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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 18
# Folio 14r, page id 00068089:0027
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f
Hinc estum haec austica proprietalem alterius habet quod nempe per suam
quolutionem aliam sanstram describat sibi similem.
 sit Parabota AG9, (figs. 6) cujus vertex A axis AH in quo punctut
est datum B., quod radios BG, Bgenibsit, qui reflexi constibuant lunvam saustica
determinandum est eiut punctuma. vel invenicna est longitudo GL sit paramete
a + AB,b. AH, r. HG, + = sar BG, y = Y (rY - rb + bb + . Ergo GS, ax.
Ai 9, adr: 2aY. addantur eorum quadrata adrdr + 4 drdr, : 4+ = Gg. qu.
est autem dy (seu 90) = 2Td 2r - eb +a, dr a ( 2 t(rr - 2rb + bb, + ar). adoque
G9. qu. = 9Dqu = Gd qu. = abb  arK a Lb- r) dedr: rearr. syb + 4 bb +4ar
dx dx constanti.hlujus itaque quantitatissumendum esset differentiale, et aequandum
nihilo. sed quia universalitar solvere niius prorixum foret samemus casum specialem
sit igitur b =0 id est ponatur B in vertiee erit caetens ut prius, BG, y-Y(eY + ar)
G1, dr. gi, adr: 2Var. G9qu = adrdr Farardr,: 4r et dyd2ra Falar:
2Hertar). ideo G9qu - goqu seu G0 qu. = adrdr:, 4r + a) = dxdx constanti
ergo differenticindo, si adx =0 fit dar = drdr: 2(r+ a). Et ddy erat -aa ardr F
modit ddus = = aa dar +2rY aer Far, drdr 4(rr + aY) err+ ar). quia
+Z (2ddr + addr) pr + ar), : 4
ergo dxdx + dys dy = drar (a + 41): 4r erit ydxx + ydydy, :, dxdx + dudus 2yd
= + 4Y) ((e tar) : 3a =GL si ergo fiat 3a ad BG ut a + 4r adquartam
erit ea GL. cur si addatur recta BG proveniet 4 (a + Y) (er +ap): 3 a pho
valore curvaelausticae AL. sputium cansticum AGL etiam quadrari potest
oronidis loco ad jumgemas determinationem causticae in cyclorde vulgari
agradiis axi parallelosum reflexis sit (ycloides
linea Avulgaris AB verlex (, axis CE, circulus Gentor CFE, radius
quae causti ca generetur
incidens GB, reflexus BH, cujus consitudo quaeritur. sit (E, 2a. CL, Y. Arcus
CF5. LB F = 5 + SEar -rr). peripheria CFE = AE = p Erit AG = p -s
- ar -1r) = X. et GB, 2a = r = y. ideo dx = - dr (2a=F : Y) qui a.
autem dx constans, ert ddx = a drar: r((2ar - YV) - dar Y (2a -r: Y) = 0
ideoque der = a drar :, 2a r = rr. quia y = 2 a+ 2a = Y erit dy - - ar 80
EArdrtadr) Cirer faer)
Cddus =Cer+ ar) Cer car Fedrari adius
serdr + adr) (2Y ar Fadr)
(+(rY+ar) (2r+ a) ddr
(rrar) ter ur)
ad (a tar: r) sta
Quae non consentiunt, Ergo est
tota AHE, est dupla diameter EC. spatium ABH aequale subdduplo semisegiento AG
si LBtransit percentrum circuli eut H omnium supremum. CausticaA AE
(r tar) r +ar) et
=rrC2aD 
nrtam Terrtan)
error in caleulo, alterutio
ddyu = - ddr = =adrdr: , Ear - rr. Itaque pro dxde + dy du, : Eddqu (+ pato - 2day)
fiet 2a -Y = y = BG = BH quaesitae. Hinc curva AH est dapla BE, et
prartaur, trr ratrarn
est etiam cycloidalis, cujus circulus genitor est etio sub quadmplis circuli EtC
dar
virtur
traar
My-) car-aa
et ar
et
fi abk = d2. 21p
pro hocloci ex pumeratore
videtur exfigum trD 2aV
scribendum C
seu dignna dxdr( arar +Erta) adrisia
sit
atat)t or
et posito det = dTdr: LCr + a)
fit 2drdr + (2r + a) ddr = Grai dr 
3 a drar F 2x drar a arar
ar, rta, 1(ertar
in Zarr = 3aaY, 1, 4, 1
seu dily = arar, Saar - 5aN - 2a3- 2Y,
 rrtar, ia, Crtan