# Leibniz Legible — https://leibnizlegible.com
# Piece: https://leibnizlegible.com/api/records/13663/text
# Catalogue record: 13663 — Johann Bernoulli an Leibniz
# Incipit: Quoties literas Tuas aspicio, toties amissa Tui videndi spes reacerbat
# Date: 1695-02-12 (95-02-02)
# Sender: Bernoulli, Joh. — Addressee: Leibniz
# Akademie-Ausgabe: AA III,6 N. 095
# Other printings: -E (nach e.unbek.Vorlage): Leibnitii et Bernoullii Commercium, 1745, 1 S.25-31 (tlw.); -Gerhardt: Mathematische Schriften, Bd. 1-7, 1849-1863, 3 S.158-164 (tlw.). -Leibniz. Matemática-Física-Metafísica, hrsg. von B. Orio de Miguel, Madrid 2025, S. 52 (span., teilw.)
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=13663&katnr=30250
# AA III,6: 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-iii-6
# In the Akademie-Ausgabe: AA III,6, p. 360 (PDF, Repositorium des Leibniz-Archivs, CC BY-NC 4.0): https://www.gwlb.de/fileadmin/Leibniz/repositorium-des-leibniz-archivs/III6A.pdf#page=360
# Work: https://leibnizlegible.com/work/DE-611-HS-958803
# Title: LBr. 57, 1 · Bernoulli, Joh.
# Folios: Bl. 15 (from the shelfmark LBr 57, 1 Bl. 15.20): 2 pages, canvases 29–30, page ids DE-611-HS-958803:0029 to DE-611-HS-958803:0030
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-958803&page=29
# Lines: 155 recognised on 2 of 2 pages, mean confidence 0.78
# 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.
# Machine transcription. Not an edition. Errors are expected: about 8% character error rate on Latin and French lines, higher on German.
# Licence: CC BY 4.0, Leibniz Legible. Machine output: do not ingest it as verified text; keep this provenance with the text.
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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 15r — DE-611-HS-958803:0029
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 20v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000029.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 102 lines, status machine
cotiesliteras Tuas aspicio lotes anisca Tui videndi spes reacerbat dolorem
c
sitat enim profundis ingenum amplam imasine sed tanto pejut qod iprius
Archelypi fuitione dertitur: Jnvitus et inoidus intelligo juniorem Hurmiu
obtinere locum qlio hicki paratus erat; s quae alia comoda occasio pare aliquando Euro
pae reconcessa eumu avide auciperem, Si modo ribi me prossiorem redderet: Hic enis quia
Lese offerres
alia onnino ayere debeo, vereor quod et isse vereris ne tandemm Audiis mathematicis
valedicere cogar
Non video qua ratime quadratura Figura cujus ordinata est x possit citra
seriem dari. Si vero in serie abilinendum ably, per ruaa methodum inbehiunt, ideu
quationes destruititia satis ordiantae, ipra auten qua inde emergit seriet non ita evidet
ter procedit, ut sepositis aequationibus dettunt lin à quovis alio continuari posset. Recle
mones quod L: adhibere velitant ipran quantitatem logaritm cam R. e. ordinatan
artiticialem rec simpliciscima ratione fieri possit, interim eliat obseroo quod sine calculis
et fortantione aequationum destrultitiarum primo intuitu habeatur et enim ly= plx
et x = 1 + 2 lX =  2 - L22 + f23. ergo muttiplicando x per lx habetur ly=
++ - = + G d. = 2 + 2 22 2. 35 2+ 74 2. qua eadem eit qua Tun.
+2- L22+
2
+24V.
Integrale termitio xdx urica operatione in venio per additionea et subtractio,
e mei
mesX atque
- 6. c- mp e
m+ 2
X Cx dx-
c. e -1  +1
m +.3
c
na
m+i
alx.
e 2 e. c - 1. C -2 m C-3
atx
 X x dx
m +1 3. F Ex ex-
e. e- 1. c-2 m n es
m +14 X atx
nam termini 2d et im u et tuo (tuet masse decruunt ob alx- dx, verum, meque hu
e
 et tuo tucet glusge. sitant sumti per concruitionem sumari possunt; fiet itaque
f
e
X Ex ds = m
X 
me
e
morse
e mar e e. e mAer eee mA e f. 
m2
m+13
e LC-1 c. C-1 fC- c. e- 1. e-2
m
m+c3
m+, +
m +, 4
x
x Hi. ubi iali patet quod si etit
cmerus  integer et positions series futura sit finita et quidem tot tertin nomen
uno subduito quot in e continentur unitates.
