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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 52r, page id 00068201:0103
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quaeritur extractio Radicis quad inticae ex gytab, yytac, sit
sit y = 1D+  seffict yy = 4619D + 20 10D, et yy+ ab, in yytac, b dabit
4, qu 0+ 5) +40D + 4 (9, D) 40. quod extrahibile reddendum est, quod multiplicatur per EG
si faciamus e=_ constanti, et extrahendum identificemus quadrato a  + 4 +0D, fiet
+
(b+C) (O+ D + bc + 2 (b+C)
1..   e      al   aa
e fi
= q = 2a (0 + D) + a Cbtc): 9 sit 0+D =X et btC = c et liet t rax + aC
Aa
cuper est qu (9+7) + (b+C) (et D)  + 4 C. B + bc = aa YG +9905, vel  reductis bumbus ad X,
fiet aa XX + aa PX + 497 + bc = 4 a XX + 4 ex +  ex +990 seu
a Xx + CX et + Aqb. N=(DD= osa co)nn + seo iuaqu n naaeu identificando fit
 10. ua
Ex X fint quuto (00 = 10) aa seu 112 00 ie t _  =9 seu 10 L = do ra 1).
aa
qquiaa
seu aasju - qua) XX + aa (99 - 4 na) ex + aabcqque = (99- 4 aa) 99 0) +  a ee
aa XX + aa ex + elD bc 9q - aaee pa = qqD et exeplicando B8, ciet
99- 4aa
910 (00 -18)XX 100 10) B1 ax = u. uaa
. 1.
a  F ex X fiet 1199 A e     eto et ex X fit 11. May = bi 99 - aaée
00- 2.10
Eggo
4aa-99
Ergo 1. 1 9t =  a e  = bc 99- aaee qq quos duos valores aequando, fiet:
ca- 19
00 - 4. 00. 10 +. 4. 10. 10
loo ta ee ( a ee qu) = 00 bc  00 a ee 40o 10be ∞t, 4 00.10 a eeque + 4. 10 vbeg. 2udaeg
seu qu(qu (00 -2. 10) bc g + 4. 10 (00- 10) ad eequs = +00 a ee
00volaa, quem valorem substituendo in aequatione novissima
fiet qu (00 - 210) bc gt + 4. 00. 00  e) = (2. 00 a ec) et fiet 00 = 2. 10 seu 10 = 
ae
superest ex X, a 49= 10 (00- 10)
et 00 -10 = 1 et 10 (00 - 10) = = et 49 = 4aa, sed ita 0. B 1199 = 90. 2,10 fieret
ae = 0 siu  C = -b_ - b. sed hinc ob n. n44 = bc 9q- aa ee, :, 4aa - 99, fieret et beg4 -0
Haec via ergo exitum non habet, iqq sane si successisset
etim nume rationalis extractio saltem quoad variantes ex yytab, gytac
identifictnas
quod non licet.
Itaque quod supra extrahibile facere volueramus aequemus quadrato a 91aq+ r sD
haberi potuisset, neglecto scilicet 8
et fiet  qV quod est 990G + rY DG + 297910D fiet G. Q-2aa ( )  raax aa e
et XX + ex +4OD + bc = AQ (49O+ n5) = qqq (o rua) raax 1 a= ) + H( (ex 16) X ua ) 
10. (9q W)  (t 2XX + 2. 1 a X + 11aC)
30do
qu.
0ve
+aa)  sentqrXX + qr ex  F bCqr 
Ex Xifiet
+49r 0B.1o XX +toB. 11 agrX  11. 1Iaa gr
-99r. 10. 10 Xx =8. 1B. I agrX
+ 10mC X
+_ (BB. raaXy + 00 ruVer) 00
4700. 00 grC + (rr. 99) vo. 10 C = 00. 00 rrase
2 (yq- rr) 40 a 400 agr + 8. 10 aqY
Plali
00 (99. 1r) ae)
Ep X fit g. p. 1 aagu + (9. rr ) aeit = 00. 0b bc gV. sou 1.  + oocqpor tae. 11 +  qu ( quaqu.
