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# Piece: https://leibnizlegible.com/api/records/48338/text
# Catalogue record: 48338 — Hermann an Leibniz, Hannover
# Incipit: Doleo, quod responsoriae Tuae, Illustrissime Vir, quarum in postremis
# Date: 1710-02-13
# Sender: Hermann — Addressee: Leibniz
# Other printings: Gerhardt: Mathematische Schriften, Bd. 1-7, 1849-1863, 4 S.359-362
# Record in the Leibniz-Katalog: https://leibniz-katalog.bbaw.de/de/extended-search?id=48338&katnr=30037
# The Leibniz-Archiv's transcriptions of the letters of 1710 (pre-edition): https://www.gwlb.de/leibniz/digitale-ressourcen/repositorium-des-leibniz-archivs/laa-transkriptionen-1710
# Work: https://leibnizlegible.com/work/DE-611-HS-973581
# Title: LBr. 396 · Hermann
# Folios: Bl. 77–79 (from the shelfmark LBr 396 Bl. 77-79): 6 pages, canvases 153–158, page ids DE-611-HS-973581:0153 to DE-611-HS-973581:0158
# Original at the GWLB: https://digitale-sammlungen.gwlb.de/resolve?id=DE-611-HS-973581&page=153
# Lines: 119 recognised on 6 of 6 pages, mean confidence 0.90
# 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.
# Wording rule: say "the machine reads it as …", never "Leibniz wrote".
# 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 77r — DE-611-HS-973581:0153
# Scan: One image of two folio pages side by side, registered by the library under both labels (also fol. 78v); this page is its right half.
# Source image: https://digitale-sammlungen.gwlb.de/content/DE-611-HS-973581/jpgs/default/00000153.jpg
# Model: FoNDUE-GD_v2_ft_Leibniz, run 321, 2026-09-10T22:31:41Z — 25 lines, status machine
Vir Mustrissime atque Amplisime, Fauror et Patrone pl. lolende,
Patavri d. 13 Febr. 1710.
Doleo quod responsoria Tua. Mustrinime vir, quarum in portremis Tuit
meministi, ad me non pervenerent, cum e contru omnes meus ad te delatas
videum. Quod ad subdubitationes meas attinet, quas in pracedentibus meis
protuleram, libens agnosco eas tanti non ette, ut doctrina Tuâ virium cenerifuga
rum et Centripetarum quicquam de certitudine detrahere posrint, quod exemplo
pereleganti eoque satis asposito luuulenter ostendis, un de factum, ut mea difficul.
tates jam euanuerint, perrpecto insuper egregio concentu theoria Tua cum ui
qua ex Augenians repertii memoriu prodita sunt.
Quod en qua nuperrime circa lurvas se mutur alternatim voloentes, quasque
fortane amicabiles vocure liceret, protuli, non displiverine, est quod mili gratio
Ler. Har inquam curvas, qua se mutuo sui euolutione describunt, amicabiles
nomino; perinde ac ulli numeri ab Aritemetiui hoc vocabulo insigniuntur, quo
rum partes aliquota in unum addita, non quidem datos numeros quorum
partes aliquota adduntur, sed alternos restituunt. Interea non inticias ibo
me in pracedentibus meus rem omnem satis imperfecce proxosuine, nam ve-
inimum est, quod observatti, non sequi tertiam curvam quam congruere suspo
nebam gnimu, secundu quoque congruere debere quanquam cateroquin auerlum
verum sit, u num Geometrice id demonstrabo. Sit ergo (eig. 2.) mrima
curva RPL qua sui enolutione secundam LHC, et hac tertiam CDA describere
intelligatur. Si prima et certia congruunt, etiam seunda etes congruet princ
ANL vel tertiu CDA, atque adeo hu curva non solum amicabiles sunt, sed perfecte.
aequales. Ad demonstrationem Lujus prropositionis; sequenti utor temmate.
Si in dubus curvii ABM et abm (fig. 1.) sumtis quiburvir arcubus aqualibus

