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# Title: Leibniz-Handschriften zur Mathematik LH 35, 5, 2
# Folio 8r, page id 00068020:0015
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sed in quan ista cadant figuram subtilius inquirendum ess, cum redeatur ad
mirabilem illam methodum Tangentium inversam. Ego vero indestigandum adhinc censeo
an non alia supersit prvcliviorue vici. Si AB lems deum decuresum et BCspatium illo
tempore decursum, utique erit AC(C) parabolica,
cujus verlex A. tangens vertixis AB(B) Ergo sic
spatia sumantur uniformiter crescentia e AD erud
tempora ut applicatae parabolae; et quolibet temporis
no erun crementa temporum Z  seu tempora quolibet spatii momento
perursa, erum ut differentiae applicatarum parabolae, nempe: 2ax  applicata
spati pra
parabola, duarm applicalirum differentin: S2aX - t2aX - 2a β. HZ. unde
t rax B Z +  2ax -2a β. et quadrando utrobique: (2ax Z + 2Z Y2aX. 2a  (ra B 34
sive:
= 2Z Irax- raßB 
et ordmando y4
Quod si reliquissemus:
- 8 a X Z2
+ 4 a B
-2aβ, sive 4Z2aX - 47 a β, 2 (- 4 Z a  ax
+4aeB 0 quae aequatio ultra dividi non potest
+2aX - TraX = 2u β H Z, fiet:
2aX -  2 V4aX = 4 a pX  + 2ax - 2 a β (
4aX = 2 a (3 - Z
 sive:
H 2 14a X- +a BX et quadrando:
16a X - 16a βX = 8 aXZ, + 4ae 3 + 4 a 3Z + 74
sive BaX 
crementatemporum lt in motu
16a N = 16 a BX
G2 + 2aB, A itaque quoliacu sint si me
uniformiter acelerato sint 3 erunt
e + 2a β 0 8 aZ.
spata in quibus illa contingunt decursis illa contingunt Z
contra: ut inrestigetur valor ipsius
Z4 - 8 a X GN
+ 4 a (.
pt64a to
= 64 a βX
+ 1642BV
cum motus est 4
fiit:
+8 a2 X2. adeoque a2 - +aX + 2aβ - 2a S2X - 2 CX
-8 arX
(+4a
4a 8
sive ZA aX = 2a β + 2a F2X- 2PX
side Z1 Vtax - 2a β + 2a F 2X2 - 2BX
Quodsi ergo notus est uniformiter acceleratus, epeca temporum seu virium crementa, Z
sunt in spatiorum percursorum , rat ratione quae exprimitur; motus est uniformiter
quolibet temporis momento uniformiter acceleratus, quaeritur cum orementa
in spatiorum percursorum ratione reciproca, seu cum ZN
ura
N
quae tum fierit ratio accllerationis, in quolibet temporis momento. Nimirum ut
aequatio 3 1,V4aX - 2a (3 + 2a f2X  - 2B3X
sit ad hyperbolam secundum asum
plotos, sive aequivaleat huic: at, fiet aequatio inter hoc duo:
Tax -2a (3 +2a Y2X2- 2βX
1
a. cujus aequationis ope quaeratur valor
ipsius X. et epctus nempe 4aX -2a ( + 2a S2X 2BX p) . sive
raft 2X, - 2BX ( a = 4aX +2a B et quadcando.
ax  (9a β3)
et ordinando:
X
a = da X + 4aB, +) a X - (16)i BX + 24
n4
na
= C X + a po. sive X = (X (+ (28
e
Habetur ergo veilor ipsius X.
spatiorum et temporum
Bnt
+ta
ent
+ aβ sive 8
d
+ 164
16a
+ L
qui insertus in aequatione
n2
4 a
aus
qaB2
16a 
+64 a
N V + C
- 43B
C
VX
23
F- 2a a +8-La
Prax Y: ex primit relationem
Immo festinandum lente hic enim alioquam
homibilem paralogismum
committeremus. Sane verum est, summam) omnium y haberi posito earum valore expresso
per X. sed non datur in his regressus; et non ideo habetur qualis futura sit y.
Quaestio enim huc proprie redit: data est aequatio: Trax. quaeritur, quomodo
nestegana explia explicanda sit quantitas X per aliquam quandam in cognitam
ut n, ut scilicet differentia duarum y relinquat aβ
hoc autem problema
culalo lui non potest, niti ex data. Hyperbolae quadratura; ineis n autem solvi potest ope per
quadraturam Hyprobolae Geometriam quae a me inventa est. quo Dolabilis isa inquisitio
est, ita enum duae methodi, diversissimae circa idem problena habuntur, et ratio Exhibetur
dstortum allos er in se restegam metroda tamentnt oneisam, rlibeandi ad simplile
et ad quad ratuas.