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# Title: Leibniz-Handschriften zur Mathematik LH 35, 7, 14
# Folio 1r, page id 00068085:0001
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de inoestigandis figurarum quadraturis e
ordinariis, quando sunt possibiles.
 aequatio (11. y + 10 20
 ubi  coordinatarum minus ascendens y, endats
subsistat in primo gradu: sd lituae coefficientes 10 et u involvent alteram
coordinatarum X. Et 10 = 30 + 31A3IX eu et nL qurqequ etc. sit dy: dx e
ifferentiando jum aequationem 1. Ht dd10 + du. y 80 unde tollendo y. 
fiet  10. 21 - 11.20 =0,  et in ea 20 = dio +ne et 21  d1. et fiet: 110d11 = 11. 11e = 0
Naut interpetenur per aequ. 2 et 3. est protera a pono dio - 1. 314 2. G2X + 3. BX 14 34X e et
d11 = 1. 41 + 2. 42X + 3,44 X + 4. 44 X et. Ergo ex ∞q. 9 explicando fiit:
+ 1. 30. 41 + 2. 30. 42X + 3, 3093R + 4. 30. 4aX et ino   =  40. 40 + 2. 40. 41X + 2. 40. 4 2X + 2. 40. 43. + 2. 4044
E1. 31. 41 +2. 31. 12. +9. 3. 43.
=1. 32. 41. + 2. 32. 42
- 1. 31. 40 ∞ 2. 32 40 13. 35, 40 - 4. 19. 40
fit + 41. 41 +2. 41. 42 42. 41 13
etrs   eseu 2 + t=Oe
4a it datr aequatio:
1aora  oL 1EX ex = not MIXrBX e
cgo 4B+hD=
fiet 28 + hD ODe et 40= 0gC Ergo
=1. 30, 41. 110 +2. ZC AENoX + 3,30. 43. 1OX +4. 30. 44 110X + 5. 30. 45. 110 X  50. 46. 110X Ck
1. 30. H11 d  2. 30. 2. 11 0 + 3. 10. 42 1. 0 +g. 1
+1. Se ALI a+2. q aune. 2o f q. M. ta o. eu. e
41. 10 a. 113. f2. 1
1. 31. 1
= 1. 31 40110 = 2 BE. gonco X. 3, 31. 40 (OX - 9 14. 90110YX - 5. 35. 0 110 X - 6. 16 o11B. X
iplicau L , 3L. A. 1 = 2. 3 40 poX
pb. 3o0.g1i 40. H sil e 3. R. H:, 4dta r mt  = l. 51. 48. 11 0 - 2, 31. 40. 111 a 3. pp. qu 1(1 1 = 4. 34. 4e llic
1 H ag2+ t214
etto reoqu
=151. 72. 12. - 5. . qus2. - 3. 37. 4 12.
ne
equ (16)
+ 40. 40. 100 + 2. 40. 11. 1or ft 2. 40 92 BX + 3. qu. 43. loB + 2. g0. 4 4  o X +2. 40. 45 1VX
4 40. 40. 1010 (2. 40. 41. 101 , +2. 40. 2. 101. +, 40. 13 101. 42. 90. 44 101.
= 40. 40102. + 2. 40. 41. 102. +2 40. q2102 + 2. qu. 3 102
+ 40.40. 103, +2. 40 4 103 +2. 40. 42. 163.
