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# Title: Leibniz-Handschriften zur Mathematik LH 35, 8, 3
# Folio 8v, page id 00068098:0016
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Ergo Y=
ers
Cva unde 4ny = Cak,  vellsqus.
e
Ergo r = 2CX. unde locisha: YaX- 4CX vel loy hic: y = aX sutola apte
ax
 es  sa pet 3
eax-
L yx
Ergo Y = 2yc unde locus ax = ya Trianus ulum.
quibus as
en
nes disscontiae absis
Ergo semper duglex tantum locus, t et ita proportionalis
Si figura data sit Hyperbola, et Axis sit Nymptotos, aequatio est:
ax a = yX. unde yl = xy. Ergo l = X, e vel x yx = Xl, ergo _ l - y.
Ergo
d
a
-
X
2+
Ergo Y =
Idemque est etsi y. assumas, suobstituto tantum y in locumX.
hinc duplex sufficit constructio, loco quadruplicis:
ca. Unde locus Xy = a. Hyperbola cubica
6
Ergo Y = Cx . Unde locus: ay = x Parabola.
Si pro ax spatii assumatur ipse Hyperbolae axis, aut basis ei perpendicularis, aequatio
es (sumta Hyperbolae specie in qua latus rectum transverso aequale:) ax + X = y. sed
quo ut calculi similis repetitionem vilemus, et circulum (vel Ellipsin) et Hyperbolam
simul complectamur, fiet aequatio: 2aX + X = y. sed antequam huc veniam
subit animum experiri aliquid quod circa parabolam tentare obliri sumus aequali
ubi fuit aX = y. Unde si abscissa est X seu cum directrix est axis, tit al= 2y. Ergo
ayr quod fieri potest = =, unde fieret
ab
l = 2y, eritue ratio , vel - =
lC = Y. sed inde non potest fieri locus quia sic una tantum habetur inco-
gnitarum in aequationes nisi ea explicetur. sed si explicetur inci-
2y7
demus in lota jam n jam exposita N. het ergo praetermissum fuit.
Ut ergo rem ad Fli CirculuusietEllipsin) et Hyperbolam nunc pergamus
duplex ineundus valoa ipsius  P. primum ex X abscissa, deinde ex y abscissa
ex X, ita = 2aX F X = 1, unde fit 2ap + 2x = 2y,, vel p = Y seu
a =X
si X sit abscissa, fict: 2 ax + 2X = 2yP, vel  p= ax +X, seu pro x sub
c
y
Jam ut pro X substituamus ejus valorem, considerandum est, esse 2aX = X + a =
=a. atque ideo a + x + 2ax = a + Y,, ergo a + x = 1S a +
v
iu vossu
haec dubits aleabno  X + a =
Et si subsistamus intra quadrantem circuli,s quia tuno X seuper minor nu X nunquam
major sit quam a, radius; possumus tuto dicere: a + X = S a Fy. et
X=
X =
a +Y -a, vel X = + ae Fy = a ego X=  v5 4a7 2al
2a + y = 2a5a+7 ego p - + a Sae +y 4a + 2a + y - 2a Say
g Ergo rupta primum ipsius p valorem erit:
W
Rax + X
a + X, SaxFF
f2ax+ x= E. Ergo
a +X
ca = Cx Y. = y
sarta
unde locu ha talis ays ax + yX = ca + Cx  2CaX. Ergo
Ca + CX F 2CaX,, vel y - a - x, c
2aX + YX
esinucirculie
EHypert
y
unde. inde seu in Erculo, figura prosine adjecta CD -Y AB a 7
2
BC- a AC. sin recto
a =k siny compl
sin. rect.
quod est verissimum, et exhibet nobis quadraturam Tangentium complement.
Cop ruxta posteriorem valorem ipsius p, erit
= 4yafy + ay
3a - y = 3a a +Y
F cuus fignat habetur quadratura hoc modo, sed ipsis
Y constructio investiganda.
sax
X- ZXCZ X Z
fat a
lan
id
spax +x b atx
at X
uende patet quod hic
dicitur idem esses quod
diffeencias invenire ip.
plicalaum divisa qua
libet per ejus tangente
productoque per rectam
constantem C. vela
mulliplicato 
circulo
sit sit 2n
3