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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 241r, page id 00068201:0481
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pars. rdams schedrasimatis. de extractionibus radicum pars 2da septb 1674C
etsi non satis universalia ubiter, s et credam abias praeterea vias superesse:
stonro ante omna
si chimomions aliquod in se multipletur
va
debin
erum radicos
sive a + 3 aYbC + habe + b  bc, sive a + habe +3a + bc sbC, qquorum duorum nommum, Quadrata
sunt: ab +bc + 9a b  et 9a bc +6 a b c + bc quorum differentiae sunt:
a - 3a bc + 2 ab7 = b C qui esst cubus ab a = bc. Genemliter Clapus
et smpliius res ita euum tiaei potest: d + e cubus pi dd + Bdie + M + C erit d’ +3de, H
3de + eH rubo ab d - e., suc d + 3C, d, H  - 3d + e,  C, B, p cubo a d e
vel d Theorema est per elegans, sed qua arte sit moculum, id est quod nosse velim Nisi
id forte per ind inductionem scontiga, observatione in numeris facta, quod sane saepe ad thecrem ata
invenienda commodissimum est. Aeud Theorema: Binomium multiplicatum per differentiam quadcatorum
partium producae aliud binomium: in quo differentia quadratorum partium sie numerus ubus ianultiplicetur
a + soc per (4) a 4) bc fier: (4 a (4Ibca + () a ) bc, bc Quorum partium quadrata a se invicem
subducta relinquunt  a- 2ab +bCa - bc 2a bc  - b2 sive : a - 3a bc + 3a bc - bE
Elegantissimuta est hoc quoue theorema. Esto jam binomium 2 10+13800SFE ex qua
est cubica; 2249 + sur 53800SI2. , hatius hoc nullum continet fractum, et alterum partem habet
rationalem, secundum artic. 1. et 2. Sed et squadcatum pantis irrationatis, rationale est; cupdum artic. I.
Quadrata partium habent  sunt  501 7b0a
-230e0L
818IV2
vdendm est ergo an ex qe17068 extrta
ex qua extrahenda radiim cubica, fiet:
Obiter annotabo Theorema memirabile:
Omnis numerus cubus Terminatur in numerum,
qui est differentia ultimi numori radicis a denario
sematium
vel idem cum ipso altirmo numtr rudici quinadium
alterminatur in unitatemretiam cubus termi nalitur
si Eosiger
quinarium, septenarium, octonalium,  rubus ter-
minabitur in ejus differentiam alimia denario.
Hinc monus cubus terminatur in seplenatium 
1488. 8
et3800512. quolum differentia: 2 121088
274 + S- 12, nise cubere dabu.
est psolum Juuma
2744 + 588 s. in +4 -  504 +  - 1728. Vem  (Y 1728  12S.1
2240 + 600 s-E, vel 40. f tror 1.3 ex qua quantitate jam radicem
Cubidam exhahemus. sane omnes habet terminos fractis carentes et altrum ex nominib,
srationale, et quadratum utrisque nominis est etiam rationale. Debet ergo radio inbica
(+ ad
14a + f- 12 bc. in se cubice ducta dant: 2744 a + (588) (3) 196 a f- 12bc - () 168 abe
exhasi posse ex differentia quaddatorum partuim
Et  tiet: - 744 a
(4-BLes abc
(2240)
B 196 a s- 12bc
(588)
+ 12 bc
(600)
+L 1n8b2
Et quadratorum
unde otrihenda fradio cubice
partium hijus binonius 33760d est cubus, a a differentia quadratorm partum vinbum, quod est
Pone sim datum esse numerum: 2240 af +b0o L- 12 BC
Extralendi binami radix, seu est cubus a 196 a +  12 bc seu es cubus a Nam:
av  6abc rgab, +A9a bc Y, 6 a V5 4b2c3
F9aVc 16a76C - b25
a + +bC cubietoens:  a 3 a V+bc ∞+3 abc (bC SZbC, beretesse XbC (+)
cum ilbeat esse a  a bc +9ae u
radicem ab
+ 2a Y 4 bc. Quadraturi prions: a +6abC +gb posteriors 9ab Fabeq
auferendo unum a sudouaorum padialou ab atterofu: a 3a bC +9 abC +63. Unde pates
sive a+ 3abc
jumsomendi )  + pate regulam supra demonstratam n procederte n binomid quae irratio en injeginatiam stiment
Sed ideogl eo s dissimulanda fucrit. Quasi iinclusum in isa radise. Anpotius uandum am
constat ipsum- in summam 2240 ingrestum) ejusque quadatum esse percnsensuens ab + 6a bc + 9ac, adeoque
quad binomium cmpio aetrum concipendum esse fingendum et 3a- bc, bc, vel 3a +d 1b Aaneltum
Dutique differenta ert quadratorum patium erit cibus, sed quia hac dignosci ex jam facto non potest
moiteile adeatuo tegerus sitonn ad vectansiven ad inignaus apescare
= 2240.
89600
4 4 8
0 760
64,
a si
i
q 
196- 1
580
125)
iret
in 1s
ado extrahend
a 8818112. nempe:
o sit cubica
xquo radio extrat
4860
69600
448
000c
3600
differentia
881 8112
360000
a b habet
a +6a bc + 9ab
F9a bc -6 a bc +b2:
+3aebb
superest difficultas, quod
a = 3 at