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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 14r, page id 00068196:0027
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sit aequatio differentialis data scrimi gradus: dus: dx = 50 + 51y,:, 60 +61y. Assumatur
alia quadraturabilis dae: dx= 20 posito 20 dari utcunque ex sola X. sit  = 30 + 31 Y + 3241: 40 41
per compendium (4t)  = C: D Ergo d0 = 5dB - 0 +,: 5D. Est autem d0 = (t) dx 40 + dy d dx assec. X)  + du alccu)
et similitur d9. Et fint du (7mo) =  dx D  sec. X) O + ay D d (ser. y) C) = dx O d ec. X) D - dy O  sec. 9) 5, : 37
Et d (sec. X) D = (8V0) 430 + d31. y + d32. gy Et d ec. y) 0 = (9no). 1. 31  + 2. 32y. E+ similiter
 (sec. X) y = som) d40 + 4 41y
Et 4 (sec. y) D = (imo) 1. 41. Uend tandem explicando tg. 7. fiet
jam ex 2 est du ax 20. Ergo hunc valorem aequando valori in aeq.7. explicat fret
amo
dx, 2057  = 8 d (ec. X)D + C ( sec. X) D dy, D d (sec. y) C = C d (fec. 7) D. quae explicendo per 3. 4, 8, 9, 10, 11
fiet 4 20. 40+ 2. 20. 40. 41y + 20. 41. 41 9y
-, 430. 40 + d31. 40 y (+ d32.907y
+ 430. 41  + d31. 41. + 452. 41y
+, 30 d 40 + 31. 440 y + 32. 44o yy
+ 30. 4411. + 31. d41 .. + 32. 441 y3
(4, 1. 31: 40 + 2. 32. 40y
(+1. 31. 41 . + (232. 419y
=, 30. 1. 91 + 31. 1. 91) y (F 32. 1. 41) 9y
jam aequation data 1, multiplicetur, per 10 + 81y tam numerator quam nominator et fint
50. 80 + 51. 80y
(14+0) dy
1B. 20 est dimensionis nullius, n est dimensionis, uno minus quam nulius it  2n sit conti 4)
(60. 80 + 61. 80y
dx
(15to)
dy
+60. 81.. + 61. 81yy  dx
+50. 81. = 51. 819y Xy
qui duo valores ipsius dy: dx ex aegg. 14 et 15 sunt comparandi inter se, et per y numeratorisfint (16to) 41d32 = 32441
seu 432: 32 = 441: 41 seu log. 32 = log. 41 = 5 seu 41 = (17mo) 32n. posito n esse numelum constantem
Nunc incipiamus a nominatore, quia ille caret differentiis, et quaeramus 80 et 81, fintque (18B0) 80 = 1. 31. 40 - 30. 1. 8  6b, et
(19no) 81 = (2) 32. 32n(= 32. 1. 321,: 61. vatores Htini ratores ex 18 ot 19 substituendo in coefficiente ad y nominatoris fiet
69. 61. 3. 32. 32n - 67. 61, 52. 1. 5en 61. 61. 1. 31. 40 -61. 61. 30, 1. 32n 4 60. 60(2732. 32n(60. 60. 32. 1. 321- (20mo)
60,61, 2. 32. 40 +60 61 1. 31. 32  60. 61 2 n. unde (21 mo) fit 40 = (60. 61. 31. 1. 32n - 60. 61. 1. 31. 321--6, 61. 30. 1. 32p
460. 60. (4) 32. 32n (60. 60. 32. 1. 327, : 60. 61. 2. 32.90 = 61. 61. 1. 31. sture veniamus ad numeratorem
ibi per y est 2d  (22d0) 20 = 50. 1. 31. 40 = 50. 30. 1. 32n + 60. d 30. 40 - 60. 30d40, : 60,40.40. Unde
per y. fit 61. 2. Y. 50. 1. 31. 40 ~2F. 50. 30. 1. 32n +2. 41. 60. d 30. 40 - 2. 4r. 60. 30. dq0 + 60. 40. d39. 40 - 60. 70d304
150. 40. 31 d40 +60. 40. 30 d 32  51. 61. 1. 31. 40. - 40. 51 61. 30. 1. 32n + 40. f6050. 2.32. 32n - 49. 60. 5032. 1. 32n
si esst semo firet 20 LL, 30r+0. Et ex 5 fl. 61. 41. 41. 50. 1. 31.90 - 61. 4 fJ. 50. 30. 1. 32n +61. 41. 47. 60 430. 40
=61. 41. 41. 60. 30840 - 6+ 6 61. 60. 40. 40. d32. 40 - 61. 60. 40. 40 L 31. 3, n +69. 60. 40. 40. 31440 +6,7. 60. 0. 103 pd 3
60. 40. 40. 51. 2. 32. 32n = 50. 40. 40.51. 32. l. 32n. super sunt exg0, aegy. 2. 23. 24
si in datis ponamus esse 50 = 0. fiet ex 22, 20= , 30:40 quodsi etiam 31 simul esset, 0. fieret
Et ex y in numeralore fieret;
eitu
q0= - 61. 61. SE30. n + 60. 60. P. 32n ,:2. 60. 61.
fcee nbdus
2. d (30:40) 40. 32n 30 32n 30n d32- 51. 39. 32n: 60, et ex yy in numeratore
fieret: (30: 40) 35 n d32. 40, +32d40 =51. P. 32.32n  : 61 + 2 n 6o (432)
d2o: 402 40d30 - 30440: 40 40=  6e n 32 = L S1 n 30, 440 = +n d6e X32 - = n + b 20
Ergo d (30: 40) = 40430 - d = 30 460 3 n 30 + a 30 :130 s8, ap + 1 n 306 d3: 40
(30:40) = 2n 60 32 d, 8 - Ln 51 30 d 30 - Ln 30 d 32 + Ln 30 d 6, o, - = n 3060 432 + L n306 d3 
4
compendii causa, sit sit seu oin) In Ce, 3at3o 3od3r