Integrale ejuadet quantitatis  l dx aliter in super inven in poteu, led perserien
intinitata in qlcocunque casu: ponatur enit mil = 5, erit xn
merum iptius 5) et x=
se
=ns (perns intelligonu
e eit autem n5 = 1 + L5 +
Liando lanc seriet habetur dus seu mx ds = ds. 1+ + 1+, L15 +, L, 53 V. ideoque multi
plicando illud per
1 813 1
(12. 3
+ -2. 34
Ix tde
s
ma
e+4
et
e+1
mactxem
et1
1.c+2
me erit f t dx-
di
m e 5 1, eti 1 et2
1.2
Hi. ergo sumendo integralia terminorum singulorum proveniet
e+1
e +1
etr
1.+2
mc. L et
1. 2.  +3
etzue
e+4
12. 3erus X.= (subili tulo loco sepuvaloremali
mc.tetu
1.2. 3. C+4
X. Hac itaque series erit etiam = illi lupra inverta
quod sin obiter tanguaa consertacium datum velia.
Vir Amplissime ac Celeberrime
Factor Honoral:stime
12

## Folio 15v — DE-611-HS-958803:0030
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 20r); this page is its left half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-958803/jpgs/default/00000030.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 74, 2026-09-06T16:05:13Z — 53 lines, status machine
Caterum speculationes irta adhuc ulterius eptendo poscunt; si nempe quantitates loga
rithn ca l, t, tx 50 cum u huherit x vel Xr, pY. vel concantibus, quomodocunque
conponantur et compositae sint comprehencae sub signio radecalibus. Poscunt exim pleruma
sumari vel per vel citra serier, vel etiam per eplensiones curvarum, ex. gr. Daxsxit
et = curva estens a cujus coordinata sunt L  et xlx - x; plura alia eaque non con
Tenhenda circantare Meteriata animaducitere posiet, qua vero brevitati ergomo
Tuae retiaquens induchea nihie effcgere polert: nondum satei tentadi exten konem
iptatu culoci x= y, num comode per Rriet vel etiam . de serie exhiberi possit vellem
ipce disquireres, meretur enim luam applicationem.
Quod concernit alteram meam seriem et quidem uni versaliscimatu pro onhibus qua
draturis et integralibus Sydx = + 7y - )
ds +. X day H.. Jam fortan ii aitic videro
ax
1.2.3 dxE
me ejus orginet honinta longé accersere, cun in ex snen ms pilcilo alio aper sit q
continua additione et subtrantione tertitorum aequaliut ier supra teci: praclarissine
laten eandem dedius et serie Jua décresteate ejurgus diffeentir primis, secundis, ter
fiit c. quo hon patun gaudeo, multas enia proprieiates dequibus anteu non contubat
circa nuteros figuealos exinde delexi: ob 6rvo etiat quod si serier decrescear tit harmon
ca hon solum coiniidant termiti a, b, c d Vi ipsis a e, l, pY. Sedetiake e, f.g, hH
ipsis b.f. m, 9X et l. M, a, 0 H. epeit c, 9, n, r V.. et ila consequenter. Hems:
progrescio harmotica +L+ L+V. continuetur ut auterus terminorum sit
X, erit summa progresciohis = X-
X. X- 1.
V. 2.2.
X. X - 1 . X- 2.
1-2. 3. 3.
X. X -1. X - 2. X-3
 1. 2. 3. 4.4
Hincotaten hondum perspicio, quomodo suna progrestionis hartoti ca finita espente
per conpendium exhiber possit, uti exhibentat sumae progrescionum figuratarum
vel etiam arithhetica et geometicu; si quem noveris modum pro hoc, mecut
Luud gravatita comunicabis.
Dum hau sceibo acripio literasi  D. Marchione Horpitalio, in quibls Muttit
novam sclutionea cujurdam problematis; quod tan jat aate bitectre comunicavit
una cum sua tam invenia solutione, quan ut quantocyus Liprian mitteren
rogaoit, quod etiat fec sine Mora, adheitent schedcormari Horpitaliano atimad
versioneta ata uli exhiblii aliat solutionet ejurdem problenati sed generalem
quam per conunem Geomehiam et sit plic scime inveni, cum tamen Hogitaliata
quae sperialio ett multo differentialium calculo opuo habeat: Quod cum Horpitalio
indicascem. problemati se de novo applicuit, invenitque etiam generalet solutionem
quam nunc millit, quanque’à meu purum differre reperio: Optaret ut priorsua
hanc.
ut et mea supprimerentur et nova sca subititueretur, sed cuea Liprit non
satis mature apputiuram sutem, rogo Dn. Menchehiuo per literas quibut prae
sentes Lac incluta, ut hane novam solutionem 8. Horpitalum prion scibhectere vel
salte subsequenti actoruta Meaci inserece velit; qua ratione ipri quodatodo se
Liferi espero. Problema autem en tale: Et pons subliciue AB convertibilit
ciria apem A, sitqe trochlea C circumdurtus funis BCM ciijus una
estremtas suminet pontem, altera pondus vel sacona MI. quaritur
qualis debeat esse culva CMN sulit ubicunque excitens pondem M im
curva, semper aequi librium faciat cum ponte AB.
Ae