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# Title: Leibniz-Handschriften zur Mathematik LH 35, 12, 2
# Folio 183v, page id 00068195:0366
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Sit Aequatio 20XX + O2yy + uXy + 10aXtoray +00 aa = 0. Ita praestanda in
proposita
rationalibus ut prodeat identica si fiat X: a = 10OV = 10Ia,: 200 P + 20. a et y: a = 300B+ hoa,: 200V + 201 a
Et secundum hos valores explicando Aequationem propositam, debet prodire atia cujus termini evanescant
20XX = 20, 100100V + 2. 100. 101 a V + 101. 101aa
Oryy = 02, 300. 3001∞ +2. 300. 301 = + 301.301 
I1Xy = 11, 100. 300.. +1.
101.300.-
10ax, = 10, 100. 200.. +1. 100201.. + 101.201..
101. 200.
O1 ay = 01, 200. 300. +1.
. 201. 301
+ 201. 301 oi
00aa = 00, 200. 200. +2. 200. 201.. + 201. 201..
compendio 20. 100402. 300 =I1. 100. 300 0. 402 et 10, 100 +01. 300- 50: 401
sit 20. 101+ 02. 301 + 11. 101. 301=00.412 10. 101401. 301=00. 4N
2. 20100.101 +2. 02. 300. 301 F 11. 100. 209 + 11. 101. 300 -00. 4012
fiet (1mo) 208 + 401. 200 + 402 =0 Ex termine summo
(20) 201 + 41. 201 + 412 =0
(3tio) 2.200. 201 + 401. 200 +4012 =0
medio
o = 33 4  - 2.10. 6) = 2.313
Adde compendia ad aeqqu. 12, 15, 16
Sed quoniam arbitrariae videntur sufficere faciamus 4020. Et ex aego. 1. fiet 200 = - 401. unde ex 0g. 3.
seu 201= 4012- 402 :9 2401 - 4 11  (ep ag. 2) - 411 + Y(4. (- 9. 412), : 2
faciamus calculi magis deprimendi causa 4112 - 4. 412 = (8)  = o fiet  ex nq. 6 et y 2t = tnoz 201 = (9) = = +11: 2
habebimus 2. 401.201 - 4012 + 411. 201 +4012
1 4012 - 401 = o 401. 411 = = 4112 ob _ qu 4 fiet 308: 100 = 11 m0 = - 11 = Yp5 - 4. 20. 02) , 02 = 12mo per compend - 102: 000
at ob aqu. 8 fiet 10. 10P + 0r. 30 P + 2. 10. 01. 101 301 = 4. 00. 20. 105 4, 00. 02301 4, 00. 11. 101.301. unde per compendium.
adeoque 301 : 104 = 1910 = - 313 + L (313. 313 - 31. 33),: 33 = (15t) - 2 00
 explicatis 4012, 401, 481 et pro 300: 100, Idem 301. 10l substitutis ex agq. 12 et valonibus compendiosis fiet
fiet 31. 1014 33. 301 + 2. 313, 101. 301 = 13tio=
4002 
G cell.
sa
2, v1 1. 000. 102
∞2104 000. 0
42. 1081.
+ O1. 112. 112...
=2. O1. 102.102.
-4. 1001. 000. 102... +2. 1081. 000112...