4ua qu
=  4 qu (00(99- rr) ae  0o bc q seu 11 + 00sq9 - 1r) ae = L-
ouqgae- vorracrquoa
Diagr
Gaa 1Y
Jon (q’anee _ 2. evogr ee + orrt aa ex L 4. cro dabc g9rr).
Aaa q1
4 aagr
Ex 8 est 10. 10 -+ 2. (9. 1) - 4gr + qu (14 11 - 2 1) 99- 1r - Zgr.) + 1699rr)
= 90
daaqus
Y(4 q F r1 _ 299ur = 493 r + 4 9r7 + 4 qqur . 69gri)  YY - 44+ 24
ex
si esset e = 0, fieret 11 =0 dbc : a et c, 49 - 12, : qr  = 2 s00 - 2. 10)  seu 10 - q
seu 10 =
00squs - oorr et 10 = 2.0 qr vo qu 00, 29r + rr - 94, : 49r
compendii causa sit q9. rr- at. et 00.
et qu
= 2at. 10, 05 -00. 00: 2
Ex X fit o  00. 00 gr + 497. 00. 10 - 49r. 12   1010-
seu X 00. 00 gr 42 (291 - at) 10 = 4 gr 10.10 = 0 seu 00. 00 gr + 2. 10. 10 gr = 4. 10109
seu 10 =  Y(1.2) = 29r + rr - 99, : gr. seu re  + (2  N(1. 2)) qY + (4 -YC. l. 2)
10 - 2gr = a t, : Tgr
H 2) qu
et fiet Y = L qt (∞ 15 - 4 d -2) -. Z q  C2 Y -  XY +  2  sit 2 - + ( 2) = Bi a et fiet . 2
sit gq. rr = dat, fie
Yr + (P: a) rq al epi aa) - d 1( (na 4 BB + 1, 49 - BB + 2aaq fiet rt P g - q B+ 4aa) seu
eB aaa = taa = l - 2  1) + raa
4a
si P =0 erit ex X + 19) e = bb = Q (990 + 1r) =(10X F ua( 99- Wr) + vOrr X, 2a2 X, quia B - 2 aa x et C = 10 x tua et D2  1
nec riq (β 42) a β,: 2a
Si e = 0 = bac sit c = - b et bc = - bb et  1 = b 1. 1
= (1. Haa. unde XX = 10 (00- 10) XX +400,  8. 10. 1. ax 1 =- 4. 11. 10 
tooaar xex
tut  sed nuitus ex X, lit 60: 00 gr + 4 100. 10 gr - 4. 10. 10 4r = (2. 00. B aD + 2. 00. b0 rY, et fit ov. 00 (gr. 211) = 4. 10. 10 - 8.10. 10 41
=4. 10 = 00. at: qr. seu 10 = 2. 00 gr - 00 a t,: 4 qu, sed munjus reex x fit
=a. 10. 10 qr = 00. 00 Cqr. 2n) seu  1.  = 00. r
Gorr 2 agrt + aaH:: 1649 rr seu 89Y - 499 ur -a 0ssr - 40 r + 49 12 4 + P- 299r1
e
41 +
00. 10 a
+ 1 = 0
46r 47 + = 12 99t b
seu OD - 10 (00 - 10)XX +(00 - 10) 11AX = 10. H a
tur sii -erb: a) sE 1. 2) etipris
 aliaie conmittenat et i  et . it  a uterintut mnm at acripeesse
is tum patullarum omprotendat etrae an letatie i doieis deis est putaria nels oam ateripaesse
- b r -2 4
10.10 = 47 00. 00, 2rr - gr,: qr00. 00
seu t + p - 1293r + 49r- 699 rr =0 seu Y