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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 145r, page id 00068196:0287
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homveoptota
sit aequatio assumtitia transcendens, unde ducatur d tangentialis simplex
6 = l0 + IIX + InXI + 12 X + l2Xy L13 X, + (3or 31 X,2 32 Ny + 33X + 3. 33 X y L 3. X
40+ 41X 3
= l gode + Iu oXxto A E. go XxAx + 2la A Ky qpa lil sop ly ) (  +. X. sp 3. lo a1 X gay + 2. la 1. 3 X ptx + li a. XAy 1 3. 13 1 X I
ydy
n ex
ydus
Im 1. a1
ydx
Xy
Z.I2  ln. s 2B _ 13. 13. H
1. 21
l30. 51 +3151 +31 5I 23251 32.51  3. 334 3. 32 51
gelu
)2
3251
3351
Naer aequatio 2 conferatur cum aequabione generalis quas data (ut ipsa aeq. u inveniatur)
(3 Sodrr9IXIG + SIXIx + Buxyax + 92 X ay + 82XZ +921 Xydy + 821 Xydx + 93 dy +93 X Ax
Vax ydys
a
Vix
Vdy
ay
Vax y’ay
rgdx xisdus
ubi  praefigo cum differentialis dx vel dus) sequitur potentiam allior inferiorem, alias 8.
nempe o 9npraefigo pro X ydy, quia x est superior potentia, et y inferior, habetur autem
et tespiciende patet fore 921 = 3. 93. Caeterum  in 8 et 9 ponatur latere 1. ita in S
 dia
comparae 19q.
comparatione omnium numerorum damtis praefixis Vivel 8velu, summae characterum sost fixarum
idem conficiunt in eodem termino, vel indicant servadem legis homogenectum, quod calculi
rectificationem adjuvat. Sed ut catculus sit tutior, ao hibeamus anteptraefixas, exprimentes novenarium residuum valoris
pro 80. 91. 81. 811. 92.92. 921. 821, 93.83
191-481 = 1. - 2. 14. 40 2811+ 2.f2 = 411. 41 69 21 = 3393 subil 8rb. 191 4 81. 8139. Debgr. 182. 293, b69
241
11= 880-30. 51 : 40 = 191 - 111. 40 - 31. 51, : 41. Et ex 5q. 8 fit. 889. H - 30.51. 41 = 191. 40 - 111. 40 - 31. 40.5)
fit 111  191.,40 - 31, 40. 51 = 889. 41 + 30. 51. 417: 40 111 (per X y) = 792 - 121. 40 -, 32.51, :41
a
et ex fit 1 = 191 - 481 + 2. 12. 40 et ex 5, 111 2 2.792 2811 +2. 12. 417: 41. Ex  et B3 fi C 121 = 2811.797, 2.12. 41. 3251, :40
Ex 1et B3 fit 12= 2,792-2811 - 191. 41 + 981. 919:, 2.40. 41 - 2, 41 cM p et 12 fit. 191. 40-31. 40. 51 -850. 41 130. 51 H- 191.90191 10; 240 
Ex A et 16 fit2 2811. 40-2,792. 403- 2.32. 40351 -2.191,40. 41 +2. 31, 40. 41. 51 + 2. 980. 41-2.30. 41. 511219140412 18,. 104
3: 2. 40t. Et aequando duos valores ipsius 12 ex 15 et 16. fint Z. 792. 40- X. 2811. 4072191. 48. 4 2. 451. 40. 412 2. 191 40. 41- 7. 31 40. 41. 5
1.880, 40. 41 + L. 30, 40. 91: 51 C2. 191. 40291442451. 48. 4)  - 7. 191, 90 31 + 5, 21. 40. 1. 51 + L 880, 412. ( 20. gi. s 1 1). . 1 su. 
