 15.35. 70
0 2.9. 25. 55
1. 3.6 10. 15
2.9. 25
1. 2. 3
65 s
51 44 - 10- 3 -
4. 32-4,-2 -2
b. 5. 4 - 20 - 4 - B
123
n
I l.q et
6. 942, 4. 3 = 1 6 =9
1. 2.3.C
6. 54. 3. 2 - 4. 3 2 - 6. 4 -2
5. 4.32 -  32 - 5. 3-2
1.2.3
1. 2. 3, 4.5
7. 6. 5. 4.3 5. 4. 3 = 21 - 10 = 11