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# Title: Leibniz-Handschriften zur Mathematik LH 35, 14, 1
# Folio 194r, page id 00068201:0387
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10X + 11 =0 20X + 2120 10. 21 - 11. 20270
20X + 2IX+ 32 =0 41B2IX L10.220 Ex 5 et 2.
equa nova
11. 20X
fit (ut ex + et 3 fit ceq 3 nem pe in 3 pro tC, ponendo 10. 21 - 11.
et pro Et pontior 110. 22, et sicC: 10l 21. 21 = H. 10.3 10 20. 2
Ex 5 et Y fit aeq. nova ut ex 2  10.20. 22,
e in 3 pro 20721, ponendo 10. 21 -11. 20, et 10. 22
et ex aeq. 3. sit 10. 10. 22 - 10. 11. 21 = 11. 11. 2015
20X + 2 X + 21 X + 23 =0. Unde ex 2et L. f1 + 1O. 2IX 4I0. VX H 10. 2
X + 21 =0 tollendo fit:
20)X+ 2IX + 2240
1Ct equ. 1 aequ. 2 dat
reg
(SH 1 X + 12 seu X + 1222
tx  22
H-2
Ergo ex aequa 3 (per 6 po 2t ponendo) 2. 22 121
fit. 13. E X
IaX +H o ax+o
fite Ex 8 et 1. sit ceq. nova, ut ex 4 et 1. fit 6. Et in 6 pro (10 X + 11X + 12 = 0 20x + 21 = 0. multiplicetur aeq. 2 per
11. 20.
20, 21, 22, ponendo 10. 21 - 11. 20, 10. 22, 10. 23 fit
 inde ex 6 fiet 10. 10. 10. 23 7010. 11. 22 + 10. 11. 11. 21 - 11. 11. 11. 20
C1) et
10X et aequ. 1 per - 20, et producta invicem addendofiet
+102IX
ita s ( detur 10X + 11 =0 et 20X + 21X + UX+ 3Y14 -11. 20X - 12. 20
fiet 10.24 - 10, 1. 23 + 10. 11. 22 - 10. 113 21 + 11320 =0
Et ita porro in reliquis
Aliter ex + HX + 12 o 2X F uXFLE8
plicando pr crucem per. 10 et 20, fit 410. 21 X 4 102
aliter multiplicando per crucem ner 12-22, fie
divisip omnibus pex post additionem, fiet: Al3. 20X + 13. 21 =0
= 10. 22X - 11. 22
et conjungendo 6 et 7. pro aeq (10, (2)  1). 2d =0 fiet alia, si pro
on e cun
si pro (10) 
20)
poratur
(21)
ponatur + 10. 21- 11. 20 10.22 -12.20 12.20-1022 12. 21 - 11. 32
unde fit 20 = (11) et pro (10) 61) - (11) (29, fiet:
= 10. 12. 21. 21 - 10. 11. 21. 22 - 11. 12. 20. 21 4 11. 11. 20. 22
+ 10. 10, 22. 22 -,, 10, 12. 20. 22 = 12. 12. 20.20
qui calculus est memorabilis, quia procedit sinem
omni destructione.
Sit 1023+ uX 4 12X + 1320 20X + 2X +X + 23 270
multiplian rucem  fiet
+ 11. 20 X f12. 20X F 132
Multiplicando in crucem per =Het zo. Multiplicando in Erucem per +3 
fit +HI. 20X F 12. 30X +13. 20
= 10. 2 = 10. 22 - 10.23
fit: - 10. 23X = _ 23X - 12, 23 (4
+13. 20. + 13.21. + 13. 22
unde ex aequ. 8 calculi proxime anterioris ses nova
(11)
(22)
(12)
(20)
(217
ponati 11. 20-10. 2. 1220 -1012 1320.10. 23 = 1023,1, 2 =71. 231,3. 2 =13. 23+13. 22
Si pro (10)
ubi vursus coincidunt 2  (12) et (20), ut paulo aute 11 et 20
Leges combinationum: si duae sint aequationes secundum
quarum ope tolli debeat x, una dimensionum a, altera
dimensionum b. Necesse est  in nova aequatione
ipsa litera X carente, coefficientes aequationis dimensionum a
ascendere singulas ad dim. b. et contra. 2 coefficientes
ordine sumti retrorsum, eodem modo se habent, ut coeffici
entes ordine sumti prorsum. Sen quod idem est si coeffi-
cientes invertantur dro a substituendo L, illi in horum
locum succedunt et ricissim: 3.  quaelibet ex  coeffi-
cienlibus aeqvationis  giic dimensionum a significeta
 B (item b) significet = a ( - b) reliquis
literis significantibus 1. aequatio neva in qua tollitur
Sublata est X, sedet sic explicuta succedet. Ratio
est, quia sic sulcedunt aequationes datae, penindo X = 1
Ergo succedet etiam ex ipsis derivata 4. Quodvis
membrum constat est dimensionum a a +b.
et summa ciphrarum dextrarum facit ab. sic si
in aeq. ex AB 10X + UX etc = 0 et 20X + 21X et = 0
0
rursus in ultplicetur aeg. 2 per
10, 21 - 11. 20 et aeq. 3 per
-20 et fiet:
(0. 21. 21 - 11. 20, 21 + 12. 20.2040
es dimentionum 5, et summae b, nam 10323 seu 10, 10, 10. 23. 23
dat summ. 0+0+0+3+3 id est 6, et 10. 12. 22 dasum
20X + 21X + 2220 multiplicando pencrucem
jnvicem addendo, fit +10. 21 X + 10.22= jam
ope aeq. 1 et 6, fit aeq. in qua absit x ut ex aeg. 1.