ita ex Eterminis dx, Xdy, Xdx, Xydx et xdus habuimus valores literarum 11, 111, 12, 12) et praekrea 69.19 A
Ex X dx est 13 = 082 - 2. M. 41 - 32. 51, : 3. 40 = (ex X. 1∞) 683- 33. 51,: 3. 41  unde fit 12 = 3.682. 41 -3. 31, 41. 51
3.683. 40 + 3. 33,40.51 3: 6. 41
121 7 293 - 33.513 : 41  (ex Xytant XI) 1841 - 4683,  2. 33 5tj 
Ergo ex  et 23 fit 2, 293- 2. 33. 51 = 1821 - .683-2. 33. 51 fit 2. 293 +6∞3 = 192)
aequando duos valores ipsius 12 ex 16 et 21 fiet , 8. 191, 40 cu . Bii. 10. g2. 51 -  680. 412, . 30, 4. 51- frigi 30. 2l  is e
(26)
082. 46. 4]- 32: 40. 4151 - 683,407+ 33. 40.5)
Et aequaddo duos valores 121 ex aeq. 17
et 22fit. f 811. 40 H - f79C q. 2t f. 3i go. n. 51 f. 191 40. 11 4 f. 31. g0 4(. 31 + L. 58D. ~ - + 38 4(51t L. 191 46. 4 - f. 481 40  9, b) 1 18,
Itaque invenimus valo Itaque ex decem comparationibus invenimus valores literarum
quaesitarum 11, 1I, 12, 12Y 1t _ (9. 7.) _ III (aeq1o) ret (qC
quaesitarum V, 11, 12, 121, 13. et determinaves datae ad comparandum per aeque, 6 et 25.
et denique supersunt adhuc aequationes tres, nempe 18, 26, 27 in quibus non amplius
Extant literae u, 11, 12, 121, 13. et ope harum trium aequationum resoli adimpletione,
habemus adhuc literas affecta 3 30,41, 4 30, 3) 3233, 40, 41 0x quibus tamen
(ue
valent tantum quatuor. Utins pro inveniendis de, 31, 32, 33. considerndo 40, 
sinfinge4041, ambae literae in eribus residuis aequationibus destrnentur et ex aeg. 1 fiet: 2792 428H gng1- 41. 51
porro est in nost a potestate salva gegervitate caccere ut sib  40=i
Etiqita in aeq. 15 e vanescili.
nec datur ejus valor, sed aeg. 15 et 18 e0 redeunt, ut sit 4= 2. 792 - 28113: 481- 191. Inventu ergo1, absolute jam
440 + 30s
restant aequationes 26 et 27. in quibus s jam hibitas 40 et 41 relunquemus sine explicatione, ne examina
r ex per abjectionem novenarii, et numeros homogeneitatis indices turbentur. Quaeramus igitur quantitates 
on endo aegq. 2b et 27. destruentur mutuo quae duobus punctis signavimus et fiet 2811. 46. 1- 79230. 41
082. 30. 41 - 683. 40 + 293. 40 et fit 41: 40 = 683 - 293,:,092+792- 2811
jungendo un aequationes 28, 29, 31, habetur nova determinatio et sit 682+192 - 2811. E792- 291 = 683. 293. 4 81 - 191
Resdat adhuc una aequatio cujus ope quaeratur si placet, 32 set siec 32 = to81. 48. 91  -4 P. 51. 30 + 4091. 3131 + 40 3 51. 33
13.40
equ querit i vulac 2 cpsidea 082. 45 4 - 683. 4 1880 N 491 4 
quia 40 = 1. ideo in valore ipsius 32 avanescet 191. 40. 414 191. 40. 91
8o 1g1. 40. 41
+Y890. 413
A.. +191.40. 41
481. qua
403.41. 51
conjungendo determinationes seu aqu. b, 25, 32. ex 6 et 25
fit 2.692 +3. 683=31821. Ergo ex ag.32 fit 082 +792. 2811. 1. 192- 3. 281  3. 1821- 7. 6921 . 481 -19
aequationes 49, 32, 35. non servant examen  uam pendent exassumtione aequationis 28. caetercese
eadem formam habiturae
afficit jam hic calcalus pro aequatione data, qua uti quam tentare debemus ita t
9 seocerit instrem calcutus, aliique elici camones si pontmu 121, 13, 33 esse 0. et ita natum aeq. 2. mntriplicemus per 10 471Y
ermare, ut praestentur aequationes 6, 25, et 32
ad vindem vuadium exattetur qui atae mnro us d paa. on ta dintite veteritines moestgenus