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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 90v, page id 00068198:0180
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dat
sit Formula quaedam
orta
alteri, explicando X p
aequales in numetatore et
sit X: a = E 10 27 + 11 aZ
Radices ipsius formulae  dat
secundum quamlibet X, ipsa Z
e quaerendo Z opeae
Causa pro 20. 2027f
et fiet tn +10a
=30
naa =3 1ax. - 120
= = qu 10a = 30X
100
Compendio sit 11. H1 - 4. 18. 12 = 113
et fiet Z - ) - 11 a + 31 a X +
se Radpes aequales, poteri
-11a +31 a l + a 12 aa + 13
2, 10a = ab 30l
=ua + 31 al = a Y(I2 aA + 132
21 10a -  3ol
sit ta C = 1 (12 a a + 132 al
il 
- 11a + 31 l = 10sR
- 11a + 3lB = 1F 13
=ua +31 n = f13.
fiet Bilutra O:  13
1311 = YaC;
Ex aeq. 6 fiet 1311
1380
per compendiu
a 2.
seu fiet 1220
1340V
10Hagoa 17.
den Eeudem hutd nigra aa
vidpamus an -13)
namfieret 15AO
si 11 et 21 = 0, fier 7= 2a L(-10. 12aa 413. 30aX), :, 2.10 a- 3. 3ox. ubi diudi potest  nominator per nuneratorem
et fiet Z- ai  mai qu. lon -3ek)
1129
= d
-10a +3oM
pa
et rursus  +.
=+12a sed ita evanescentet via,
210a +3ol
omisso 21, unde et 31 et 32 evanescent
-10d + 30n
et fiet Z7 1 a ta s aa + 4. 12 30 aX), : 2, 10a - 30X. Ergo ut habeamus duas rudices aequales, fiet
sint 10 a -3ob = 150 a
10 a - 30m = 160 a
104= 3on, = 1704
AH. II aa = 4. 12. 30 a M)
N, aa = 4. 12. 30 a l (4) 1 aa t aY (= 4 l. 12
2, 104 - 50l
ia 
2, 10a = 30m
1. 1012.
i
0a + 30d =  250a
10a + 3om =  2t0a
104 + 30g = 270a
et fiet  1a F e Slum = 4. 12. 20) B) na + a ssum - 412. 260) et La = spril - 4u2 250) (5 1a Fa + (ug - 2. 270)
2. 150
(2, 10a - 30n
2, 10 a - 30l
Exaq. + fint  l1. 160 + 16 aB ~ 1. 150  1170 L a 2 N et ex ag. 5 filq 11 170 - 170 faC = 11. 60 + 150 
160
vel pro ag.6 scribemus ta ta 2 - 16L o 160 Ya C) - 150 1a 2 = 1 1 (19 - 160)  se 160aO - 15 a 2  3am- ) e
2. 150
 sit  l.em = e l+m = f (ln = h. (n = 9 et fiet 150 a4 - 160 la 5 =1. 30e. Et pari jure ex 49.4∞
fiet 150 a Y + 170la C 2. 3bh. Ex aeq. 8 fiet 150a4 +16074) - 2. 150. 160a504 - 30 ée. seu 
 150 a 2 +16040 -(35 ee = 2. 150. 160 a 102 et quadando utrobique 150 aa 42+16 aa 30 + 2.150. 168a02
150a 2 + 30  t = 4. 150. 160 aa (32 vel 16+1 150 aa 2 2 16  aa )- 2. 155,160 aa2 
Ca2 - 30. 30 ec) Et pari jare ex a.9 fiet 150 aa 24 + 17 abB 150. 17dac = 30. 3ohh (2. 150 a  - 30. 30 hk)
ibi ut melius obtineatur e calculi contractio inclusa patestthesibus in 99. 10 et 11 aequemus quadrati o a sit
= 2. 30, el 1160 a0
 et ex eplicemus a27 a0, a  fiet . 62 Et ex parcathesi aequationis 10,fiet  fiet
= 30 30 ee C3. 160 a
+ 2. 150: 11. 11aa - g. 8150. 12,3(030. 30. C =100. 100  seu 12aa = 18a  (11. 302 -2. 11. 1504- 2. 11 160 a 
2 158, 260 + 160, 250
+2. 150. 112 a - 8. 150, 12. 270 au - 30. 15= 205 at seu -412aa = 200 + 15. 38hh - 2. 1I.  (150 + 176)69
+2. 160.... . . - 8. 160. 12. 259 ract, odi
Sed ex aq. 10 ope aegB fict 150a24 -160 a0 30. 1100 seu 10 1N = 4. 150. 12. ∞ - 165. 11 + 4. 165. 12. 250 = f 1B. 30. 1oaC
X, 138. 270 = 1702.250
Et ex parenthesi aequationis 1 fiet
e v = 150- 160 - (10 a5 = 2. 10. 304l + 301= 10 a) + X 10. 3e am - 38 mm = 30. 9 d. VF, - 2. 10.30a
