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# Title: Leibniz-Handschriften zur Mathematik LH 35, 13, 1
# Folio 202r, page id 00068196:0399
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CL A55I5 FRIMA
alerilus 1 10 X + 1 2o 2oX + 2a = 0 sp. 2ox u nA = (o2oB + 10ua fit 10.  - 11 20 = 0. elter 1B 217  11 21) = 1 20XF 11 21a) fiet idem
lalus son 10X L Ha =0, 2OHCFINXF aA Do (IB 2OXXSH I. EOAX = (B. 20KDHIB LI AXHIB L2aa fit , 1B. H aX + 102 aa 20 quae conjungatur aepatim 1. frot 10. 10. 32 = 10. v. 21 4 111. 2020
eq. A si pro 10 gubstitua permutet 0 et
Aliter 1B22A XFH. ma) = 11. 2oNT + 11. Uo VanD (., ve 12oi =  F quaesi compimgatur 5p
q lum havetimar te di ddiu
puitaeq. 4. Hinc duplex processus her irit
at esqu. llmnam callauli (m posito 22
ciluus 1eXtIa = , 20XPXt 2NX + 23a=0 (10. 20X + 11. 20 axx = (10. 20V Ho. 2axY F 1v. VaaX + 10, 23 a  seut 10. 21 ayY + 10. raaX +. 1B. 13 a La consendi 2op tyax   aa 
putiontum 
un l nlepire s
ncauli prrecedentis, si ponis ho-0 et pro  l p2, 29lice, nonas 2o, X1, 22. Eotem modo ex aeque ultina pasculi pre  b.  
3) calculi
pentinaequ. d
Aequationes quartae calculi (2, 3, et. consentiunt inter se in eo quod ex Ata calculi alicujus fit quarta calculipraxim e a u sipliando ilam per X, et adento   10. 2C a. Et ita paht etiam quomodo quintae ex se invicem criantu, et ita pur
10. 10. 11. 22 + 10, 11. 11. 21 = I1. 11.
10X . H
sequationes 3tiae  calculi 1. Ed, 3ti, consentiunt inter se in eo quod ex Htia Ealuli aliujus,  sit otia calculi prorime sitelianto vtamm per X, et adendo 8  B
poncido (20), (21) (22) calcili hanus, et fit
remotae a primus nempe 3tiae, quantae, quintae _c. a se invicem ducantur Vibeamus et quomodo a se invicem ducantur Uttimit
Era
suamncaliani
Quaramus praeterea residuis
tet berdiaa
s
iit aua
divitor 10X + 11a
noru ntrcdae rattis 2tit  ua
e Resid. )- poero
penmetimae, et antepenultima
comite decurem lessum t hae mat
et Residuus Cum
tione altina salub
pisisur 10X + 1
uinistox t LlaE + DOaX Y2
e - meaet -e ut 
Caeterum aliter et brevius poterit hic tolli litera X, si ex aequ. 1 fial X = - 118 et i valor
unc aequationem conjongamus cum sequentibus
I0XX F 1af + 1244 = 0 2OXX + 2IaXF 22aa = 0 (10207X + 11. 20aX + 12. 10aa = (10. 20Xx) + 20 21 ax + 10. 22a4, fit. F10. 21 ax
sibstituatur deinde in aequatione secarda
CLASSIS SECVNON
Sed tunc transeamus ad aequationem 10 XX + ( IIaX + 12 aa,
alculys primus
tit ex lalentosecundo clastiis praecedentus sidio 22 (oi) 2t 1 (t 1) 20 = 0 _ ubi explicando 1ou lt, habelatur equatio fine p.
