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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 92r, page id 00068194:0183
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Ergo r= 8p es quad atque
et p 129 quadratus
seu ponendo p4 aequ. s. fiert r - 8f et r= 12f duo quadrase
certum est disserentie horum duorum quadralorum esse 4 (2
a O Y aequ  ergo Yr. Bsi 4 Pr - 1 s quadratus
Z-9-Tr- 12s
ut t ret s primi inter se
mn- nitn quadratus
amn 2m en quadratus
J Enn + man quadiatus
quadrutus
nn + i - n5 quadratus Ergo
Mhnn + m- v quadratus
jam
Emn +m au quad ratus
A+ B. aequ. amn.
efrmnanean quadratus
F 2noituit a quadratis
) inc demonstrationem ita absolvem
differentia
 duo quadratorum A et B a s midiem subtracti ex hibens
n
differentiam duorum quadratorim ani et qri
jam a A= B, seu
Amn. - 2m +2n - 4mn = 2m + en aequ 4n - am
rgo necesse est vel A+ B ex his quatuor A+B, A- B, 2n + m, 2n - m
posdam habere communem divisorem, nam in integris si eaeu
quantitas duobus modis diversis producatur, ut a per fg et per hl fiet
fg aequ hl. et A aequi f9 ergo fg dividi potest per l, ergo vel
saltem
f dividi potest per l. vel g. dividi potest per l. vel posito C. l compon
ex duobus in se ductis po seu habere divisores, f per unum g per alterum
ipsius factorem dividetur, ergo necessario A+ B et
ponendo ergo A+ B aequ. h. A - B aequ. l. 2n + 2m aequ. F. 2n - 2 m aeu
necesse est A (BC et A. eI A- B et 2n +2m non esse primo
item A - B et n = 2m, et pari jure A +B et 2n + 2m
et Aa B er 2n. Em hhabere divisorem communem
Ergo hi quatuor A+ B. A- B. En + 2m. 2n -2m. sunt habent
divisorem communem, ergo et hio .
+ A + B +A + B CAB, + 2n + 2 m
+ A = B
= A + B
sive A sive 2B
2
2.s5
sed eum utique alium quam 2
niam etiam 2A _ 2B
omnanem Duxo
amn -am +2n
concludimu:
 +m
Ergo. et
Amn +2t -si
+2n - 2m habent divisorem
sive hi quatuor
+2n +2m
an
n +m
n + m
commune
+2n- 2m habert divisore
communem
am pari jure habent
n -
n - m
n- m
divisorem com
habere divisorem
omnunem
habent diviso
Communem
2 habent divisi
cy his omnibus et quidem initro statim manifestum est ipsos
rem comm
n et n. neoessino habere divisorem communem contra hypothesiu Res
 e cutane
Em