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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 73r, page id 00068196:0145
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sunto
it aequationes besbinas habemus
31o 31 30] 302
1B.312 20. 31 30 31L 40.312
suli 10, 20, 30, 40, sunt
arbitrarii
ob 312 erit 10. 51 - 11
20. 61 -21
30.91 -31
302
20320 11. 310 + 12d34 + 21. 2 + 22 d 3 + 31. 300 + 32 d 3D0. F 41. 3d + 42 d301, = c
51. 350 + 52 dex +61 3 +62 de + 71. 310 + 72 d3ID + 81. 3)B + 82d301 (70
10. 11. 312 +10. 12 c312
20. 21
30. 31..
40. 41..
g1D. 12..
d20. 22.
d30. 32..
dao. 42.
20- 22
30. 32
40. 42
13
3
10. 51 312 + 10.52 d3120 Hae binae aeque pdnantur edentisicar
20. 6
30. 
40 g1
d10. 52.
d2o. 42..
l30.52
d9u. 62 ..
20.62 Obdy ) Erit 40 = 10(52- 12) + 2c(62- 22) + 30 (72 - 32)
30. 72
40. 82
+ d10. 5et = 0 se 10. 53 + d10. 59 = Hec arpuntio
d2o 64
430. 74
c0. 03
30. 5
d20. 64
d30. 74
multiplicet
per 84 et
- 82
compendio 4  10. 54 + 20. 64 + 30. 749: - 84
et dq=- 8 4F454. 10484 54 d10. _P + d 34 ( 52. 18 3: 84.84
ec
64. d04. 20884. 64 d20
40. 83 440. 84 Explicndo cuiernofirt
34 d74. 30 84.74d30
(7et. 30
. 82. 53. 10 +94. 5951d L 83. 54. 10 + 84. d54.10- de. 5410 f 154. 547di0
84, 63. 20 84. 64 d20 83.64. 20 +84. d6g. 20- 1. 64  84. 64 d 20
84. 73. 3084. 14430) 8. 74. 30 +8+ d7430- 184 3130+84. 7 4d30
superest ergo ut eq. 1o applicetur ad data. Ante omnia videamus an a2- 82 non evanescal, ut haberi possit 40. dende videamus an refficientes ipso
40. 81-41 d40. 84
30 et deo non evanescant.
11
12
22
31 32
41 42
valent eost 11B. 202 00) 12 V0 IBEs -lN2oL 120.210 0
+83. 74
3i 32 61 62 71 72, 81 82 Eyo gr = 84
003 19 202 0o7 1a24 10122 0 na2oI - 1eo. 22
+ 120. 120 202 (201-210)
Bd. 8 fit 42 - 82 = - 84 = 120. 202 73= 71 - 31 = 202 (101 - 110) itaque 30404 74 = 30 f- 120. 202 (1 2 0 d 202 + 202 d120))
81- 41 = 83 = 120 (20(-719 74 = 12. 32 = 120. 202
6C = 62 - 22 = 120. 20]- 120. 210 = 120 (201 - 210) =64
63 = 61 - 21 = 00 7 - 003 =63
4= 52 -12 = 202 (101 -110)
53=9
Ergo tantem ex aequ. 10, fiet , -120 (20- 201) [120. 282. 30
= 202 (110-101
Cadit
+120. 202. 202 (101 - 110) 
(1120. 202 (rodror + 2024120))
120 (10 1. 201)
A. 1B3. 64 (o(2ot. 20) pol vor. ud
d. 20 64 = p) =nonor ano cnom no ao holusaro- drc
463
F12∞ 20200(7- 3)
aguta licccon) eroio   sid
(83. 54 = 120 (201 - 210) 202 (o1 u0)
 82 d 54 - 120. 20 CRO2Cdioi - ducKtd2o2(1oL oB)
323 = 120. 202 0V (7 - 51) - (X. 202 C∞0. 0.) (notrdi r)
s h
Eo 20. 20 ( 12o (u. 1) 1B Zo2. 10  2o lador sq. vorbu 1 robrot en u   Catioinl toB)
sed quia tota aequatio dividi potest per 120 (20-201) + 202 110- 101), fit tandem - 120. 202. 30 + 120. 20, 20 + 110. 202. 10120 vel 30 = 120. 210. 20 + I1c. 202. 10, : 120. 202
22
40 = (120. 202) 202 (101 - fi 9) 10 f120. 202) 120 (20) (=210) 20 + (120. 202) (120. 210+ 29) + 620. 202110. 209. 10),:. 12072
Et ita putehre satis fit 40= 101. 202. 10 + 120. 201.20, ut 30 = 110. 202. 10 + 120. 210. 20 Hi jam valores substituendi sunt in aeq. 3, imiidentificata est aeq. Y
20
200
120. 202
Ex coefficiente ipsius C d2 12 multiplicato per 10. 202, fiet 120. 202. 110. 202. 10 + 120. 202, 120. 210. 20_ = 120. 202. 110.202. 10 - 120. 202. 120. 2020= 0 Epqu. Crefficiens ipsin) 312 evanescit mate
Quaemmus nunc etiam eoefficientem ipsius 312, multiplicutum per 120. 202
tiet 202. 1b. 202. 10  2or. 10. 20 L d1 20 ,1 20. 202. W01 2b. 1DF 20. 202. 202 d 1o. 10)
Vi
ae (F110. 202. 10. 202. 10) (= 12o. 20. 1el. 202. 19
120. 202 =110. 21026 =12V. 120. 210. 201. 2OC1o. 12 0. 2∞9dior 2 ruo. 202. 120d20. 20) (+10. 202. 120. 210. 20) =12O2C. 120. 201. 20)
H
