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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 158v, page id 00068201:0316
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Reperio, etiam magis ordine procedi et observai melius duplices homogenectum
leges, si ponamus in aequationibus datis et formulis assumtis non negligamus
Indefinita
Sit ergo aequatio data ) X + naX + naX= +Ba X +a N nsa X 1 e 2
Sit assumta conjungenda primi gradus (2d0) 20X + 21 a =0. multiplicanda erit illa per formulam
(3tio) 40X: Haec per formulam (4t) 30 X + 31a X + 3 a X  + 3 a. X . 2 etc. Et ex fiet
ex 1 et 3, (5to) 10. 40 X +  11. 40a X  + 12. 40 a X  + 13. 40 a X Z + 19. 40a X  + 15. 40 a X = et = 0
et ex 2 et 4 fit (440) 20. 30 X + 20. 31 aX + 20. 32a X  + 20. 33 42X 3 + 20. 3+  X  + 20. 35 a X  et  -0
+ 21. 30a + 21. 314 +21. 3243 +21. 334 + 21. 344 ete
Et aequationes articulos 5 et 6 identificando, fiet.
AE egminis
(7mo) 12. 20. 30  = 10. 20. 32 + 10. 21. 31  Ex quartis fit  (800) 19, 20. 20 = 10, 20. 33 + 10. 21. 32
s
3 totns
cotepsis pon
Ex quintis Terminis fis (9no) 14. 20. 30 = 10. 20. 34 + 10. 21. 33 et  pon
jam ex art 6. artic. 6 estim +) 30 = 10. 20. 31:, 19. 20 - 10. 21. Hunc valorem substituamus
m articulis (7, 8, 9, 10, Et ox artic 7 peru fiet : 15. 12. 20. 20. 34  = 16. 11. 20. 20. 32 +18. 11. 20. 21. 31 - 16. 10 20. 21 1 - 1 6. b. 1 2 1. 31
(12 mo) 4. +6. E1.
-12. 20. 20-
=10. 21. 21 a
10. 20. 21
aequatio solita haberi per alias vias
Sed succurrit omnia ordinatuis et problema propositum reduli.
ad simplicius, nempe ad solam sublationem
literarum, ordt quasdam priores seponendo, tot scilicet terminos, quod commide accidit, quia pro illis sunt
literae 40, 41, etc. et ita vel in hoc exemplo licet sari s supposita aequatione 10.40 = 20. 30, quospe
pro qua ad est litera 4e, supersunt pro reliquis item aequ fiet
30 = 11. 40 = 20. 31, : 21 et 31 = 12. 40 -20. 32,:21 et 32 = 13. 40 - 20.33,:21 etc.
et 30 =  F 11. 21. 40 - 12. 20. 40 + 20. 20. 32, :21. 2)
Imo rectius oporcet alliores literas
31 = 11. 40 - 21. 30: 20 et 32 = 12. 40 - 21. 31, : 20  seu 32 = 1. 10. 40. - 11. 21. 40 + 21. 21. 30,:20. 20
quarere nempe
et 33 = 13, 20.20. 40 = 12. 20. 21. 40 + 11. 21. 21. 40 = 21. 21. 21.30,: 203
et 34 = 14. 20340 - 17. 26. 21. 40 + 12. 20. 212 40 - 11. 21240 + 21230,:20
Et ita non est opus pergi patet enim progressus.
a
la
eaer
+  ZIaX + 22 a = 0, ad datam
F15aX 5+. 6
oet 30 haec vero per 40 X + 41 a Et hac quidem multiplicatione prodibit
possunt haberi
pro unitate 10. 40 XC
Expriore vero
Exposttriore
18. 4 a + 1. 414
20. 30 X  + 20. 31a + 20. 32a
121. 30a +21. 31 a
si egonoes finitae intelligantur, vertus +22. 209
vit iuniformita
in fine ru finem rum
duabus prioribus
. +12. 41a
+ 20338
+ 21. 32 a
+ 22. 31 a
+ 13. 41 a erc
+ 20. 34 at
+ 21. 33 at
+ 22. 32a
2= a. 134. 13HdX
ubi sepositis (aequationibus 10.40 = 20. 30, et 11. 40 +10. 41 = 20. 31 + 21. 30. (reliquae uniformi-
ter procedunt, sed quia suppono jam inventas 30, et 4 per duas Praecedentes, ideo possumus
compendio ex primere terminos quantitates 12. 40 + 1. 41; item 13. 40 + 12. 41, et ita pom.
imo ut omnia procedant rectius, ponamus pro 32, 3, 30, 31, 32,  33, 34, 35, etc,
et habebimus aequationes: 20. 3 = 52. 60 20. 33 = 53. 60 20. 34 = 54. 60. 20. 35 = 55. 60
Et ibaporro Et fint 32 = se = 21. 31 - 22. 30 75. 60: et 33 = 9 be + 21. 21. 31 + 21. 2230 - 21. 52. 60 - 20. 12. 31 + 20. 5t. 60,: 20
26.34 = 213. 31212, 27. 30 + 21. 52. 60 f20. 21. 22.31 -50. 21.53.60 + 20. 21. 22.31+ 20. 22. 22.30 f-20. 22.52.60 +20.54.60, C.20
Et ordinando melius. A 20. 32 + 2131 + 22. 30 = 52. 60 =0 28. 37 =2F. 31 - 21. 23. 30 421. 52. 60
35 = +2(7. y + 217. 22. 30 = 212. 52. 60 - 20. 2. 223] + 20. 25. 57, 60 - 20. 25. 22. 3] -= 20. 21. 25. 3,0 + 20. 21. 21. 52. 60 - 20: 2. 54602: 204
= 20. 212, 22. 31 = 20, 21. 22. 30 + 20.213752.60 + 28. 272. 31 = 207. 22. 53.60 4 207. 55. 60 
2035 - 21. _31 - 213. 22. 30 = 10. 2F  +212.52. 052. 60
Iaveniendis per 41 et 51. posito 4 et se unitatem
et 13. 40 +12. 41 sit = 53. 60, et ita porro,
e . 153
id. 1
fran... +15 21...
1+203ax
2233 234
ponamus 39, 52, 53, 34, 35, 56 etc
ut scilicet finere liceat 3E assumere 50
itaque 12. 40 Y11. 41 sit = 50 5t. 60
28. 34 + 213.31 + 212. 2230
o2no L 20. 22.
+ 20. 21. 2 
20. 22.