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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 76v, page id 00068196:0152
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resolvendae
11Y
200 troZ + 20 q a 0
dio ie
2
100l + 101Z + 101 lv + n2ol dZ 1 102l dv
nodaque 0 per lmin, p addamus in unum fat fit aeq. 5.
202 ddv
200m +2omZ +201mV +22m 7 +202 m dv
dioon Fdnoq + dtnV + yon du ton d + 1on ddque + 1o2nda
dnon dZ do2n de
42oo pa ducop Z Fdrospo + 210p AZ + 2oip do + 220 p dz F 202p ddr
Aerop sqZue idrorp dr
t ooah + 310fzue 1301 fV + 320F dZ Y302 fdv
=dyogz + 13o1gP+ 310g 4Z 1 201gdv +320gdZ + 302g dav
aequatio tractabilis)
Assumatur jam fO + gdO tah = (6 - 0 
et sit  = 310Z + 301V + 320 dZ + 302 dv
fet de (8 L d3oZ F d3o,. B + 31dz +. 30. de +3rodtqe Bodar
 fiet eg. 9. tractabilis
identificentur jam aequationes C et . Utetiam ipsa N95 fiat tractabilis
Habetur h per terminos absolutos aequatos
Aequandli daque et ddo coefficientes habemus n etpperus
(10) 202. 920 - 220. 202 
fit p = 3209 -120n, : 220 = 302,g - 201n, : 202, quos duos valores aequando fit 2c2. 329 - 202. 129 = 290. 3 org - 220 0 2, adeoqe N = 20 170- 210107
dgrogaqe +d3olgd
quo valore ipsius n substituto in prie valore ipsius B, fiet p = (202. 120. 320q 2 1907. 3209(- 202. 320. 1209 + 220. 302. 1209,: (202. 120, 220. 102)20
enim _ p = 102. 320 = 120. 302gquod non diffeit a priore, nisi
102. 120 - 120. 20X mutationesignorum umeratoris simul et nocaleris
in
permutando 10. et 200 ex 1 fier
Compendu 1eq. 1B  t communis denominator, 120. 202 = 102. 220 sit = 13 = D0. 1002 fit 1 2 L . 3o- 20. 10g et p = noqua tor e9
inde eodem
modo
110- 0 t -1000Y
120. 202 - 102. 220
eodem modo 3015 + hoju -dioin - droip = 17 = 00t et ob Z, fiat nol + 20m = x 18 = oor et ob  fiet 101l + 20im = 19 
quo supra invenimus n et p, invenus l et m, et et el pro 120; 22D, FD  102, 202, ndo 3 20), 302n, B, fit
101. 20
201. 110 - 21010, = t M = 10. 201-101,210
et compendio sit 110. 201 - 101. 210 = 22 =00.2001, fiet 1 = 23 = 201 Y - 210t, : 2001 et m = 24 = 110t -101Y, : 2001
superest ub aequemus oefficientes ipsius dz, et coefficientes ipsius av. ubi compendii causa fiat 10 +d120 = 25 = 1120 2+ + 220 = 6 - 2120 101 + dp.A. = 27 - 1210 201 4 42od - 18 -240
et 30r I +dhon g - 1210n = 2210p = 30 = 00 . Ergo obdue fili2oltiom =  atgult2oem = voV
1002
paeterea fiat compendio 320f + (310 + 4320) 9 - 1120n - 2120p = 29 - 004
Quos duos valores ipsius L ex aeqa 23 er 35, et ipsius M ex aegg. 24 et 36 aequando fit 201. 1002. Y - 210. 1002 t2 202 20018 -220. 2001 ,+u0 100rt - 101.1002 R 2o. 200 p 102 201 
120. 202 - 102. 22
1B. 202- 102. 220
110, 290, Y, 101, 201
ubi numeri sunt dati sed literae debent explicari Quae libet harum quatuor literalum novissimarum continet n et p, quorum valores supra inventi continent  320 et 302 incognitas
Unde rur sus habemus l et m, pro
sed non continent 310 et 301. itaque si ad calculum absolvendum sufficerent hae binae, pro duabus 849. 37 et 38, non opus foret devenire ad explicationem num  seu valores lite
rarum n et B. itaque majoris comprndii causa, in aeqg. 16, 17, 29, 30, faciamus dion + d 2op  00 β, t _oin + d20ip = 00V et 120n + 2120p  00R et 12on + 2410p = 00 p
Et ita jam explicando Y, t, 8, I in aequ. norissimis 3738, ope aeqq. 16, 17, 29, 30, et 39, 40, 41, 42, fint ex aq. 37. sequens aequatio 43. et ex aeg. 38 sequens aequatio 4 
1002. 201. 310f + 1002. 201 d3109 = 1002. 201. 00B =1002. 210. 301F- 1002. 210. d3019 + 1002. 210. 0072 2001. 208. 3205. +2001202517209- 2001. 28 00-200+2 20. 3 12 f. 2001.20  39 . 00 n
1092. 110. 3015 + 1002. 110. d3o1g = 1002. 118. ∞) - 1002. 101. 310 f = 1002.101 d3199 + 1002. 101. 00 so = 2001. 120. 3025. + 2011 1205, 701 (9-2av1. 129, eo p- 2007 1 1 2. 30 f 2ogi. 12 f. 1. 9 420o1. 104 0 9
troons  BA
Quarum duarum aegg. novissimarum, ope invenire oportet 210 et 301 possunt autem haebines aequaliones ad majus compendii redigi, compendiote comprehue
omnia, in quibus abest 310 et 301, si hae binae literae incognitae suffictunt, quod tentabimus. In aequ. 43 fint: 1002. 201 0β3  vor. 210 vol 4 2001. or rof t 3roy). 1001. Wul eot. 2em. la ro
Undi ex ceq. 43, fit (1002. 201 f- 2dol. 2029) 310 + 1002 1019 d310 400 1. hrog - 1002. 20 f )301 - 100 2. 219, 1301  p0  ili aivor map Meni te e 2e t. pe a i ut 4 20or 100 t 2 1 a
 A
Unde exaequa fit. (200. 1079 -1002 101F) 310  1002. 1019 d310 + (1002. 10 f - 2001. 1209) 301. 4. 1002. 110y d 30,  007 V
Videamus jam an ex his duabas aequationibus una componi possit, in qua absit tam sot, dat tam d301, quam 301, ita ut supersint solum 310 et d310.
