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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 202r, page id 00068198:0403
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RyX+a
X, X+ a, X+2a. El
XPXFa3 X+2a5X+sa
g. 4 5..
Al
t
(n 41
1. 3 2 5 3Z 4. d, 5
1. 2. 3. 4.5 6 7 89 10 11 12 13. 14. 15, 16.
+ 2n 1
st 2
X) XIXTa) X, XFa, XtLa) X, XFa, X+ 2a, x+ 3 a (∞ X +a, Xt 2a, X+3a, Xt +a
X, Xta X+2a
(XI Xta, XF2a, Xt3a, X + 4a
X, Xta, X+2a, X+ha, XF 4a
Numeri A, sunt numeri producentium in termino quovis seriei A, B, c.
BC, et
ubi:
sed pro serie interponenda producentes sunt numero 3: 2, 5: 2, Ctc. set (non apparet
quomodo numerus ipse terminorum possit esse fractus
in serde A, sint termini m, fiet, (M): m = x+ha  (mD: (m) = X + h+i)a
fit na = (M) :m =X et ha = (m): (m) -x -a unde fit
(m) : m  X) = (mB: M) (= X = a seu (mD: (m) - (m): M = a seu S= a
inserie B sint termini, C, et fiet (n): n = X+ ha, xtihtda seu fit
ninseu
(2) n = Xtha + a (X+ho sit X+h = Z fiet (ZZ + aZ + L aa = Cn) + L aa
 (n)): (n) = Xthatra, Xtha +3a = Z+ 2a, Z+3a, seu fiet
32 + 5aZ +6aa = ((n) et fiet en - (n) = +ZZ + sa Z + baa = 4 aZ+ 6aa
2
= aZ
= en) = (5) = 4aZ + 6aa fit 16 a41 2Z + 4 8aZ + 36  = 6 (5) (5). u unde
Es
16 Z 16ac (2) = 16Z + 36 et fil Y32a3 = 13) (3) - 36 a -1 6ac )
a
n +484Z -(5)18)
et fit a (5)15) = 815) = 16(1) - 12 aa. seu
(n) (n)
(1) (1)
( (n) (n
nn
= 2()
(2)
n
L= 8 aa 6) - 48a
inter BL, eN, med. prop. D. medio loco collocata, das
eqao samicam media harmonica Hyperbicam, media arithmetis
eat rectam. Quo si nar et si bc = dd m est med. geom
si b + C, : 2 = d est med. aritem. sit L + L, : 2 = d med. har
cC ( 
2b 4 ZC =d
quitan
d ved. aritem. +
2∞= + Z =  =)b 0bsc =
2 b
e
quid si B L fab +L fac = d seu
ac aabb + L (60 CC) =dC
bb + dd = ae seu apbb + b + C + 2bbcC 
bb + ee = af se bb F seu a bb + a’ éc +a = af. undem
quid si bb + cC = ad
abb +  (aa b b. +  (Ib + cC)) = a)
libt _ = 49 seu a bb. + (2((a. bb + (2) saabb + 13 (bb + CC))) = a a
Y sy yy _ + (y) 1y) = ay) se y ag si e
60C sit (y) = 3+ du et (7D = y + dy + (dy)s et ficf.
gy + gy + 29dy + dydy + 2y(dy) + dy(dys) + dy (di) = a y + ady 117a
Hic dubitati potest, an tali ratione desinatur un quam in veram curvam
id si sit CC 2y + = Ry) = (y) fiet 2y + L. y + L dy 4+ L (dy) = y + e dus
ubi nemp
differentiqu  tosur sut
ubi evanescente y fit tandem L(dy) =  dy seu dy est coustans.
et reducendo
(2yy ( 29 dd 5y lly
L
L
+ 2 y+dy +(dy) = y+ dy
posita dx constante.
Edus ay-rtyatus (it y  rus (9yAt 2ydys + dy dys (+ y Hy s dius ay) t 7y 
lydes
 tyd et sit dydys + dyldbys trodus = y dy) quod seu d9: dy)= y2 y +dy +ldy)
us
s
+∞
an liceret Ent invenire curvam
y: (9) = (dy): dy fiet (y) = yy + (7)(7V, :, y + (yD quae
Et
ut desiderabatur.
foret progressio auti harmonica de qua ad curvam redigenda laborai
si
ZaX = yy
ago curvis valgaribus possunt duci tates prosrietates sit
non vapat
fit 2aX + 2a dx = (y) (y) et 2a X +4ad x = (y)()
follendo
Baec gearqsi a
rax ex duabus prioribus fit 2adx = -yy+ (y7(y) seu 2adx = 2ydy + dy dus E
et tartus 2adX = 2(7) (dy) +(dy) (dy 1X sea Ergo 2y dus F dydue = 2(7) (dys) +(dys) (dy)
seu Eydus 4dy dy = 29id9) +2dy dy) +(ay) lay J. nen 2y ddu = 2dy(dy) = d9) dy) = dydy
ubi in infinite parvis patet esse ganududunam A dyu sunt
intrartisixfte pava
fit
e
yddu Lte ininfinide anai
C
onlicis
i infims bb
et
ad, quia in epsa
terminationis
Etoce
alles
boree lit
c
=d2C, 10
sd - b1
si
d=
-b
cb
= cd -c
bbac
bC
in
s
d
gdy
Fddyddus
agdus