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# Catalogue record: 58288 — De progressionibus et de arithmetica infinitorum
# Incipit: [Formeln]
# Date: 1672
# Akademie-Ausgabe: AA VII,3 N. 006
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=58288&katnr=39679%2F%2B39679
# AA VII,3: the volume in the Repositorium des Leibniz-Archivs (CC BY-NC 4.0): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-bd-vii-3
# In the Akademie-Ausgabe: AA VII,3, p. 94 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/VII3A.pdf#page=94
# Work: https://leibnizlegible.com/work/00068195
# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folios: Bl. 129–136 (from the shelfmark LH 35, 12, 1 Bl. 129-136 + LH 35, 12, 2 Bl. 163-164): 16 pages, canvases 257–272, page ids 00068195:0257 to 00068195:0272
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=00068195&page=257
# Lines: 929 recognised on 15 of 16 pages, mean confidence 0.80
# Catalogue: Leibniz-Katalog / Arbeitskatalog der Leibniz-Edition (https://leibniz-katalog.bbaw.de/), ed. Arbeitsstelle Potsdam I der Berlin-Brandenburgischen Akademie der Wissenschaften, CC BY 4.0.
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# After the first blank line, page by page: a "## Folio <label> — <page id>" line ("## Canvas <n>" where no folio label is recorded), "# " lines naming its source image and model, then its recognised lines in reading order, one per line.

## Folio 129r — 00068195:0257
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Habito centro grvitatis Habito Centro Gravitalis Convendis
Parabolici habebimus ejus superficiem. Eodem modo habito (entro
gravitatis solidi cyclocidalis ex revolutione sphaerae, habebimus hujus quoque
superficiem videndum an haberi possit centrum Hemisphaerii
ejus cum summa e area ejus et superficies habeantur. Expe
Idem est de Hemisphaeroeide. Ad
Ex probanda ess hinus doctrinae veritas in superficie Exlindrica; ex supposito
pera
centro gravitatis eylindri. Rectengulo abccirium voluto  fiat cytinder, cujscentrum
gravitatisdentis duobus cons ex revolutione Demantur duo coni ex rerolutione
hiangulorum dae. deb. facti. Quorum aream ita investigabimus, estre basis
estoX. altitudo ad vel cd esto a. erit conu unus Aax et ambo 2RaX.
at totus cylinder est 2aX. ergo duo coni sunt aequales condtololi maximo
qui cilindio inscribi potest, et residua cavitas demlis duobus conis aequatur
hemisphaerio seu est 4ax, et proinde si rectangulum ab esse
quadrutum supponatur aequabitur hemisphaerio Videamus an
idem 4aX producatur ducta superficie cycindrica, qu. est, 2aX posito bc esse de ac. sen a
vel x in tertiam partem A L a - + a. fit Bra - 4ra. 
seu rxa quae duplicata, quia non tantum superficies cylindrica eb ducitur inbe.
sedet idem fit. In ergo idem productum utrobique.
c
Esto parabola altitudinis quibuslibea seu D axis X axis ab.
et ut species parabolae. determinetur applica semiapplicatae
cb aequalis x cum axi ab. Centrum gravitatis parabolae
est in axe ab. quod ita definiemus, constet ex demonstratis
a Cavalerio, solida, figurarum ejusdem basis atque actitudinis
revolutione genita ut sunt clinder genitus ex figura rectanus ulo
Cd et conoeides geulum a parabila ead esse in composita
rationa figurarum generantium, et distantiarum centri gravi
tatis cujusque figurae ab axe communi revolutionis. Ergo circa
batin parabolae cd. volvatur rectangulum cd et, ei inscripta
parabola cad. Cylinder productus ef erit ad fusum parabolicum
cadg. ut 15 ad 8. quae ratio 15 divisa per rationem
figurarum 2 dabit 30 (E rationem distantiarum Contro
quibus centra absunt ab ax columumi. Cum ergo Centrum Gravitatis
dr rectanguli ed nempe h. distet a basi recta bh. dimidia
altitudinis ce seu quinue derimis, Ergo COntrum Gravitatis parabole
Cad dimidia seu A decimis aberit. ab axe cd et seu recta ib erit 2 de ab.
Ergo Ih. erit 3 de ab. Jam ex centro gravitatis i ducantur rectae
ui extrema figurae ic: id. Et nunc tandem calculus ineatur. Altitudo
ab esto (a.) vasis cd (2a) distantia centriguae a l Cylander Rectangulum
ex = (2a) parabola cad = (4a) iadistantia centrui parabolae
a basi ib = (2a) Ergo driangulum cid erit 2a ra2
hoc est (24) Ergo residuum, mixtilineum cadi. erit parabola demto
Triangulo seu +a = 2a seu 200 6a (144
Quod mixalineum cum intelligi possit fieri ex basi  curva parabo lica
Cad decrescente seu evanescente use in centrum gravitatis i in ratione
altitudinum sumtarum in recta ia. hic= (3a) hinc intelligi potest
Mixtilineum produtu ex basi curva parabolica ducta in dimidiam altitudinem
3a.) et contra si per 3a dividatur mixtilineum, 14a 3a, 10a 28a
9
10
10
45a
Poducetur 
18a) = Curvae paraboliae cad. Ego
tass fulso

