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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 2b
# Folio 83r, page id 00068198:0165
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Sit aequatio generalis ec non excedens affectionem alterutrius ordinatae duplicatam
o= 12 + 15y + 18gy + 2idus + 20ydus. Videamus  quio requiratur; ut reduci possit ad quadraturas
ope generalis proxime inferioris, in qua nempe affectio alterictrius ordinatae non est nisi simpler
quam nempe jam reduximus ad quadraturas. Assumatur O= 52 say + 5oyy et scribatur 600 + 70dCb
quae aquatio coincidentia aequatom 1. multiplicatae per 30, explicando autem O fiet
70 dst aa +70 dsj ay + 70 dso yys +70. 5ta  + 2.70 5oydy  30. 12 aa + 30. 1ay + 3010aa yy + 30. 21 a dy + 20. 20y dy =0
60. 52aa + 60. G1 ay + 60. 50yy
Exohus 70 = 20. 20: , 2. 50 = 30. 21: 51. Ergo  51: 50 = 2. 21: 20. Ex gy fit 30. 20(:2. 50) d c + 60.50 = 30. 10. Ergo
fiet 60 = 2. 2030102. 10.20. 50 = 20. 30d50, :, 2.50. 50. Ex ay fit 20 30. 20(250) d51 +60. 51 - 30. 11
seue 60 = 2. 11.30.50 = 20. 30d51, : 2.50. 51, et pro Si substituendo valorem 2. 21.50: 20 et pro d51 substituendo
Itidem valorem 2 d (E1: 20)50 + 2 (21: 20) d50, ex novissimo ipsius 60 valore fiet 60 = 2. 11. 30.50-, 20. 30 d(E1: 20)50- 22030 (21. 20) d50
: 4. 50 (21: 20). Ilaque duos ips ales quem ipsius 60 valorem aequando cum primo,  jam invento, fiet
2. 10 (21 : 20) 30.50 (= 26. 30 d5D) = . 11. 30.50 = 20. 30 d (21: 20) 50=20. 30 (21 : 20) d50), et ita evanescit 50.
ostau et fit 2. 10.(21: 20)  = 1 = 20. d(1: 20) est autem 0 d (21: 20) =(970 d21 - 21 d20,: 20 E0)- Ergo fiet
de 51. 71 aa + 60.52a a 770 d52 aa = 12 aa
aetere
21. 20 sit constans, debet et 11: 10 esse constans 3 21: 20 Item
tandem 210 e + 20do 2. 10, 21 = 1. 20 = 20d2) + 21d20
si I1 esset = d21 foret et 2. 10 = d20 sed ila facilis adiunde esset summatio
80. d13
Idem jam tentemus alia ratione, loco  sit O = Siat 50Y et facis fiat 60 +70BdOraa
s
O0 = 51. 51 aa + 2. 51. 50 a y + 50. 50 yy
20d0 - 2. 5i d5 lar L. S dsoay + 2  dro pu t2s medis ho nyay
25045lay
unde facile habetur quantitas arbitrara 61. 71
qua explicata coincidentietur cum aequ.d multiplicata, er 3o fiet
=o coincidenti 30. 1 aa + 30. vay rio 10  gy + 30. 2 alys - 3020 pay -0
q. 90. 81. S0
Fiet 20 = 30. 2 20. 30:250.50. Et rursus 70 = 21. 38:, 2. ∞051. Ergo 20 (30) ( 6d 51 = 21(35)(2)50(50) seu  51:50 = 21 : 20, vel
+60. 51. 31 aa + 2. 60. 50. 51 ay + 60. 50. 50 y)
2Y0 SIdst aa + 2. 70. 51 d50 a y + 2. 70. Godso py + 2. 7050 stadly +2. 70. G0. 5o ydy
Ex vy fiet (20. 30.59, 30)50d5o + 60. 50 = 30. 10, et
fiet 60 = 10, 30.50 = 20. 30d50,: 50 superest ut conferamus ay fiet (190. 50S1) (51 d50 +50d51) +260. 53 (2. 20)
290. G0d51 ay
S1 = R50O 51 = 50. (21:20) et d51 = ,50 d(E1: 20) 4 (21: 20) d50
fiet (20. 35: 5050) (50d50 (21: 20) + 50 d (21: 20) + (21. 20)50d50) + 60. 50 (21 : 20) = 11. 30
unde fiet
seu . 3 d5ff 2. 21. 30d50 +50  20. 30.50 d (21 : 20) + 2. 60.50 (21: 20) = 11. 30.50 Unde fee un
60 = 1I. 302.50 = 2. 21. 30d50 = 20. 30.50 d(21 : 20), :, 2. 50 (21: 20) quos duos ipsius 60 valores primum, d hunt
aequando fiet 2. 10 50= 2. 20. d5.  2. 10 (19 (21. 20) (2. 21 d50 = 11 (50) C 2. 21 d50 = 20 (59) d (21. 20), quae 4
Si esset O = 5( a + 5by, :  41 a + 40y et esset aequatio 600 + rod0 + 80 = fieret
quatio coincidit priori determinatio jam inventa
+ 80. 41. 41 aa + 80. 2. 41. 404y +80. 40 401y
+ 60. 41. 51 ad
+ 70(414. 451a
70
+ 60.541. 50ay +60. 40. 50yy
+70 41a deoy +70 40450 yy  + 41a50 dy (+4
dElatody
50. 40yd9
+ 60240. 5l ay
+ 70. 40d51 ay
-70 S1a d91a - 70 7 51 adg07 70 50d4oyy
tians formare datam aequationem
ubi 60,30, 80, 51, 50, 41
sunt 6 comparatitie
eum tamen sint tantum
 compaatationes
ut si sit 12 aa + 100B + 2 d - 0
4. faciamus
et fiat  = 30a + 201: g0 ubi qu ostris solim ad supplendu t in purgenhitali 50 .    aa + aB+ od  E4
180. 41. 41aa 4280. 40. 41  ay + 80. 40. + y7 + 70. 4510 gdy =0  con 12aa Fuay f 109 + Eta a -
60.40.50 .
70 d (9050)
-70. 50ad419
praetereplicet etiam
+ 60. 4501
20= 21. 30: 4510 90. 40 +60. 40. 50. + 21. 30 d (4. 50) 4516 = 40. 20
2.80. 40. 41 +60 4901+ X (41. 30 r 40. 31) 2130.: 4510 = 30. uay m CC
duabus agy. dantur 80 et 60
+70 d (41. 51) aB + 70 d (11 50 + o 31).
+60. 41. S1 ala
a
e
 316 +321
10 11.6 + 12. 7 + 138 =0 20+ 116 + 22.7 + 23,8-0