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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 28r, page id 00068194:0055
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ut omnia si licet gmmodius tactemus fint primm
IEamx L nx +pX aequ SsXX. seu l l+ mX +uXX + pX aeGossX. seu Saego LAnet EX + pax
fiat Hap saequ v + ex   siet vet  V s x fint Saeqo c+ v + ex et
et fiest: (aqu cc +2CP+ VV + 2CVX + LCYX  rCX, X a qu L +M + nX + p XX
M
+
e R estabi
20
Unde XX +, ∞g + nrtacvrace aeq. A 00+ 4v + cC + L quantitas aequanda quadinto sed nondum
habetur exctus. si liceret facere in dato boquae l aequ.0
sed video his omnibus nihil agi, semper enim oportet tantem, ut du
si qua formula ( aequanda quadrato, etiam formutas producentes
eam Det e quadrato aequari singulas debere quando sunt primae inter se;
Aflia ergo via alamur, qua codx, et sadio simul miscentur,
quia per solam coda rem consequi non licet, et videndum an hoc fieri possit
sine calla mutatione literarum co et X. Nempe scribatur:
Cd Lx + C  Xdco +  Co d x + g co deo + hX dx  + R dx + l, dcoaequid
seu 1x + (2 + 3x + 4 + 5c0c0
ai
6+1C0+ 8000+9X + et quia ad eam rem
aptius est adhiberi 7 quam X, rem totam de integro resumamus:
Aequatia ad secrionem e Conicam, generalis a me inventa haec est:
_ 2aX + a XxX aequ gg.  erit yaequ Prax + a x.
ut verd hubearus et X faciaius ex facianus XX 4+ 2b X ( bb) aequ. + B yg (f bb)
s
seu e X aequ. Sbb + Bgy = b
seu dx aequ. dy 
a + a x
a+ax
jam a + 4 X  a. ex ago +. eeques (a)  4 5bb + l ru  L a b
Erg. dx aqudus + a L1t +
seus dx aequ. dyt vy
seu a +a x aequ. + a sbb + 2 yy
dy 3
seu dx aequ dy
+2Sbb + yy
Paa ae tahyy
Saa + a yy
yy
bb..  bb a
aa + a y7.
(10)
aequ Elementum Curvae Conicae
simplicissime expressam
et faciamus Et faciar
Scribumus  compendii causa faciamus +1 + a aqu e senempo b fa se aequ
Sive coco aqu aa re gy
 a
Baa 10 + aa b y7
13
aa + a gy  Ergo dx + du aequ. dy + du7
10)
Sx + du aequ.
Saa +  yy
saa fayy
= . vel denique  aa coco + 2 corogy aqu aa B yy + at
aa  aa+4 yy
quae est aequatio ad figuiam clerais cua planam curver
Conicae proportionalem 
(13
Ejusque aequatio differentialis est: aa ce deo f  cocoy dys f a coyydco aequ ageydus
seu dys aequ. aac0 +  cocoy + E cogys
deo
i y  ca y
Scribamus jam aequationem differentialem 
pro ut e re visum effictam
seu d C. 4) ) 6a ab co + cocoy + coyy
dee
(15
du
egu
 acy
2 + 3y + 4c0 + 5YY +6 coco +7 yyc0 + 8c0
de  12 + 13y +14co + 15yy + 16 c00 + 17cocoy, + 18y3
quos duos valores 5 et 16 aquando fieret aequatio  assurgens ad sextum gradum, it in qua tamen
deessent C0 et y et ideo ut comparari posset aequationi deberet ia multiplicari per formulam
tulem 91 10 1  9 + 10y + nco + 19Xy unde fierer aequ (18.) comparanda cum aequ.
si verd nolimus assurgere ultra quartum gradum loco oqu. 15 scribemus: ey (.9) 2 +3y  4co +5 gy budo
qu
qui valot aequationis 19 aequandus valori aequationis 14, et fiet aequ 20. aequanda comparanda cum
12 + 13y + 14c 9
lequatione 12. sunt autem termini comparandi cy. coyh. cocog. c0. coy. cogy. Co7. co. coy. yy. 10. y. 
unde sunt decem instituendae comparationes arbitrariae autem sunt tantum e et ad summum 9. sed hui
malo mederi poterimus crtificio in hac scheda invento instituendo comparationem quodd licet
incipiendo ab altioribus et ubi deficient mmbitrariae sub finem comparandu residuam formulam forruut
e eri ctuu   su e oeur s ie
cy cos
cons goy
coe y5.
o