## Folio 129v — 00068195:0258
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2ata - 3a
Ergo si in parabola, cujus semiparabola a Ergo si basis sit
a. curva semiparabolica erit 14a. Ergo si basis sit 1. Semiparabolica
erit 14. et si basis sit 9.  semiparabolica erit 14. Regulam
ergo hanc constituere possum. Si In omni Parabola, cujus
Semiparabola, eousque cujus basis sit aequalis altitudini, latus
rectilineum, (id est basis vel altituds) est ad curvilineum, id est
curvam parabolicam, ut 9 ad 14. Eadem  methodo rem
refinie facile est, quaecunque sit ratio baseos ad altitudinem
Nunc age de Centro Gravitatis Hemiphaeni ratiocinemur:
est illud sine dubio in radio AB. semicirculi maximi dbea. ponatur esse
in f. et distantia ejus a De radio e basi esse (X.) radium autem essea
ductis rectis FD. FE. et Triangulo FAE, circa axem FA. voluto, gignet
conus cujus basis circulus maxim eadem quae remisphaerii, circulus sclicet
maximus, altitudo X. cujus quantitas p 
= (a2X.) Residuum mixtilineum denito cono quod scilicet fit
Arixtilineo plenus DBEF circa axem BF voluto fiet ex superficie
hemisphaerica DBE. = (2 a7) ducta in tertiam partem altitudinis BF.
= (a = X) erit ergo hoc mixitilineum concuvumo = 247
lonu
a X
s = (24Z = 2497) e Cin si addatur aZx. fpet totum
247 a7X = toti Hemisphaerio 2aZ. Quod est absurdum ergi
enoum necesse est fusse in ratiocinatione. Et difficultas ocurit
ingens si centrum gravitatis, aut in solidoHemisphaeud quantur interposition
remisphaeriorum continue proporsionaliter, id est lateribo homologis crescentium,
quando hemisphaeria interponentur, hemisphaerio datos etiam semicirculi echum in terpi
nentur semiciaculo dato, et si remisphaeria sunt homologeum cerunt et semicirculi laterum
homologorum, et eum simul evanet aut, idem erit Centum s Htemispharii et
Temiciculi anasoris quod est absurdum neque enim planum basi parallelum
per semicirenlum mnajorem transiens, alios snicirculos minores bisecat.
Ex amnanda haec difficultas in polusedris. E. Ns.
sed ista methodus generaliter ita refutatur. Curvae aequiparandoe sunt
polygonis caterum infinitorum. Jam in polygonis, nisi quae regularia
sunt, qualis ex curvis est circulus tantum; non procedit illa methodus
iucere perpem dicularem a centro oravitalis ad superficiem, sed separa tim
mensurandum est omne Triangulum ex centro formaturu duobus radius
 uno latere formatum. Idem ergo et, can curvis ut Ellipsi, prarabola, etc.
fieri deber. In circulo res procedit, puto et methodum in Ellipsi haberi
posse, etsi fingulae langentes considerandae forte enimi certa ratio
progressionis, imo forte in hparabola itidem. Quandorergo
hiaaea rectiis a puncto gravitari a centro ad tangentes ducties
ut ex a ad b. c. saltitudines ca. be. df. erescunt in certa
quadam ratione, velut ut padiata, al titudines aut ut quadrata
pertae endam lineae, tunc ex ipsis inveniri ob rad ambitus, data
decrescere proportionalitur
area. Porre istas altitudines continue
certa eadem rutione, figura fiet ex quarta parte radii maximi
et peripherda, si scilicet ad evanescentiam usque procedatur, ut si loco contri
sumati aliud sed quia id non sit centro quippe posito intra figuram
ideo residium, ab evanescentia detrahendum. ita etiam si cresiant alti–
tudines, ut applicatae parabolarum, vel ut earum axi parallelae, vel alia quadam
ratiohe summabili ita excogitari possunt certa linvaram genera, in quibus ea sit qua