ita enim habebitur solutio desiderata. Nempe aeq. 4b et aeg. 4 dividatur quaevis per coefficientem ipsius 3ot nempe multiplicetur aequ. 4b per 10 et 48 per 2∞
et de inde in unum addantur ita ev anescente d301,   per se, superest, ut evanescat 301, fieri X (10, 220. 21I - p20. 410)9 +610.210 - 110. 20)51002 f = 0
Itaque reen tentenus aliter coua ex aeggu. 46 et 48 facere tractabilem multiplicvndo ibam per b, hanc per c et dende
Sed hoc non licet, quia ita eo unisuit s. g, et urire aeqvatio in datis
Comporendo et efficiento ut coefficientes ipsius d310, et 4301 sint aequales, et ipsius 310 et 301 sint etiam inter se aequales iun
enim aequationem fier tiae stabilem alibi ostensum 3t. Et fiet (2ag 12 ) b Evrofc
(1002) 201 (9b = 1002, 101GC =  (1002 2109b + (1002) 11oGC et ubi cum evane seat g, habetur ratio b: c
Eiquemus jam et coefficientes ipsorum 310, et 301, fiet
1 corf 1002fab - 101C) + 20019 (102C - 2026) ) 1002 f(110C - 206) + 20019 (220b - 120 c). Ubi loco b et c substituamus
Nam fit C: b = et = 210 + 201,: 110 + 104
quae ipsis proportione respondent e et fet 1002f (1 29(F 10I. 201 = 101. 210= 101. 201)) + 2009 (02 20 (t o2 roL 10 2o2 (101 20D) = teoz ff2o. 2, 9ruo 20s = 110. B 14. 200,
1019 (110. 220 + 1. 210 - n0. 10 - 1 20201)
cesse copstantes, se
itaque faciamus non quidem coefficientes ipsorum du dio ut dio
  eorb oC:, 2t0b + HdC =  13. 1
m  2 + 13. 201:, 12. 10 4 12. 101 L et in aequ 50
sed ita cum utrotius destinatur f, ehunescit etiam, 9
pro C et b, substituamus proportionales fiet 13. 1002f (201 12tC) 1 12. 10079 (102. C - 20 6) = 1. 1002 f (Hoc - 210b) + 13. 20079 (220b - 120C)
3 HC
in fiet
itrtob
ay + Y = 0 quae rebditur
12b iotc
tequatio 49 ex 46 et 48 (piost multiplicationem pe respecive per b et C) compositis formata, compendio scribatur fin im 3Id + n 301 + p d310 + 4 3 ee t h = (aB.. 
2e:
 ee
3P1 - 181. 1.
naaum ntculi instituto inveni enveni prodire et in datis efficiendo mpZ n. mg - np. = gdp - pdque si jam ratio p ad 4 fiat constans, se  veluti 9-
his positis reperi posse tractabilem, seu aequationem istam  aequivalere huic sd +  ndB + Y. si fat O = E310 +  A 301
= 12p dp: B3, seu tandem 12m = 13n C et 13 12 p = 3
2 12p: 13
quod epistemcis in
100if (. 110. 201 ( 13. 101. 208 - 12. 210.104 (B 201. 10r)
quod tentemus efficere in aequation qs, composita ex aejus 46 et a8 Ept autem m = ex 5eg. 4b multiplicata per b et 48 multiplicata per C, eritu
9- voeg Eferne F. anf 210
s et mp.  n 12p: (3 = np =  12dp 113  er ( 12 p dp. 13
m = 1002f (201 b - 10iC) + 200) (102C - 202b) n = 1002f, (10c - 210b) + 2009 ( 20b - 120C)  p = 10029,  201 f. 101c 9- lvorg,
-= 12. 201 +1, 210,:, 12. 101 413,110
super est jam al faciamus 12M = B3n
-1 20. 11
1002. 1f (C (01. 12. 4091 201. 1, 1. D =101. 12 201 -101. 15. v0) + 2 0v1. 129 (102. 12. 201 + 102. 13. 210 - 202. 12. 181 - 202. 13. 10) = 1av2 1f(no. 12 o (+V0. 17  9 210. 12101 f21O. Iu 10)) 12os. 3 (ervo 11   .
