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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2c
# Folio 138r, page id 00068199:0275
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Quaeritur Sdx xx mm, nx. nns quadratura ratinali. C facere posimus ex um = B0D et xx. un - D 84 poterit
radix extrahi ex Axp- um, PX -nn) = ODQ. Quia radix extrahenda ex Xtm, X- n, XAn, X. n, genralitatis causa faciamus
rudicem extrahi debere ex X+ L, XtM, X+M, X+Co. Ad sint O et Q simplices ut..  +. ; sed X et D duplices, uti th6. 
faciams Xr = XFt, 1st = eid Xtm, B = D- hB X faciamus XtM = β,F B0 X; β B X+ m B= O
X+L, : β= _Y C_ Xtm, β = 07
XI  QN Xrn   Qs X+c, V = 22.  ex xel, Xtm, Xtn, XtCo,
sit e = b+ Atb siet 3 gt = bit + +h6 = Cod pst +9 +6
Xs XGl it afiGS= C, est 1 d96 + (cgt s) HtS ubi noto par ex literis ajumtitis unum videri inutile, nam si fecisemus
O c+dNS X 5A+916
2Zh I + H 1b. Q=
() = sccq F td YddB, et postea faceremus cch = t et ddB = S, macaeterae, f, g, hiK, p, 4, b, statum suum retinere possent
fiet 17 X5234E
detineamu, tamen has literas vel distinctioris calculi causa, eventus enim quid in iis superfluum docebit.
 Ex aeq. 1. per 6, 4, 8, fit lB + bYG6 ~ CfZ+ d96 + (C9 + d f) Y5, seu (11 mo)   = B cft + Bdus - l
et (12 mo) b = β (c9+ d5) Ex aeq. 2 per 7, 9 fit 24 + Bm + Bb 16 = ChEFdKB + CCK+ dh) 14B seu f
Ex aeq. 3 per 9 et 10 sit  +n + bYL6 ps  19961 (9+ 9() 1t6
unde (i5t0) Q = ypft +1498 -n et 16to) b = V eeg +45) Tandem ex aeq. 4 per 9 et 10, fit 18 +) cox bLBa PhiL +  B 14 9(XGB
(3t) 8 = L cht. +  dk6 - m et (14 to) b = B CCK + dh).
conferamus jau valores
xtam ipsius b, qui sunt mere constantes, ex aegq. 12, 14, 16, 18, quam ipsius  ex c99. , 13, 1s, 17.. Et quidem
sumamus primum conferamus primum omnes valores ipsius b quiape cum his insint merae constam
unde tste) (17 mo) Y =   eatue + J. gk 6 = a et (1Bvo b = = (PK + gh)
ex 12 et 14 fit (3( (cg+d s) = (19no) CK + dh. ex 14 et 1b fit Ex a9. 16 et 18 fit (20mo) YV (p9 +qF) = BK +Dh. Ex
20, 21 tollamus literas d et, q. Ex aequ. 19 fit d = C, eBg-X: h- BBf
 posea substimendi etien in aeq. 2t. t
sergamus nunc ad valores ipsius 4, ex ae49. 11, 13, 19, 17. Ex 1 et 13 fit (251) 8 = (βh) Ct dm:, K - P(9, 0
aequ. 12 et 16 fit B3 (c9+d f) = V (p9+qF. ope harum trigt
Ex aq. 20 fit (23tio)  P, WV9 -K, : h - )VF = 9. qu duo
et ex 1) et 15, fis (2 to) _8 A pis  B(h. ets)ChF C BCm n),., (3g.  ex 15 et 17 sit  E6to) C
aeq. (24tu) p: C - . . . . Adero
VV9)4. Denque ex 13et 17 fit aequ. 11 et 15 fit  (27 mo) 6=   (c. 19) f. ((- n); (3. P)
cncodentiai cum itio, sed brevi aliuud provit s aequentur inter se et fit
siprimt 2dus  coincidentientur interse, et ob constantes fit (28V0) g. d - (H BCg: K 1 9) V. B (2 co, lum)
=(99 f -h) . - 7 (n - Co),: x (F
reg. K, pss ): (Bes h, VVg-K) patet
oiants fit 2tahic srgnot ses t pnt itaque hec equ una collatio vaoum ipsius e nit producit novi
princt et secundus valor ipsins 8 aequ
sulstituam valorem ex aequ. 1t, f  qui est h. BB: C egK
4 cq. D K eeo
fieri S- (BBS A) et  B( ritr k BBg C (BCD K h es) seu fiit 6 = qu Bes- 1) C- B(P5t) (l- m,:, C, que lg
efiau, ex arp i3 sit d. P - Et -h
etiam hine si in ratore F  1s aequationis 25
per 2, f et 9 per 3, h et R per 4, faciemus BB9-
= 1ZCt - β(Lm): - 140
et 1 5 = 23 pts - V (n -co),: = 249. et pro d et 9 ponendo valores sit 8 = 13Ct - (3 (la m), : - 14 (.. 14: 13)C feu 6 = 13. 13 ct - 13(3 (. m). 149c
et f = 2ypZ - V (a- co) :   s - 24 (. 24: 13 p seu 5 = 23. 23pt - 23) (n. c0): 24 24p. quos duos valores ipsius 8 ab a d et 4
netinso per l3 3. C e. 13
liberatos inter se coincidentiando, ob variantes fiet 13. 13: 14. 14 = 23. 23: 24: 24 seu 17. 2 23: 24. = + 13. 14. Ante explicationem
13C: = 140.
