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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 96r, page id 00068199:0191
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eag - kh
+ eag + eh a β compendio rp = A q +   59 + Bβ
eas N =K
rursque efgu que) = h (9. B) rp 4 - rrpp fiet L t ef qu (9. ( - hh Bq = h (9 - ) (54 - B() = 554q - BD B - 2 d8 4B
erat Ip =
=hh Cq
ev V =R
adeoque efq4e + efBB + ref gB - hh β9 = hd gq - hq3 - hsqβ + hd B(3 - 5149 - 60B3 - 258 9β
tiet 44 tet 40
+ +Zet) 4 + et = 0 Et habetur 44. BB
0 = f + gy + hdy +KBB comprobemus
= h
= dd
dolutionem prioris aequationis, quae ubi est o= f+ gt hapr KY sentemus  in exemplo
758
+250
sit XX + _ aa d + yB = 0 fiet f a x 9- q b = =aa et K= 1 = L Faciamus v -e +Ey
fiet de = de + (C: a) y +(C: a) ly et fiet aa XX P a dc y B ac dy + ccyy =  eep ubi G
hdy +KDV
h
tahy
jam debet esse f+K = g adeoque  aax -adetanee +a CC ace = de Ubi tres sunt variantes, X c, e, et duae incognitae C, e
a de +2Cy 
Pitaque dicelli potest in duas, et  -a de = -a de et axx +  F aa xX + aa ee + aa cc = 2nace, ex priore fit
e= C ex posteriore sit  XX +2C = 2 e = C + b unde exx ex posteriore substituto valore ipsiuse
fiet; XX + ce + 2bc + bb + ce = 2cc + 2bc sed ita fieret Xx + bb =0 quod absurdum, nam nam b est
VY =0 quae aequato
=aa a dp -20)c Iaande  2BCC destruentur
+BBCCipsi oat gemina est,
constans, fiet ergo e = BC + ay, : a et fiet
au) Y +2apC
Hinc fustra tollimus cc, et C, s enim
.tollamus omnes terminos in quibus dx fiet aa XX +aayy - 2aaVc +2a BYC tua cc - 2 a BCC + BBcC, = 0
ita simul evanescet dx, et supererit - a dy - dBC - Bdc + adc adeoque fiet  ab- aY = BC sed ra fit coest
recidemus in idem absurdum
devellamus aliter, ut sit au Bdc + 2a (Y C -  fit et firt ab 4 a) - βc + ac = 0 seu e = ac + ab
Aa sic 2 0 aVV + 2aY Bc +B BBCC, habetur   ay = A BC, ubi A pendet ex D. et est onstans
sed ta  ex UAXX- a de tanee +acce = 2 a ce - a de fit aequatio absuria.
umfiat aa cC = 2aβCc + (1 - D) BBCC = 0 seu aa -2a (3 + (- 8) eC =. hantur V, et C, sed B est ostans
XX - aa d9   2VC - 5 ( 7 s(Bdc - A - 23 BcC, quod recidit in priora
sed sic licebit faciendo -a dy - d βc - Bde + ade  seu   L( e +C) + D e - etc
fiat  aa de - aa de =
et dβ evanescit, et superest
eit c - e = β et fiat e a dβ = (Bh x et a dβ : βB = hdx et ab -  = shdx
Resumta ergo aequatione  XX. aade + ee +cC = 2ce = aade
et fit β = ab fhim e a: ab -sMax et βB = a:  qu (ab - stid x) =  xx F per seq B, et fiet
 
a:, ab- shix = +  et Ab - Jhax - + X et et hde =  +  seu h = + a: x