## Folio 130r — 00068195:0259
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ratio lanentiu ad altitudinum seu intervallorum intertangentes et centrum
figurae. Sed in parabola id invenori maximeo perae proetium foret.
Astunto quodam in Prrabola puncto intervatl Tangentium a puncto dato, 
et quidem punae Tangentis basi parallelae intervallum est ipse axis ad
Linea de esto (a) - enit eg =(a.) Ch - (6.) ergo gherit (A44 + b) ea a
ah = B b + a 1rq. ga etro ) erit ak (b_) et gh nqa
gk (aq a +b) et V AGH = (a b4 L4) idem dividatur per gH.
et productum duplicetur habebimus AF ab vel ab aba
2
abaaut
Zoregatb aga
En ergo regulam generalera calcutandi intervalla puncti dati maxe parabolae
oupeatangentibus applicata  ad puactum contactus clucatur et diametro ducatur inequndentam
rationeus majoritatis inter diametrum abscissam ab applicata et abscissam a puncto
dato, productum dividatur per latus summae quadratorum applicatae et ejus dametri
Quotiens erit sum Intervallum tangentis a puncto dato
porre si duplcetur a. ut df fiat dl (2a) fiet Cm = (44) et lf fiet.
R42 b) et non fiees 1b a + 262) ma erit a a (si longius procedas a a
aba
a0 (b4) no (a Rq a +b) et A, ab
_ 22 = 6kq a +b
a Rga +6
Ergo prim altieato est intervull. posito aba
aba
tertium erit aba
2 R4 a +b
secundum erit:
 Ergo altiuines ita decrescent.
b. R4 a +b
intervalla, ut L   etc. Quae si duplicentur omnio het:
3 = = etc. in infinitum si a. supponatuo esse punctum
primum seu aba, erit ipsa diameter. At sequentem statim
quae non nisi puncto differre debet esse partem ejus idam absurdum
IR4 a +b
est. Enorem ergo calculo inesse necesse est. Nota aa est ipsa
diane eter, Cum licet enim a sit punctum, a tamen est infinitum
Redinchoemus: de = (a) ch = (b) da = (a∞) ga = (aa ta
seu (aa+1) g3 - _ a + a +2a à vel (a  a +2a + 1., + b 8g.)
tam ratio (R 4a + b). Jam ratio ge ad ga est ut 2a ad a’ at1.
seu nt 2 ad a +1. seu ut 1. ad a + 1 ergo N est (ba + b)
et gk est (a + 1, Aq 4 a +b) seu g 4a  Triangulum gaZ est
a+ 1C a0 + a, ba + b (baa + 3ab a + ba) dividatur pergt.
hroci divisum per Gk producto dorrcduptia+
et productum dinsidis
dabit, allito AF
vel sic: a a +t. b a+1
fiet. bac 2aba +ba
a + 2R9 4a+6,
vel sic: ab a+I inter vallum puncti dati a langente quaesitum
2R4 44 + b
tum duplicemus de, ut fiat dI. dl erit (24) In en. (A92b) ma erit (a + a+ L,)
N  Rq iba
2ba est n C eut (4) ratio nl ad ma erit ut 4 ad a + 2.
seu ut 1. ad a + 2 ergo A0 est (R42b a+ 2) et MD est A9 16a + 2b, at 2
triangulum M Ad est a a2, hab a +2f dividatur per
moducto dupliento
tret; a Rqb at
2 a 16a + 2b
si dl sugponatur ore 34. fer a R4 3b a +3
6 A9 3ba + 3b
24. 692b
sa R93b
ta 2b
ga 36
iba 4b
a+
+1

## Folio 130v — 00068195:0260
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a. 982a. vZ. EH b. Gd a
2d B G2 R9 +
R. P Duq = ab
MD. 24. Eius H. ad H Ds. pet:
tq +a + Adb = MS. porro
Plum Mos = 2 RgB, b vel
2ab Ra 8. dindatum per 19 +a +b
Habuimus ante ab
Rg4a + b
2ab
aqe
si jam LN ponatur R43b. DM = DL 3a
ML 6a- Ds R43b vel R43, b, vel
R4
A412, b. MD. 3a. Eius 2 ad  Ds.
fiete ga + 12b = MS. porro Dtum
Md  = 3a R412, b vel 3ab Rq12
dividatur per R4 9a + 12b, fiet
3ab
ab
108a+ 144b. A4 36a + 486,
ab
1436a + 4b
6
pte
=DR
LN = R92b. DM= DL = 2a ML -4 a.
Ds Rg2b- Ng, b , A4 
29 10+32b
1
oh
Aga +b
s
n
Ry 3a+ 3
G+ 6a - X = 3b+a
le
Methodus in extralendis radicibus alique
a+ ax = x = b + a
a tax - x =b = a
utilis
rovamfingere
terminum
a.b
extrahere ex
parte pro nsoduo
= d + L - aX

