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# Title: LBr. 56 · Bernoulli, Jac.
# Folio 37v, page id DE-611-HS-958800:0074
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in aequ. d
graduum ad inferiores possint reduci ulteriorum differentiarum reduci
possint ad differentias citeriores. An vero quod sane si hoc ponlicet,
novae circumspiciendae sunt construendi artes. Cum dico nii ess non
eHse desperandum, intelligo, quar quamdia non adest Demonstratio impossibilitatis
esse desperandum, id int elligo dum nullum adest argumentum inpossibilitatis
dz. Z= X dx solutum haberi suppono, quoniam reductum est ead logurithmos.
suffecerit dZ. Z Uti dZ: Z = X dx summatu   aequatio est quae solvitur
per quadraturas, ita fieri potest ut aliqua arte ddz: Z= X dadx etiam solvatur per
nec reforre
som seers quadraturis quadraturarum  inaequo
23 itx AaA cB
aZax ad Fa  du tubus ta,. Z2 cxdx
-fdxda Zu + dedy dx) d Z ddx =( Z didu) F Xp Zdx
Cax- yy +pe gu ax 1 el fiat g.- dZ: x Se
padraturas
dxx dadydix -2Xx dx3 =0 s addx: dxd si esset 
dZ dZ (: Z2 dxdx) dx, + Xx A- +.p
si du opdis
lapeod
Aliter
V dxlaazue +  dZu ddx. Ergo exaequ. C fiat -axditz tdz ddx Zxxcx
dxdx
am _d 
Axdx aa ddy = gyddx F2ydydx + XX ddX + 2X dx dx
Xx ddx + 2xdxdx ponatur =0, fieret Xddx + 2dxdx = 0 seu dix: dx = - 2 dx: X
= - dZ: dx = sZXXdx
seu loglx = log. XX. unde aa ddy = gy ddx +2ydydx seu aa dely: yy = ddx + 25. 2dy dp. y7
si px = o fitet accddus = 2 ylydx 4 2Xdxdx aa yddus aadus e : dx = E gy L XX
(smo 2a nali , 2Xd  Z1 xx a 2xda + rgilus 2 ado eA = av. gy dx = ado ryag: 2 Favgu
tq firit dz = 11 212 xdX + 3. 1∞+ 4. 14X et et
Het gadus  ado- 2ydys)= av seu aadusa aabus
qn diZ f1. 2. 12 dxdx + BR2. 2. 3. 13X dx (+ 3. 4. 14 XXdx ax  4 4. 5. 15 X dxdx + 5. 6. 16 X dudx etc
proaequ. Da2. Z - X datix
oaaox
() Z = 1 11X + 1XX + 13X +19 et
+ 11X3 dxdx + 12XY Axdxrec
Erit 12 et 13, ut 4 = 0, sed  1 = - 11. 6). sed 16, 17, 18 = 0, at 19 = - B 15: (8. b)
et ex Ad Z+Z p dxaxfins  
et 20, 21, 22, = 0 et 23 = = 19: (12) 13) sit 1 = p. fiat 5 15 = - 4. 5 t 
et 19 = 4. 5. 8.9
48
et 23=-
4. 5i 8. 9; 12. 13
X = 45i 8. 9: 1215
4. 5; (5. 9, 12. B;16
seriem tuam pro aequatione ay: dx = gy+XX,: aa vri qua ex primis valorem
ipsius y, stideris duxisse ex aequ. - ddZ: Z = XX dxdx ponendo y = - dZ 2 Z dx, inveniendo
pe per seriem, et ejus: ope y. quod si  ipsa prima aequatione uti maluisses
+ 15X + 1626 etc. (int 
L 10 10a a + 2. 10. MaX + 2. 10. 12 aXX + 2. 10. 13aX + 2. 10. 14a X + 2. 10. 15a X + 2. 10. 16a X erc
12 XX + 13X
+11. I1 XX + 2. 11. 12X + 2. 11. 13 ,4 + 2. 11. 14 X + 2. 11. 15X
+ XX
 4. 197
+12. 12X + 2. 12. 13X + 2. 12. 14X
+13. 13 x6
- 5. 15X - 6. 16X
 2. 17 X
= 2. 12X 4 3. 13XX
si istet 10 = 0) firul et U  12 =0, sed 13 = Z, d 14 =0 et 15 =0 et 16 =0 sad 17 = a  9. 
ec
Raisus :. 18 =0
 3. 137
19 =0 20 =0, sed 21 =
 obtinuisses aequationem y- _ x + - X
3.3.3.7.11
si autem non
et linea regularis re regule
non posset haberi perfecta aestimatio
ctis compar comparari possunt casus inventi, et reque observati,
ducenda
i ex ille
Si utae menstrua excerpen Germanica
c sumahebat, apud ine fonta idic pensionar, de Wic liballus exiguus est ad usum ubi communiuli aestimatione
in pirica aestimato
in praxi n.D
siur qua possibibitate cascram aequalium aequali, et hinc ostendit ia pro reditus ad vitam sufficientes
oltus Beegire scripsis est, ut aequitas rei in vulgus appareret.
ti scrptis casu reperi in schedis simili me aliquanto artificio usum in solvenda 
factum potavis solvi
emperus exemplicausa
ydu capdx = XFg: a, fiat dy: dx = Z: a et Z= ∞ +y et dq= ex tdy unde Az= dx + dx C a) seu
cujusmodi d sit Xty, da utcunque data ex dy: dx dico problema solvi pus
AZ. 9 14 (Z. a) 
per quadratu
seu E per X. positis quadraturis, seu transcendentali
a et sooi potest di dus  Xty: a
betur X per t velhodo ulta tase semper Inventiquidem dim semper
liac quoque y per
oblitus sum notare,
semper posse separari inceteminatim jam dudum quidem ne invenisse
dissu equationibus iit rictibus differentialin ac
ad inferidres Sed non ideo posrem non posse redigi ad quidra-
heteregenedtates quadratorias qualis est Podax b quoniam
semper posse separari incognitus pro
niuscunque gradus defenivendo adhac
xp ax + xdax summar
iest et dat meraliter et dat Coatu Cddu)Xd
latio problemabum
differen  vel saltem
On fratium tuum ita loqu
exemplo
ed vides ex hocippdeam non
e in separatione incognit. in determinata rum
daze . d2 = XX axax; eam quod est adeo simplex
o cangii cave ranto mmus in magis compisitae optatam quidem und
Pimatio
idet minusc
turas quia sole
et Vialtum cous