si fecissemus aud (3 = βhax et calou B = shax et  (3 = b shar: aa = e A X et β3(3 ed
eo dba Nam saequatione aXX (= de + +  cc - 2c e A = (ph - BBh, 
seu axx + a dB ta B3 = -3βh + BCh fiat a d β = AB(h seu seu a β. BB = h. 1x se
tade
- a: ( + 50 = shax = - a: β + ab = shax et a. B = ab = Shax seu β = a:, ab - stax
  superest axx +  a ββ + BBh = 0 et fint ββ = axx:, a th = a . qus ab - shax)
(ab - Jhax) ex = hx dx
 qu ab a cet ha) = ax ax (ib - st dx) dh
Estraias
aXX et dh. Rshdr = abdx = hax a H) sed ita sh vocando Z diitur ad dtZ nam hax = dZ
ab = shax itaque opus foret loco aeq. _ adB = BBhax assumi
ad
aliquid ex (3, voc quode voc abimus H, et fiet q aa dB = rchax, adeoque seribe c
2f(..-
et dhdx = ddzue posita dx constante
aequatio resolvenda sic scribetur _xBB = Ghax - ahdx, et 
aa dB = rh dx posito r = B. Et fit aa fd B. H1) = Thax et faβ. e = 
rursus XX + BB =  - rch adeoque 1 ( Xax + 23d β3 = - ndh = hd, ubi H pendet ex B et h ex X
A idem jatem ent generatius adhuc, si nh = W, fiet a al B = 1 dx et 2XdX + 2β.X = - Y dx. seu
xx +  (3, + Y = 0 et  nx dx + 2Bd3+ d p et ( =  ( + - X X) it d β = - a A I - 24 ax
Y_- ap - XX)
si assumatur aa d B = Ahdx et 2xdx + 2BdB = - rdh - hd F
sit du = d  dB adeoque et dh = Ah dx et divellendo in duas 2Xdx = - r dh dx (seu 2X = n dh)
et 26d B = - h dn dB seu 2B = - hd et 2 d B = dh lX = 2 nh Lx. Ergo ZH = log h
= y dxC
et +Xdr = logh dh seu BNX LXX S loghdh
Rresumatur a d 3 = r hdx 2x dx + 2βdβ = - rEdh = L_ hdr. et divellendo fiat
et 2B eth dx iaa + hdH =0 fiet
et
destruendo h fiet 2BdX + aa dec: H = 0 seu log r = - 2sβdx et H = B PF
2X dx + H dh = 0
et 2BdB + s dH
a d B =  b et 2Xdx + 12(Bdx dh
et qu-d =
2X dxdx: b25βax
et -dh = 2x dx; B2s x
EXaxAX
et log(I: h) = ad
= 2X Ax: lsde fint ad B:  p.s  = 2dx
a d β =
dh = h 2Xdxdx: ad 3
F. SPdx
dd β = 2 du 2βdx bFe
et dd B = 2dx  2dx ad B D 2da seu  dd ( = 2dx
del
seu 2X = sad (3: b2S Bdr
sin I = 2loq dB et B log X = - 2 (log. d 5.
f 5 d L B: dB
et a dβ: b. 2 sad B. C = hdx. Ergo h = (a + β : b. 2(.). : de B: 2 dB β
Ergo X =
seu h X aad (3 = rhdx 2xdx = - rdh 2β dB = - hdF. Ex 3 per 2 fit  onhcx ina = h dN
Ergo fit dH ua d M: A = =  2(3 dX et g log(1: 17) = 25 βdx et log r = - 2s dx  i - (25Bd
Ergo ex 1 fit aad B = b. 2 h dx eto 27. dx = = b 2t dh et 2B +B -h. B dx b 2s da quae
aadd β = dx, bs dh h=Bdx Bs
seu hdx = aa dB b2S. coincidit ortae ex 1.
dh = t
aa c (3: 2∞d X = _ 2X dX : aa d)
Y.  2XdXdX: aad B = dh: h. Ergo t Bh - s. Eaala Af.t B3)
io  ponsem +. rtlarae tae hi nes aa pre Bueim 1 et ta poneo
ad quaesitam pullo fractu
Valorem- m.B
P. C
- )
Se = constans