## Folio 131r — 00068195:0261
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ac + b
b
e
bd +c
Ebc sa cd cd
bi ac+b
cd vc Xd
acd + bd + bc
bcd
In tali ergo casu, cum fractiones summendo sunt, q ita comparatae, ut
se nommator praecedentis sit numerator antecdentis, fiat fractio, cujus
nominator, sit factus ex omnibus terminis demtoprimo numerator compositus
ex tot terminis quot sunt fractiones, et quaelibet pars ex omnibus terminis ita
it
ita componato, ita ut prima constet ex omnibus terminis, demto se secundo, secunda
ex omnibus terminis demto tertio, et ita porro. ita terninus pri pars prima numeralori
differet a nominatores quod loco termini 2di, habet primum, (quo nominator caret)
pars 2da Ao, quod loco termini 3tii habebit idum C.
dam ut talia: acd +bd + bc  4 + h +E
d
bcd
breoder symnes ita orinor agendum erit: differentia imer acd+  b
uud sic potius; nominator habet ses demito primo, primus numerator habet oes
demto, ido reliquinumeratores habent suum terminum bis, et carent praxime
ante adentet proxime sequente. differentia inter acd-
summam ita investigabimus: acd + b d = ac + bd
bd + bc = bd +E b
Hini apparet si b. sit media proportionalis inter a et c. vel C inbib et d. tunc
ad summam inundam, tantum primum terminum multiplicari possi per numerum termonorum
et facto velut numeratori subscribi posse nominatorem factum ex omnibus terminis
dento primo v.g. acd + b d + b facilius perveniri posse, quia ac = b et bd=
1 2. 4. 8. 16. et sit  q l B ibi mendi poterd oes temminos esse
Quid verd si temini siil Arithmeliae proportionales, ita stabit:
a + 2aX
1+37
act, t a a atir, 4 at or  atr, atn aror, f an acin atx
a + 2 X + 3aX +6 aX
a lax + ran = 3a X + 3X  6aX =
= a2+ 3X +5AX + 7 ax
atx a + 2X a+ 3X. (bd) arxat
Ar  + = +  + . sintque a. b. c. d. e progressionis arihm etiae.
a +  = 4t +b (B huc aequita ulilis observatu ad reduchines) i 
B+C
ee
a +X.
a +b, 1 ab + bc + ac  + - abe + abd 1acd +bed. Et (hoc schema unversalcest
ab NC abc.
abcd
in omnium fractionum additione, in quibus Numerator Unitas. Cum jam numeri su pr termni
sint progressionis Arithmetuae, sunto a = a. b- a + IX c = a + x d = a + 3x ex
1+ = = a + b = a + a +X 2a +X
ab
a aIX ara
+
1 E
L. abtbetac. arax + a + Er FSax F a trax (3 a 1 6ax 1 2
abc
quod si jam X- a. erit 3a X Ha et si a = 1. erit 1
ad + 3a X F 2aX 9 a +3aX134
math. X1B 2 131
ac + b +, bd + c acd +bcd + +b d tbC-
cd
DCd
acd +bd + bc
bca
Ca a a +2K =
a + 2aX
abc a aFL atEs
= a + aX  a + 2X.
+ a X + 24
F ax
30

## Folio 131v — 00068195:0262
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2o(20
CZ+
50 
44 16 50)25
24.
24 12
250+24 274
24 _ 120
+X 2
2.
qu
120
25 + 1
12 F.5
125 + 12 137
60  60
11 96
a ia Ba a sa
5
it rto 1Bo ta 1o
10
240 60 14 12.
14 2.
a:
duasis assantis sequbiamons iu
iat
120
3
n5
(7 psa
720 , E4
120 (1445
siB
5er452 234+ 15+
239+ 234 
nortoiana:
 26id
264
i  l it L - 2 , 1 + e2 +, 1e 1 ( + i
2a
X7 4 X2
pt1 44 +6- 50. + 1 250 + 24- 274
A29 720xab
B)F F. 9 i 2 + 1, 3 + 12
a
1t vi
2X3 = 5+ 2+ 2+X5
d. 129a . 1644 + 120 = 1 b4 ) 2 ~ 24 (h Z f2
120 26
1
 4
e
manet.
nos
1etes q5 er 34b.
1et + 12 +1 1erZ + 13+ 123, 1- 11334 1 24 +14534+1254 + 1
220324
1
20 Eto 60 120
+2,: es+ 2 + 3+ 23 , 1 = 53+ 24 154+ 234, 1 184+ 2551 145 (454 5 45 + -
1128
25345 26.
12 2
gro Cuso  240 + 380 + 14 4+ 120 = no b+ 
425.
240 120 9660
o 60 36
et
11207)
(3) 3ea5o. 
2545~346+2356+2456+ 5456+ 23456 Hac ut sum menus opus graeguiture
r mn β)1ise
eorum seu reductione ad aequalitatem per mutuas compensationes
e3 45 4 2346 = , 2343+ 24, Y   56 = 23
23545 + 2346. = 2345(2) + 2334... + 2556= 23545() + 236 +(2) 254. X...
n + 2456. = () (4) 23+5 + (3) 2341(2) 236 + (1) 256.... +3456 =
(5) (23545+) 234 f(3) 2326. +(2) 2516) (1) 456. .... +23456. =
6) rerae 5+ (5) 22 3 4 + (4) 2e 3Cb + (3) 256+ (2) 4526+ (1) 3560
-20
20
I atset
t
436 560 8120
12o jaqe
100
240
360
ppe
(13) 23 45 +() 54 (3) 23 43 (1) 2345.  e 1
(3) 2545 (2356 (1) 3(456
Arta
(3) 34
23 + 29+ 34 + 23. 4.
la
d
2
vil an
2

