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# Title: LBr. 396 · Hermann
# Folio 78r, page id DE-611-HS-973581:0155
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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