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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 345r, page id 00068196:0685
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Quaeritur pro valore ipsius X, formula irrationalis extribus quantitatibus
irrationalibus  et una rationali, in unum additis constans; ea lege
ut sive ad X addas b, sive c, sive d, possit ex formula proveniente extrahi
radix quadratica.
Forma hujus Formulae sit
a x = pim + qun + rImn + a5
ubi l = 10Z+ na M = 20Z+ 2a R = 30Z + 31a. Sed has significa-
tiones non erit opus substituere nisi in calculi fine, ut scilicet compendium laboris fiat
et identificandum
itaque debebit esse
poem +q in + Y Im + a5 aequale, quadrato alicui a radice, 112 um = 13 sn  
ab
quod et
2.13. 123 Mrm + 2. 112. N23M sb +2. 112. 13B rmm Fom
psidentisicandum
2. HS  (. .. +2. 100. 43f. . . + 2. 100. 123f.-. p 25
pilm +q un +r Smatas aquitosbadrata alicu a vabla, 212 stom L. B n+ 2e5.) mh + 200g
quod erit
s
2. 313. 31ypl 4 3. 202. 223mhr L2213. 29 blmr r 2
2.200. 2129.. 2.200. 2139.. 2. 20022491. 213
125
m +qn +Y Int+ as aequali quadrato alimia radice, 312 m + 3 13 Vn+ 33 sum + 300h
perit
2313. 313 lm Y 2. 3 12.323M fln +Z. 312 913l f  312 
2.300. 3uho. 2.300319.. 2300. 323.
BIn
32
30
axtac =
et identificandum
comparationem astitula boippat, cibsvis nempe ex tribus quadratis
+X +ad
Habebitur P ter, item 4, item Y, item s. Nempe p tuim sertificantur termini affecti per
ut
cum eo cuj est identit
irntio Nempe p dum identificantur termini ass per irrationaem m
n
sic
sun
10899
evelipsinus q un ipsius a secundo (tertio
mere rationales
2. 312. 3c1
 2,212,21
2.300. 5
F31. 3
V
313. 2
3Et. 3e3H
30ei
prigmqvalor ipsius p in eo tantum differt a posicricti
p. = 2u13. 1232
 2. 100. 112f
q
p l 3
2. 1B. 1135
p = 2. 1lt
(10)
a5_ + 112. (12 lm
113. 113 In
123. 123 un
100. 100 85
=c213. 22n
2. 203, 2129
212. 2231
2. 200. 21 9
P 2. 212. 2.1
2. 200.8
(11)
ar22bn
23. 2h In
223, 213MN
200 2004
ac
Hactenus exhibuimus duodecim Aequationes identificatorias Fundamentales, seu praeliminaria
Nunc id agitur, ut hinc inveniamus, tum ipsas
in etiam nislatendo tumun ne tum sas l, m, n
calculi, quae nullo negotius scribunntur
ex Z pendentes. Quod oltinebimus opeae savendo ne unquam valor talis indeterminatae habetur
i et
indeterminatasq s,um etiam ipocot
per mepas determinatas. Id vero obtinesimm ope aequationum auxiliarum destructitiarum, per
10, 15
quas habebuntur ipsae valores determinaterum pro arbitrio assumtarum, isu
212, 213, 223
314, 315, 112
quod pro 100, 112, et C. ponitur 300, 212 etc et pr (300. 3, 24
isius P, quod pro n ponitur M, et  012 ac 013
Dalores ipsins
Valores ipsius Y tantum in eo differunt a valoribus
permutantur
psio
quod pro B ponitur l, et e12 ac 023
permutantur
primus valor ipsius as prs dispert a 
secundo (tertio) valore ipsus ejasdem quod pro
100, 112 ete ponitur 200, 212, ete (362 313, CH
et pro 5 ponitur 9. (h) et pro. 6 ponitur C, l
utum
H pro 5 ponitur 9,
eo
differunt a valoribus
20, 21
30, 31