Ergo si 12p = 139 fiet 12. 201b = 12. 101C = 13. 10C - 13. 210b seu c
enimus proportio
et in valoribus m et n, pro b et c, substituamus quantitates quas mv
Fietque
itaquex solis 310 et 3oc rem consequi non licet, sed adhibere postet adhuc 3 320 et 302. Itaque redeundum est ad aequa 43, 44 ibius explicant (), A, er
sed video ne hoc quidem prodosse, et similiter eancdestiniae f, unde et evanescit
1do2
Et 00B = D d1o1. 202. 320 = d 101. 220. 302 + d201. 120. 302 - a 201, 102.320, : 00. 1002. Compendio sit  120 d 201 - 2e0d 101  2dD, et
d0p. 302-20d1.3le: 00. 1002, et 90d = 1129. 202320-1120. 220, 302 +2120. 120. 302 - 2120. 102.320,:00. 1002. compendis sit H20. 202- 21208
per 39, 40, _1, e2, et per 14, 15, fint  00 β -7 d10. 202. 320 - di10. 220302 + d2go. 120. 302 - duo. 102. 320, : 00. 100
. 2 20. 320  2020. 302: 00. 1002. 5t 00 s=, 1219. 22. 320 -mo. n2o So2 + 2210. 120. 12 - 2210. 102320, : v0 10o2. Compendio til 1210. 202 - 2210. 102=- o2so, et uo tuo. 121a no = 201
001002
29 + 100 2. 201. 10d2. 302 = 1002. 2001. 202. dyrog F1 2v2. 2001. 2w d3 029 = 0
et fier 00n O210. 320 = 2010. 30∞: 00. 1002. H. jamvalores inventi ipsorum β8, γ, A, e substituantur in ae q9. 43 et 44
8.
et fier s e
der 22o = 10d2302,
et 103 1701 2o d101 =od1 et fiei Y
+ 1120. 290 = 2120. 120 2020 et fiet
et ex aeq 13. fist 1002 (1o2 201 f - 2001. 2∞ 9) 310 + 100n Vo2 10N 7 1310. + 10ve ( a mrogu 301u ar a E. veg 13a=
410c
L200i
2001. 202, 202036.
e ex aj. 44 fiil 1002 (2vo1 1029 - 1002. 1015) 310D 1 ver. 10l9 d 310 +1002, 00 2. 105 ,30j u 21uc2. 10, l341 +1 col. 1. 2081. 715) - 1092. 110
2ool. 120
5i0.310 + sol d31o + 52o3B1 +5or. d 3o1 + 5 30. 320 f5(3 d320 +5∞. 2o2 1 504 dzo 63)
(10. 310 + 601 43, 0 4+6i0. 301 +602 1701 + 827. 320. 603 432 +640. 302
et 301 = 21320122302
30. 90,320 +50300d320 +540. 00. 302 +504.00 dse
At in 1501. 12 1.
502. d22.
2
+
502. 2.
+604 dy 23
et in aq G Huimus 510.H. 
vates, ut haten
psisnt
Q tor roq ivrdvus -Morar reci tvoiseo
630. 00. 320 +6038320 +640. 8.302 +00t. 00. d3026
610. 1.1
601 di ...
120. 2
ot du.
quatinae ripnict
4 10
11.
601.11...
602. 4...
finni ras coefficientis est
lieis olsitns
o u o
Ex org er iomprdu fint
15964 fint
5ol. di . . 4301. 11
L a2
502. 27
onanus ce ad ad siat S . E puirae fl 1542
pic
a susicere   d
1 310  p. 320 + 12. 302
6o. oo
210. 3og + 2og1. 20. 2010. 3
1201d8. 3c03
o ec aiio 
50i deZi 601. _ ... -0
6or 822 . 602.22..
multiplicotur aerg. 68 per 60
im ab abtera tollendr eomnesce
teram aeguatua
43e0 in aeq. prod
s 40. 001 510 m2 25odi  d d2 C, . A40. v0 + 610. 12H Bo, 
604. 00 +bd1 12 ++ 60Z. 22  Py 5oq. 00 +5o1. 12 + 502. 2
sini 60r. (526 - 50d= 502 (620 -602)3
ant alisele
o1. 12
C+C
ti eo
o pir. o
en aoormat  sond. onr ey) (.
ior vornog
ner 320 et ag  et habebitur ita 320, Ergo et 50ct