ne sit 301: 100 = 101: 100. Et priora duo videns c jege eeter Neque huie malo
videntur hypotheses nostrae restringentes esse obnoxiae. (301: 300 = C112. 108: 000. 100 per 
multiplicando scilicet tum 301 : 101, tum ejus valorem est aeq. 15 per 101: 100. Ergo si 301: 300 esset
idem quod 101: 100, foret 112:000 = 1: 1 seu hoc saem (nisi in certis casibus specialibus
quos bransformatione evitares licet) non patitur aeq. 15.. superest ut ostendamus negi
gereratim saltem) esse 301: 300 aut 101 : 100 = 201: 200. Nam ex ag9.5 et 9
habetur 202201: 200 = V 4 (119n0 = 411 : 2401 = 10 101 +01. 3013: 
2(10. 100 + 01. 300)= 10. 000. 101 + 01. 112. 101, :2 (10. 000. 100 + 01. 102100 quod 
(10,000 + 01. 112, : 2 (10. 000. + 01. 102)) 101: 100. Ergo non aequat regulititer) psi 101 : 100
sed nec aequatur (regulariter) ipsi 301: 300, seu per 18 ipsi 112. 101: 300. 100
alioqui foret 10. 000 +01 112, : 2 (10.000 +01. 102) = 112: 000, quod utique regu-
Veniamus ad exemplum quo iudigemus sico =a (C +aXX, :, aC + ag XZ)
lariter locum non habet
quaeritur (CD)dX. siat   ZZ =  thX et fiet ( 2+Z = hdX L ch
ot du = h dx: Y2RX et Cix AZ =hat (aE + 2FhX, :, (a C + aghX) 2hX) debet
ergo a C FrafghiX, ac + raGhX, 2hX aequari quadrate. Quoisi ergo res praestari
otest faciendo X = 100V rioIa,:, 20ov+ roia, su effici potest, ut  aa C+ rfhX, aact 2h X
Liet 02 = -A1., et 20 = mn: a et n 20 et 01 =0 et 10 = ac con + t et  men,ic et 00 = aa
quadrato quod sittjus fietque
per 5 erit 200 = C=(M+n: C) 100. Et per 9 erit 201 = - (mt 2 (m +n: 2C) 101
33= = 4. 44 : CC sed 313 =0. Ergo per aqu. 15 fiet 301: 101 seu 112:000
10. 10 seu 101 -C 2mod + nirueumqus + ammus - m M L- a nin (bmmnid 41e n +(3 mmuB)
seu  majus quam um + ns + Bm + B n. si MLn hoc non proc
10X = - 4 11 4 3 (1. 4.10. 13)
+12 mmnon resse majus quam 44 (in + n) quodsi alterutra m, vel n sit quantitas negativa
saltem nmmi
10. 10XX + G. 10X. 11 +10.12
(A (aa : CC)q (m -n: C)):  (= 4aa : cc) seu A (m - n,: C) a(a: C): (- 4da :cc) =  f m. n ). 2a = 1 : a
Denique ex aequ. 16 fiet + 4. ma 1100. 181 - 2. si + _) 100. 108 tq.  F) 101. 101 =0
rp. velbreniter A. 20 - 2. 10. 10) 106. 101 +  101. 101 - 2 = 0
Et 101  = - 2. 20+10. 10 + ((4. 20 4+102 - 4. 20. 10. 19 6 - 2 10t)
ubi videndum an fieri Bossit ab irrationalis sit quantitas reali
non iideo quid obstet, sed si ambae sint affirmativae debet esse
d obi mo
nam soli
sed quia Quia 11 =0 fiet mn. 100 = a a. 3 d = cc. 402. Ergo si 402 =0 fi et 300: 100 = Imn: a = 102:000
sed 412 = Mn. 1012 - 4a. 3012, : cC 401 = mn (m+n,: C) 100 41 = (m +n, : C) 101 Ergo
Ergo 201: 200 = _. 101 1 00 101 : 2. 100. porro 31 = 4 mn- mm - hm - 2mn, : Cc se
seu 31 = qu (m - n, : C)
=4 mn (mmenn)
jus
10XX + 10X. 11 + L. 11
aequetur quadrato, facile effici potest, ut deinde et iha aequetur quadrato. Par fh scribemusn
prorfg scuibamus am, et pro 29h scribamus an, et ac + nX, ac + nX debet aequari
mEX ju + mnXX +acmX + aa d  =0. itaque
unde 101 : 100 = (B, 2) 17mo = - 59, + Y  50. 51): 51 oe
ubi tantum cavendum ne sit 101: 100, neque 301: 300, aequ. 201: 200
 qu et denique ne sit 301: 300 = 101: 100.
Ubi tantum cavendum ne sit 101 : 100 = 201. 200, neque 301. 300 = 201 : 200, ac demque
42 001. 1o2 12 ...
40. 100 + g1. 102 +,GD1. 100. 101 - = d
100. 100