## Folio 77v — DE-611-HS-973581:0154
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 321, 2026-09-10T22:31:41Z — 34 lines, status machine
AB et ab, duclisque per terminos B, b tangentibus BD, bd; anguli DBC, dbc
comprehensi tangentibus et ordinatis ubique aquales sunt, ipsu curoa ABMet abm
sibi mutuo congruunt. Et si loco arcuum aqualium AB et ab arcus utrinq auipi
antur in data ratione, et anguli DBC, et dbc semper aquales existant, euroi
ABM et abc similes sunt. Hoc posito.
Si eurva ADC et L N R (fig. 2.) congruunt, sequitur omnes lineas CR, 98,6P, B
qua connectunt arcus aquales AC, LR: AE et LP, . tum parallelas etre tum aequales
ipsi AL. Unde ducis per puncta quavit 9. E, D receis gH, E), Dx perpendiculari
bus curvi ADC et per totidem puncta prioribus respondentia in prima curva B 
N rectis eH, P1, Ns Tangentibus cuream, qua prioribus perpendicularibus ad curru
ADC occurrene in punceis H, 8, X H. quu ex hypothesi sunt in Curva CX L. Jam vero
anguli perpendicularium curvii ADC et parallelarum ipsi AL eo majores sunt quo propiores
fuerine eorum vertices puncto ultimo c. nam RCB est recous, sequens vero egH auutur quide
sed major qua n angulus PCt, et sui major adhuc est angulo ND N, et tandem angulas fit
nullus, quando DN coincidit, cum AL. Hoc sequitur quin curva ADC cava est versus eunde
rectam eB vel AB. Si jam angulus Pei statuatur semiredus, erit Et = D), vel curva
DNL = Curva AHC, et angulus E17 = arg. T1D vel alterno 1DW. Porro reclâ q8 et DN
ita ducea intelliganeur, ut anguli 8qH et NDK simul aequales sinae recto, hoc est ut unus
horum angulorum alterius complem ene um sit ad recum: atque hinc atubo eriangula rectangula
2gH et NIK propeer hypothenus as aequales q8 et DN aequaliu sunt ee similia; unde qH = NX
hoc est curva CH = curva NL, et angulus gHG = angulo XNO; et sic infiniti alii arcus
aquales in curoit CIL, et L N R Sumi possunt ita ue unguli tangeneiu m et ordlinataruntdue
tarum pes puncta contacens utrinque constant er aequales sint. Unde per Lemma superiu,
Curva estL congruit curva LNR vel aiquali ADC. 2. E. D. Calculum quoque pro luroin
huiu modi multum contraxi cum is unica analogia perficiatur, in qua ne quidem
seundorum differentialium aue exprenione radu osculi opus sit. Nam vitando AL vel 2N
= 2a, AT = X, DF = y, o D = 5, Dd = dx, Dd = ds. Fient Triangula Ddd et DN X sini
lia, et cum Nh sit = NL hoc est per hypoth. = AD erit NR = 5. Adeoque
Dd (ds): Dd (dx) = D N (24): NK (5). Hinc sds = 2adX, et 55 = 4aX. Unde constat solum
Cycloidem se inverso situ sui exolutione generare.
et naly sii grro curvis similibus parum differe a pracedenti. Si (fi. 3) Curva LN D sui
euolutione describens curvam Lte C, similii sit curva ADC descripta euolutione seunda CAL
ac prius
eodem fere modo probari potest omnes tru dus curvas similes esse, Unde linet siL, DN, CK

