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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 80v, page id 00068201:0160
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(7 () Si rs aequationis radices positivae sint a, b, c, ete,  at privativa sint
l, m, n, et c. aequatio canonica ex ipsarum inse invicem ductu orietur talis:
Aa
y + a Y + ab y
d
0c0
per ab intelliges ab acabcet. vel Bea abceteinsune in 1. l y
sicec 1. a sipui fiat 1. a =+ l: b est vere  qeabemt  +18: a.
vell oeaftab. em b
+ abc y
v- 2
 + 1. lm t
A- 1: da
+1. ab
+ abed V
tabem
+ lmu
ilma y etc
+ 1: lma 0
= 1. lab 
+ 1: abc 
nelalabeli  prior exprssio incipit a termino
vel 
Summo, posterior ab ultimoi
significat A vel - proul numerus radicum
privativarum l, m, n’est jar vel impar
(ED ins puni aesquitione sicuis amnes padices sit possibile necesseee tertrum
terminum lesse diffectum signa politiea i proinde si tertius cujgeunque aequationis
terminus sit affectus signo
(B)In aequatione cujus
omnes radices sunt reales s numerus radicum negativarum est
pprivativarum est pari  tunc eaedem regulae generales
quae crca  signa  aut privatuva terminorum
secundi, tertii de ex se invicem dependentiam condi possunt, nulla
non considerata radicum qanitat aut ex ipsis compositorum
quantitate, etiam de termins penultimo antepcenaltimo 
condi possunt quia ut illic a, 1,  ab, al, ln etc. ita hic
1: a 1: l, (: ab 1: al 1: Im
sjusvem cta. seriet
13) possumus rectaa culat exparmere per potentias, idque si faciamus tam
in serie radicum negativarum quam in serie radicum positivarum
et ex his preductis, facilius judicari potest, quid quo majus minusue
sit, quia non tantum exe posito a +b ex esse  quam (+ m X
constat eliam a +b etc esse majus quam
(1)Si2aF b +C et  quam E+manete.
(9)  si summae laterum
sint aequales, et summa rectangulorum major, erit summa quadratorum
minor; quia summis cilerum existentibus aequalibus, etiam quadrata a summis latem
st aeqalia, quae constant summis quadratorum et cuplis sumaris rectangulorum.
(10  summa laterum vocetur 1,
bi re Tangulorum. 11 trirectangulorum 111, &c.
12 aequ.  2 + (2) 11
vel 13 aequ 3 + () 11. 81 - (6) 1
i’ aego 2 + (2) 11 
13 aeq 3 +(3) 11. 1. =((3) 111
quadratorum 2 cuborum 3, eC
(4 aeu 4 + () 1. B1 - (6) 111. 1 + (4) 1II
aliae formae e
et 1 aequ 3 + (3) 21 + (5) 111
X + Cx + ex +)
=Ex+ Ex + Se
Ex inductionibus Harrioti ut a vallisio exhibentur, apparet
e ber ecx e
sit aequalio X a b x + cx + d (1) aequatio  - ducta p in X - l dat aequatione
qua sig
2 si in aequationem in qua signa alternantur rucatur aequali X-l orituo rursus aequatio in qua signa
maltialicetur in qua suma alternantur; si in aequalit
aeternantur
 in qua omnes sunt radices entes, signa alternentur
X L X = bX + CX ∞d
A7 - b ∞3 + cX
ex + b x = CX
Nota per a intelligo a+ b etc
per l’incelligo lxm ete.
ita port
(3ETC
ubi sane res manifesta est, ejusque ratio
in aequatione in qua omnes sunt radices
omnia signa sunt +. Hoc facile es
si qua aequalio constans tormula constans ex meris signis 4, ducatur
in formulam, in qua figna alique alternantur, prodocetur aequatio
ima ac est pro nopto
ti
ioties
Si asquatio in qua omnia signa sunt affirmatuea ostendi debet (2) nempe per calculum actualem
e xt 481