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de 
i e) ds
e a
720
0 5040 25 2 vssop q
st
1680, 420 1 84
12602
nl +
620952
1008 10
840
10
ee -1
b intt
n eta
e15
417
el
=H
0
Es
8
I
lo

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de su
Nous soubsaignez confessons d’avoir réceu, de la part de Monsieur
Boinebourus
Monsieui le Baron de dhonborn, et de Messieurs Leibniz, Schick et
Schiiz la domme de deuxcens trente Ecus blanes, par avane
à bon conte de leur pension e,
 aquelle
omna. les Menbles de la maison Ayant fait acheter pour cela
pour cette somme les meubles de la maison, ils en demeureront
les propnetaires jusque à ceur satisfaction. Fait à paris, 
Ce 16 17 ce 16me May Anno l’an 1673.
Pierre

## Folio 133r — 00068195:0265
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aa + Xx aequ. yy.
XX XXF 1X (. B aep V2 2ay B
y+ 9
360600
173
77
Ergo + X ax aequ y dy (et rursus dy aequ. + fxpx
et iterimus aa + ra qu. g. seu ag aequ. aadly exader 2fclys
37
Var +x
TaX AE aequ a. AB, X. BC,y
yy aego 2ax + x + da e
I dy a
y sit aequi e+ fX + g XX + h. X + K X . du aequ. f + 29X + 3hXX + 4 KX eto
ydy aequ. fe +HX + fgXX + fh X + fK X ec
+ge + 2X +2ggX+ rgh X +  K X
aequ. +3AEX +fGC +  + 3th X + 3h K X
+aFex +AKfX +GKGX +AKhX + AKK X
fe aequ. a. seu faqu. a. et aa + 290 aequ. 4 1. seu 9 aequ 4 e 28
A. a7.
a
et
212
rce
2C
13 aec +2a K aequ. 4. 3. a e + a + 2 e 4ceaaFa
4
205
16
seu firt. K aequ. 2. A +  a + 3. 4 a C f 3 aaee + 3. 4. a0+ 
206
AC. CB. Bd aequales si AF FEd ciculum langant
erit AF aequ. AE
Traxrqua
sit AG aequ. X.  AC aequ. a. sit: AG AG. ravegua
x aqu
fier AE aequ S2ax, et GEaequ Trax - XX
jam AF:  AD:: EG: GD. seu AF aequ. AD. EG
G7
Ergo AF aequ. 3a Vrax-. Accedente jam
3a = X
tangentum proprietate: est DG: GE:: GE: + X + a seu 
seu DG aequ. _ 2 aX - XX. Et AF aequ 3a, +X - a
X = a
srapux
equando duos ipsius AF valores, fiet: rax (=XX ee qu 3 a X - J aa fXY
et 2a Xaqu3aa. Et X aeqo 3 a quod verum est
+a x
Tentemus in idem accommodari possit ad Hyperbolam
F AFaqu AD. EG aequ 3a Y 2ax + XX
Gd
.3aH +X
ya xta
aequ
sract
Ergo (aXtXXeqo (3 a X + 3 a a (F XD + ax. seu X aequ 2 a
se
L
EGaequara + 9 reqo E L2 ga AErgo Zaa + 2L aa
se AE aequ : a (30; et ACaqu. 43a t21 , 3a + 2a
seu AF aequ. a sed inde oritur impossibile, fit
Cum EGaeqe a Y23a +gaa sunde
patet nullam ex d posse duci tangenter ad Hypertolam, quod
est et verum est
X

