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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 324r, page id 00068196:0643
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il
tanvier 1676 Requationes tetragonisticae per calculum
Y)
differentialem
y+ YX. Ergo b+ SCB y+ 3y EX p, b + tc.
9)X, 6yX + X3H C. et qC x
3xy
X
orebyr et X⊓ B b- y 
sit BF tage portio axis inber tangentem et ordinatam  +, B bXZ,  et AD
eit
e
b
bx
brE. AY + bx =,
 Y b 4 cX + dX + aX arc, fieri sit A
2ft X? b + cy + dy + eyte uide-
2XF 1) b + cy + dy 4 e v ea tp cy 42dy + 3ey ox
VLV +OdV +Cy
b + ca + dX + ex B Y. Ergo c+ + d Xt +  2y
et fiet.
+ p, 2 b + ec X + d X + aCX et Ergo quando potentia u ipsius  ordinatae ration
+
est rationalis modo
c C+ 2dX + 3 ex. er.
usa incognila
cum _ d F=K  1 = = seu dp y - = y. Eruntue DEr X~.
a
erunue De ra habebunt eandem potentiam rationatem quam habent y. verbi grata
si y est radix quadralica, erit: dDE
2b+2CX+ 2d X t 2eXC
posita b 4 Cx + d X + ex ex , y fore DE B   St ob +C+ 2dX+ eX er.
A. SDEH SX dy ut ex alibi  a me demonstratis constat, quaeramus ergo
CAudxEe e Cop (ZY
dy. per X. Z 1 . et + X 7i2
c+2dX +3 ex ex
utra Jam seo semper haberi potest, e . Ergo Explicando Co et y, habe dimus theorem.
Sequens Arihhmeticae infinitorum promotae, sane memorabile et mirum
Sinur Erdato brCK+ Dx rexSobetet Exex A Yb + CX+dX + ex eu B ScSy
Brez V + Cx+ dxtexeu
(ilem (DE + S y ( VX. Unde  Bheorema Arichmeliae infinitorum:
 b+ CX + dX + ex e. A + X V ob R 1 + 20X+ ; d H X VbACX +X Fex
Ergo Ex hoc posterione Reoremate habemus s commodi, ut smper cummar C possimus redicere
ad summum omnium D quae utique simplicior est.
Si sit X2 ( _ b + cY + d3 cc. erit at pphy +2dy et res fobrcy +dy
21
Ergo vel CZt  A vet jam S + dy B Sydx
sit aequatio:  Xb+ (Xy + dxy    e + fY+g  erc ponemu):
4b + cAy + d+y erc B eo + fy + 23Y ex,  = CXy = 2dXy e
Z+XY
et LB eet fy + 393  cy - 2d8 e+ fyt 9Y Quorum in dy summap Sar
b+c7+47 b+ cy+dγ c b+ cyt dyt e
si ad sumum tdus addatur summa ady, habebitur area productae quantitatis, Et S+ + X temper haberi potest
Ex hoc loco. S ec =153F.953 b +cyrdy  = Cy- 1dy 5 e+ fyt 9y o 4 C rsy tgy)  b4 cy t 27
(2t b + cy+ dy
HC + fV + 93, 2 b+cy+ 2y, ay
b + cy + du 2
jum pB Z. Ego P)
 si b+ CX + dX + ex er ⊓ y eut t⊓ e 2Y
C+2dXY3 ex
sive Et quia semper sp) γ fier c+ 2dX + 3 ex er
imo aevenium ab enore
+ ex + da tex
CAOEHOe
A 2brcerdirer
 brCxrIN rexer
(Ex+ er cr bc a B:  bcx  
Ses
2
Ergo
Sesu invonitur.
(XVdu
us in
2