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# Title: Leibniz-Handschriften zur Mathematik LH 35, 15, 5
# Folio 87v, page id 00068207:0174
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sit Lia, XFL, XEm , XIn _) = ay quaeritur asy ax quadratura rationali
Quaeritur modus efficiendi ut factum ex X+L, X+m, X+n sit quadratice extrahibile
ex prae
 si AB, AC, BC, ducantur invicem produci Quadratum AABBCC
Manifestum est
fiat ergo. a,X HI = AB aixtm = AC, A _ aXtn = BC X = AB - at = AC - am = BC - a
Videndum quomodo assumi possint A, B, C
B = 21 SaO + 221a
Tentemus facere A = 11 s( + 12 fa D
Erit FAB = +11. 21 aC. + 11. 22 a SOD +AC - +11. 31aC + 11. 32a SOD
=ab +12. 32 a D + 12. 31 11
C = 31YaO + 32 VaO
+BC = +21. 31 a C + 21. 32a16)
=an +22. 32 a D +22. 31 .- 
-00. 00 an
00.00 am
+12. 22 a5 + 12. 21 4a0 in
Coincidentietur primus valor cum secundo, et tertius etiam secundo, irrationales interse, et rationales inter se
2 Ex secunda +21. 32 ) + 11. 32. 
+22. 31 + 12. 31
+12. 21 +n. 31
(2) 2. 11. 32t 2. 1231- 21 (12+ 52)
= al
00.00 al
Ex aequ. 1 et 2 fit 11. 22 = 21. 32 seu 22 =21 (12 - 32, :, 31 - 11)
Rursus a)q. et 2 eas in unum addendo fit +11. 22 + 2132 = 2. 11. 32 ande 22
12. 31
12. 21 7 22.3
+12. 21 4 22, 31
Et aequando duos valores ipsius 22 fit 21, 12 -32, 11+ 31 = 2. 1. 32 (131 7 - 215+1275+317
3252 - 111
D.
11. 32
+12ZC+ar, s+n. s + 31
fiet 21 = t12 31S1- 115 =, - 32t13i5 +27325- 1
11
2fF11. 325+317
- 12. 31)
tn. 3F2- 31.
Ex coincidentiatione prima
in aeq. 3
= 12. 32..
Secunda +21. 31 aO + 22. 32 a D 40000a C
I1. 31
Et et aequio per 2., et ita aequentur inter se
=HI. 31
- 12. 32
 2125 +217O + 00.00 h- 1t
eequentur Termini absoluti fit 22 B. 12
ol. 31 52 = 12
fieri 21() 1l3Cl- n). si teata
coinsedentiatio
 at 1  231912. 32 (- 17 ) l. n)
si tentaremus 2A AY 3+d aY feret 22 8
32s - 12Y
 et debet pari jure
12. 32
l. p. 72 et nr. - 11. 12. 31
sen 32= 2t HV FAV, . H. ande (ir. B. 31 3 13. 31. 3dCIB. (. H_L 11. 513rL
31 A = IIV
271EIaD
et evanescente  fiet aequatio nova
132s ~ +21(+22212310 400005+2712V
- 12.32
11 + 31
3  g1. 32 -u. 32
Lu32ZCM2L. SFII.32
et erit 21  +11. 31, 4VII. P + 11.32] quem Valorem subsitundo 22 = 41. 31 3l- 325
fiet
ex coincidentiationibus rationalium ponatur m- 1 A et pL n = V
0
2-12. 31
12.71 - 11 12FII31
(12. 1 = W I)3: H + 1 +2 124- EL. I1. 12V
Ne qui valores 21, e 22 sunt prioribus aequandi, nec puto permissum ut aliquid pro arbitrio
seu 2. H B. 3t _ = 31. 31 (12+ a) + (12 11) r (FF1)SIB + ( c V2
asumamus, quia ex coefficientibus 1, 12, 21, 22, 3, est duae aliquae tnutiliter quo ad b determinationem assumuntur
cum absorberi possent ab  et di
a0. V. 31 + 3. s1 .