comparando, fiet ponendor gu et co esse datas aequales umilutiu 
 312  -30  1oS, 130. 16 31 1 = 100 +10G 21O seu 41 = u0 3t . 100. 30. jem ipsas 100, 11, 112
Sed hic eundium per gravus primum enim nonamus extare 30; 40, _, non ultra, et fiet 116 = 40. 40. 100, 30. 41- 40. 31
comparando inveniamus
100, + 101, 102, et
si extaret tantum 4 30, 31; 4D ghecet e = 430. 41 + H, :, 40 + 2. 40. AX + H. 91 X
i extucent tantum 30, 31, 3; 40, 11, 12. ficet C = 139  -2. 3
sit jam ( C = a:, 20 +21X +22 X, fi et 20; 2, 1 40 45. 1  3 
rt 2t 12 a2F
-2.3. 4 adeoqe e = a 40 + a 41X + at2X:, _10 r 10 9 X 
30= a 4 + 2. 3240, : 2. 42 fit 24 40. 42 +231 40. 92= au p2n. teg
a 412 (+2. 32. 40. 41) -= 2a40. 42 = 2. a40. 42 +240 (2. 32. 40. 4D seu fit 41 = 4. 40. 42. seu 21 - 4 6. 
31 = a 412 + 2. 3240. 41 = 2a40. 42,: 2.40. 42 31 = a42 L 71 41  +32. 41, :, 42.
et dantur 40, 41, 42. restant 30, 31, 32. X 30 = a 40+ 3140: 41
  Ergo e = a:, 20 + 2. 190. 49X + 22X. sed ita rursus nominator est quadratus.
comparemus cum e = 21 +22X + 23X + 24X + 2GX,:, ( 40r 4, HX = 3E fit 30. = 2 +31 40,: 4 et hoc 121. F2. 31. 40. 12 - 2. 32. 40. 41
(+2. 30. 42 X 1 + 3.50. 43X 333
-2. 3
Se ex laret tantum 30, 3133 33; et 4. 4142, 43 fieret e - +3
= 22. 41 seu 31 = +22. 41 + 2. 32.40. 419: 40. 42
. 40. 40. 41. 42 + 41. 42. 22. 41-2.41. 12. 21. 42 +
3. 21. 40. 42. 4313 31. 40. ate    q0. . hitart
91. 2 F  40. 41. 42 + 7 33. 49. 40. 41.42,: 40. 41. 42
31. 40 91. 21. 42 = 40. 41. 42. 23. Ego 3
2= Em 32 = 4. 21. 42. 43 - 2. 22. 41. 434 29. 40. 425e + A. 33. 40.41. 427: 4. 40. 41. 43
et ex Xt fit 32 = 25 + 33. 42,: 43 et fit 4. 21. 42. 43 - 2. 22. 41. 43 4 24. 40. 42. = 4. 25.40. 41, 5, (1340. 41. 42( 3. 40. 41. 42) =33
+241. 9240. 41 a
Ergo 32= 43. 2 41.
Etrursus +9. 21. 40. g2.43 - 3. 22. 40. 41 453 + 13. 40. 91. 42. 43 - 25. 4041. 42,:, 40. 41. 43 - 3. 40. 41. 42. 43 = 33
43 - 22. 41. 41. 43
14. 21. 41 4113 - 2 u. 41. g2 4 + 24. 70 M 5 -4. 25 a0 31   32. 
q0e aytars. 1 nalt3. 3 aeid.
21. 40. N. 45 427 2.40. 1 47 57
+15. 2 avin 15 - 9 12. quaits 1 23 4 4. 42 13 + 24 40 AI. 4Z + 2. 25. 96. 4(. 42 = eA. et  aeres
Et 2........
3. 24. 48. f2. 43 -7.425. 48 41. 4.
   iuo
(Ex A. B. 0 ost =,  C 10 matm ese 1 . 2 B t qea
322. B 28. 4o. 1 3+ 291. H. G 3C. 5. Ap. 1
10, 840
mu ultina melioiem quam
hactenus viam sum ingressus
Hauui, sed circa eam fui bagatus
impatientia prosequem
Nom angpusa eq. 16. suficit oeuir
sen valorim ipsius edatum iu
derivari ex est con fugere
iri us  t  abaant
ob 30, 31, 32; 40, 41, 42 fue. 31
Ob, 30, 31 N2 33; 40, 41, 42 asto 1 2. 40. M X e. 4
uo isiit i i . uarini
2243. 2. D- 4. 121. 42
2.4  3. 4 = 4 40. 4