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# Title: Leibniz-Handschriften zur Mathematik LH 35, 2, 1
# Folio 265v, page id 00067993:0532
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Y L aX
Sed quomam Xiquaeritur ut methodus duplicum radicum aequalium commodius adhiteri queodu ab initio statum
de integro ordiendum, quaerendtumque Ellipsis tangentem artitior, non y, sd x. assumto. aque 7. eliminato.
( Nimirum S= X + v - 2 vy + y. Jam quia 3X XX9 = ryq- ryy. pondo
a ex aequatione priore sequitur esse Y = N = Y S - X) hec ergo ipsius y. definitio in locum
ejus in secunda aequatione substituatur: fet: XXq = r9B - rgFS - X, = rv Y5 + + 2r sS. 
vel X4 = rav- 15r4 - r X= rv+ VErS Hr
2X9 = 0. + rgX.
0 + 2rx 
DENVK
Ulnie ficiet . 2x 4 = rqX + ZrX = 2rVX. vel 2X4 + 2rVY = rqX t2  X )vel
vel: ZYVX = xaX + 21N - 2Y 4 sive = Aa + +X - +9 = v
2YX.
Unde sepoeutiu: q 4 X - 29 = 7. + y - Yr. ser X - X9 - X + Y- 7 - 2
et x = Z + 7. X - 4. _ vot i sit absudum, errorem in cale et necessrege
Eredo in eo quod sundosper exponentes ipsus  multiperlavi
possimus autem eliminare sinditates, si ponumus Y5Y4 rX cum S4YB5- 4r
ab una aequalioris parte caeta abalta, et utmmus quadremus. reliquum appellem 
fied O + D - 2 + BD = 8. ive Unde duae surditates reductae ad unam quae dem
eliminatur, n 0- 270D = f - O - D. ideque 21OD = + - C - D, C sed haec
proligiora, quam ut iis insisla opus sit brevius cum ostensum sit X v-  + Y
PM = Y- vr  substituendo X pro Y. Quod ut fiat consideranda aequato: X = ry - = 1
Ergo 0 - X = Y Y _ - ry p 7= 4. crilrro - x = Y p - ry + Y Ergo Tr - X = 
=1= Y Ergo Y ineo casu X sr - X, +Y a y.
Memgrabile hoc vioetur esse quoddam Ellysos i ua pateo circilum iqua tam
asplicata ad abscistum appicuta, quam abscissa pura relatione explicari potstu scblicet
quandu latus transversum recti quadruplum est. Idem in aliis figuris aoplimu potest.
huo idem fieri potest in omni Elles si, nam si data sit aequatio: 0 - X = Xy = ry,
e
addendo utrolique  Y = 47; se q-X = s - s  Non tai se ductadant:
Cs-ry +4 Ergo 18. X + Yg7 = ∞ 1 . Eip 54- x + 14 t
ive sF - X4 +  4 = y 
a
s
Sona mynos figuram seu locum applicatarum, quarere ex dato loco Reductarum, idque in illipseos exempo
du
cntemus, piinum posito MA = y. SCM- X et PC = S. et AP = 4 V et (M = v - y. erit PM = v + p - 2y.
et CM = X. Ergo S=  V + y = 2vy + X. Cum autem quaeramus ratio relationem X ad y.
ponatuus s  X esse EY. fiet S= Y+ fy- 20sis + Ys In qua aequatione cum sint duae
radices aequales  fiel.  0 = 0 + 2y - 2vy + 1725. Ergo 27 4 27 = 20y. atque ideo
V= y + Y7y = V. At jem aliunde constat esse ergo PM seu V- y = 13y. Alqui
dem PM = Y = 37. Ergo 57 = Y - yY. Eugo 5 Y vel X = rY - vr debent esse:
X Yy = Y. X enor sergo, nou calculi sed mettodi, quia scilicet inisto Zes. non potest sciri quot
unitulum sie exponens ipsius y, quia in ipso  latet quoddam _ y.
Ergo not ad Cartesiii methodum duarum Radicum aequalium potius quam Huddemanam recur
u
reuriere debere arbitrir. Nimirum s-v- 5 + 2vy+5y = 0. et ytearye, ctano.
ergo: 5 - v -  + -) = ZY = 1C - 29 -  21 = 0 y
Ergo 5. X-  4 2vy 457 =  +e  sis. Ergo 5- v + 2vy +57 = y
5Y4= 2y
verissuma quidem, sed quibus vihil explicari patet. Tollatur ergo jam V. ex aequatione, quoniem ejus
cum  comparatio cum Y. cognita est Est autem D = Z + y - YY C Ergo V erit
=t 1y- y + y 2yr + Y
Nel 152
et Lvy - ry+ir. 28x. 1
-y+ V + y-
+x + - 2et + Z5= 4(0
Sed uli exhac quoque aequationes ducipotest, quonium si s. Explicandum sit, ounia tolluntur
ei 2
X X. tin G2
Ergo 431 +   zoy.+y 57 2