8.32V Fo 71
Sic ergo (alculum resimamus A = 00 da  + 00 laD
ax = AB - al AC = am BC - an m l=
Lam est
391.II 21+32 = 31 +32 = ov. 315 + 00. 32X
0∞uf. 31  00V-321
B= 21 5ao + 2 la D C = 31 la O+ 32 +a
00 s unitas A V v - 5 = X
+99+106. 24 +00. 22  9D 40051 a F00. 32aSOD ~ + 21. 314O + 21. 32 a L0D
es
= +00, 22 aD +00. 21 a 4) = +00. 32 45) +00 31 a Y() D = +2. 32 a5) +22. 31 a YC)
22= 00 72 (31- 00), 3132
31200). 31-34
Ex prima identificatione irrationalium, fiet  21 + 22 = 31 + 32 Ex secunda fit 21. 32 + 22. 31200 (31 + 32)
- v0. 00 a R
-00100 ab
-00.0 am
Ex 1 et 2 fit 21. 32 +22 31 = 00 (21 +22). Ergo 22: 21 ~00 - 32, :, 31 - 00 Et ex 1rt 4 fit 3t +32 31. 3 fit
21+ 32 - 21, : 21 = 00 - 32; 31 -00. seu 31. 31 + 224 - 21. 21 _ - 0c 31 -ogg + ee 21 = 30 21 s 3221
seu 2t seu 31+32 -1 = 00- seu 31432 = 0s-3 + 31 - 00 ac denu 21 2 31 - 00, 31432:, 31 13231 32
-2,32+ 31,:, 32831. Ita enim
0
31-00
21 et 22, differcent solum permutando notas 1 et 2 dextras, ut opoatet
Ex prima identificatione rationalium, fit 21O + 22D, A 231O + 32D Exsecunda 21. 31 D, r2. 32D  o 00 310+00. nD
21. 31] +6000Y,: 100.32 quos vatores
=n. 21
Hos valores coincidentiando fit, ex constrantibus A: Y = ∞ E XC   00. 32 A - 21. 32 X = 00 321 - 00. 22V 
et  per 9; D 1270 + 00 A, :, 32. 22.  (per 10)- 00:
(13)  10.  00. 32 FR:, 22 = 00. 32 (r - 1):, or 32A seu 23 00. 32H:, ooV - 32 X
tiq
3 - 21. 22.32 421. 31. 32 00- 31. 327
EEd
Siut 31 et caleg
60. 31. 31 (+22. 31. 32 = 21. 22. 31 (+ ∞
eu 1021. 72 4 ∞3231 00. 31, 32 00. 22. 3, 15 21. 22 (32 - 31) fer 21 00. 32, 2t :, 00.  21. 31 f 21. 32
Ex C pro C variante ( tollenda fit +00]5 +21]32 =-
abe filfev. 31i3 2t. 3
13 e ad 2 f 209 213 =60 (31 +32 + 21. 22 (33-31, et
Bu Etpo C variante tullenda fit in quae qua aequatione ut et n2 servanter inter se legem justitiae, seu
d-. a(e - 2 31=C
0) f (22. 00) 325) + 0000. 00V
=(31 - 21) 00V, seu  21 = 0031 (V∞A) :, 00Y - 31X
duos t sulore tq 15, aequando sit do
M2- 00. dor 4003 = 22 (32
seu 2 00. 31T: ,100V - 312. sid idem debet prodire ex 14 et 15. Nan ebe 15 es Videamu  examen
unde si
accut
30 21. 31.92D
e
Ex 9 fiet d2 (22 - 32) D + 00X1 :, 31 -21 
Et ex rationalibus sit mdentificasis, fit (t ) 31X
equa
60.00. 21. 3cV = 00. 21. 32 32 = 00. 5t. 3 ∞. 90.5  H (100, 21 31 5) Etutle
frei (00 -3)2 (00+21) 3FN
Ex 15est 22= 00. 21. 32:, 00.3] 71. 7. 31 t ( 32 = ( 1) 31 + 32f2t, et fie
+
re
1- 21.3.
00. 21 32 = 00 3131, 0931. 32 - 00. 31. 21 ( 1 31  51. 31. 37 f rl. 31 3 ((21. 32. 3( 1 31. 32
60. 31 (32 - 1
fiet (00 3c)Y t
00. 2IV = 32. IY + Z1. 3IH = 0031 T. fil 21 = ( 31H. turt 31V
(X =
22- Fd, 1
32.91