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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 18
# Folio 9v, page id 00068089:0018
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et
id est, ut tres abscissa, veltres applicatao correspondentes punctis interse
Mrerum fianl aquales et secundum methodum (crtesi prodibit equatio quae rela
sionem ostendit inter AH et HG, et proinde nafuram survs quam desiginnt
puncta screntorum osculantium. Operatio est alis qu. BG_-qu. Bp
- qu. HG = rect ang. 2BF. HG = qu. F ( id est vn - qy - 2yt  =
(5 = 25X + xX _ vel Xx + yy -25X + 2ty + 5s + H -vv = 0. ope aqualia
curo 6 tellatur X velu ut se curva sibstarabola, ubi x = yy:a fiet
y4 + aagy + raaty55)
=2a5y G
ve
20 si ergotres radice saequales
comparonda hoecaequabii cum
producto ex y- e, y- e, y-e, y-f undef
comparanda haec aequabro it - 3ey + 5Cegy - 3 Chy + e2f =0
L 3
3efusys -zeefy
quam aequationem priri aequando fit. -3C f20 Hec + 3ex 2aa -2a)
C3- 3eef = 2af et e3f (5 +H-v) aa perprimam F= - 3C, ande
in 2da6ee =aa -2as. Et in 3tia erit 80= raat unde juptu illudes
e= 1(2a5 - aa): 6 et juxta hoc C= 2V raaf. Quas duos valores
quando f et ponendo, 5 = Y+ a, fiet ar = raat set
16Y3 7aH. Unde patet curva naturam, in quam cadunt centra
circulorum parabolam senlanbium esssem parabota cubicali secunda
cujqus parimeter est 27: 16)a, et cujus vertex distat a vertile para bolee
data quantitate La, quoniar s+ a = it si raditem eB quoque inve-
nire libeat, tantum supram saequ. 4ta ef - (55+ H- vv) aa substituatur
valor ipsius f et e fiet =3C(5s+ tt - VV) aa et quia ee = (2a5- aa): 6 =
ar: 3 eut = (aarr: 3) F id est - 39. 7 = (55 F id est rr+ a+ + aa) ett - vn) in ad
et reducta e aquatione invenitur Vv seu qu. G0) = Z rY + ar 1 aa
+H. Hinc si r = 0 erit F20 et v=ha. id est centrum circuli oscudtantis
parabolam in vertie dipatcenro quara ab eodem vertice quantitate ta
similiterfaciendum in omnibus curois Algebrinicis, sed quia sartespanaG eo
pietiase ad Transcendetes nonextendit, quomodo Cago centro lirculorm
seulantium inveniantur ex calculo differentiatium nyllo negotro ostendi poest
non osculantium centra qatuntus
osterdimus ea esse in concursibus duarum perpepdiculariam Bd, od infinte pam
sit ergo cupua data, ABdenius cu
ap antitate distantium imcenenda enjoest longitudo, sd ex abscissa AE, et ordinata B
ducatur Be parallela ipsi AG, et dF perpendrcularis ad BC silt
2X, EB,
AF dx. F0, dy. erit FC = a dy: dx, et BG dxdx + dydy3: ex et quia .
di +
ydiidu cujus differentiare posito dax =D es
HG quia autem BC: HG = Bd: dH auldevidendo
ACHGi BC = BH: Bd, i ess- Yadys ddx dp + dy dus) = G H(d Z ext dyduy eli
F. F0 = AE. EH eus EH
des dy + y ddy): dx=
ad B, d Ergo Bd = (dx ax+ aydis: adue  in Cdx dx + dy dys): 
demaliter inveniri potuisset dicendo B0: BF = HG: LG. ergo LG erit
dedx + dy dy + y ddi:, S (dxdx + dydy). Nunc B0-LG, : B0 = BL
per alios adhuc
dem provenit quod aute.
(vel BH) 2Bd pro quo ad cognitionem rectae B3d. dacantur nempefius q
dats tangentes pM BN, et arculus vel perpendicularis MM erit
rerfferentlalis ippus AM et quia B0. OF = NM. MP. aabetitur Mp.
modos peroenri potest
eoM et Bdo sunt trpiasula similia fociendum ut p M ad
del adhocaliter ducatur perpendicularis
AG eut Se disferentialis ipsius AS, et quia B0: BF = 59. sR, ets R: B0
20, ita AO adBd quaesitam.
quia ante
=DS: Bd, inveniri potenit Bd, quae semper erit (dxdx idydus,: cidus) t
in Scaxdx + dydus): dudx. in qua quanltate so subestituatur vator.
nont natura curvae requirit, proveniet seu pers Bd
in quantitalibus ti ordinariis et absque differentialibus. Monstremus
ipsius axvel dy
7 obi AB,  AE
BEy = F ax, et gni Talore
st autem in farabola,
dy = adx: 2 lap. proinde dydy = adx do: 4x et ddu = adc ax
excuplo paribalo
oclor dus et adus.
axtax. Ergo fit Bd = a + 4X sa + 4x,: 4 a) sit jam A sed more
Bd dicto substituendus
Cartesi sit sit jam AL, s. et Ld, t. quia EH= ia erit BH
ob BH :EH, = Bd: Et fit Et = 2
axi. =aq
a + 2X et AL = 2 a +3x= et sit
=3X vel X = L Y et sic 4X: 4) sax
qu e: 3) HaT: 3) = + et reducta aequatione 1673 -27 af ut supra
al supraposuimus (= 2a +Y erit Y
in parabolucabicali prima parameter a AE,X. BE, 7= Baax
dy = adxe: 3Yax fit tandem Bd = (aa X= YY aaxX): 2aYa
saa +gr saa: gxYaax. in Hypibola diapneter ex panmettib
AEX. EB,) = (aa X + axX,: b) unde facile habetur Bd. sit
AB Cycloestalis linea, AGH Erculus Gentor (fig. 8) AE x. EB, y= EGrAG
= Rax - +x) +s: Et dy = (a dx - Xdx : (2ax - XV) - ds = quia
ratx-xdx: EaY -x. et aly = -adx dx: XHeaxa
Ladys.
c1( ax- 7. P
atran s
epera pretum ert ostendere particulariter quo pacto percom
s
un
F2 ag = 2 ax)=
renum inveniri possit distantia a vertice centri circuli qui curvam
d
osculatur in vertice. Sit AB euva data cujus Ayus AC