## Folio 78r — DE-611-HS-973581:0155
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 321, 2026-09-10T22:31:41Z — 40 lines, status machine
qua supra parallela eran, nune concurrere debent in quodam puncto z, rectu vero omnes
A2, D3, C2 dula ex puncei quibur vis curvâ ADe secabunaur à curoa LNB in data ratione
Unde vocando A3 = a, LZ = b, AL = C, D3 vel 5Z = y,  D = 5, D d = ds, 2d = dy et dd = dx.
Erit DN = C
a et arcur LN = NX = b5. His posit is, ertangula similia Ddd, et WN N dant aqua
tionem bsds = -cydy et bss = aac - cyy. Atque hinc elicui A.. dx = dy 1Cayy - aab : ab - by)
Hl anc aquationem esse ad Epicgeloidem cujus diameter AB circuli generatoris AMB sit aequalis
a = Vab, et DZ = V ab = radio circuli immobilis sic ostendo. Ponatur Epicycon quadam
ADC cujus B aiis BC arcus circuli descripti centro e intervallo Be = m, et circulus generabus
AMB. Sintque ne prius DE = FE = y, AZ = a vc. Requatio differentialis Epycycoidii erit.
B.. dx = dy (2 yy - am): V(. aamm. + Caa + mn)yy - 4). Jam prima aequat io A, mulsiplinar
do termines fractionir surda, per numerarorem Vcayy -aab), facliu debitii reductionibus,
muvatur in C.. dx = dy ( yy - a vab) : V (- a b + (aa. + abjesys - y 2): jam Si loco Vab
ponatur n in hac ultima aquarione, aequatio C plane coincidet cum aquatione B,
qua est aequatio differentinlis Epicycloidii. Ergo etium A est aequatio Epycyeloidlis
ut dictum est. Atque hine iterum Constat solas Epicycloides sui evolutione sibi
Similes Durons describere. Quantum ad Curvam se ixsam directe evol
ventem altinet, talis erit omnis ea cujus radius osculi equalir est curva
data linea cuutá. Quod facile ostenditur. Sint dua curvi AB et CD, quaru
illa describatur euolutione huiur ita ut ambi sibi mutuo congruant.
Si arcus cD ubique à qualis est ariui AB, et cum ex nacura euolutarum, angulus
quoque FDB conrtunter aequalio sit angulo GBe, manifestum est per superius Lemma-
curvam CD congruere alteri AD; adeoque si radiui osculi BD curva AB fuerit = AB + RC
curva AB produueur enolutione euroi sibi aequalis. Et si curva CD ubique à qualis est alteri
AB erunt voordinata CF, FD a quales coordinatir AE, EB. At deoque vorando de C = a, AE=
C7 =X, BC = FD = y; erit BM = X + y et MD = HD - HM = a + y - X. Erit ergo
BB Cax): bB (dy) = BM (X + y): MD (a + y - X); unde Xcly + ydy = adx + ydx - Xdx a quatio
differentialis curva AB; in qua si indeterminata cum suis differentialibus separari pote
runt, habebitur constructio curva quasita; hanc vero sepavationem nondum obtinere
porui. Si radiu osculi BD ponitur aequalis curva AB + data AC Rabetur aequatio ubi differex
tialia separata sune, sed nancitur aquatio tranicendens seucndi gradus. Teneavi etiam Dro
blema inverrum vinium centripetarum quod adhuc intactum remansit, inveniendi nimiru
Curvas illas in quibui mobilia habeant vires Ceneripetas juxta datum legem progredientes
ut juxta reciprocam rationem quad ratorum distantiorum a centro direuionis; nohnulla jam
anequntur mihi videor de quibus fortane furius agam in schediusmatis Aciis vestris Berolinen
sibus inserendir si modo ranti videbuntur, tenues speculutiones mea. Hirce vale vir Illustrimime
ee favere non detine
demplimime Nominei Fui
Hinvantinimo a
J. Hermanno. 1

## Folio 78v — DE-611-HS-973581:0156
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# Model: FoNDUE-GD_v2_ft_Leibniz, run 322, 2026-09-10T22:35:29Z — 7 lines, status machine
Xal ydx - Xdy = Xdx - ydy = adx. An posito y decrescere crescentibus X
fiit Xy = EXX - Lgy + aX seu Lyy +Xy = LXX + ax
Xig y: x = B LX+ a, :, 2y+ X v yy + 2Xy+ XX = 2XX + 2aX
ta
seu y = Y(2XX + 2aX) = x sed falsum est y decrescert, crescentibus Y
Ay: dx = y-Xta, :, X-y = =1, +a
cy

## Folio 79r — DE-611-HS-973581:0157
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eg. 1.
e
- - - - -
sd
e
fig. 3.
fig. 4.

## Folio 79v — DE-611-HS-973581:0158
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