aeque. 25. 26) unde aeprndo fi
p. - 249, seu 13:14 13. 24: 1423 = dp: ca
 qu B . unde tanerm qu 13. 2t = qu. 14. 23 seu
  u s  h) pt F (3 B- 1) Cnaco) i qu (VV9. K). P. Et  Contidentiturt veores 2b, 12
: 1
valori
+ (oBF-h, in ((3g- k
)DCC. m): e X qu (eBBH)C et ex 
obque ipstis  
compendio, B,
E 1 B(L. m): qua+ 14. 14C = 23V (n - c0): 424. 29 seu β((- m): 14c = + V(n- c0): 24P, seu p: C =
23: 24 = +  - 13 + 13: 14 ut ante
ecundum variantes fit (30 mo)  Vyf - h, : VVg- H = +, βBf- h, :, 3(39.k
+, (V: B) (n - Co, :, l - m) (14: 24) seu tandem simo) p: c = +, (V: () (n - C5, :, l- m) (+ 13: 3) seu tandem
Etiam ex 23 fit (32do) g = + p, B3G -k, h- βf
proportionales, quodu
(c: p) D foret = C sed si quod (in quam nollem
se si + ~ + foresit C et Q
et ex aeq. 26 fit
fiec (31 mo) p: C = (V: B) (n. co, :  l- m) (βBf - h, :, Bβ39- H o
ed si f ~. - fiet G = p sq - (pd: C) 15 et (C: p) Z = cst - d6, quod placet. Ergo  essu pro habeat
unde ex 32 et 22 fit g2 β33tio). q: D = _ A  d:c si ic
tollem. Nam C C posito O = C Ih+ de  fietit Q = p sB + (pd: C) 16 unde
24,  sit et ad aeq. 27. tollendo q ex aq. 21, fiet ex ea
c(4: 13)f = Ypa R + ps (4: 13)), seu
YP, :. A3C = 13g-14 f,: 13g + 14 f
sane si esset 139 = 14f foret 
fg f. g = 13.14 = . lid s  in toret g = si la . 11 5, d: C) 5 et foret s = 5 1G - d: c)FB et (C.  B =  ct- d6
21 (unde fit Xqu Z ~ aq
4p5) unde tandem 
+ sit _ p: c = (1: V). (139 - 24f, :, 139 + 145) seu
5)F): CePs - h)9 = (BB9-K) F
qui valor comparandus valori
Ergo B+t) = sit =
3. C13cg - 14c5) =) (13p
=e prasit 
Unde rursi ne sit)
cincidentiari debet alius valor veluti repertus aqu. 2. Sed pricus
=(C :p) Z per priora) quod non placet. Praestat ergs
eu 35to) p: C = 3: V, Kf -hg, :  (Bβf -usg 4 (Beg- X)f
S- (BC - D PHG - (L - n), : (- Y P (24. 25 + Bc (+. 13)9 se
bto) S = (Bc - y pi fH. - (1 - n1, :, (+ 3 p+ BC_, 14: 13)  qui valor coincidentiari debet valori 28; qui erat _
l- n, 14. +C = _3  (1- m) 3 L  pF BC, 14: 13
m- l) :, 47 3Cl-n
q
Restat aequ. 27. Cujus valori ipsique e
ex aequ. 27 tollamus d, 41 R. fut ex pti e
Calculo dubiu contra es homogenestm, nam l, et BBL, non sunt homagenea, quia 14 continet e
13H P +704 BC 7
(3(3gm (329933+
Ex varventitus comparando _q. 36 et 28 fit
g. l-m, =Y + C. pis p. c a R
(3C- V p)814f = 139C + )p+ BC) seu (3  - 3(gVp Kf BC + Kfpp 
est p L(BBg
BBg (m - n) = K (1 - n),: = VBC L- m)
B  1)9p  14Vfp = 14f fB 1329C - 145 pC et sit p: c = 3g (9 v) 13 g (, 139 - 115, :  1yg
quam aeq. compar ido cum aeq. 35 fieret 139. 4 14f = = 139 -145 seu et foret posito + 2 = feret
seu 139 = - 145
seu (37mo) su p: C =
139 + 12f 139 + 14f = - 139 - 14f seu. 2 139+214 f -0
compurando cum eo ex qua per explicutionem facta H aeq. 35 fieret eliam 13-145 =0
non consistit, nisi fiat 14 =0 et 13 = 0, ande elim foret 23 =0 at 24 =0 sed haec non licent