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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 19r, page id 00068194:0037
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1687. Formulae aequati locorum seu aequationum
ad lineam, tertii gradus, in se invicem
2) envun. s yd tr loco podd solasunt.
CX, H. ((X1 (∞ + yAy. XL ytx  PYedrei
stye  parr
Xur X y y MyLyIX Kyay yy
. xyt
Xy 4yAY +X
enus adneae  egvg eine ps erioix
ec cucc mni
y.   . Hyr eyexygax e vara gexieria
ZKgyi. y Xyr (Xexy ay Kye t ytynty
Ls nis
  i
GXXyZy pnym. Kryis. XEXG Ly Kxy EX. Xuxyee er
Zxorum)
Hgiga ga. Hy rvex hyreita
Sind tos. Zeg  e  1s. enz ayes suregagox X argusita omin
4 d in Bi eagy xexsty 
7Naat 1y e rger. p aara Y) _rry aye. altys ryek X xp sr eti erie
4u regy
  aos e  osi
y ry y xs (X 
este ere. B yAyex V(Prryurine
2s + 2 +3+ 2 +7+3 +7+4 +6 + 1 7+ 4 +6+ 2176
 Nempe formulam generalem  tertii gratus,
∞Z3+ Z v + ZV + v + Z + ZV + V + Z+ V + Y = N
oscissam et ordinatam, caer
et signnifisant oriing
certii q
formari generalissime in aliam formulam 
ponendo Z = aX - bH + aa td Saa +bb et v = bx+ ay+ ab
saa + bb
raa + bb
e n
 et curva posita CC AB esse X, BC, y.
et curva existente Ce; EG, Z CG, V. AB, X. BC, y
AD, a. AF, b. ED, d
Nam ope trium arbitrarioum a,b, d, possum tollere tres
terminos formulae generalis completae. Se
Sed sciendum est non tres quoslibet terminis tolli posse
am x et y non simul possint tolli in parabola,
multo iminque x etii in una secundi aratus Ultimus
minys
eodem modo se habens ad X et y, ut x9, vel XG. Ut
tres terminos tollere possimus semper tollendus est ultimus
seu absolutus; nam alioqui occurrunt solum et 
et sunt in speciem tantum tres literae, item est in
a pluribu
illud manifestum est semper
posse tolli terminum absolutum seu cognitum
unum ex x vel y; e (ex cepta recta)
et terminum xy. itaque, quaero quot modis
variari possit formuta:
y. X. Et. X. Xy. xg. 93. et viteo hinc posse
Secundum extalorum supra positum) formulas 
∞4 vel plenas vel defectuosas omnes exprimentus
lineam tertii gradus, et harum formulamna
non est generaliter in alteram reducibilis, cum
enim reductio fieri non possit, nisi ope arbitraria
rum a, b, de, quas jam insumsimus in tollendos
tres terminos _Y, X, Xy. non possumus (generaliter)
aliquod aliud per ipsas efficere, nisi has quaes sistulim
terminos, restituendo; quos tamen volumus a nostris
formulis  exulare.
Modus indagundi omnes formulas hic fuit ut
eon horums quatuor X. 1t Xy. Xyt. (:qui omnes
simil a formula tertii gradus abesse non possunt:)
iiniones biniones tertiones et quaternionem duucerem
in hos septem 9. S X Y. y +X
y. x. y. yts. yAX. y aX. y+y4 X
cavendo tamen ne unquam formula comperetetur
ex solis  divisibilibus per _ x, vel ex solis
divisibilibus per y.; item cavendo ne diversae
form eadem formula bis ponatur tantum permutando
X et y. verbigralia formula x. B. X
eadem est cuni formula X yt. 9. Et hoc modo
era pretium
dus esse 66
ormulas tert aiidem
a
prodiret numero alias tres formula
completa tollend formula tollendo, quam
quam quanquam id non viderturs