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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 124r, page id 00068196:0245
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Concipiendo abscissas x procedere in progressione Geometrica; eo ipso
omnes auae figurarum simplicium fiunt quadrabiles. sit enim
V = X e quaeritur sydx = X dx. sit dx = x deo posito ed esse progressionis
Arithmeticae, fiet sydx = X da. Et reri sunt progressionis Geometricae
ideo summari possunt aecrescant in infinitum Ce q
b bb B3 1
cob _. b, b lob,sbb 1.bs3
 =s. 1+b +bb M.. et s, xete
1
AX, X XY, SA XX Xxdx = A Xdeo
1- pert
8
deo sR = W X + X f X + X et = X, 1+ X + x + X et c, du
= XZ d  quid est rectam FA:, a X duci in rectam co
7 x3d  an = X
AB unilas est major quam X ultima nam X debet
esse minor unitule, ut potentiae ejus decrescant
hr logarthmica, ubi ACd est co, et covse
est AX seu X. et coY suit X. ita Y in du
ae quantur ipsis areolis ad curvam Y Y. 2 laque
omnis quadratura  signit simplicis recncitur ad quada–
turam igo areae Logarithmicae qualis est V4. sed omnes loga-
rithii co sunt similes. Ergo res omnis reducta ad unam quadratura
Et reperietur logarithmicas inter se, ut numeros respectice
sed a si esset ns(it) x ponamus itx esse progressionis Gcornehrilae
nec
Nonidem erunt et d (1+ X) progressionis Geometricae. Ergo et
) Ergo net dx
1+ B 1+ 2b + bb 1+ 3b + 3bb + b3, 1+ ab + bb + 4b3+ bt, etc sseddemto b 2b + bb 3b. + 3bb. + (3
1+ b
1. sunt
b, 1+b
1+4 2b + bb
1+ 3b + 3bb + 63
b, 44 2b. 1.
1 2+b 3+36 + bb
sint X progressionis Geometurcae eo et ax et A X progressom G00
et d (1t X) sunt progressionis Geometri cae d (1+ X3) est (- 2) A (1+ X  (1GX)
sint 1+ X, progr. Geom. erit et (14X) progi. Geom
sed qui e quaerend cujus progressionis X, sit _ B = 1 =+ X erunt
d
1+ X numeri, et cd logarithmi. Ergo si deo constant erunt 1+ X progress. Geo
2
a
o = deo b
dx = de (1 = X.): n. X
si esset = = - 1 faect m - - 1
dx
-1
Et 7 (1 + X) ax =   ( +1) 1t X2:n
- (de 
n. X
si esset B=X fierent X et X progess Geor. puosita dD contante et fiet
et erit s
1 =XH
d n, Y
deb= nxax  et B do =  n
sit
= Y
fit do = de.
nm
I: m I (X
m+2)  de.
du
(9 +r. + 5) etc =Y
(19.. MY + n5) = F
SyAx (vax = Co fit tySvdx + dx V sydx = deo
(l. + m +n, et) = C
(yvax = Y fii yu du dp
C D = flq. +
( yy qu (Suax + vn qu (Sydx) + 2 vy (su du) syex
fam deoded data = es, fiet = yy que sude -voqul sydx dx = ge vyax
Axax
deodeo: dxd
gy soix ve sylx dx = vydx sit y = ((1 + X). fint 4X- L
2svdx sydx
2sydx 2sotx fiet sydx =
L. e aZ=  L.+ = +A F. R23:2
incipiendo a Z = 1 sit V = Ex fiit sd 2 incipiendo aat
seu.
sudx sydx
vel Sydx = Z (I +X) 3.2 et f sit v - sx sit sudx = L X
anpo (et L(X 1 X) d = Cd = = (X X +X) 3. 2 et deo = 4 (XX + X) : 2  3 sexcx t H