## Folio 133v — 00068195:0266
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 134r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000266.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 86 lines, status machine
Logarithmus rationis b +n ad b est sit 
 ete
2b 3b
n aequ l + V + 13 + 14 etc
12 b 1. 23b 1. 21. 4b3
Logarithmus rationis B m ad b sit aeq
Logarithmm b = m ad b sit aequ. A
erit
 aeq. m + m + in + m etc
m aequ. A - N + N - 1 erc
2 1.1.3 123. 4
Quaeratur curva cujus area sit l+ B + V ete
14
1. 1. 2. 3 1. 2. 3. 4.
A. 2 123. 4 F2. 3. 4. 5. 6 etc.
area sit 1 + 1+
At ququ a + bl +Cb + d l + a l etc.
dq +q. b +rcl + 3dl + 2013 etc
ACaequ Y, ABaequ A.
AE aeg. n
BD aeq. 1r. a VA BdVay. S d
ron
EF aeq Vr et VAEFrS da
rn
Si Y +n sint progressionis Geometricae, Sre du erunt propossiGeom
rtn
ut logatittui
Nam si ra progass. Geom. erit de siedu purge Arithm. Eit  in aeque ac
rin
quae co ponitur contans ergo dn aequ r+n co. seu eist da ipsi r+n propor–
tionalis, ergo progress. Geom.
sit co aeu. d. quaeritur relatio inter
et sit naeqo a + bl + cl + 1 + f12 etc. eut du
erit du oy
rrsus du y.
rursus duaeque bAn aeque
ece
YL M
Si AB sint pro pessionis iui geomelticae seu numeri,
ut Nuin eri
posito raq. 1.
x. Ergo eut
b +2c1+ 3el +4f13 etc. dl) quos duos ipsius du vulores compa
+1. + bl + cP + e13 etc . dis
+a
fiel ergo naeq. e l + 1, + 13
rando, fiet: et ponendo
a a9.0.
frt bag. P. et. Caequai et eaequt et sayt, 19q
17
123125 et. posito laeg. Ser aa
Ainc AB aq. et, si AB sit n, R YABdV, E, eut naeq. l + 1 ∞ B etc
1.23
AE sit n et VAEFIAL, sit n aeg. l - V + 13 e
1.2
+n
1. 25
Sit AGaeq. BD. seu GHaeqe CB erit VABdY pu VAGHV, quod sit l, et AB, n, et AG 9.
e
3
2 1.23
14
 et 9 aeg. l - 12 + 13 - 14
 seu n + 9 aeq. L + 13 + 15 et.
1.2.3 1. 2.3. 4.5
7.2 12.3 1. 2. 3. +
et n- 9aeq. l+ 14 + 16 de
Sed quidi si n.g sit quantitas egativa orietur absurdum.
1.2 I.2.3. V 1. 2.3. Qs 
Reassumto ergo calculo faciamus: frran aeq. 4 l ita scilicet, ut si + significet-
rin
unc et logarit huu ponatur quantilas negaliva fiet F aeqn I
t fict. dn vu b + 2 + 3et + 4fPer dl
annu In  aeg. 1 F bl + cl + e B et al, 7 et fier brg. 1 cag ea
+ a1
Ergo n aequ. I + V + 1 etc sive sit  sine
1. 2.3
soe sit l’aego san sive ago san quod est absurdem neque enim haec
+1
1- n
duo possunt coincidere. Itaque puor methodus bona mnebit nisi
quod non habet locum quando, q majod quam m, nisi forte velum
scribere esse:
ry l+1 seu
1. 2 1. 23.4
et alia cujus

## Folio 134r — 00068195:0267
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 133v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000267.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 66 lines, status machine
N + G vocrmus b, et n - q vocemus C
arit b+ cl ae lb + hc aeq hb + 5 aeg. hb + et
erit hb + el aeque hl + h13+ 4 hI
1 +
+ h1
1. 2.3 1. 2.3.4. 5 1. 2 C)
12
+
1. 4
+
1.. 6
et facre ado h aeq. - (3 fiet  - 3b + c l aeq. - 2l  + et
et omissis reliquis faciamus: _taeq 3b + Cl∞9. El frit laeq 3 b
2 +C
x
est autem Bqu  bg bag. BGBA, et 9 aequ.  posito esse
AK aeq. AtG et CG09. Bd. Ergo fiet tru
Est autem Ray AG AB, et 4aeq. AG posito CGaego Bd..
baeque HX B A + AG1 seu BG seu BK: et si sumatur A (in AB) AM vy. AG
fiit fier BM aeq. n. 4 cut dimidiur n - 4 erit AN, est autem CA, 1. erit rg
erit ergo n-4+1, (s CN) aq. e seu CF c eut CN. et fiet
laeque 3BK.nin produ
2+eNV
Constructio in producta Ac sumatur CZ dupla ipsius CA et
erit QA, 3 et QNverit 2 + C. sumatur NZ paralleta AV aegits si
pragatur Zp secans A H ver
cujus reci
aequalis ipsi BK
alita
) Buviter: CB est Namerus equaeritur AL Log aribhmus sumatur G aequ Bd, et
3t et AN dimidium differentiae inter AB et AG et Np. parilleta Bd aequalis dimidiae BG
3) et  CZ dipla CA juncta QP, secubit AV  (parilletam Bd) in LC
quodsi velimus explicare difertius laeq.
qaqui + l = 1 + N
+n
b, 36
2tC
1et K
bagu ntqe Gta
n + q aeu 2n + nn 
+ n
et baeque 2n + nn  n = 4 aeg. n - n aequ n en -n aeq un, et n -9
in
e
I+n
2+ 2H 1
1+n
1+n
et Caequo un r + 2 + 2n, et 12 + errgo nn + 3n +3 et fiet:
laequ
 1A t 20
2 + 2n
6n + 3 un, son  aeu = 2+ 32
nn +3n+ 3
n
un +1 + 2+3n
Alier l aego n = n + nd et El 4.  1 + n ay 1 + 2A
A sy. n + n + nt
(+  aeq. n + n +ns
l.Aag n4 nt +n5
L. e 2+ l aeq. log 2+ 2n aequ ly 1 +en, ut D est  loy ian
ergo 2 +l aequ L Ergo 24l fit ex 1+2n ut l’ex n
seu 2+ laeq. 1+ 2n - 1gnim 4 1+6ntA+SN ea
seu 2 + laqe 12i1i eu A 22- 2 + sn  = 8 1470
na
2 722

## Folio 134v — 00068195:0268
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 133r); this page is its left half.
# No recognised text on this page (recognized).

