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# Title: LBr. 57, 2 · Bernoulli, Joh.
# Folio 186v, page id DE-611-HS-3653099:0372
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 185r); this page is its left half.
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§ + 3Q + 9R + 275+ 81 5∞ 243 V4 729W p + 39 + gr + 275 + 81 F + 243 u) Qvia igitur P+ 2X+
8 + 2Q + 4R + 8 5 + 16 5X 32V + 64W p + 29+ 4r + 8. 5 + 16 + +32 u RXX et: ponitur esse
D + Q + R + S + 5+ V + W p + d+ r + 5 +, t +
s
∞ - 2 + e - 5 + 5 - V + W p - 9 + r - 5 + + -
u divistorlis per p +qu
+ XXX etc: oporiet p
U esse partem aliquocam
B - 22 + 4 R - 85 + 165 - 32V + 64 W p - 2 4 + 4r - 85 + 16+ - 343 u ipsius 5 + 2 + R +5
P -32 +9 R - 275+ 81 5-243 V + 729 IV p - 34 + 9r - 275+ 81 +- 243 u) + 5+ V + W, ec p + g
=r + 5 + E + u partem aliquocam ipsius P + QR + 5 +  S + V + W, et p = 9 + r - 5 + t - u parcem aliquotam
ipsius 8 - a = R - 5 + 5 -Y + W. et p - 29 + 4r - 85 + 161 - 32u parcem aliquotam ipsils 8.- 22 + 4 R - 85
+ 165 - 32V + 64W, etc. quia quando radices datae et quaesitae aequationis utrobique cadem quantitate minuun
tur vel augentur, aequatio ex illa resultans adhuc dirisibilis est per aequationem, quae ex hac, resurgit, adeoque ul
timus terminus hujus dividet ultimum terminum illius aequationis, oporeet enim p + qu F rxX p. esse partem
aliquocam ipsius PF QY + R XX. p. quicunque demum valet pro X substituatur, sive 1. 2. 3.  sive o, sive
= 1. - 2. -3 e. Unde patet inter divisores ultrinorum terminorum semper debere esse aliquos sc: hos p+
39 + gr etc p. + 29 + 45 etc p F 4 + r etc. etc. qui ita progrediunturut supra dictum est, quemadmo
dum videre est ex hac tabella
P. + 39 + gr + 275 + 81t + 243u
P + 29 + 4r + 85 + 162 + 32u
differentiae primae 1 disserent: Edae per (differ. stiae p diss. alo differentiae quin
4+5r + 195+65t + 211u
p+ q+ r + 5+ t+ usq + 3r+ 75+ 15+ + 31v
p - 9 + Y = 5 + 4 -
p - 29 + 4r = 85 + 16+ - 32 u
q+ r+ 5+ t+ u
 dirisae
rt65+25+ + 90u per 3. divisae per
r+35+7+ F 15 a + bt +25u viste.
g- r + 5 - + + usr +05. + = + 0 u5 + 21 + 5 u t + su
q -3r + 75 - 15t - 31u
p- 34 + 9r 275 +81+ 293 19 - 5r + 195 - 657 - 211 u
F- 35 +7 + - 15 a- 2l +5u
+ ou
65+25+9015- bt 125 ust - 5 u
taper 5 divisae
u
Quod autem in qualibet serie differentiarum medius terminus aut primus ex duobus mediis sit summa omni
um possibilium valorum, qui hac expressione continentur, mn = im - IN
12. 3.M, Ver. 2. 3. . M -1 etc:  N. sic patet:
Finge radices aequationis minui successire his quantitatibus a b. c. d. e. f.g ete, quae denotent numeros pro
gressionis itrithmeticae se inricem unitate excedentes, ita quidem ut medius rerminus exgr. d sit =0. et ul
timus g = primo termino sed signo - affecto h. e. = - a, radicibus igitur ita minutis ulrimi termini
aequationum resultantium erunt.
differ: imae, differ. 2dae per 2divisa
p+ ag+ aar + a5 + a 2Z + a5u.5. = (posito nempe N pro ) s: an A N) quorum disse, s: an - b n = N
p + b4 + bbr + 635 + bEt + b5ug. = pi q1riSit. aG. h. e pro s: bn
p. + cg+ cCr + C35 + c2t + C5 Ws = cujusvis termini aequa) s: cn
p + dq. + ddr + d5 + dEt + d’us. = tionis inveniendae coes s: d
p + ed + cer + C F + ett + e ug. = ficiente, et n poro nue s: en
p + fg + Ar + f35 + fEt + fSup. = mero dimensionum e) J: fn
p194 tgr +95 +get +g5 uc
dif: 3tiae per 3. dirisa
s. a2- 36n + 3Cn- an
b -3Cn + 3dn - en
Cn- 3dn + 3en fn
d - 3en + 3fn gn
diferentiae 6tae per6 divisae
jusdem termini
a2- 4b2+6C1- 4d2+ et
6en.
a -6b + 15c- 20dn + 15C- bfn + gn
2. 3. 4. 56.
et
N.
sunt s: cn - du H N; S: cn - 2dn + en
Ni S.
GN
es
cs
se
S. g f, N
diff. Ita per 4 divisae
bn -4cn-
Cn
6
n
n
tenriae 1mae S: Bn -c r
2dae 3tia
4ta etc:
sunt
l. ca
S.
(. ar 2b + n
bu 2c2+ dn
Cn- 2dn + em
d
2en + f1
N
Ns. Cn- 27 n + g X N
differentiae sta pers dirisae
an -56n + 10C2- 10dn + se
Sc + 10d= 100n +5 fn-
4.5
in
n
In his medii termini aut primi ex duobus mediis
bu -3cn + 3dn - et _ N, S: b2- 4C2 +6d2- 4e2+ f
3. 4
qu
etc: et generaliter positis M pro numero differentiarum, m pro = M (vel pro. M F1 Xsi Mn sit nume
rus impar) h. e. quia d = 0 pro c, b, d. f. = 1, 2, 3  medius terminus in serie differentiarum cujus
numerus est M, est = S: m - M m - 1 I + M. M. =L m - 3) n M. n = 1. M -2 m - 31 + etc, H V
1. 2. 3
Divis. per 1. 2. 3... M, h. e. si divisio reipsa instienatur = (. mn
in - 2n
1. 2. 3.1. 2. 3.. M = 3 etc:
in = Mn
1. 2. 3. 4.. 1M1
C
in =
m-2
1. 2. 3. cM 1123. M-1 13 (128. M. 2
t ac V