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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 209v, page id 00068196:0414
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10 a + na X 1 tanx 1 17Y + 1X et 20 a 1 2t aX ManXX 1 23 aX + 24
communem. is sit  a 3ad +31 a X + 32X X
debent habere divisorem
D Dividatur  20X + 2 ax3 + 22 aaXX + 23aX +24 a per 10XX+ Maxtnaa
 t proditit 20 1oxx + 31aX +g2 33 + 34 at. et si sucedit divisio fit 33 = 0 et 34 - 0
10
 10XX + MaX f12aa
fit 10.20 + 10. 20 10. 31 L 10m - 11. X - 10. 32 - 10. 2 - 11. 31 +n. 30 10.35 = ,0. 25 - 1132 + 12. 31  131 = 10. 24 - 1. 3 + 12 3t -.
est autem 37 ∞0 et 37 20 fiit 1052y - 11. 32 +12.3120 et 10. 24 + 12.32 =20 x, fit 32 = - 10.24: 12 et 3
rt 31 = , 10. 12. 23(61 0. lt. 24; : 12. 12
= 10. 22 -1. 31 + 12.30(3:10
Ex 1, 2, 3, firt 30 = 20 31 = 10. 21 - 11. 20, : 10 et 32 = 10. 10. 22 = 10. 11. 21 f 11. 11. 20 + 10112. 20, : 10. 10
32 = 10. 24: 12
et fiet tandem: 10. 12. 12. 21 - 11. 12. 12. 20 = - 10. 10. 12.23 (10. 10. 1.24 Ft 10. 10. 12. 22 - 10. 11. 12. 21 + 11. 11. 12. 20 +10. 1215 20 = 13. 14
Ex Ad3 fit..
31 = - 1 12. 23 - 10. 1. 247: 1212
+10. 12. 12. 21 410. 10. 11.24 =0
+12. 10.10. 23
12. 12. 11. 20
1oxX Fuax tiaal
IOXP + 1ax + naa
20X +aX +2aaXX +23aX +24 a
(10 xx + 11 ax + 12 4a)
gecap-  aum
= 10. 1. 21) = 10. 12. 217
+y. ll. 207 + 11. 12. 20
010
1010
= 10. 10. 11. 227
+10. I1. 12.20
+ 10. 11. 11. 21
= I1. 11. H. 20)
103
+13. = 24 = 10. 11. 12. 21 20 miror in his duabus aequa-
+ 10 12. 72 120
4 H. H. 125
tionibus nonprourilagem
121 - 11. 20 ax + 10. 10.22 - 10. 12.20 + 11. 11. 20
20 xX +
= 10. 10.n 2
+10, 12. 12. 20
410. 11. 12. 2
=II. 11. 12.20
103
10. 10
fiet
10. 41. 21
213. 23 - 10.10. 11. 22 = 10. 1012. 21 + 10. 11. 12. 20 = 0
+ 10. H1. 11. 21 =H1. H. 11. 20
et 1824 - 10. 10. 12. 22 f 19. 11. 12. 2 + 10, 12.12.20 =0
= II. 1I. 12.20
-10. 12. 20 = 102 14 = 10. 24 - 10. 10. 12. 21 + 10. 11. 12. 20
Hinc 10. 10. 220
= 10. 11. 21
+11. 1. 20 seu 11
D 11. 12, 23 - 10. 12. 12. 21
seu oen eite dietus scunt res
paec aequatio servet legem justitiae
Hic valor substituatur in aequatione e paulo ante posita fct vel potius in
10.10. 24 - 12. 12. 20
Ejus ope tollatur n, fiet 11 =
1010. 12. r2. 21. 21 - 10, 10. 12. 12. 21 21 f 16. 24 - 1p,10. 2. 22 4 10. 12. M. 20
12. 20
4. 12, 12, 20. 20 4. 12.12, 20.20
10. 12. 21 + f(10. 10. 12. 12. 21. 21 = 4. 10. 10.10. 1. 20. 24 - 4. 10. 10. 12. 12. 2. 22 + 4. 10. 12. 12. 12. 20. 20)
illa quaeratur _ est 1. 11 = 10. 12. 2L 11 +
12.20
Unde fit: 2. 10. 10. 12. 12. 20. 23 - (2)10. 12. 12. 12. 20, 2 = 10, 10.10. 12. 2. 24= 10. 12. 12. h. 202D, + (V. 1. 24 - 12. 12. 20) Y (1010 et)
e
mo
2. 12. 20
et fiot 11 = 
= 10.10.12. 22. 10. 244. P. 1031. 22. 10. 12. 20. 24 -. 18.12 22. 157.28 + 19, 12. 21. 10.10. 12. 23, 1070.24 L10. 12. 21. 1010. 12.
+10. 24, 110. 24 = (2) 103. 24. 70. 10. 12. 12. 20. 24 (+ 103. 24. 12. 20. 10.)
(t10. 12. 21 ( (9. 12. 12 21.12. 12. 207 = 10. 12. 21. 10. 1212. 21.10. 10. 1 =0
H = 12. 20. 1816. 1. 25 fp. 20. 10. 12. 29. 29 - 1 (1 2. 20. 172
(+10. 12. 12 2010.24)-(2) 10. 12p2 20. 10. 12. 20. 24. L. 10. 12. 12. 20. 11.10
2 42  1. 24 - F 10. 12. 10. 24.  - 18. 1. 22. 24 +2. 162 15. 20 2. 24 + , (, 1 23. 24 -10. R. 2
WV
4 14
 20
+18. 7 21. 13. 20 = v. 12. 25. 26
Hic  inter 12 et 10 non obserratur lex homogeneorum justatiae, deest sed in teliquis excepto tomino, nam deest 23. 24, quod mitor
=13. 125. 22. 20
pro 10, 11, 12; 20, 21, 22, 23, 24
= 18. 12. 25. 24 Unde habetur 22 rationalites
d = V. 18. 203 = 102. 12. 26.24
Et divisis omnibus per 10. feliciter prodit aequatio in quae servatur ex justitiae pon
9= +106.243 - 18. 12. 20. 4, -10,12. 1. 24 + 2.103.13.20. n24 + 18,. 12. 21. 27. 24
+ 10. 15. 2. 23. 20 = 18. 123. 23.20
itaque si  sit 20X + Y ax3+ 22 a X + 23 aX + 24 a divisibilis per 10XX + 1ax + naa.
+12, 203 = 18. 127. 20. 24 -10. 135. 22. 20
habemus aequationes unam O. quae dat valorem rationalem ipsius II, siune 22
D1. quae dat valorem rationalem ipsius 22 sine I1
atmm  esset lex sustitiae si
 dt non esset aequatio
alteram vero
plnuntu lo enim respon dentium is in priore manet endem aequatio, sed in posteriore
steque d) servab legem institiae formularem.
etiam eadem formulat
s