## Folio 135r — 00068195:0269
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000269.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 139 lines, status machine
Lavenantius quotutat qualuor
triz sunt faciliora. qualnor assurgunt
invenire qualaor numeros progressionis
Geometricae, sic ut summa quadralore
summa cuborum sit duta:
LeXG X y L
XZ= yy
X +s + Z b x+ yr BC
+Fya
sit xy+ XZ+ yZ = m. fiet: at 2m.
(9)
b+2 m = a. Ergo b + 2ya = aa
X3 + y + Z3 + 3Xy ∞3XZ + 3 5Z = a3
X)
x3
93
(11)
seu  C. + 3 Xy~X = 3XyZ = a
Z
seu C + 3am
093
 (n)
393 = a
seu c + 3aa y - 3y3= a
ex ex g et 10 inter se sunctis habetur  aequatio
pro y. vel si univis ex 8hraberi
potest aequatio pro quod optime fiet collendo
prigum a, fiet enius:
y LaC  b a: 2 - b: 2a, ex aey.9
un in aeq. 13. fit
a = C + 3 3a : 2 - 3 a b: 2
 - 3a: 8 + 3. 3 a b: 412a - 3. 3 ab: 2. 4 a 4. b: 8a
1b
seu 8 a =8c+ 12 a - 12 a b - 3 a + 92b- 9 ab +, b3
2
seu a 9a - ga b. a 9ta + 3b0
et inventa a facile habetur y per aeq 9
seu a-3b a +88c a - 9bb a + 3130
1N 729
o is
2d
648
problema duvena
5. 4x x
sent ra + a + a
Ergo V a N
 spril. b6ss
,X  aN
ax- 3aH,  F + 3axX
beE 108
tam idem  aN. ergo aequando
a N= 3a X, X  + 3axX -
1 Na
b.
E
seut cubus ab aV = X  X a 
seu
E 1 10 nane - a 
Quod si summam cuborum a
quatuor terminis. X. a x 4 X. a x
putes equalem dato a
tunc eadem orietur aequatio tantum
pro  substituendo a. et pro
X ponendo X, et pro aN ponendo
a B nempe.
cabus ab a B X a X aB
s
r3 B cubus ab a
eB aB
X3II a cub. ab et.
o sive
à B- X3 
Unde:
aB3 - 7a 8 X2 4 2 + 3a 8X 43 -x  px a B
sive  saB- aB a D neoque conferendo aeq. Z et E
sue areo avran f xaa )- a   a
 d. (BB- 2 av   4 2- a uZ- 0 a
EIta X + L X
a X à
a X + a X  a N. problema davenant
=b v + X+ isP = C. fiat v+xty+ Z =a et v exG=n
(3
fiat b + 2m =aa C+ 3am - 3  3n= a
L aa - b
3(107
fint c - 3ba 3n 4 3 da a. sent 2c - bn - 6ba + a -0
n exy + vygt vyz+ xyZ fint Cexs per 1. E4. () X gng.
x +Ky + xy +G = n
b (∞)
b
ryt 2 a aβ t. Pn. lin m
si suat quatuor R. y Z. v. X. y. Z ex v
sit V + X+ y +Z2
N Ay  = XX ~Xy + gya
uty
0
3
. -729 +, 648 = 729 + 81 = 0 cubi X + a x + a x + a)   a B ad lne 12 Apuil. 1695
probatio per calculum posito X-y- Z =1
G1b
inge
solvialibi adde ubi de homoeositotus ex. XaX
ad gradum nonum
tres
m = ya, (per 4. 5. 1.)
ton
M
X. Y
P y
py
vvare a
n  et virav
 Ve a give: C)
at
X
ixa
sive d’ a
adeoque: yak- atNax
Xay 1
sive yar-a xaya
hioc V aN
aX. XX a
Ergus aex
 calriti
+ 3a V X
XoaDaS
unde conserendo aequatione
C et D tollitur y
(3) Xa NM 3a.
Na X  Z3 at
Quaeritur X. et b
istota oldenbum
ex vetdstm= a
ex
 2y 39
XX +Ny + hy + ex + 9Z + vZ + xy
artut niutque
vra.
Ox + XX + xG Fgy + yZ = m = _ aa- b