41 20. 2d2. 110. 202. d 10). L trorcoL 1. 21B. d2O 120. 202. 10 d 292. 10- 120. 202. 202 d lio. (0= 120. 202. 110. 202 d10) (̸ u0. 202. 1204 202. 19( 10. 202 202 4 1o. 10)
P2o. 202. 120. 20 d 20)
C120. 202. 120 dio. 20=120. 207210 d12020)= 120. 202. 120. 210 d 20f 120. 20. 120 d20, =0120. 21 6 202 d120 20)
Ergo coefficiens ipsius 312 in aeq 3 vet a, etiam evanescit
postet fai ao aers et 3o uo elanta ad
U ah + e bZ + cD + B dq + fdo sit 5:  eibo fiot abi 4 bg +rv + ce du 4 (aV =  e ta a h + bZ - cv + l.p 1q7 + C 1 = 0
sit bZ1 cV = gW fir ah + guo + e li+ e gdu. sed ita oporti esse constentes, b, c,9
Mitar sit Ha Z F = gt aZ at Z + adv = oduo a codque fist de  gdr + dg W
et
sit qu =  9Z + rv adio = qZ d9 Z + dr V + qu2 a rdio Ergo (W Vdu = (3 aW + Pa duo = Bg + CpV =
Vd Ndr V +) 9Z + Vrdo
qu’od ut coincisal data abzue + ac V +ae dz tafan, fiet β. b et Y  Y af: Y = ae: q
ab = B9 + pdque seu ab = (t aedque et ac = Cr + af dx et fit B = acr - a fdr, : 1Y = abq - aedq, : 99 r - 54. C di- tc. gat
et fiet abq ab - aedq  ab9 - aC dg. 5: 49) = _ aC59 - 4(4 - af - a ss d9 - af 9 d5 ,: 199)
Repetamus. Esto formula bZ + cD + edzue + fdx. videamus an reduci possit ad hanc ) BW + dW.
q E sit aW qq + rV erit adu = at. d9 Z+ de V + gdZ + rdo et aBu +du ext  ent  B9Z + βrV 
(6)
Identificando teminos deb sit 9= af: Y et teminos dzue sit V = aC : q. Ergo r= qF r - 59: C et di . 41o s posta V dgZ + qaV =+ Ga Vrd
Vaeztevtetz efdo
constante quod in arbitrio est
Ob terminos  identificandos, f ope aq.7 fit BY + a fdr. r= ac seu β acr - afdr, : H. Rursus opt ae 4. 8 fit BOb termnos Z, ope ag. 5 sit B9+ aed9 u9 = ab
seu B = abg - aedg,: gq. quos duos ipsius B valores aequando sit acr - afdY, : rr  abq- ae dque: qu et ex 9 et 10, pro Y et dr substituend
1
valores, fiet abq - aedque, : 49 = ac fq: C = a ff dq: e - afqdf: C, :, sf 49: ee, seu  b fgC efdq = ceg ( fe dq - egdf, ergo
tandem fiet in da evanescente q siet in datis  edf = ce  ce - bf. sed haec non est ipo in pte
1op) + updus + 12 pldus an redigi potest ad fC + gde posito () = 20y + 21 dus ubi d0 = droy + 20 dy +2, ddys
a2 du
Gdroy +920 dy +921 ddy  10p y. + up dus + 12 p ddy fit p = 29. 12 = 205 + duo 9: 10 seu _ 10219 = 12. 20f + 2d209
f20y + f 21 dy
superest aequatio 21f + 209 + d 2 g = 11 P, ubi substituantur valoes f et p. Multiphicet hoc aeq. per 12. 20 fint 21. 1. 21 - 21.12420 +12. 20. 20 + 12. 20d21 - 11. 2021
fiet:
fiet f = 9, 10.21 - 12d20, :, 12. 20
Quodsi fiat 21 = 2m 20m: a, fint 20. 20. 10mm (= 12. 20d20) am + 12. 20. 20aa +1 20312. 20. 20dma + . 20 dio and = 11. 20. 20 a m
Es d21 dy
10y + 11 dy + 12 ddus, divitatur in duas partes Bato + = 10 ay + 1 adus +oopdus + 124, ddus prior pars sit differentia ipsius osus, poserae itus gda
qua divisa per 20, habetur ita ve evanescit 20. nec profectum est quicquam
liet   0s df y + oif dus ~ 10au + u aydys siet f. f o. 00 et f  et  a Agdus + aoa gdus ~ vop gdus + 12agibus sit pg = ar
d
vop d
r = dg g = 00g: 12 = df 00 00df: 10 f.  oof q r = 19 - 00 5, . d
119 - 00f 500 fit dg. 9. 10 = 25.5 + 119df = - af5 + 11 dtdf00: 10
200 a + 210Z+ 201V + 200 dz +202 dv =o Ex 1 et 2 fiet ngo 200. 2oa (+ 210. 1207 ) 2o1. noV (220. nodZ +, 1no. 202 dv = 100. 220 a +10 220Z (u0. mo 
c
n4 13
arg
= 10g: 12
ldg = 119 -005, :00
fiet ad prro. 200) + _ 120. 201 dv + d120. 201) V + (120. 20+) ddio + d (120. 202) dv =  (10. 290 - 120. 210 ) dZ = d (10. 20 - 1B. 20)Z andent tolltu Z et aZ, fiet
00. F29
1005
il fdZ
100 a + 10 = + 120 1Z = 0
=20oga = 2019p - 202 9 dv
21947
+ 110 fZ
+21097
F= 120, 210 - 110. 220-2209, : 120
F = d (120. 210 - 110. 210) - 21093: 11C
27