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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 6
# Folio 15r, page id 00068051:0029
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ot musur hinc
si Recta AC in plano ujus paginae rotetur circa
Centrum A; continueque inter rotandum diminuatur, atque
ita describat lineam EE; via centr gravitatis 
et ita post se relinqual partes EF tunc via centri
e line ae AB
gavitatis totius hujus linea utcunque divulso, ducta
in ipsam totam AC, aquabitur sectori CAEC
ut vero Khanc viam. Uentri gravitatis inveniamus
quaerendus est unus ejus situs indefinitus, ut in casu A+ E
quo linea _AC divu sa est in partes A4F, 3E3F, 2E2F, 1E1F
sit AG, X
et GE, y; AE,Z; EF de
quaeritur centrum gr. hujus situs.
ynE Xx + g et AC, C EFes  fit et Zdz = Xdx+ yduu Et ut centrum gravitatis omnia
ZZ habeatur, fiet quaeritur  quaeretur fdq y et (due 
qua sit omnium EF, centrum gr. sit p. et fiet p R  (distantia ab axe)
AE (ut A+E) bisecetur in H. Erit
H centrum ipsius AE. jungatur HP. seceturque in puncto q, ita ut PQ sit adAque
(dqy: c - Z et AR= (dzX :, c- Z
ut AE ut AC -AE (quae est summa omnium EF) eritquequeCentrum gravitatis
praesentis siliis divulsae rectae AC, sit Bisecetur AC in S, et erit linea sq
via centrigravitatis rectae AC, usque ad rasum 4E, et rectangulum sub AC, et Sq
aequabitur sectori CACC CA+EC. Verum Ex _ H ducatur ad axem normatis
XL: RL :: C- Z,: Z 0bf
HK, et ex Z normalis gL. quaeruntur AL, et LQ
1Z Y seu AL - DK: ARL AL:: c -Z:Z Et similiter +pE
 Ex 6 fit (Z.A E - Z. AK = AR. C - A L. C e
seu fit ALL +C+ H F A +C. Z AR + ZAK;: C ( similique ratione
LQ = KH RP = LG . c- Z, : Z
ex 7 fit (ZLD - ZXH = cKP - c t Q - Z RP (+ ZLq) seu fit LG = c. Z R P + Z KH;C
(4
jam AR  ZX et KH Z  y. Ergo ex 9 per 4 et 12 fit AL =  (dz. X + = ZX,: C = M
taque ous
Et ex 11 per 3 et 13 fit Lg= f 4Z. y + +) LZ7,: C =n
est mo quadaturas summari ipsas idzue  et ipsas Ha est antem per 2.
ipsas  dZ = xax tydy,: ExX+yy et fiet Xdy = XX dx + Xydy, : XX + 9y Hinc
m= E SXxdR AE Sagdu + + ZX
(21)
sxatyus
raxtgus
yq= Xydx +gydy: Sxxtyy Hinc
ubi sumcmae sumeniae decrescendo
+= (Xydx +1Zy
incipsendo a prim maxima z. Quod si velumue inciperae e infima, loco seu
2c
Cyydus
finc n=.
C rtgus C saxtgus
mi a minima, tunc pro sectore supradicto; sumi debebit DAIOAET
si esset XX+yy = XXyy: aa
Et
quin _ ex ryus voletur 2Y, hinc Xx  qu gs et iu =
Xa + =y + 1 Xxy
ex speciali curvae natura, fieret m = E) sy
 feret n = a sudy + ~ X + + xyy
adeoque fiet n = (Inx-aa dx + a y +  Xxy
Xx =  ay aagys e:, yy-aa
seu X = ay: ryy -aa
et HH = XXaa: XX- aa
seu y = ax: Ixx - aa
et smuliter n = = (Syg aady +ax +  Xyy Hni pateo m etn pendere ex Est. Hyp. Taegus -2
(22)
postremo (C Sdmam + sandn = Areae dictae
dm = L xxdx 1 + xy dy +  qla 1 L xtq + ic tax +qu dx + Xxdx g
tercyus
Nec dubito quin hinc quod querimus etiam calculo facile demonstrebur
Sagy
e tkxcgus
nostram theorema.
rertur quadraturoe quaesita
Tar et cyntaatro ad
etur ali
Ex coptemus exemplum pmondetur 
fit XX ~gy = bov seu Z= Bi et Xy. Z- sp - a: A XXd. Z ax: Y b X
requisitae ipsius vero sareae soi
Ngu x
aa v (od rrxcoruto Erpo YY dx. Z es summatilis  yydy: Z bv- a dus
va aa- tol,: 
X _ 2Hy = 2a∞ = Xa L , _ na- vv = ya
y
veaaa + ygaa = aa + vV sit v - +x f vy v
arvo- yA co =  v ~ XX + Hy = vV
XX + Yy =geri X = K K sec m WB+ nB + p y = fvV + gB +h Jin. 5.
2np V + 
+ ffW + 2fg0 + 2fk VV
+ gguv +2g0 + hh
rumv + 2mnv + 2mp VV
fit m 20 et f20 sit. p = 0 et AC0 et mmVB + Emno + nn es quadeuto a Lmmiss V + Tnntgue
2 mns,
+ nn vr
seu fiet mmante + mingue +Hfun (Hgg)- (mmun) H99) +2mntq seu mg - fn = 0,
adeoque x = mbP 19m: fB  y = f4 9b (hinc g. x H = v mult. per est.) Ergo xxagus = Sum+VV mmog. H+99V
X = 1v, 1+ g. f V
3 = su, 1+GFV ExRFy = V 1t g. V Ssm Z axb. Z = mn ∞, 1+g.+ B (. F 4yA - A 1d. s0 entm
haque isitum
NoE ngus Xy. Z = Sasv, 241. v
itax et y forent ad
rectam
Cas prayrlu
rXtV
Cm +7
=aa av ma-c0a
apsatv, a
ua eco + aud
an- vy- at, a 
6a-