Unde ex eae C et 2 fet
 10. 22- 12. 20  ( 1021 41. 20d a X ad xarsus = fi. 2l 2
in hoc calculandi modo id est laudabile, quod hoc modo non a
Lode
calculus secundus 10xX. + H1aX +1 2a4 =. 20X + 2XX + 22 aax + 23 a =0
unde Ex his priorem applicemus primae datae inversae, fiet 18. 16. 23 p
rrunt depressiones
x)Hisorvaret (2. 20 aaX = (10. 20X1
ax, 
et pradbit (18. 12. 20XX L 11, 2. 20 aX + 1212. 20 aa = (0. 12. 2oXY) + 0. 12. 21 ax + 10. 12. 22 aa, sed aequatio quae ita prodit coincili
Rarsus et posteriorem Minvensione _ (1) ex inversioneortam appliaemus primae datiei
 quod si ex aegu. Micte data, et rda inventas ada modum
Calculi rdi
nec 2dae ditae ntra gradum 2dum Ego aequationem pristem datam, et ap. ) conjuaga
+ 10. 10. 22 X +10. 10. 23a 20
sinilite raeqrationem priorem datam et aequo 2) conjana
10. 12, 22 X +
seu = 10. 1
= 2. 12. 20
- 10. 12. 20
+ 11. 12. 20
I. conjungere ejus inverso
1o t H.2 B
(10, 12. 22. 27). 13. 1 2 = 10. 12. 21 22) . 10. 11L. 2i C 10. 12. 16 23  10. 1. 1 20. 27 + 10. 123. 20. 2 1U15 20 22 16. 11. 12. 21. 23 X10. 18. 21 5 410. 4. 12.  -10 15. nat. 2e A  ie  tv s. litivor   . 
eeu. -10. 1.25 (+18. 12. 22.2 (15. 12 2. 13) + 10. 11. 12. 21. 23(-157, 12. 21. 22 10, 15. 2824) 410. 112. 1223 + 10. 11. 12, 2822 = 11. 122.20. Qu. vatione defantuse esanctis destuendis divisa per 1 prodit
FI. l. 20
+123. 202
418. 1E. 35
. 2 2
a3 + 11. 20. 23)=0  tatem videlum
1X. 12. 20.2
sererali vin, lie iti dita aeqpuatent 104 rX = B e roA A 20 puX +2EXL + 127d + 24 X retis pstumus touim X, eoum in aequ. postriore pro a prerer =  unde fiet a
= 20 - 19  + 1gue n - 1 13 4 1 24 -  1825 + et e vel  18. 20 - 10. 18 al + 1. . 22 + 18. 10 3 33 - 11 10 24 + M aente periveniatur au  plae qui ex terminus altimus.
 se sint jam dutae aequs. (D. +. HX + 1 BXX - 0 et 20+ 2VCL LZXX + 23 X + 14 X + 25 X + 26 X + etc = 0. Eit XX + = 4 g0  ader -  u+ sini
quo valome substituto in aqu. aleradata fist 20 B. v + V2- N 23+ 1 2 1 - 1 25 + erc 
20 - 1 21 + 1+E 12. N 23+ uit  18 25 + 1tg s 1 17 + et = (a2, 21 + 12 23 + 12 75 + 92 27. Quo si jam detimus vnere trationalem, fiet
20 -  11. 20. 21 + 2, 1 + 42. 20. 22 - 2. 18 20. 23 + 2, 1 4+ 4
16. 104 8  2 11 2025
+ n 2121 - 25 .F2 212 + 2
Unde t deptrum latus aequationis in sinistrum transfereudo, fiet:
+2 1ror 286 - et
+ qeate vu - 2. 1 + dta 23 + 2, 1E 3  v 24 - et
20- B: G8 1. 2021 + 2 ( 19 62. - 2 (L. 13 20. 1.
