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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 1
# Folio 293r, page id 00068194:0585
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X
Laura cinca rectum
y
2XY
Hypoteunsa
Demynstrandum est aream Trianguli rectum
aut in immens non sisgma
N + y Latera Trianguli Rectungali Rationalis C esse quadraetum
ypotenusa lateri uni fit Quadratum, si alteri quadrati duplum
Area Dl
Cubxaiatur
Si X. et y. primi inter se, et Xy D exy. sunt quadratus, erit tam X=y, quam Xy quadratus.
Ergo eddem modo X. y. X+ y. X - y totidem quadrati. Sro X et y ponamus b. et C. ,
erunt b + C et b -c quadrati.  et si b. atque C. primi inter se. Ergo b+c. et b -. eruat quadrati
Habemus ergo quadratos tres: b + c p 42.  b+ Cp, e. b- c p f2
Erge e 2C7 e f2. e 261 C + f2. Ergo b p 4 2efF e CB  2e5 4f
ergo + C et + f. Ergo c f+7 28 item b- c4 quad. Atqui jam fuit
 Spadf X y et b=  1) quade e f
Hoc amplius: e2 fC e f seu C2 f 4bc.. Ergo d f  2bC
et
FB+CFFD 2 d 4 4bc. 8, 1b7 4bc. Ergo eH bc ut supu
b  4 - c. Ergo d + C quadratus. et d - c. quadratus. Ergo 2d - e4 f quadratus g2.
et 2d 48 = fr quadvatus h. Eodem modo C d- b, Ergo d +b. et d - b quaerati.
Ergo 2d +e + fp et 2d = c= f quatat p.
adtendo
Ex primo dH 2g + C = f ex secundo dpu 2h e+ f Et 7dYG+ Zh
et aequando 29+ e- fr  2h - e + f. fiet: g + C h+ f
rursus dpy 21 =- e - f 1 2m + C + f
(d- d 10.  =9 4- f = h+ C2 f
Ergo 9r+e f+ 
Ergo de 1+  rupsus ex his 29 + e( fA, 21-  (fd). Ergu 5 + e P. Ergo St S 1
29.() 2m 6e7. Ergo g Y m+f
2
2hCC2 + f 2lF ef0 Ergo h + fp po
2h- e (5) ei 2m +C(Ff). seu h p m +C
Habemus ergo aequationes quatu novas: 9+e. fth BV. MAf r9. nterti
ex quibus sed exhis concludere, desideratum difficilius
Illud jam ex his pobatum habemus non posse tria latera Trianguli rectansuli esse
quadrate foret enim Xy quadratusE et X. et y. etion quadratus ergo ba-
Omme problematis cond propositi residuum in eo consistit, ut efficiatur in numeris haec
quadrats
aequatio: c+ pd. Sonimus autem C. et f. et d. binas quasibet esse timos inter se
Eodem modo 149 et 129 quaduli. item lF h et l - h. et g+ m. et g- m
et h + m et h - m. Ergo l⊓ n - g p p + 9. Ergo GH n = p eodem modo hps)
et mp s-F
Se nullum ex eris exitum vid eo
Videndum jam quonndo ratiocinari liceat, si X. et Y. pro numeiis sive noleguis sive fractis generalita
Sumamus: Xy CNy. quadratus. a latere Xy (3. fier X -y, XyB XF y(B
Fingendae quaestimes numeriae, earumque quaerenda per analysin solutio; ut invnire Triangularem
qui unitate auctus faciur quadrato quadratum. y + Y +1 p Z3. seu y +y + 2p, 274. ubi
Y)75. et Z H 2. sedid per analisin invenia hec opus hic labor est foi 5+ y + Z B824.7
et y +  H (oZ. 7. Ergo 822 7. quadntus. 6Z
P+Z. seu Z p ~ EV + 14
Ante omnia 2V + 14 aequetur quadato. sit v18+1. fier 29 + 4G(+2)  quadrato ab r. q. quod r BrB
pone H cr. fiet, 20, + 4erp r- gr. sive e2Cr + 4C, r = 8. et 4C + B
24 - , BV. jam debet 4d. 8 1 - 2C4
 sit quantila aegnerva, faciendo -CBd. fer 4d  +8
 et fiet 4d8) - 8dFDY 2d e - e
Ar vel ut
2d 4 rursus aequariquadrati