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# Title: Leibniz-Handschriften zur Mathematik LH 35, 3 B 5
# Folio 66v, page id 00068255:0132
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(14) Ex articuli 13, a99. 3 et 4, quaeramas 513: 113, et 713: 313. Et quideme
aequ. 1 3 fit 513 + 4.112. 123. S12. 523. 513
11223. 112, 123 113
4. 112. 1232. 512 +23 = 112233112. 123,512. 523
seu 5B + 4112. 123. 512. s2Z 51
115 11223. 112. 123 113
seu D12. 123. 512. 523Dhh-S) ()Uh = (3+ 0) 11223 sh et ex 1 et 2 fiR extractioni fecta fi
defunctis = 112, 123, 512. 523 f  112.523hh + 123. 51 hh = 2, 112. 123. 512. 523hh seu 51223h= f~- 112 123. 51. 5C3
sed ut in eg. s succedat extractio sit Hinc sequitur 112. 123. 512. 523 (7 mo) esse quantitatem negativam
utjam amm a qu S succedat extractio, fiat
ex aeg. 5 firts b(f. ath)  11. si h.  Ex 9 est 123. 512 = bb (2h- H): M. 523 et 
ex 10 est 123. 512 =  (f - 4hh) (b: h) = 112. 523 quos deos valores aequando fit 12. 523 - (, 4hh) (b: h) 112. 52
(11m0) = + bb (H- 4h4) seu 112. 52A - (f- +h hi bh) 1 2. 523 +  A stlbe bbf - 8bb ffth + 6bbh (: ahhil
=at bbf obbsthh (+1bbh) - 4bbhh ( 16bbh s: atk  bbff (5 - 12 hsC): ath
seu (12m0) 112. 523= 65 Fs5- irtih qbs- att):  = bb bs hc 166h aF f - 166d: +tih
textratendo f 112. 523 = (12mo) bf ss-ath + _b (f5 - 4hsi): 2h. Tantum ergo superest, ut fatiamus
H- 4his  aequalem quadralo f- 2f8 + 85 seu f = (13tio) 4 sh + 88,: 28 et fiat sf- +hh=
V 16ht + 8hh 48 + 84-88hh,: 484)- (.26.) 4hh - 94,: 28. Adeoq ex 12 et 14, fiet
112. 523 -fista) by 16h (  1bt2 - 8titis P : 8995 seu 12. 515 =  ah-  h8,: q
Ergo per g fit substituendo valonem in 10 fit 123. 512=)  (( og, f1)  16 + 4tt9 f. 4h 8
seu 132 123. 513  (b ) Clot) ) (b:h) , 88 - 4 hp. unde 112. 123. 512.5, 16= bb  4titi: 8
unde vera reperitur ag. 9 per 14 set ex 15 16, fit 11223 7 (1800) (b; ahg): 16h otha, quod itidem
consentit cum aeq. 10 seu fit 1122 (b: h) (T (+th - _8 : 28). Et ex 6 per 17 fit 51223  b. + 1bt : 484
am M 23= S1223 =, Z. M. suq = Lt (b: h), as 2- g + 16 H + (9  l 1 5h2,: 44q
+ ∞ + prodit valor ipsius i 112. 523; sin sit + ∞ X, prodit valor ipsius 23. 512. Unde patet n
in 3r esse 51223= fb, h), 16h- 8: 4g non mutati signo. posito ss aha esse
4kh - 8: 2; spe inen emne non contraria signa ut in aeq. 14. non vero contrario signo.
