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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 320v, page id 00068196:0636
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Ad promovendas Quadraturas magni momenti foret, reducere (do (ovab, i, von ac) ad rationale
quod multis modis tentavio nec tamen satis hactemus successit. Tandem placuit hec via:
avv- aab, a vv =aac aequand. Oto, quod esse poterit 0OY 8 DD, si sit: avv - aab = 008
et a = auc = ZDD. sint l, mn, indeterminatae simplices pendentes ab x, velut. X +..
Faciamus V = 12 Am + 123 Imn fiet v = 112 lm  + 2. 112. 123m sIn. sit  = 204p + 213In
sit 0 = 310 sal + 303 fan fiet 93 = 315 al + 2. 310. 303 a sIn, et 00G = 204. 310 alpfr 213. 31 al Iln
Ubi notandum Aumerorum fictitiorum notas initiales, 1, 2, 3,  esse ad hibitas ad indicandos.
quatuor aequat valores faundamentales assumtos quantitatum v,
204. 303 arpZ 213. 303 an
2. 317. 1 9. 3 5a  2. 204. 310. 305 ap..
204. 405 anp 213. 403 an
8, 1), D. Et quidem ac  ini 2. 13 10. 4ch alm 2. 204. 40ta ap.
3 ac 4 initiales  eodem se modo habere. Porro  post initiales potas, eas
quae secuntur indicare 1, 2,   literas l, m, n, p. Et 1 et 3 prorsus ut l et n se habere eodem modo, sed 2, interponi prom
cum abest m, loco 2 ponitur 0. id tantum interest inter l et R, adeoque inter + et 3 post initiales quod l’etta praeponitur
ipsi 2 ve m et n, sed n postponitur ipsi L et m; eodemque modo 1 praeponitur iplis 2 et 3, sed 3 poseponitur ipsis 1 et 2.
omplremus jam primum
ob cum  valorem ipsias avv- aab cum valore ipsius ODQ, et similiter
Hinc 1, n se habent ut iuc Compare R
Coincidentiemus jand
coincidentiemus jam valorem ipsius aVB-aub cum valore ipsius 0O_ secundumq irrationales
Ex rationalitus ob sarv- aab = a 1D Z piet  1  Im = ab 2 204. 1 6. f ip + 2. 1. 310. 303 In
lavo -aac = ZDD fiet 112 Im = ac = 204. 410. Ip + 2. 217, Mo. 403 In
ope harum duarum aequationum n quaerentur m et p et primum quidem Tollendo (si placet aequationem
ab aequatione 2, tolletur m, et fiet 204 (4102-310) 1p +204 (903 -303)n p  2. 213 (410,403- 310. 303, In + a (c - b) = 0 seu
p. (312. 213 (310. 303- 410.403)17 a lb- c),:, 204(410- 310) l + 204 (405 = 303)n. Addantur jam in unum aegq. 1 et 2. et ex summa
quaeratur m, fiet m L 204 (410 +318) Ip + 204 (403+303)n p + 2. 213 (310.903+ 410. 403) In + a (b+C):, 212l + 13n
avv- aab = O04 fiet 2. 112. 123m = 213. 310 al + 2. 204. 310.303 ap
Exiralioraibis db q avv. - aac = QDD fiet 2. M. 123m = 21. mo pl + 2. 104. 41040) ap
unde ubi rursus inveniri poterunt m et p; et quidem subtrahendo aequationem 5 ab aequ. 6. ut tollatur m, fiet:
p (7) 213 (410 - 310) + 213 (403 -305)n, :, 2. 204 (310. 303- 410. 403), aequando duos valores ipsius p ex a99. 3 et 7, fiet
213 (410 - 310). Il + 213 (403-303) nn +. 213 (410- 310) (403. 303) 4. 213 (410. 403-310.903) In = 4. (10.903- 903. 310) a (b- 1) =0
( in jam (418. 403) 4103 303=- 310.403 ft 310. 303)-918. 403 318. 303 + 4. 410. 403. 310. 303 - --(2) (1B. 403 -. 11030) - () (+0. 303- 43. 1 1)
2. 2132) (210. 403- 310303) In +2 (410. 303 - 403. 310) a (b - c)  0
Ex aequ. 7. cum sit simplicior quam prior ex aequ. 3.
2. 213 (2) (40.303 -403. 310) Ln
substituemus in aquvalore ipsius m aqu. 4., et firt m = 213 (416 - 308) H -+ 213 (410 4916) (405 - 305) In 4. 2B (410. 403 - 310. 305) In
Porro ut rursus inveniamus m Praeterea novum valorem ipsius
(3 (405- 303) M 213 (405 +305) (410 = 3102) IN
jam 410. 405 (= 410. 3032 4 403. 910)- 318. 303 +
qu. qui 4032. 310 + 410.303)- 3101. 305 = 4. 410. 403 +404. 315. 303 est 2319. 303 -2. 4102 4032,
2. (410. 403- 310. 303)  a (b+C), :, 2. 1 l4 (310. 303- 910. 403) (1 2 l + 1231)
m 213 (410 -3107) l + 213 (4037 - 3057) nn 42. 213 (410. 403 - 310. 305) In =2 (410. 903 - 310. 303) a (b+):
4(310, 303 -410. 403) (113 l+ 123. n). Rursus ex 5et 6 in unum additis fiet: M = 213 (21543,5) fl + 213 (405 4 3032) an  2. 204(40. 40 3 + 310. 3) B.
