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# Title: Leibniz-Handschriften zur Mathematik LH 35, 4, 4
# Folio 10v, page id 00068001:0018
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in priuntivis
Tianguli rectanguli
latera
latus impar
X
1 y
qDE
Xt Golsitat
atque par
Triaamuli redangili  y
secudi latera
latus par
2XV
20072
inzempr
lcuu
E
m e
2 mn
mn
Trianguli rectangulitertii n
 Latera:
Emm
m-n
me
n
3+V aequ X
W + N aequ. Z
Ergo Z aequ. m +n Ergo mn aqu. C0
Ergo Trianguli area ma quadratus.
et Z minor quam x advoue et minor quam
e Xey2
ato ergo triansulo rectangulo ninumeris, cujus area sit quadratus, dabitur
aliud striungulum rectantulum in numeris cujus area erit etiam quadratus
 hypotenusa autem minor priore; et sic in infinitum. quod in integris
est absurdum.
Ergus in integris impossibile est dari ud
rectangulum en numeris cujus area sit quadhatus. Hinc demonstratur
iem etiam impossibile esse in fractis.
L-3 = X -y X xty
∞2 12
E
S2
12
22 5+0
22
iuvemre tres numans 
invenire tres numeros integros quadra
progressionis anthmetrcae quoruque
sumun quadratus differentra quadratus
X equ.  st. _ X2 - y eq 5. yrego 82 51
Eip
3
25
1349
t 2E - h aequ 2 seu 252 aequ s2 + 3 seu  ponendo st aequ2 + β. haequ 5+)
fici (18) aequ (E2β5+ B + (3) 275 +V fier 5 aequ 3+ quod
et
impossibile est in integris. Nam: Za B e + 2
BL
3+V
2(3+2)
26V
β+Y. sed impossibile et 
sin 22 aequ. 6+)-
67
sed iumpossibile est (Y esse integrum nam B+Y est fractus nempe  +
 + ell. Nam (37 aqu Y - got. jam Y est fractus
le
q
2C
3+)
β+7
(+)
(3+)
1 1 V
B) e+) s est fractus incdncibilis. Ergo et inversus ejus.
demonstratu m ergo rest dios netres numeros quadratos integros
quiat
progressionis arithmetriae dari non posse. Quare nea
multominus dabuptur ties quadrati integri progressionis farthmetur, quorum
differentia sit quadratus quod tagen ad problema resoliendum
quod tamen postulabatur; ad habendum triangulum rectangulum, cujus
arta esset quadratus in numeris plegris
mpossibile est ergo dari striangulum rectangulum
in integris cujus, area sit quadratus a
 R a
x
5 5hc
4
X = h 2Y
G2
2
5=G
n
st
22
X+hV
cnca
2
52 y aequ h2. vaejuos = he
524y
ergo impossibile, est, quadrabo quadratum ess differentim
ductum quadratoruu summam
∞2+y aequ 2
5 Mua 2E aequ 22+ B2
S. Bt et aeque B
3 aequie 4Z
E (   C(x
45 aequ
l817
+6(
Hypotemis
sit a y in exy quadratus erit ste
yr aequ v
X + y sitt quadrusi Xtq k g X. y xaequ 32
X -G  Z WoY Y aequ. co
m
c
m+n
=X - X
D
C