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# Title: Leibniz-Handschriften zur Mathematik LH 35, 10, 7
# Folio 2r, page id 00068160:0003
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ponamus lineam in curva ferri motu composito ex descersu versus e
et ex circulatione circa C cujus velocitas sit in ratione reciproea
distantiae 15 eauet roi sit temporis incrementum seu ared
sqo seu tuingulam 162G C expositum per constans ad. fiet utique sicus radius r = cG
19L = aB: Y. i porro vis centrifuga circulantis est ut oGtet ad D: a, ut a: r
seu ut YCB P seu aD: B. Ergo et  fal  sit in vis
puvitatis, fiet: dde M = dar + aD: r3 nam vi centrifuga
detracta a vi gravitatis, fit demum ve a novus descendendi conatus prioribus
Superadditus.. Sit jam ut =  a.  Br: a seu in relatione mtermina, sive
patione simplice aut multiplrcatam directa aut reciproca radiorum. fiet
B an3. = adr + a B: r: seu fict: O r’ dr: a = dr ddr + à r: p3
Ca
prodit
ergo fiet: Or: nt a = dr. 2 + ab: 2r. cujus posteri cujus aequitionis dextrum catus secux idem
Ergo D- ddr : r: a: nN = a. r. sit jam n = -2 seu m - Ba: Y
firt D = drr:  -ar: 2 - a seu f tempus vel areaintegra + = aror: S- ar: 2 -a
Huc ergo res redit, ut inveniamus aream seu summam talium: Sdrf. FrFt (drr: ErEI 
sit Y + 1 = v2. fiet r = v -1 et dr = 2V dp. ergo fiet inde in rationalibus f2Vdo v=1 : v-dov.
F4 v - 25 do: v quae pendet ex quadrature Hyperbolae non ergo curva GG potest
esse Ellipsis, ubi area ex radiis perquadralum Hyperbolae non determinantur
sit ua gravitas in ratiare ratione distantiorum reciproca seu sit n=  triplicata,
s
fiet n + 1 = -2 Et fiet B = dr Y  coust eruntque tempora in ratione radioru
fint (B seu t et ut r seu tempora t dlinsumta erunt
De 1GC
in
se
uGC
duplicata ratione radiorum. in restigemus qualis sit figura, in qua areae sint in duplicata
ratione
ratione F ut quadrmta radiorum C1G, C2G
sit BG- y et AB = X
et AB - h. fietit f Sydx + 1 yt , et y + tartix a  seu Sydx = ati atruta
e firt (9 4∞) +  L h dy ( L y dx) = Xdy, ydus a 2bx +2x dx
Ergo (X du = _ his - 2gy + 4hx = 2 X
+= hy
Ergos d syd hec estai (yzati( ih Xt X- 1 yt t L yad ( hy (y 9 tx B3
Ergo ag ix
Er quoe est aquatio ad Hyperbolam o sint nor in calculo
Rrepetimus  Sydx + Z y h- X = 15 + h = 2hx + X. seu Sydx = y +ti - 2tx +X- ryh. yx
cujus aequatio differentialisuest (9 dx) = Eydus - 2hdx + 2Xd - L hdy ( L y dx - 1 X dy) seu 4d
yd
Lxdus
seu axdx + aydus = 3yda +exdus + 4h x + hay. seu 4Xdx. 3ydx = 4 hdx = Xdythdlys - 4ydu
Si quaduta semporum scribatur Odx = Ddus  C dx F Day
et fiet  sit ana ldx +MXax ydx = pdus t qu dy + rydy
et het aX 39 ah in F + query Xth ay in Assum atur
l+ mXrny + pxy +qX + r  fiet mdx + ntus t prdus 1 pudx + 19X axt  r ydu =
Seu fiet Max + Pydx + 2qXx = n dy pady- erydy
Y= at bx rcy+ exgt sxrgy CFhXy Clr + nXAn ec
dyam
si tempora
Ad Neut. A. 50
ZR nocemus M et CA, 4,
k EX. PV: CA: CP  porro GV. PV: QV.: BC: C
QN
LH XXXV, Z+
24 CDr: CB sit L tatus rectam
6
E sit lotu ZX: qY :: AC: pF
Ergo L. eR= 2BC: AC, in PV. CA: CP. per 1 et 5.
B1. 2
+ 2ligX 89XG
Atum +nX -am
sit primaru = 2 BC: AC.
rursus PrE QV. PC: CD G tfiet L. Gp  2BC. (C) in qV. PE(CB: CD. G, VD
ali +a
at CD = QX. CB: q F ex 3 et 4; Ergo ex aeq. 8 firt LGR = 2BC i PC. QT: Ex (CB7 GV. 
seu  L. gR = CB 2 2 PC. qV. ZF:  GV. QY. Est autem 24. 
2+ = ad.  sP sit F eR seu pX vocemus m, et PC, C  et PY ∞ et GV = 2C - v
 Et QX seu AP = n, et _W XV  = n H ZV = ALX h + n et QT = K
porro per Ergo fit e LQR, GV.
LX 2C V L.M. EC- D h l + Ehp + nt
2C. K. ubi ipsique n infinilies infimte
pdoae quadratum (ut infinites infiniteespirva negligere possumus.
porro sine
bb trangula PVX et rCE similia fit PV seu V : VX seu n,: pX seu m: PC:CE: PEseua
Y = Cm: a et n = em: a. Ergo  a L. m. 2f- fm: a h = n + rhem: a 28. K
Ergo 24. Me tr 2 Lmh - Lm h: a = 2t K A HCM 4 2RhK M: a
et fiet
Unde jam collegit N si jam rejiciamus, moti: a itemque retik m: a tanquam
caeterarum respectu infinite parvas fit, utique YLm_= 2K2. seu memit ipsorum
nmvodrais proportionales; ut coclusiset Vertonus. Verum talis argumentatia
 ep  tellisi  isfcadtu
x- du
t h-+y
in+ py
 9 349X
h+ X - 4y
m+29X+ PY
taqu
Ahpus pxy ap
= 3y
2ry
brs