## Folio 135v — 00068195:0270
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000270.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 88 lines, status machine
 y -, 1 yt, v.1
1
q -, 36uI 100 -) 196+1
li 7A 2. bqd. 16 etc.
en
Momentum hujus serici exet est:
1
y-  2-1 6-1 10-1 is etc
seu L
Momentum ejsdem seriei ex B est
erc.
.. = = n. Ergo
Y -1
7 N 3 +  + d   + 1 etc.
et1
3n + 1 - 1   etc.
v yH HC
19-1
+
y + b y +β +b
2y + L 2y+ 2B+
Unde Eb β + dB
L.y + 2cdy + d
+E. + .dB
Quaeratur ergo:
2 + day +d. Ergo conseondo cum  fiet:
+ B. + = d 2d a + BC,70 sive d = βC
et jam d + B1  ergo de B et d p B
C
quando ergo differentia ipstm y et 2 summa
omium L habetur hoc amodo
V+b.  Ergo sa+ba  EyX +da Unde:
C9+d a
y  da - 1a
A S a sBy+ ca + dy ea
de
atsgit hy + la  atf ) +h
asby+ ca,  Fy+s inda tsy B sy+ 
ahis by+ ca , 
CL
la
cur explicatio nnt mutet quia ineeroallum hassbus +bB (+dh 1) ACbb vlarbus abs
+ lad).
Unde Trah y Ybyica  hav Nby +62
kb a
l’a
+ la
Unde quadrandi causa ordinando.
+. ha by +ca + coa - Za V , + haY Yby + bB
atii
e
rab
etirs
la
C ec
+ ca
sive
loaca 12a1
ca
a lar 
by + ca + 2 (a coa dt coa arca
+ Elartiba
harato
+h ba
+ 2 hbaahque
ah
etiba- ZaB - (FZ a Y
rahiy
ha y by +b(3
+rhales ica
Calculus fit nimis prolixus
+
 1 vivy1
si tollationis causi loco  fei:
y +2yf+f
confernio fiet 
ergo fr fiet: da + da+
Unde pet luisus (37 +. et patet
determinatum mutari explicando
non potest.
1
1
 2. rdy . de
+ la dβ
ea)
+ eahB.
etc
+ladB

## Folio 136r — 00068195:0271
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000271.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 52 lines, status machine
l’a 7 1. 1745 67 89 111 133 155 117 199 2u 243
0 b Y 2 24 4b 58 90 m2 1 156 190 200 2n 244
n CX 3 25 47 69 91 13 1hC 157 179 201
1B 145
 d u 4 2645 70 92 119 135 158 180 202 24 24b
225 297
5 27 49 7, 9u 115 137 159 181 207
f5 6 18 30 72 99 ab 138 160 82 204 21b 248
7 in 81 73 95 17 13g 161 183 215 227 249
Bg 8 30 52 24 9b 18. 190 167 184 2ob 2 250
a i p 9 31 53 75 1Y lius 14c 167 185, 207 224 251
lo 10 32 54 76 Gy 120 142 164 186 208t 230 252
dm n 1 3) 55 27 G9 121 143 165 187 209 231 253
n in i 34 56 78 100 122 144 166 188 2o 22 15a
b
1 13 35 57 79 101 123 145 167 189 21 232 255
14 36 58 40 102 124 14b168 190 2 234 256
h 15
3
8
103 125 147 169 191 213 235 257
in Y g 1b 38 60 M2 104 126 140t 170 1gr 214 2 3b 254
5 17 39 61 83 105 128, 149 171 163 15 237 aS
18 40 62 14 10b 110 in 194 2ib 28 2b0
a d 19 41 63 85 107 n
X c 20 aV 6+ 86
b. 21
4
8
151 ,B 195 217 239 261
105 130 ist
10g 13
18
 1gb 2t ra
in
2b
Z a 22 44 66 88 ab 13 154 176 198 200 242 264
fs ist miae enne ziyheo etrea non ohugrglese 12 Zeile gessickt worden darans
abgunehmen, daß dat Alphabet deb scribent, aud zweg hundert iidetzlichen zas en
bestehe, weelche unteo die 22 bachstabe ause gerweiste mubecandte weniso aub getheil
Uerilus abeo dirge staet au,D die 12 antroschindtliche zahlen vinem bouchstabs assignicnt
vervdn stomins, so sind ob in dio thal 12 Alphabeta. Allem ene Zixhro gb
Fraum aub 12Z pien bestehet  und mit 12 duaes ein ander gemiphete Alphabetu
gerhuirben, Zumase da des augang und dat ende dar wooshe aus dirdichher
ncst abzuduesmen, ist e andend ds ohugecahohu solvion obnanglich. Uad tran man nihee
auderes darauduehum, als das einslcher discars hu suchu darinn gewiste binhstaben
O und so ost und vn solchus odo solchus stellu wurder bommn. Es konnus aber viel
tansend dergleihus wronchmlnhe disenche ge aundus aeerde denn aben dinse zighre zud
appliciorus, also de nncst audros als casu dio crchte zu findus waerum man abrr
welistr in voas oodumnus dim hisfroz im Alphabel tandus
LH RXXSICIC
B

## Folio 136v — 00068195:0272
# Source image: https://digitale-sammlungen.gwlb.de/content/00068195/jpgs/default/00000272.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 2 lines, status machine
WILHELMLEIBNIZ BIBII
DERSÄCHSISCHE LANDESBIBLIC