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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 3 (shelfmark LH 13, 3)
# Folio 122r, page id 00068200:0245
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Sipiora haud dabic succedo ut; videamus tantum nan brevius sic.
t
a sit unus numerus. deterhatio a quodrate, X, fiet ab eo detrahatur quadiatum
X, fiet al = X numerus 2d9., a quo si detrahatur rursus quadratum, fiet
al = X - 30 numerus otius, atque is iterum detrahatur a prumo al
fiet al - al f X + Z2
tan quaeruntur tantum duo quadrati, quorum summa sit quadratus.
seu X + ZY  (, W Ut ergo radix extrahi possit ex X +.
fiat  y + b 5+ X, et X + C,, 7 unde V + 2by + b
seu X + Z1 F)
3
+ 27C +c
+ 2C + c
fietque X  27C + c et pro Oponendo X + d,
2
fiet X  2ZC + X + 2DX + d. Ergo (ZX + 7Zd + 3d0
s Unde patet obiter, quomodo duorum quadrato rum summa fiat quadratus, nam
si q sit A 8, erit Nam si datus sit X, erit Z e - + iideoque
ponendo et dopro artitie, seua et minorem f majorem d t
tivum, A seu ( - 9. et g majorem 2X, ponendo -23, et 2x -917
fiet
d
idque exemplo probemus.
eus
 17 +
2X 2g
+
Esto X e 8. ponatur g 1,  10 erit d5 do, et, 2d p800, 
et = 2d2  + 800, ergo Z p 800 200.
cujus quadratum 40, 00b. ergo 40064erit etiam quadcatum
Si verus
ponatur g F) 10, et v d 10, et Fd r, 100, et
 - 100s, et Z 1 = 100 17 150.
jam
n
2500 + 64, seu 2564, debet esse D,
x 54 Erravi in calculo est enim 2ZX + ZdX,
Hue
sive -d
2712
+ 2X + 2∞ d
a
27c + 20+ d e0, 8. pro
C, ponendo X + d fiet 27X + 27d + 2d X + d ,0
7 = 2d∞ - d
et ut signum negativum evanescat, ponemus:
-adx d2 2d 5 - 29., et g pex
jam ergo esto Z 0 X ( 8. et d B - g ⊓ - 20. erit
vel 64
+160 - 400-40
+ 16 = 40 17- 24
575+35 1 60+
5 5 Jam 25, 64
o
N= Z7X +2dX 1D X  , Se 1 _ qL 2ZC + C H Ergo X H 2ZC +C. ponendo
CCe X A d fiet ex ( 2ZX +- 27d + X - = Xd +d 3E
et a Ergo ponendo X esse cognitam fiet Z e C- 
20
sit penim quadratus aliquis, 16  et alius eo major, 36 pC, erit Z17 2015
jam 25 + 16.
id
14
16
ac 
t
 Eraf 1E4
a
1Z, et 46 p, X et Z + X F
1
169
13 13
12 3
utergo uniora resuamus quie Z ep C + X..
erit yu pC - 2CX + X
ergo ut habeamus propositum, id scilicet  ex tribus num eris duorum
quorum libet in tervallum sit quadratus, erit primus a. pro urbi trro,
secundus al - X, sumto X rursus pro arbitrio erit tentius al X-
erit tertius al - x A = C +2CXX et habebi; intentum.
sumita C, rursus pro arbitrio et patet.
4
differentias duorum quo rumbt
se quadcatus. Num estilcicudum, ut et difflrentie quadhatoru snt quadrats
Eodem modo Z
Ergo -d-2
1d
Ergo X
si
 X
d = 2d7
+d
d