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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 52r, page id 00068194:0103
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gntqe ader arquela etiem
Diameto coningatae DCE
vertet 
)Dianeter vel
Sit Hyperbola Aequilatera BY(Y) cujus vertex B,
Axis (BXIX), centrim C, latus transversum (CB aequale
rectoi, cordinatim applicata ad axem  YX constat
esse rectangilum P X B aequale quadrato ab YX
Et ut alia Hyper bolae reipondent Ellipsibus, ita hac respondet
Circulo, centro C, latere diametro, t PBd. DB descripto,
qui et ipse  simul lateri transverso et recto aequatur.
Nam et in circulo PXB aequatur quadralo ut YX, tantum
interest, quod in circulo X cadit in Latus transversum, in Hyperbibla extra
Porro jam ex Apollonius constat (21 primi) Hyperbolae aequilaterae
Asymptotos rectas facte C Z, facere angulum rectum
in Centro C et transire per A, E extrema alterius divgarnolalis
Ejas quadrati cuius sragonalis est BC, et hoc quadratum esse potentiam hujus
Hyperbolae ad Asymptotos, seu ex puncto Y ducti ad Hy
Asymptoton CE, recta YZ parallela  alteri Asymptoto CE
atque adeo hoc loco angulum rectum faciente ad Asymptodom in Z
so r rectangulum CZY aequala quadrato constanti a CA
Vocetur CB, b CA, a Est autam a = b b = as
BE, BX, x, XY, y. AZ, Z _YZ
erit 2bX + XX = gy et aa = Zp = 2bb
sit linea ip plane de ducta quaecunque Y (Y) ex
cuius punctis Y ad rectam quandam per A punctum aliquod
constans D tramuntem ag e  ex et ex
iisdem X normaliter ad curvam ed ucta YF, vecurrant AXi dicto in F
denqus in XY sumatur X V aequalis ipsi YF, et dacatur linea Y
va ut imgona e va ()et spaticue s X (1( () 4
volt  ma e am x qualle rectaagulo sab HG et rica ae quali curvae Y Y) aequaliter det
menta car
ram respondennum, qualis VN). in tangente curvae ad Y
arcuum
respondentium Y(Y) sum sumatutur curva puncteta V elementariter et p) rclementar at tistanaa en prabio Y, ita ut
Re qua erit tens polygont  currae etem aequisalentis, ut error sit dato minor etqe V et (V) paralletae Vd issi AX, (∞) d ies, YY
distans a puncto V. et ex V in XY agatur normalis vd, et jungatur VY
erit ragitus Pr rectus, et triangula V d Y, YXF erunt similia
seu rd Vd ad NY ut YX ad YF. itaque rectangulum YP in 6S
aequatur Erit () latus polygoni curdae aequivalentis ut
error sit miner dato. Ex V et e ducantur norment ad DX normales
vA, (V)M, qtuae producantur in AB, M X aequales iphi X 8. compleatur
rectangulum s VA nd. Ob triangula  Y XF, v8(∞)
simitia (ex quibus hoc characteristum volare soleo, quia constat
ex Elementi, cordinatarum et curvae) erit YX ad YF ut vd ad vY
sed recta et rectangulum YF in  vd, (seu X in dM, hoc est
rectang. SIH) aequale rectangulo Y X in ~Y, quod est momentum ipsiis rectae
ipsius v (V) ex axe aequilibii BX, lum sit distantia vacentri gravibatis ipsius v(V) (punctiscilicet medii)
ab axe. Itaque si polygonum arcui Y (Y) circum scribatur, constans
ex rectis qualis v(0) erit aagregatum rectangulorum quale A FT,
curit
HV) (X) X4
rectangulorum quale AB aequrvate 
lumna laterum polysgoni aepuntulet arcum Y (Y) Cum  possis
ostendi minor dato. Ergo zona est momentum avcus ex axe AX
si linea Y(Y) sit hyperbola aequilatera cujus axis AF BY
iisot positi
Cerum C erit linea p () Hyperbola alia, cujus centrum C, vertex _ axe-
Semilatus rictumtrans versum CL aequale ipsi est, semilatus
rectum ipsis CA vel CF duptum, ordinati normam
ordinata ad axem, ei decarrens in B, nemne BM
erit b aequalis ipsi CB. Et Zona Hyplrbolicae
VY X BH erit ert ut momenta arcus BY ex axe
liberi _quabitur rectangulo ex arcu BY in recton
extenso et KH distanti a cente ipsus K, ctuhi gravitalis arcus a recta
vatis jam admonstratum est esse XF aeq. CX, et YF = CY = 7
et qu X( = CR B + qu CX = CB BX t CBB) t qu CX CB. CB + CB. 
 icee ittar
=HB GH distantia eius
a axe aequilibrii BX (X
punctr )  ipsa XY bifariam secetur,i
concuerant
d vinique ex et et s normalu
in AX dutantur vsC DX pot producenda
iltae in B, haec in EC, ut X β, ( A sint ipsi X  aequales
juncta Ae icmret curva () in 
2. bl2 X
= si temper ipsae L (V) des
continue decrescant, ut fiant
minores data quavis; ita
enim et differentiae decrescentium
ab his in quae desinunt, datis
minoris fient
2bx+ xx = yy
Rodx + xdx = ydy FX.
dy : dx = b+X, : y
dy = dx, b +X; +(2b X LX)
dy:. b+X = FX = CX
bb +xX + AbX +XX =
Ax + 2XX + bl = qu X Y
Ha  X = b - e1
ri -bC +261 +2CC - 44C
in ee eb +
fint 4bd + 2eC + bb =
crbb, seu e= bH ( bV. et fra
ut e er eai t