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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 44v, page id 00068200:0090
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C atd, aud
qdqui
a+ b. a - b
quad
quad
Problema upossibile
Invenire triangulum rectangulum in numeris, cujus area
sit numerus quadratus
in nam aris
duobus lateribus; impossibile est differentiam quadratorum ab illis aequari
summae quadratorum ab his.
d
Nimirum sunt duae aequationes ab aequ et et a+ b aequ e f. Eris
faequ. ab. per aequ. 1. et a +l aequ C- a b per aequ. 3. et 3. Ergo
ae +bC+ ab e
P. fi
loco ab vel et scribamus lmn. et a aqu. l. et b nqu. mn.
et elago Im, et faqu Unde patet necessario aliquem ex his duobus
a vel b. alicui ex his duobus C vel f esse condioisibilem
C
Hinc in equ. 2. pro a. b.  e. f. sunstitarendo valores fiet:
()
r
pelnar ncer
 Pono autem l. m. n. esse primos inter se.
Hinc divisa aequ. 7. per 1. patet nen+ n, id id est vel m+1 vel n
dividi posse per l, sed non D. Ergo m+, dividi potest per 12. Ergo M+ pB
a. b. c. d.
se
Rursus aequ. G. dividatur per m. patet l’+ n divi di posse per m. Euo L +ri p m
Ergo,
Denique dividendo per t n, debet lm- P, id est M = dividi per p
adeoque mr n +1
(10)
perdelerminationem 8 fervitur patet 1p l- m quod in integris
imposibile est, nam impossibile est aliu unitatem esse duorum integro
rum diffeartia majorem esse per n 9. sequitur (24n p  l1. Ergo
 npat seu n + par quod patet. m Ee p m n G. 10  C+ p ∞ ,
n + 1   alh  2hl +1 hifferentia quadritorm unitate ancta
m = 1 + 1 e quadato n 1
12 s m +1 p h 12
m + ut hp
J2+ mn +n- Vm B0
M.
9. I M- g. m.
fiat M eq. + () n
m+g.) non possunt simul esse
arttameti lae
quadrati, seu non dantur tres quadrati progressionis geometrice
quorum differentia sit quadratus..
nei in- g⊓ h
m+gB H. Ergo mej h  B kX Ge K
M 5 K2 +teo M. Ergo K - h est quodratus, d et K +h etiam.
Ergo si dantur quatuoq quadrati 9. M. 9. M. Mtg.
b+c b- 2
seu si datur
eritque
quad quad.
N-
N1
il
4
M- G+ aequ. quad HX datur K= h aequ. quad
K + h
FH4
et 2G2 k. H riph Fg
qu
Ergo quolibet quadaato in integissarretu
Eliut trimus quia M t
aeb ab +ac (b bC quad B+ 2bC +C) 1- 2). 2aC +2bC ) e 2, C, att divisib per Varb, tc, + Varb, 160
ab B - ac + b- bc quad
ab + ac = b - bc quad 3
ab - ac - b +bc quad a
Ergo LCF) 1b +c4 1b
----
.- -
brc - 5b-C F, b+c - b.c
72
quia l et n. sunt
pumi inter se.
a 
l
plimm inter
M
1
vec
G
1I. 15 +15.
u S2y+ Sc St