substituemus 10. 21 - 11. 20) et pro 21 subistituemus 18. 22- 12. 20
et exaeq. 4si et+ 10. 19. 13. 22. 22 - CB2, 10, 10. 12. 20. 22+, 1212. 20
 = 10. 10. 11 21. 22 + 1.X 10 11. 1220.21
∞ 10. 11. 11. 20. 22  = 11. 11. 12 20. 20)1
. + 18. 10. 12. 2p. 29 -. (2.)10. 11. 12. 20. 21851 9
et divisis omnihus per 10, fit tandem aequatio (s)
(10. 10. 22. 22.5-10. 11. 21.22 9.f. 21, 11. 220.12,=0
12, 12, 20. 20 l = 12 1 21,20
10. 12. 21. 21
seu ordtuando 10. 276- 10. 11. 21. 22 + 1 20, 22
2. 10. 12. 20. 22. Groba: d 4
sit 10 X + 11X + 12 = 0
(2)
2.19. 12. 20 22 = 11. 12. 21. 20
ponendo 10.2 - 14 20 10. 22. 120, 10. 23
calculi
Itaque ex ag. 8. fit
 in nova aequ. 10 ascende
combinationes possibiles. ita quidem ut mumirum habeat
at productores ex aeq. dimensionum a, et contra. Summa
id dim. 3, et 20 ad dim.
naecidentis fit
10.0+ 12323 oto 1. 1 12 20234 10. 10. 1212. 2C. 20
1B. 112 12. 20,20 2. 10 11 1912. 2021 1 ( 10. 12. 13, 21. 21
10-10.10. 10. 23. 23
1o 13. 12, 12. 21. 21. = (2)16.11. 12, 12. 20. 2 (+ 11. 11. 12. 12. 20. 20
10. 10. 10. 12. 22 2L - 2. 18.10. 12. 1220, 22 + 112. 12. 12. 20. 20
= 10: 10, 10. 11. 21. 27 + (. 1. 10. 11: 12. 20. 23
= +0. 10. 11. 12. 21. 32 (+, 10, 11. 12. 12. 20. 2V 418. 11. 11. 12,,20. 22
redditur dimensionum ratio redditur ex eo quia aliotivi in vgq.
(I1. 11. 12. 12.20. 20)
10. 1c. 11. 11. 21. 23 = 10. H. V. 11. 20.23
Et odinando
10. 23 ∞ 10. 11. 22. 23 B +10. 11. H. 2. 23. F 11. 11 12. 2022
 - 2.102,12. 21. 23 0. + 510. 11. 12. 20.23 =11. 13.12. 27
+ 102, 12. 22. 22
10 = 10. 11. 12. 21. 22 412. 1252. 36 1
. - 8, 10. 12. 12. 20, 22
12, 11. 21. 20 8 = 2. 12, 10, 22. 20 12, 10.31. 21 12. 51. 11. 22. 20
+ 3. 12. 11. 10. 28, 20 manet, teu  si ibisi respondet. Nemper
duabus datis, adeoque nec in datis servarentur s homogeneorum
tet 2 sequuntur
a = 11. 11. 11. 2022
4(0. 12. 12. 21. 21
Examen antistrophes coefficientium ex Lege=
pro 10. 11. 12. ) 29, 21. 2223 debent prodire idem quod nunc
23, 22. 21. 2
Notavimus duobus punctis  membra sibimes avr, S pEFOYTa.
(+ 18. 23
1132122) + 1∞. 20
autem seu valor dimensionum manebit a+b. cujus valoris rutio
22 F1X. 20
et 20X + 21X + 22X + 23 =0
∞10. 21 X + 1022X + 10. 23 =0
- 11. 20 X = 1X20 X Cu
pro (20)
(2)
 +10. 18. 12. 29, 23 +2. 10. 10. 11. 12. 20. 23
2. F 3. dmonda est rufg 6.
Et secundum hanc legem fiunt omnes  nembrorum
= 11. 20X = 12. 2C
et 2 tacta fuit aeq. 4 et in aeg. 4 pro 20, 
O =
Nola suendi ommia - 1. Acepto 1- - 2 et 20- - 3 + D Bg 3
nata cum a sit 2 et b sit 3, fiet a +b, 5 et ab, 6.
ideo ni membrum quo dlibet ut 10325 vel 10. 12 20. 21
seu DtX R 10. 12. 12. 22 dat summ.
Et excrac lege 4 Hic exglicata pce
5. Lex Signorum + 10) -1
(3)
musti- aeq. 5 per 10, et aeq. 1 per - 20 et producta
20
O3
po
coordinemus et secundum hanc legem
(-10. 11. 23,22
(+ 10. 11. 23, 21
l- 127. 1 20.21 l+ 1R. 112 20. 22
(-2. 10. 12. 25. 20 + 3. 10.11. 12. 20. 23
l- 212.10. 20. 22  O
 10.12. 22.22 = 10. 11. 12. 2. 27
l + 12.10. 21. 21 
- 113, 20. 23
absunt combinationes: 10. 152. 22 113. 21. 22
2. 112. 21.
perinde est igitur ac si adesse ponerentur
praefixo signor dient ia sividi alliqus posse 12
posseper 2
ratio h23 Ssi sepriui a t reiqueque p racius 20=0+
fiet. 123. 20, 10. 123 = 2 124020.70 detent
12 H20.2t omnia
Hinc excluduntur obinades, in qua situnt ab sunt 12 et 23
vel in qua simul absunt 10 et eo, sigua autem et
numeri praefixi utique ut in calcalo praecedent
Hinc ex
rcte
peurmut
d novus, supplendo tamen eos in qurs
X + 2. 21 X + 21, 2
+ 1. HtrX t H. 21
+ 12
stre a