lldeta Ex his jam s. )B = ubi possumus sponete in aegq. 15 et 16 repertas esse 512
et 525 5EG. per 12, et (12. Et ex aq seu ita in valore ipsius 51Z: (iC3 ex aeq. 4
seu 513: 113 = 112. 173. 512. 523(f - 2h). 1122. 112. 123h ficc
erguees 2t. 8
etr 1 = 8 bb,   glih - 4
seu dema SB3 (21)  2ta - b, 2h-9
jam in aequ. 3 articuli 13 pro 51 subssi hanus
bb ima. ge, (49.  s
hunc numeratorem, pro 13 hunc denominatorem, et fiet
 + +th- 4g v 6 b Et. G
quibs valoribus +58 h 21, 19, 17, substitut
nes qu. 3. articul
res 112. 113, it
aeg9. 15, 16, 17 19, 20, 21,  pro quibus continentur vasores rationales ipsorum
b,- ric2. 10 b4 (i 1123
1 1 )m n m t  ug sinae (1
mucon
pari jure ob 9, D- 0 t, ut praestetur rationaliter aequatio 4 articuli d, fie
L =uo retinendo h, et pro  f, b, 8, ponendo 9, C, V fiet 713 (1m0) = L. 31. 323)
(c. t- hY, et 123. Sir AB to) (b. at) gg 32 1C do) (c. YY) asch- 1)
et 2 et (200) 312. 723 = (Ch: YI), 4hh-  et (3tio) 323. 112 = (C: 4h),  +hh - Y)
et Ato) 312. 323. 712. 723 = - (C: 477), 4hs -  et 31223 = (5t0) (C: 4hs), Anh - V et denique
(610) 12 23 = (6. 4671), 1 166 7 4
Extractione igitur facta ex aequatione articuli 12, fes 
S1223( mm ). Etpum jare 32. 712 B F 33. 721 2 = dp2s f rrera un Amm)
12. 512 V + 123. 523Z = 5l223 + 2. p23. 12413
seu compendio 112 12V + 123. 523Z 4 600100 a=0 et
312. 712V + 323. 7133+860. 800 a3 (4to) =0
et sibtulis a tutis pro tollendis absolutis fit  p. 71. G0 aV- 31. 714. 600 a2 2 600. 800 a
quam reddendo identicam
fiet 805 2600 = 3121712: , eL 800: 6052 33. 723 :, 127. 523 unde fit (800) 312. 712:, 12. 312 = 323. 123:, 123, 523
seu (gno) 112. 512 313. 723 = 123. 523, 312. 712. sed aq. 9. sequitur ex ipsis 6 et 7. ideo, duae ex his
  t 
112. 12351 + sir. 525 H3 F 1BE3 NB. SiZ quod multipl. spr 112. 13: , 113. fit 115. 12y. s1 P us. s2 siB  vES. N2 rB  ortcissed Iter
bb 4- 8 16f +9
859mon acu p 17
 in aequ. C arsic. 16. potent explicari (. 1212. 1231) Aer 13, e et per VE. 123. Nam vir C 12 A 103 
 = bb A  . b atih. ps (112. 1 -b  
ahhrgaligh aeguets
in Bs S gl g Ahsi +b. G(, 4 Ah 29 a haer  tq  b si (+ Pha q d citrh. 212
unde U D seu (ir. 125 bb1  
adeoque 11213 a 23B =. = 113
Seu (1m0) S1 = Au 2. 123. 512 125 = 11223 112. 123. 512. 527, - 2, ME. N3. 12 S3) Ergo Aa3 esB3. M. 13. 13 = bb. ih. q Eh.  ist  - 4thr 9- +h r  chi q -  n.  pats. it il g.  . q iB
Ubi ut irrationalis reddatur extralibilis fiat 4.115. 123. S12. 523 - 11225. 112. 123. SA2. 523 = (2do) i. 123., 12. 513 f: hh,
513: 1132 (112. 123 512. 523 (f - 2h):, 11223(. 12. 129h Sed quia in 11225 latesit explicando 11223  ut
in aeg. 3 fiet 112. 123. 512. 523 (4hh, 5f) = 11= 5() 112. 5237+ 115. 5125, + 2. 112. 123. 512. 523 hh seu
ut imaginariae evitentur, et ptroinde in aequ. 3 debet esse smajs  quam 4hs
SM Si 11Z. 13512. 523 = bb (Ahh - ss et
ahh - 98, pi2. 113.