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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 4v, page id 00068196:0008
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(x +b, ∞+C, X+ F)
quaenitur co sydx posito y (, arf. X19). Hoc optima praestabitur
si formula sub vinculo fiat extrahibilis, etiam si radix ipsa effet constaret ex membris
irrationalibus, sed, ubi v sub vinculo non nisi binae variantes quantitates
in se invicem incerentur, ita enim quadratura quaesita reduci posset ad ratio–
aatem, quia omnia hujusmodi membra quadraturam rationalem 
aecipiunt. Hanc extrahibilitatem obtinebimus si quibtis ex tribus
factoribus, x +b, X+ C, X + D,  explicato ut oportet
X reddatur
extrahibilis. Et sane si ha factores sint primi inter se, id ipsum fieri oportet
ut productus ex ipsis sit quadralus. Neque id tantum in num numeris ratioL
nalibus, sed et in reductione ad tretio quam intendimus ad irrationales
simpliciores locum habet. Sit ergo Bto) ax= yrongiv ras
sit (2 ax F ab = (4t0) (I (112 Lvy + 113 AVZ + 12 3 tyz + P
ax + ac = (5to)  (212 foy + 213 10z + 223 syZ 1q)
ax + a = (61) () (312 Nvy +313 EyZ + 323 yZ +. V)
siet r o +2112p NDy +21SBZ
2fisvus 4rgsoZ Achsyz taK
 =  F 100. 12 
113. 123Z
400. E
213. 223Z
r
1313. 2Z
=300. 312V
313. 3237
+2. np Syz
Ct2A3, 13 Z.. . Fr. 12. 1237 ... t2. M2. 13∞. . . .
()100. 113p
U2. 123 hy
() 200 21
212. 213 y
2300
312. 323y
f100. nP
112. 1131V
3. 313
vy
=II5 VZ
100 pp
uit 
Bre vy
+ 112 Vy
Habemus duodecim aequationes quibas est salis faciendum  Et primu
habentur valores f, g, h K. Deinde primus valor ipsiis f aequetur
secundo, et ut tollatur Z ut habeatur Z, et Entius itidem aequetur secundo,
ut habeatur Z. Et aequando duos valores ipsius Z, habebitur aequatio in qua supere unt
P, q, r. _ similiter talis aequatio habebitur ex valoribus ipsius g, et ex valoribus
ipsius h. Habemus tres aequationes pro ipsis p, 4,, sed ubi sublatis daabus tertia
evanescit, neluti 9 itaque jam  omnes  ZG, , Z, , v, yiZ P, r habebum
tur per q. Idque optime si in valoribus  aequatione ex primo et 3tioualore
ipsius f, et primo et 3tio  ipsius , substitutio inventorum et ipsius h, substi-
tutio inventorum fiat. Valores jam in ubi opus em prodibunt tres aequationes
in quibus merae determinatae. Valores inventi substituantur in val
u supersunt jam duae aequationes aequando  valorem primum et ulfimum
cum medio, in his subssi aequationibus substituendo valores inventos
9,Z, P, Y, habebitur aequatio, in qua sola supererit in determinata q
m u H Cujus destruendo dignitatibus 44, 4, 4 4, 9, g. et destrutudo priores
sn (i9no)
habebimus binas aequationes novas merum indeterminatarum ac habetur quaesitum
xequamur
Ft q - 100. 12 p - 200 2129, :, 217. 123 - 113. 13 = 300. MEY. 200. 219, :, 25. 2, 313. 325
itio
Ex 4 et 5 itim 6 et 5 fit (15to)
(21 mo)
Ex 7 et 8 ilem 9 et 8 fit (17md)
Exaequ. 14 fiet 4100. 12. 213. 223 p
-100. 112. 313. 323.
D00. 23r. 200. 2139., 212. 23. 3103 23
V = 100. 123p - 200. 2139, :, 212. 213 - 112. 113 = 300. 3137 -2002339: 212213-31.313
+ 113. 123. 300. 312 Y = F113, 123, 200. 212
= 213. 223. 300. 312.
= 213, 223
212. 222, 200. 212
∞200. 42. 313.313
in 1n 1uo n9 (+ 200. 211. 215 213)
 1 213. 2232100. 1Mp 9 +113. 1232300. 12 V=-
Eodem modo ex hac aequatione, permutando n et. 13 fiat
313, 323
Siu
L112. 1273Y300. 313Y =- 1.33
+212. 273 2100. 11p
opposita
uae postr
+ 212. 2137100. 13B + 112. 1137 300. 3137 - +112. 113 2200. 2139
ovis lutirim
Aenembro quod semper
trificum est, tres notae
priores, et tresquoque
notae posteriores
Et rursus ex hac aequatione roma, permutandooB et . 23 fint
222
repertuntur
antitates = 212. 213)
vocemus eas  et c0
P. g et r. q, qu tollendas ope trium aeg9. 19, 20, 2Y
Compendio ergo habemus duas   +22coa  31+32C0 = 30a, hinc erit
212. 313
31211 = 30. 219 - 32. 2100 = 36. 20a = 31. 220
uernapiva =0
siet Cd = (10. 21 - 11. 20) a:, 12. 21 = 1. 22 = (30. 21-31. 20) a: 32. 29 - 31. 22. Et multiplicando per crucem
1.p = 10. a - 1 = n. 20a -11.22c0
16.71. 26. 4- 10. 31. 25. 22. = 11. 32. 20. 21 (+ 11. 31. 20. 2- 12. 30. 21. 3 - 1.39 4. - 11. p. 9neti19. 20. 22
10. 22. 31 ∞11. 20. 22 21. 27. 36  12. r31 _12, 21. 30 =0 ubi sigu
us Reione ouenim
sifint H3. 123~ 315. 723
113 13~ 31213Bipsiu
fit m
Erg 313 + H
um
ibu 2
fil nt
1 f300 m
rn
(0 mo)