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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 84r, page id 00068200:0169
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Addatur postea scriptia scheduta in 8. majore
calculum quem jam ante multos annos institui nunc repetum, et videbo an elaborare rem jam inventam
vacet quaeritur Methodus agnoscendi quaenam Quadraturae possunt praestari per ordinariam Geometriam. Loquer
auteuide quadraturis figurarum ordinariarum. possunt semper vduci enr potest semper io reduci quaestro, ut quae
ratur sammatrio irrationalium; sapponendo omnes aqualns affectus posse ad radices puras reduci. Ex his radicibus pures
ea sunt simplicissima, qua quarum contenta saltem sub vinento sunt rationalia, seu quae sunt potentia omefirabules atronales
Quaeatur ero Ab his ergo incipiatur, et quaeratur (dxVO, posito e esse numerum integrum, ete esse formulam
 rationalem   datam ex X. posset quidem scribi. (dx 23∞. sed quia hoc idem est quod Pdx Y9. D et
poni potest 8. D = 0, ido res semper reduci potest ad Fdx es, sit jam hoc = ZV. neque enim aliter esse potest, cum
summarum eaedem sunt radices tot quot diffeaentiarum; iter tit  = (dq es + ac 80, seu fiet
sit rationalis quantitas: pqu eut,  ps lou eonc 
sit dxi q= dp + da + cr etc- dsit   et fuit pq. r:, s, t = dC
 = 2 X it 1
Ex - 1, p. ) x
d e = (. C) Cbed. 1 +0 = (1. C) (l et. 0) C: e
lx 
(dx Lx = 2X laX fit dx s
 S d, X. = C XC ax + (X ax
   x
+2 dx x fic c1 bet.
t Xet et SAxa x = L Xar a s
f seu (x 10 = 430. posito   X et C -x Hinc differentrando
5C se Ax( = d1C + Z  60. et ex = a + E
dx 9 = Z 0. fit dx O = dZ Oe +e f. O 40. B seu dx (∞) = 24(0) + e 8 d0 (3e) in exemplo
dx 80 = ag 50+
x re = L x. X fit dx XC = = x XC + C X. X dx = dx B dx (X  ax (X + e X du (X
in exemploente X:  X = lx: X + Cdx: X sea  six:  X = log (X. X ) seu  et log  = log. xet quod verum
E x: f = d8: 4 F e +B: B. seu Six. 4 = log, (402)
Sit C = 1XX 10 + uX fiet C = 10. 10 + 2. HIXF 
10. X
fit  = 1010+ 21 + A AX X , 1B. 18. X
10. 10- F 2, 10. 11 X + 11. deX X
0. ∞ + 2. 10. 11 X + 11. 11XX
fax: q -log.  10X + Z. 210n X + = 1. X3 = log(p. 13) = l0g,
L(0X L. 2io uXA 1 C, 10 +2. 10 1x + 1 + s
102 + 2. 10 ux + 
dx = d3 + + f dO: C. unde s,ax:  = 109 8. 16) und sequitur quot modis efci potest ut S, dx: 8 sit loga
Tichinus quantitius ordinatiae, haberi posse quantitatem C) sic ut (exIC sit suminabilis ordinaris. Et si non
m
uuu illa
elitu  osiunt i
sappono artem hic C et 8 esse rationales per X
stiquit modiss
i esse qogaculimus
talis, ut
et sit Pdx Y = (X) 2X Lx
serepas ub et
tax (2) et eti  de  1 X S
Licretne sic procedere data
Cqualio 10rv+ I1aV + 1aa ZC
summatu V, seu dyo dy: dx= v: a
quadram summanda v, ejus
summatiix 7.q o ejus aequatio
et aequatio 4 20y5+21y+22aa
pro aeg. 1. pm sumamus radicem 30V = 31 a =0 et pro aequ. 2. radicem
407 + 41a =c ponenda hanc illi respondere, nam revera cujvis casu res
ambignlati summandae espondet summatrix
Ex 5 differentrando fiet
40dy + yd40 + a d 41= 0   seu 40 v +ay d40 + ad 41 0 ergo
per 4 et 7. fit - 31. 40a + 30ayd40 + 30 ac d41 = 0
30. 40 a y d40 + 30.40aad41
quadsi 4 sit constans fiet
= 31. 40. va 
Cx 30. 4047d40 + 30. Maa d40
seu fiet
d41 = 31.40 : 30ca
Verum quia 2 et 31: 30 est quantitas iratio
quodsi 40 ponatur constans 3ofiet
+1 = 40
3e dx
nalis, qualem exhibet data aequabio nullum
in hoc calculo lucium esse potest, et revena res reducitur ad radices
summandas
10 VV + eaV + 12 pa = 10. V3 + naV + 12 a a V + 13a =0
2013+ 21aV + 22 ay + 234 =0. fild 20y4 d21 ay + du ay + de3  
Alia via
320 yV+ 22a)V + 1. 22 a v
Tollatur jam non , (etsi is calculus facillimus, quia ita assumitur ad v sed tollatur
potius
opelet
y ope 2 et 3 ita v non assurget ultra 3, et aquatio habebitur
g  VY+aa =0 et ov 4b ∞+ cc =0 fit y
yy + by + aa = 0 et yy + vy + CCE fit bytaa = vy + cc
+ vv- bv. 44 -cc + vv-2bb + bb, cc
∞ . 2Vb +bb
vp - 2Vb + bb
coincidons data
i  a - CC,: v - b aa-Ce
vp-20l. +bb
seu aaa ce + vvaa a braa cloiecbu x ccvv- Bcel v +cebb =0