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# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folio 10r, page id DE-611-HS-958803:0019
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Sumtis itaque integralibus per partes duabus linelis intercluras provenit
int. dx = X
int. xlxde = L XxlX -  axX
Eat. pal dx = f x1 -= ap (7 + aap
int. XL dx = pElX - Z as  (p + 3.2 auxlp - 3.2
. 4. 
4. 4. . 4
12x4
ru
int f  lap =  F. Ep L = ap E + Q. aux Es - 9.32 a x x +   
(. 5. 5
a
5. 5. 5. 5
Multiplicalis his e integralibiis per correspondentes terminos lujus seriei
bi a 1 ab 1 12 3 a bb 12 1 4263 x et positis seriebut quae sunt verti-
cales Lorizontalibur proveniet int. fde  seu spatium CFs11
X3(∞
1. 2. 3 aab
X3IX
1. 3.3ab
+
XYLX3
X5lx4
12. 2. 4a bb 1. 2. 3. 4. 5a Ib3
X1(C)
1. 2.3. 5. 5 a2 L3
XIX
1. 2. 5. 5. 5 aab
X
1. 5.5.5: 5ab3
ex
∞
ex
ex
5. 5. 5. 5.5b3 
Hinc si F = b = 1, erit lx =0, proinde quaque lX, 1X, ( x = 0. spalium autem
CFHI degenerabit in CFGA, quod itaque eoanescent bui singulit serverut ter-
nis exceptis primis erit = 1- +  - . + - ix.
modus hic quodraturas determinand. per serie intonitas non solum succedit
in curois percurrentbus, sed non nunquanin aciis quoque adhiberi potet
HI. Qccaritur subtangeat curvae c = y. Sol. flc = ly, ledx = aly = adque, yledx = adis
orgoyle. a:: dy. de:: y.s id eoque s== conrtanti; quod indicat iprat cuvvam qlin
sitam este Cogarithmi cam, et viceversa cogarithta eam esse culvam perorrente.
III. Detertcihanda est subtangens curva X =; sol sumatur utriurque diffecentia
le et habebitur X’ix + Exd = leds, vel (ob cl = ∞ (5) de+ lxde ledys
aut ad +xd= lecly; ideoque a + lf. lc:: dy. ds :: 7. s ergo s= a +7
facile ita invenitur: quoniam yle = XlX, erit y = 2F, et yd = Xa=
alxde + L ppallX) - L apat ideaque integr. yds = L ( - = axx
c
Methodus hac applicari etiam potest ad curvas percurrentes quarum aequa
tiones pluribus quam duobus terminis condant qualis ett haec f +X = X +y
sumlis enim separatim differentialibus cujurque tertini per modum generalem es
hibitum, prodibit aequatio differentialis, in qua si fiat ut suma terminorum
cum de ad sunam terminorum cum dus multiplicatorum ita y ad s, habebitur
valor subtangentis quaesita: D iff.  =  de + X 1fc
(7 - 1
Ilxdo
1
-1
yxls1x + ay
_= X Olpdz +dy, ideoque S a    acti n ayt
quae quantum vis composita ope logarithmicae conctrum potent, quia quaevis quantitas
percurrens separatim sumta per illum constrim poten.
Eadem hac metlodut eliam locum Pabet in curois percurrentibus altioii gene
ris, omniaque quae duta sunt deoi peccurrentibus primi generis applicari possunt
ad cjurscunque gereris percurrentet.
Diutius quam par est, vir Celeberrime, his imoior, qua forsitan jam me
Meliur noni: interim meac super hac materia meditationes lecum conunicare volui
ut quourque cum Tcis conspirent videas; id unicum adhuc oddere liceat, quod
catera qua circa Rujurmodi curvas invenienda redant, ut earum applicatae masima
et h no, panita flesiu, caulicae et evoluta aliaque ope calculi percurrentis etiam faci
le eppediantum
bX + FX(Y
e
XFlx
1. 2. 4A aad
XlF
1. 4. 4. 4 abb
1.3.36
4. 4. 1. 46
qlcadratura