(24 3 2
81 ) L. p8 Zu2e
A sertet sujus lex habetur ex conideratione numerorum 20, 21, 22, 23, etc. quea quorum
hic ponuntur  quadrata, et biniones. e quoties hujus numeri inota dextra est
impar, ut cum est 2t, 23, 25, ex. pefficiens habet factorem H, H3, 1, etc. Cum
Quoties nomerus cujus neta dextra est par, ducitur in cum nume
64. 196 23.23 - etc
+ 2 511t 12024
1610 e 3 223
Numerus  cujus nota dextra est  impar, ut 2, 23, 25, habet in eoefficiente 11, 11, 11, etc
par ut re. 24,26, ex habet in eofficiente t, 112 412, 114, etc
Hine si combinetar par, v. g. 22. 26, coefficiens est productum ex 11+ 42, in 115+ 42
Si combinetur pari par m et impar, V.d. 22, 25, prodnctum coefficiens est productum ex 114+ 42, et 155
eisere
denique si combinetur impari, productum est coefficies est Vg. 23. 25, paoductum coesficiem
quidem deberet esse iP in 15, sed to seu 118, sed tamen ab eo detrahitur A2, quod homogeneum illier
ox 18
(A 121. 25
 Fi pu 2+
+H 277
n1 Y21. 27
 fr. 26
+111, 723. 25
(1.   3 4 28
 rt 113, uto revera assurgitur nelra 123
videndrem ergo in applicationeq generalis (anonis eo speciem temper justo altioea destruantur
esculica. ita semel cubando et sen qua uno aqpunturem ramonem tiberam a  logesitum
ntepemaltimae, etc. Nempe ulima calculi props alicujis dabis
nea et ut sucetit divieo fit l0 usante
unde per
tt ex se aegrm alissuno exp
F1ClI. 23aae)
Coincidet Residu  issue a
easse  HeumX
 12. 20a
potens sero compo-
netur ex on
equationtl ultimatur
calculorum praecedentium
diplicato fie)
primo sollere velimus nimis alt
hultie fers
io. 12 ux t un ax ( H
et 0 ita eleganter evenit, ut dum in und
et fiil +10. 10. 22
in 20X , 21 aXX + 2aX+ 232, item cum 20X + 2X + 22aC f25 a X 124a
eaa stio
XX) F 10. 11. 21aX
pt avierus n
rexFua
paprer
 dutu et leo 2. inaentus
tionis prim
22) +10102E ax +10
102 at
simplex in altero 12 assurpt ad
nota ulaeti palicata
modum classs
Eratum et co est simplex
sed quia it utoblu alagit ad p    a  b
tui vel attqui ternuni destrut
= + 10. 10. 237f- 10. 11, 237
2d a 20
divitentur per
 1220 redit
 raa 2.
2. 2) = 12 1 29t 12.1. 21 ax (F 12. 1 22 aa)
3 savit ad quadratum et iL es
habentar nili 103 et
=0. se
seu fiet
destuentat aeliqua
abest H, et renqua
a  et fiec
si esset 11 20 fiuet  Et proinde
X = 126 = 10. 222
- 20. 12 ac + 21 ax + 2 -  0 seu
XX = qu 18.2 p rgaa = - seu 1  1 o. 2. 20
1o  aa o exsecta eia sndet quanmenticnem
10. 22 aax + 1B. 23a. fit 10
10 1.33 ax (( 10.12. 23a4) = p 1 0. 12. 21 XX + 10. 12. 22 ax (F 10, 12. 234) Unde pront - 0 12.
mcedenti
uddettis proe
do danir per 12. sel nam caleati paeceduts
ido omcnia per 13.) sed est dificullas, quia ibi est
2. 10. 12. 20. 22, hic autem est solum 10. (2. 28. 22
42, 2i + 2. 72 2 23+ 2. 85 2125 +2. 42) 727 +2. 147, 21. 29 + ex
vs +  3as
+2. 48 2327 +eu
155 8 2525. + etc
(115 20.25
3210) 15
1.425 E1. 24
18 101) 1. 20. 27. Liib. 8. 14
ut pu. 26
1 ,t) 3. 24.
Sed videtur hic caltius assursi quam oportat, ex. gr. in cum hon assurgi tur ultra 23. videtur
perveniri ad I16, et 106, cum tamen non debeat assurgi ultra