: 4 112. 123. Et pro  substituendo valorem ex sq. 7. fiet m =(213 (41 + 315) L 4+ 213 (405. + 305 )n B 10. 303. 910 403) 2 3 (a p0 403 + 316. 303 (40. 310
34 4 ) 213. 46_310 303(213, 910. 10. +o E D + 217. 315. 3a 3o H 2336: 40. L3l
+213. 416. 410. 403) + 213. 410. 319. 30) - 213. 310. 410. 4u3 ( 31 . 213318. 310. 303C F 213. 405. 30. 203 ( 213. 4032. 91. 403  + 213. 30 f. 20. 30113  13. 303. quo tin
F213, 9u. 410. 403 + 213. 4032. 310. 203n - 213. 303, 410. 403f213. 3032. 310. 303n), :, 4. 112. 123 (310. 303-40. 903) vel coutre sive
4 213 (410. 403 + 310. 303) (403- 303)n,:, 4. 112. 123 (310. 303- 410. 403)
M 2+ ∞, 410 31. 303e 213. 410. 310 (410. 303-403. 310) l + 213. 403. 303 (403. 310- 410. 303)n  M 12 (2:, 2. 112. 123(310. 303- 410. 403
Quos duos ipsius m valores ex aegg. 9 et 10. equando fre + 112. 123 (48 - 21 )I+ 112. 123 (403- 303) nn-  112. 123 (4V. 405- 315. 305 )1n - 2. 12 1 9(0 4oh 310 3os seb HC)
=2 2. 1122. 410. 310 (410. 303 - 403 31) l. + 2. 1292. 403. 303 (404. 310- Ho. 303) nn +, 12. 403. 303 (403 310 - 0. 305) In + 2. 125,. 410. 310 (40. 1 0) 40z 310)1 12
de denique (112. 123 (410 -315) -2. 12 ( 410. 30 (mo 303- 40 3. 30)) l + (112. 13 (483 2 90, G - 2. 143. 403, 303 (90 4. 910 - 1o. 103)) MM =, 12 1 3 (40. 403 - 210. 305)
(410. 403 - 310. 303) a (b+C
I 2 coincidentiandae jam sunt duae aequationes
uae aequationes quadraticae ad easdem binapliteras sint coinci
Li mo i eoio to. vom elea
dentiande, absitque utrobius utraque indeterminata simplicis gradus ut hic l, et n hon posse eas habere unam tantum radicem aequalem, sed
as io. 403- 303
secundum l et n nempe 8 et  H 12. Ubi notandum si
iicum
seu
Embas necessero coincidere identificari, adeoque terminum termino esse coincidentiam
utrobiu sit Unitas. Sed praeterea opus
tervinum suum ultinum seu alsolutum, itas inmentibilem, ad alives l’et n, cr dudeutur ex se invirem
610. 603 =   ito
tamen ut neque 600, neque 603 sit 0) et fiet   (2((410. 403- 3,0.303) A, 13 () (40. 363- 403.310)L (410- 310) (403- 3032)-
513L ∞,: 2.603. debetque fieri
indeterminatis, ateque fiet sit aequitio 8 per compendrum, Sioll + b03 nn 4613ln +
Cun coincidentie Cum identificentur) e
si + sit +, fiet (210. 403 + s1o. 305, 2 2. 410. 403.310. 303 + 410. 315 + 405: 315 - 2. 410 403. 10. 103 = (410. 4034316. 3055 - 415. 305- 405. 318
seu fiet et n = ( 1(613 I -4. 610. 603l L4, 600, 603aa
et divisis omnibus per 2, fiet 418. 303+ 403. 310 = 2. 410:403. 310. 303 = 0 seu 410. 303 =403. 310 seu 4  403: 303=
seu coinciderent in determinatis. Ergo oportet  ess
2. 410. 403. 310.303 = - 410. 403  =316 7 0  410. 307. 4 40 5. 310
et rursu dividendo per 2 fiet 410. 4032+ 318. 303 2410. 403. 310. 303 =0 seu denique 410. 403 = 310. 303. seu 1410, 403- 310. 303 =0 sede
sed haec aequatio non est licita, ita enim ratio C ad D foret constans
et fiet 410. 403 +318. 303 - 2. 410.403. 310. 303 (4 410.305 + 403, 31
ta alia oriuntur in commoda, nam in aequ. 10. ubi esset valo
e ipsus
son non llue ob laudem
403 = 303. seu 403 = + 305. Et pari jure  qui aequationes dua detisina
deubificantur fiet 410 = ()B = 310 ne scilicet 410 sit ad, 310, ut 403 ad 303, quod prohiberi diximus. Adoque foret hoc modo
iderqu fieret, ex aeg. 9: sentemus de A13. quo facti fter
dentica. Hinc sequitur, ex is solutibnibus aequoe esse duabus esse unam, ut
id ita alia orimntur incommoda, nam ex in aeg. B evacescentibus H et un, restacet In de
b-c . quod esse
vel fitet 410303 403. 30= 0 quod jam rejecimus Ergo n per l rationaliter in ipsis determinatis haberi nequit. Atque edeo
 ita  jam foret per 8,
410: 403= - 310. 303 seu 410, 403. + 30303 = Reret 600 = 0 sed hoc mucto minus licet nati fier
cum cistante quod non licet
hac via ad exctum perveniti ae non potest