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# Title: Leibniz-Handschriften zur Mathematik LH 35, 6, 26
# Folio 14r, page id 00068071:0027
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be ad R. Jac. Ber Roulliz
aris
Quadraturae orrationalium simplicium
Quaeritur SQVDdx. posito D et e ex esse formulas
gitur
rationales integras quoad X. Quia coportet quantitatem et
ejus summatricem ejusdem esse ambignitatis, et quanbitatis
uno irrationali vinculo constantis, ut VD differentiale
siea hanc ipsam irrationalem multiplicatur; ideo
facio, (qYDax = eZDYD + (2 YDdx. ponendo sormuta
habe
+ veluti si D esset 10 1X + 2XX + 3 XtCk
H eamem ambiguitatem praestantis
irrationalitatis. et difcerentiale
et quia
habxiorem quam e quantum oposest etlicet, ut quae sita quadratura
et habenaar absoluta
ed Ergo utrinque differentiando fiet
e reducatur ad simpliciorem ejusdem ambignitatis
eD d  + (er1) E d D + 2 dX - Gdx = 0. quare
cum formulae 8 et 2 sint arbitrariae, abeoque potentiae ipsina X, in his formulis
iptos in ipsis habeant arbitrari coefficientes ex
eo determinandos, dum quivis terminus est destruendus;
quo
hinc posito gradum ips Exponentes gradutum ad quos
et, 7) ponendo a, (3, V, d, fore oportet
sC
ut quantum satis arbitrariarum obtineatur, et 2x tamen
assurgunt D, E).
fier
maneat quam licet simplicisima, fore V = β+ 1- a
et  constare ex numero fui
in minor expone ipso a  posse esse habere terminos
A X, X, X, etc, vel X, X X, XX, X, retc. vel
XX+X X XX, X, X et  eodem manente numero ipsorum a = 1
sed primum tamen casum esse simplicissimum, quo posito fiet d -( _- 2
Esto jam in exempo unde reliqua aestimentur,
DuX 1aXX + 1X + 14X
praeterea considero pro formula , composita
A ex maltis terminis  X
Sit formula _ _Quantitas & posset quidem esse
los
Formula composita ex multis terminis, velut si esset
sed quia
Q = 20+ 2IX +22XX F23x + et c. sed si nou quodque
ex unusquisque ex his terminis, velat 20X in  ductus
qualis X s qu est ordinatae est valor, cujus figura quadraturam uodrati
madmodum
potest gulos tormines data et
si nonermo, 
rumquales Xit
 dufetermint qualis X, X, ductam Y3
contrahamus, sumus exemplum ponatur exemplum
unde reliqua ostimentur.
crmulatione
ipsis conflata sum mationem haben
ulorum  etermingmre fractari
indenta sipta extalibus foruum
etiam 
ta
e
3) = 10 +uX  2x + 19 X L SAX aK
D= 10 +HX H1KX+ C3X +4X K X H H
 Y y XX Y Y X I + X data  - X X x Y H X X +XB
 8 = 30 uC 1 133 X Sx , 3sN H X X, ∞ , 8 - 9C + 31X +3 1XXt 33 X 13H + HX X X X
9 onaxt aet _ X X 1X AY X. X q5q. h h, =40 + E, quxx X H X X X X
Fiet a - 4 B- 8 f6
d= 2
fiet a = 4, β = 8, 7 -5
Compendii jam causa assumamus numeros
novos, quales 130, 152, 173, et similes,
qui significant. 3C +0, 10 vel 5C+2, 12 vel 7C + 3, 13
abi gabris e, 253. os de sunt poneri veri
continuossedtamen
Timo et non contiiu sed sempe
+0.
vad _x +CXX