 30. 14e
Et pa
+2. 17
8. 170, 12. 250
honto seu 30e,   305 - 2. 10a = 30C  ap. 41 e, et fiet 10- 168
gure
rit 30g 2. 10. 29
et ex ag. Hop5 aequ. 13 fiet 150a8 - 170 aC = (+) 30. 11, 200ah Et faciendo 30g- 2.10a = 24Y fiet actiqus ad imitationem aequati
purs 17 fiet = 4. 12a a  (4) 1. 30. 200. a h - 30. 2e41 1.H F, 3 175. 250- 158. 270. Resolvende sunt ergo aequationes 12, 13,10,19
1 168. 250 =  1 10a + 101030 aal - 240 504 n - 2. 10. 30 alm + 10. 3 eamm + 38m. Eodem modo 180. 250 tan-
2. 10. 36 alm = 10. 30 all + 38
tum pro M pbicuod R
190. 260 + 1002. 250 = 130 310 150. 270 7 170. 27 -  320
150 260 4 1602. 250 = l9V et 150 20 + 110. 250 = 90. Et ex aqq. 12, 13, 15, 17, fietit values ipsius Aurna
102 a + 10,10. 30 aa M = 210. 20a
mesta 3C +168 = Eib f 4. 1030 af +30Utmm) Sed compendio sit 150+1602 = D et 150 + 170 = 
L23
olo Sad quiac in hisc Honrinuolviture et Et pari compendio s
150. 260, =
Easo ex aegq. 12, 13, 13, 11 fiet - 4. Laa = Lob a +11 30 ee a 460  aar: 2
-4. 12aa = (4) 1p. 30.200ah - 30. 241. 11k, : 520 Aefe
-4. 12aa =3200 a + 11. 30te - 2. 80. 1 a - 2. 8203 se
unde ducendae sunt tres aequationes, intervenientibus lise quanditatibus _ 1c, 200, 31, et passim latentibus 10, 30.
11
15) + 11. 30. 1002C - 30. 141. X C,:  si0
+. 1005203 ae() 200. 5103 ah,, 14 1. 520 241. 518
Sed ex aeq. 12 et 13 est 11. 113=108. 3203: et 30. 320 e 1. 2d0 310 aa. Et ex ag aequatione
=4. 12aa =
Ex 15 et 17 fit
quaddandi fiet
3. 180, 320
11. 112100.520aC, f (() 2. 100. 200. 518. 50a eh, L H 41- 520 a  = 2. 141. 241. 510.51 a eR
1760. 310 a
=30. 03hh
 ratio inter 100 200. Sed opus est ad huc gna aquatione
224 S aahh
Ex 12 et 15 liceret r iN. 11. per 1002 et ex 13 et 17, R pera 200. Aempe ex 12 et se ea
4+200. 5108hb
quos duos valores saquando habetur
compendio 3oceur 180a
2.30141. 318. 11C
sit (aa et  th 2.  Buq 1ra Ex 12 e15 fit 51D. 10a  51P.616I = L 303103. 1o0aC 
seu t a2 3 .5 ta C. 10030 qu (61  100. 100   d. 30 (318: 510) 114. 109 + 30 qu (313: 510) 11 d ee
(+) 220(32D: 520) ah F (38qu (3103 520) 2th- (20 d - 2. 30 241 (3 W: 5203)
et U, et firt tandem. L 30(3 + 30. 340 (5202: 5103) a ee = 529C (30 qu (3103: 510) a
30 qu (318: 310) 11 lee 610.14
pror modo 200aa
in aeq. 18, eva. 
19. 3163 (38: 510) a hh ( S1h, 5(30 qu pro . 510) ath - et,
ione suo et it
quos valores suc gitue
14152e +241 510 ah
50e = 510252, 141. 520 de -241. 5W a h
Hac aequatro moltiplicetur utretique per sioine ut tollgntu ra
30. 310f2874a CC + 520 CY (30. 318 a= 610 a = 2.39(4 5103
30. 2b. 51 aahtil H) 516hr (3 . 320a h = brex 5ec aa f 30. 241. 320. 520 
31= e 40 +1
32= 41. 41
31 = 2. 40. 21
32 = 21. 21
Ma2
2 1
te
per +13AMI.
a (T3A (1a L. 131
 simili (13AM- 2.1
aq fiat
caeteris tracturi potest
tur extra HL
2. 13 n5. +naa + L. 13
ura
x aeq1+ fict
ex his quatuor aequation
tractabuntur
erit aequ. 12
1.
vae praestandae
13XM
13A02 + 13105
et ex 13AM seu
Ergo 13A (I= 10.31
pe 200. 2dt
oo 50.
50+68
Umme  al = I + runsit ak. 1
298. 201c
lit et
a ex aeq. 14, 15, 16, 17
A+3e hk  2.G
2ooroe . t inga-
14, 1
. 102
3.7
+) 81.200. 201
a 2  6 ud et 
el.
eto
a
Sia q 
= 13
E et fiet ex 8 et 10
Uonna